diff --git a/.cursorrules b/.cursorrules
deleted file mode 100644
index 5b54998..0000000
--- a/.cursorrules
+++ /dev/null
@@ -1,4 +0,0 @@
-Run the code in a conda environment called bispectrum.
-Activate the environment
-Also activate the environment in the .venv file.
-Do not use thick section separator comments like `# ----` or `# ===`. Let the code structure speak for itself.
diff --git a/.github/workflows/tests.yml b/.github/workflows/tests.yml
index 248c0dc..08e2ff4 100644
--- a/.github/workflows/tests.yml
+++ b/.github/workflows/tests.yml
@@ -41,4 +41,4 @@ jobs:
         uses: codecov/codecov-action@v5
         with:
           token: ${{ secrets.CODECOV_TOKEN }}
-          slug: geometric-intelligence/bispectrum
+          slug: anonymous/bispectrum
diff --git a/DESIGN.md b/DESIGN.md
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--- a/DESIGN.md
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@@ -1,516 +0,0 @@
-# Bispectrum Module — API Design Document
-
-**Status**: Accepted
-**Author**: Johan Mathe
-**Date**: 2026-02-23
-
-______________________________________________________________________
-
-## Overview
-
-This document specifies the API design for the `bispectrum` Python module. The goal is to provide a clean, mathematically faithful, and composable interface for computing G-Bispectra across different groups and domains — while keeping the library as simple as possible, but not simpler.
-
-______________________________________________________________________
-
-## Core Design Principles
-
-01. **Simple is better than clever.** If there are two ways to do something, pick the one that needs less explanation.
-
-02. **`nn.Module` only.** No parallel functional APIs. One way to compute a bispectrum.
-
-03. **Naming encodes the math.** Class names follow `{Group}on{Domain}` — a reader unfamiliar with the codebase can immediately identify the group and the domain it acts on.
-
-04. **Raw signals in.** Modules take real-valued signals in the natural domain (spatial, discrete cycle, etc.) and handle the Fourier transform internally. This keeps the interface clean and the math self-contained. `SO3onS2` accepts a raw spatial signal `f` of shape `(batch, nlat, nlon)` and handles the SHT internally; `nlat` and `nlon` are constructor arguments so the `RealSHT` transform is pre-initialized at construction time for efficiency.
-
-05. **Selective by default.** All modules default to `selective=True`, computing the minimal subset of bispectral coefficients sufficient for complete signal reconstruction. `selective=False` is available for debugging and comparison.
-
-06. **Complex output.** `forward()` returns `torch.complex64`. Users who need real features can call `.abs()`, `.real`, or `torch.view_as_real()` — this is one line of code and avoids losing phase information.
-
-07. **Inversion on-module.** Every module exposes `invert(beta, **kwargs)`. For modules where inversion is mathematically available, it returns the reconstructed signal (up to group-action indeterminacy). For modules where inversion is not yet available, it raises a precise `NotImplementedError` with guidance.
-
-08. **float32 throughout.** For compatibility with GPU training pipelines.
-
-09. **Minimal dependencies.** Don't add a dependency if standard PyTorch/numpy can do the job.
-
-10. **Code is the math documentation.** Every module docstring references the exact paper theorem it implements. Every non-obvious operation cites an equation number.
-
-______________________________________________________________________
-
-## Naming Convention: `{Group}on{Domain}`
-
-Groups act on domains. Both matter. The class name encodes both, in mathematical notation:
-
-| Class          | Group                                       | Domain                                      | Description                                    |
-| -------------- | ------------------------------------------- | ------------------------------------------- | ---------------------------------------------- |
-| `CnonCn`       | $C_n$                                       | $C_n$                                       | Cyclic group acting on itself (discrete cycle) |
-| `SO2onS1`      | $\\mathrm{SO}(2)$                           | $S^1$                                       | 2D rotations on the circle (wraps `CnonCn`)    |
-| `TorusOnTorus` | $C\_{n_1} \\times \\cdots \\times C\_{n_d}$ | $C\_{n_1} \\times \\cdots \\times C\_{n_d}$ | Discrete d-torus acting on itself              |
-| `DnonDn`       | $D_n$                                       | $D_n$                                       | Dihedral group acting on itself                |
-| `SO3onS2`      | $\\mathrm{SO}(3)$                           | $S^2$                                       | 3D rotations on the 2-sphere                   |
-| `SO2onDisk`    | $\\mathrm{SO}(2)$                           | Disk                                        | 2D rotations on the unit disk                  |
-| `OctaonOcta`   | $O$ (octahedral group)                      | $\\mathbb{R}^3$                             | Octahedral symmetries on 3D space              |
-
-This convention is deliberately mathematical rather than verbal (`CyclicBispectrum`, etc.) because the mathematical name carries precise meaning and avoids ambiguity as the library grows.
-
-______________________________________________________________________
-
-## Module Interface Contract
-
-Every bispectrum module is a `torch.nn.Module` satisfying this contract:
-
-```python
-class {Group}on{Domain}(nn.Module):
-
-    def __init__(self, ...):
-        """Initialize with group parameters.
-
-        CG matrices and other precomputed quantities registered as buffers
-        (not parameters — they are not learnable).
-        """
-
-    def forward(self, f: torch.Tensor) -> torch.Tensor:
-        """Compute the G-Bispectrum of signal f.
-
-        Args:
-            f: Real-valued signal on the domain. Shape: (batch, *domain_shape).
-
-        Returns:
-            Complex bispectrum tensor. Shape: (batch, output_size).
-        """
-
-    def invert(self, beta: torch.Tensor, **kwargs) -> torch.Tensor:
-        """Attempt inversion from bispectrum coefficients.
-
-        Default behavior can raise NotImplementedError for modules
-        where inversion is not yet available.
-        """
-
-    @property
-    def output_size(self) -> int:
-        """Number of bispectral coefficients in the output."""
-
-    @property
-    def index_map(self) -> list[tuple]:
-        """Maps flat output index → (irrep indices) tuple.
-        Allows interpreting which (rho_1, rho_2, rho) each output corresponds to.
-        """
-```
-
-**Key invariants:**
-
-- No trainable parameters (`sum(p.numel() for p in bsp.parameters()) == 0`)
-- Output is `torch.complex64` (complex float32)
-- Module moves correctly with `.to(device)` and `.to(dtype)`
-- Deterministic: same input always gives same output
-
-______________________________________________________________________
-
-## Implemented Modules
-
-### `CnonCn(n: int)` — Cyclic group $C_n$ on $\\mathbb{Z}/n\\mathbb{Z}$
-
-**Mathematical setting:**
-Signal $f: \\mathbb{Z}/n\\mathbb{Z} \\to \\mathbb{R}$. Group $C_n$ acts by cyclic shift: $(T_g f)(x) = f(x - g \\bmod n)$.
-
-**Bispectrum formula** (commutative case, [Kakarala 2009]):
-
-$$\\beta(f)_{k_1, k_2} = \\hat{f}_{k_1} \\cdot \\hat{f}_{k_2} \\cdot \\hat{f}^\*_{k_1 + k_2 \\bmod n}$$
-
-where:
-$$ \\hat{f}_k = \\sum_{x=0}^{n-1} f(x), e^{-i \\frac{2\\pi k x}{n}} $$
-
-is the DFT.
-
-**Selective coefficients** (Algorithm 1, [Mataigne et al. 2024]):
-Full bispectrum has $n^2$ coefficients. Selective version needs only $n$:
-
-$${ \\beta\_{0,0},\\ \\beta\_{0,1},\\ \\beta\_{1,1},\\ \\beta\_{1,2},\\ \\ldots,\\ \\beta\_{1,n-2} }$$
-
-These suffice for complete inversion (recovering $f$ up to cyclic shift).
-
-**Usage:**
-
-```python
-bsp = CnonCn(n=8)
-f = torch.randn(batch_size, 8)          # signal on Z/8Z, shape (batch, n)
-output = bsp(f)                          # shape (batch, output_size), complex64
-
-print(bsp.output_size)                   # n = 8 (selective) or n² (full)
-print(bsp.index_map)                     # [(0,0), (0,1), (1,1), (1,2), ...]
-```
-
-**Constructor parameters:**
-
-```python
-CnonCn(
-    n: int,                # Group order / signal length
-    selective: bool = True # Use selective O(n) or full O(n²) bispectrum
-)
-```
-
-______________________________________________________________________
-
-### `TorusOnTorus(ns: tuple[int, ...])` — Discrete d-torus $C\_{n_1} \\times \\cdots \\times C\_{n_d}$
-
-**Mathematical setting:**
-Signal $f: C\_{n_1} \\times \\cdots \\times C\_{n_d} \\to \\mathbb{R}$. The discrete d-torus $\\mathbb{T}^d = C\_{n_1} \\times \\cdots \\times C\_{n_d}$ is a finite abelian group acting on itself by componentwise translation: $(T_g f)(x) = f(x - g \\bmod \\mathbf{n})$.
-
-**Bispectrum formula** (commutative case, [Kakarala 2009]):
-
-$$\\beta(f)\_{\\mathbf{k}\_1, \\mathbf{k}\_2} = \\hat{f}\_{\\mathbf{k}\_1} \\cdot \\hat{f}\_{\\mathbf{k}\_2} \\cdot \\hat{f}^\\\*\_{(\\mathbf{k}\_1 + \\mathbf{k}\_2) \\bmod \\mathbf{n}}$$
-
-where $\\hat{f}$ is the $d$-dimensional DFT (`torch.fft.fftn`) and $\\mathbf{k}\_1, \\mathbf{k}\_2$ are multi-indices with componentwise modular addition. All irreps are 1D characters, so no Clebsch-Gordan matrices are needed.
-
-**Selective coefficients** (Theorem 4.3, Algorithm 2, [Mataigne et al. 2024]):
-Full bispectrum has $|G|^2$ coefficients. Selective version needs only $|G| = \\prod_l n_l$:
-
-The algorithm performs a BFS on the dual group $\\hat{G} \\cong G$ using generators $\\mathbf{e}\_l$ (standard basis vectors). For each element $\\mathbf{k}$ in row-major order, the selective pair is $(\\mathbf{e}\_l, \\mathbf{k} - \\mathbf{e}\_l)$ where $l$ is the first axis with $k_l > 0$.
-
-**Inversion** (Algorithm 2, Appendix D):
-Bootstrap from the trivial representation along each axis independently, then fill remaining entries column-by-column. The recovered signal has $d$ free phase ambiguities (one translation per axis).
-
-**Usage:**
-
-```python
-# 2D periodic image (28×28 torus)
-bsp = TorusOnTorus(ns=(28, 28))
-f = torch.randn(batch_size, 28, 28)    # signal on torus, shape (batch, n1, n2)
-output = bsp(f)                         # shape (batch, 784), complex64
-f_rec = bsp.invert(output)              # shape (batch, 28, 28), complex
-
-# 3D periodic volume
-bsp3d = TorusOnTorus(ns=(32, 32, 32))
-vol = torch.randn(batch_size, 32, 32, 32)
-output3d = bsp3d(vol)                   # shape (batch, 32768), complex64
-
-print(bsp.output_size)                  # 784 (selective)
-print(bsp.group_order)                  # 784
-print(bsp.ndim)                         # 2
-```
-
-**Constructor parameters:**
-
-```python
-TorusOnTorus(
-    ns: tuple[int, ...],       # Shape of the torus (e.g. (8, 8) for C_8 × C_8)
-    selective: bool = True     # Use selective O(|G|) or full O(|G|²) bispectrum
-)
-```
-
-______________________________________________________________________
-
-### `DnonDn(n: int)` — Dihedral group $D_n$ on $D_n$
-
-**Mathematical setting:**
-Signal $f: G \\to \\mathbb{R}$ where $G = D_n = \\langle a, x \\mid a^n = x^2 = e,\\ xax = a^{-1} \\rangle$.
-$D_n$ acts on $\\mathbb{R}^2$ via rotations ($a$: rotation by $2\\pi/n$) and reflections ($x$: flip).
-
-**Irreducible representations:**
-
-- 2D irreps $\\rho_k$, $k = 1, \\ldots, \\lfloor(n-1)/2\\rfloor$:
-
-$$\\rho_k(a^l x^m) = \\begin{pmatrix} \\cos(\\tfrac{2\\pi lk}{n}) & -\\sin(\\tfrac{2\\pi lk}{n}) \\ \\sin(\\tfrac{2\\pi lk}{n}) & \\cos(\\tfrac{2\\pi lk}{n}) \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\ 0 & -1 \\end{pmatrix}^m$$
-
-- 1D irreps: $\\rho_0$ (trivial), $\\rho\_{01}$, and (for $n$ even) $\\rho\_{02}$, $\\rho\_{03}$
-
-**Selective coefficients** (Algorithm 3, [Mataigne et al. 2024]):
-Only $\\lfloor(n-1)/2\\rfloor + 2$ matrix-valued bispectral coefficients needed, corresponding to approximately $4\\lvert D_n \\rvert$ scalar values.
-
-**Usage:**
-
-```python
-bsp = DnonDn(n=4)
-f = torch.randn(batch_size, 8)          # signal on D_4 (|D_4| = 2n = 8), shape (batch, 2n)
-output = bsp(f)                          # shape (batch, output_size), float32
-```
-
-Note: unlike `CnonCn`, `DnonDn.forward()` returns `float32` (not `complex64`) because all $D_n$ irreps are real-valued, so bispectral coefficients are real.
-
-**Constructor parameters:**
-
-```python
-DnonDn(
-    n: int,                # Polygon order (|D_n| = 2n)
-    selective: bool = True
-)
-```
-
-______________________________________________________________________
-
-### `SO3onS2(lmax: int)` — $\\mathrm{SO}(3)$ on $S^2$
-
-**Mathematical setting:**
-Signal $f: S^2 \\to \\mathbb{R}$. Group $\\mathrm{SO}(3)$ acts by 3D rotation.
-
-**Bispectrum formula** \[Kakarala 1992, Cohen et al.\]:
-
-$$\\beta(f)_{l_1, l_2}^{(l)} = \\sum_{m_1, m_2, m} C(l_1,m_1;\\ l_2,m_2 \\mid l,m)\\ a\_{l_1}^{m_1}\\ a\_{l_2}^{m_2}\\ \\overline{a_l^m}$$
-
-where $a_l^m$ are the spherical harmonic coefficients and $C$ denotes the Clebsch-Gordan coefficients for $\\mathrm{SO}(3)$. Each $\\beta\_{l_1,l_2,l}$ is a scalar (real for real signals). The full bispectrum has $O(L^3)$ entries.
-
-**Selective bispectrum** (this work, see `docs/selective_so3_on_s2_completeness.md`):
-$O(L^2)$ entries suffice for generic completeness, achieving the dimensional lower bound $(L+1)^2 - 3$. For each degree $l$, up to $2l+1$ triples are selected by priority from four categories: chain entries (linear in $F_l$), cross entries (linear), power entries (quadratic), and self-coupling entries (quadratic/cubic). At $l \\geq 4$, only linear entries are needed and recovery is a direct linear solve.
-
-**Clebsch-Gordan matrices:**
-Computed analytically via the Wigner 3j symbol (Racah formula) with log-factorial numerics. No external dependency or precomputed JSON required. Cached to disk after first computation. Any `lmax` is supported.
-
-**Usage:**
-
-```python
-# Full bispectrum: O(L³) entries
-bsp_full = SO3onS2(lmax=5, nlat=64, nlon=128, selective=False)
-
-# Selective bispectrum: O(L²) entries
-bsp_sel = SO3onS2(lmax=5, nlat=64, nlon=128, selective=True)
-
-f = torch.randn(batch_size, 64, 128)    # signal on S², shape (batch, nlat, nlon)
-output = bsp_sel(f)                      # shape (batch, 35), complex64
-
-print(bsp_full.output_size)              # 69
-print(bsp_sel.output_size)               # 35
-```
-
-**Constructor parameters:**
-
-```python
-SO3onS2(
-    lmax: int = 5,          # Maximum spherical harmonic degree (any value supported)
-    nlat: int = 64,         # Latitude grid points
-    nlon: int = 128,        # Longitude grid points
-    selective: bool = True  # Selective O(L²) or full O(L³) bispectrum
-)
-```
-
-> **Breaking change from v0.1.0**: v0.1.0 accepted pre-computed SH coefficients. v0.2.0 accepts raw spatial signals and handles SHT internally.
-
-______________________________________________________________________
-
-### `OctaonOcta()` — Octahedral group $O$ on $\\mathbb{R}^3$
-
-**Mathematical setting:**
-Signal $f: O \\to \\mathbb{R}$ where $O$ is the rotation symmetry group of the cube ($|O| = 24$). $O$ acts on itself by left-translation: $(T_g f)(h) = f(g^{-1}h)$.
-
-**Irreducible representations:**
-$O$ has 5 real irreps with dimensions $(1, 3, 3, 2, 1)$:
-
-- $\\rho_0$ (trivial, $A_1$): 1D
-- $\\rho_1$ (standard 3D, $T_1$): 3D — the natural action on $\\mathbb{R}^3$
-- $\\rho_2$ (product $T_2 = A_2 \\otimes T_1$): 3D
-- $\\rho_3$ (2D, $E$): 2D
-- $\\rho_4$ (alternating, $A_2$): 1D
-
-**Selective bispectrum** (Appendix B, [Mataigne et al. 2024]):
-4 matrix-valued coefficients: $\\beta\_{\\rho_0,\\rho_0}$, $\\beta\_{\\rho_0,\\rho_1}$, $\\beta\_{\\rho_1,\\rho_1}$, $\\beta\_{\\rho_1,\\rho_2}$, totalling 172 real scalars.
-
-**Inversion:**
-Unlike $D_n$, the bootstrap method does not extend directly to $O$: the symmetric square root of $F_1^T F_1$ introduces a continuous $\\mathrm{SO}(3)$ phase ambiguity that the discrete CG structure cannot resolve. The `invert` method uses bootstrap initialization (exact Fourier norms, approximate phases) followed by Levenberg-Marquardt correction steps with multi-start phase perturbations.
-
-**Usage:**
-
-```python
-bsp = OctaonOcta()
-f = torch.randn(batch_size, 24)   # signal on O (|O| = 24), shape (batch, 24)
-output = bsp(f)                    # shape (batch, 172), complex64
-
-f_rec = bsp.invert(output)         # shape (batch, 24), up to group action
-```
-
-**Constructor parameters:**
-
-```python
-OctaonOcta(
-    selective: bool = True  # Selective (172 coefs) or full bispectrum
-)
-```
-
-______________________________________________________________________
-
-### `SO2onDisk(L: int)` — $\\mathrm{SO}(2)$ on the unit disk
-
-**Mathematical setting:**
-Signal $f: \\mathbb{D} \\to \\mathbb{R}$ on the unit disk. Group $\\mathrm{SO}(2)$ acts by rotation: $(T\_\\phi f)(r, \\theta) = f(r, \\theta - \\phi)$.
-
-**Disk Harmonic Transform** (Fourier-Bessel basis):
-
-$$a\_{n,k} = \\int_0^1 \\int_0^{2\\pi} f(r,\\theta) \\cdot c\_{nk} \\cdot J_n(\\lambda\_{nk} r) \\cdot e^{-in\\theta} \\, r \\, dr \\, d\\theta$$
-
-where $\\lambda\_{nk}$ is the $k$-th positive root of $J_n$ and $c\_{nk}$ is a normalisation constant.
-
-**Selective bispectrum** (Definition 4.2, [Myers & Miolane 2025]):
-
-$$b\_{0,0,k} = a\_{0,1}^2 \\cdot \\overline{a\_{0,k}}, \\qquad b\_{2,n,k} = a\_{1,1} \\cdot a\_{n,1} \\cdot \\overline{a\_{n+1,k}}$$
-
-This reduces the bispectrum from $O(N^3)$ to $O(N)$ coefficients while preserving injectivity up to rotation.
-
-**Usage:**
-
-```python
-bsp = SO2onDisk(L=16)
-f = torch.randn(batch_size, 16, 16)    # signal on disk, shape (batch, L, L)
-output = bsp(f)                         # shape (batch, output_size), complex128
-f_rec = bsp.invert(output)              # shape (batch, L, L), float64
-```
-
-**Constructor parameters:**
-
-```python
-SO2onDisk(
-    L: int,                     # Grid resolution (L x L image)
-    selective: bool = True,     # Selective O(N) or full O(N³) bispectrum
-    bandlimit: float | None = None  # Explicit Bessel root cutoff (default: auto from L)
-)
-```
-
-**Bessel root computation:**
-All Bessel roots are computed internally via `_bessel.py` — a pure-torch implementation using the interlacing property $j\_{n-1,k} < j\_{n,k} < j\_{n-1,k+1}$ with Newton-bisection polishing. No scipy dependency.
-
-______________________________________________________________________
-
-## Selectivity Roadmap
-
-The central value proposition of the library is the *selective* G-bispectrum: a minimal subset of bispectral pairs $(\\rho, \\sigma)$ that suffices for complete signal reconstruction, reducing coefficient count from $O(\\lvert G\\rvert^2)$ to $O(\\lvert G\\rvert)$.
-
-This is proven for finite groups (Mataigne et al. 2024) and extended to $\\mathrm{SO}(3)$ on $S^2$ (this work, generic completeness). Here, `O` denotes the finite octahedral rotation group (not the full orthogonal group). The table below tracks the current state across all group/domain combinations of interest.
-
-| Class          | Group                | Domain                             | Selective?                                      | Inversion?      | Status            |
-| -------------- | -------------------- | ---------------------------------- | ----------------------------------------------- | --------------- | ----------------- |
-| `CnonCn`       | $C_n$                | $C_n$                              | ✅ $n$ coefficients                             | ✅ Algorithm 1  | ✅ Done           |
-| `SO2onS1`      | $\\mathrm{SO}(2)$    | $S^1$                              | ✅ $n$ coefficients (via `CnonCn`)              | ✅ Algorithm 1  | ✅ Done           |
-| `TorusOnTorus` | $\\prod C\_{n_l}$    | $\\prod C\_{n_l}$                  | ✅ $\\lvert G \\rvert$ coefs                    | ✅ Algorithm 2  | ✅ Done           |
-| `DnonDn`       | $D_n$                | $D_n$                              | ✅ $\\lfloor(n{-}1)/2\\rfloor{+}2$ matrix coefs | ✅ Algorithm 3  | ✅ Done           |
-| `OctaonOcta`   | $O$                  | $\\mathbb{R}^3$                    | ✅ 4 matrix coefs (paper App. B)                | ✅ Bootstrap+LM | ✅ Done           |
-| `SO3onS2`      | $\\mathrm{SO}(3)$    | $S^2$                              | ✅ $O(L^2)$ coefficients (generic completeness) | ❌              | ✅ Selective done |
-| `SO2onDisk`    | $\\mathrm{SO}(2)$    | Disk                               | ✅ $N$ coefficients (Myers & Miolane 2025)      | ✅              | ✅ Done           |
-| —              | $\\mathrm{SO}(3)$    | $S^2 \\times \\mathbb{R}^+$ (ball) | ❌ Full only                                    | ❌              | Open problem      |
-| —              | Compact $G$          | $G$                                | ❌ Full only                                    | ❌              | Open problem      |
-| —              | Homogeneous $(H, G)$ | $H = G/G_0$                        | ❌ Full only                                    | ❌              | Open problem      |
-
-Sources: Mataigne et al. 2024 for finite-group ✅ entries; Myers & Miolane 2025 for SO2onDisk; this work for SO3onS2 selective completeness (see `docs/selective_so3_on_s2_completeness.md`).
-
-### Mathematical TODOs
-
-The following are open mathematical problems whose solutions would directly extend the library:
-
-**~~TODO-M1~~ DONE: Selective bispectrum for $\\mathrm{SO}(3)$ on $S^2$**
-
-Solved. The selective SO(3) bispectrum on $S^2$ uses $O(L^2)$ scalar entries instead of $O(L^3)$, achieving the dimensional lower bound $(L+1)^2 - 3$ for a complete invariant. For each degree $l$, up to $2l+1$ bispectral triples are selected from four priority categories (chain, cross, power, self-coupling). At $l \\geq 4$ only linear entries are needed; recovery is a linear solve. Completeness holds for Lebesgue-a.e. band-limited signal (generic completeness). See `docs/selective_so3_on_s2_completeness.md` for the full construction and proof sketch. Implemented in `SO3onS2(selective=True)`.
-
-**~~TODO-M2~~ DONE: Selective bispectrum for $\\mathrm{SO}(2)$ on the disk**
-
-Solved by Myers & Miolane (2025). The selective disk bispectrum (Definition 4.2) reduces from $O(N^3)$ to $O(N)$ coefficients while preserving injectivity (up to rotation). Inversion is available via a bootstrap from the $(0,1)$ coefficient. Implemented in `SO2onDisk`.
-
-**TODO-M3: Selective bispectrum for $\\mathrm{SO}(3)$ on the ball $S^2 \\times \\mathbb{R}^+$**
-
-Same challenge as TODO-M1 plus the radial dimension. The bispectrum formula is identical to the $S^2$ case (same equivariance); selectivity is the open question.
-
-**TODO-M4: Inversion algorithms for continuous groups**
-
-For finite groups, inversion follows from recovering each $F(f)_\\rho$ via a bootstrap from the trivial representation. For compact/continuous groups (where irreps are infinite-dimensional in the $l \\to \\infty$ limit), the analogous reconstruction strategy needs to contend with truncation and stability. Conditioning of the bootstrap in the presence of $l_\\mathrm{max}$ truncation is the key question.
-
-______________________________________________________________________
-
-## Public API Surface
-
-The top-level `bispectrum` namespace exposes only what a user needs:
-
-```python
-# Main modules
-from bispectrum import CnonCn, DnonDn, OctaonOcta, SO2onDisk, SO2onS1, SO3onS2, TorusOnTorus
-
-# Rotation utilities (useful for testing/data augmentation)
-from bispectrum import random_rotation_matrix, rotate_spherical_function
-```
-
-**Explicitly not exported:**
-
-- `clebsch_gordan` (internal implementation detail; current placeholder removed)
-- `compute_padding_indices`, `pad_sh_coefficients`, `get_full_sh_coefficients` (SO(3) internals)
-- Any `_private` helper functions
-
-We do not depend on `escnn` or any external library for Clebsch-Gordan coefficients. Instead:
-
-- **$C_n$**: No CG matrices needed. The bispectrum for commutative groups reduces to scalar products — no change-of-basis required.
-- **$D_n$**: CG matrices are computed analytically from the explicit 2D irrep formulas (given in the paper). These are closed-form rotation matrices; no library is needed.
-- **$\\mathrm{SO}(3)$**: CG matrices are computed analytically via the Wigner 3j symbol (Racah formula) with log-factorial numerics for stability at large $l$. Results are cached to `~/.cache/bispectrum/cg_lmax{N}.pt` after first computation. Any `lmax` is supported — no bundled JSON limit.
-
-This keeps the dependency footprint minimal and makes the math transparent — CG coefficients are computed from first principles, not treated as a black box.
-
-______________________________________________________________________
-
-## What Goes in Each File
-
-```
-src/bispectrum/
-├── __init__.py          # Public exports only
-├── cn_on_cn.py          # CnonCn
-├── so2_on_s1.py         # SO2onS1 (subclass of CnonCn)
-├── torus_on_torus.py    # TorusOnTorus
-├── dn_on_dn.py          # DnonDn
-├── octa_on_octa.py      # OctaonOcta
-├── so3_on_s2.py         # SO3onS2 (refactored)
-├── so2_on_disk.py       # SO2onDisk
-├── rotation.py          # random_rotation_matrix, rotate_spherical_function
-├── _bessel.py           # Internal Bessel function utilities (not exported)
-└── _cg.py               # Internal CG utilities (not exported)
-```
-
-Files removed in v0.2.0:
-
-- `clebsch_gordan.py` → merged into `_cg.py` (internal)
-- `spherical.py` → folded into `so3_on_s2.py`
-- `so3.py` → replaced by `so3_on_s2.py`
-
-______________________________________________________________________
-
-## Testing Philosophy
-
-Every bispectrum module must have tests for:
-
-1. **Output shape** — `(batch, output_size)` for various batch sizes
-2. **G-invariance** — `bsp(T_g(f)) ≈ bsp(f)` for random $g$, random $f$
-   This is the *most important test*. If this fails, nothing else matters.
-3. **Determinism** — same input → same output
-4. **Device/dtype compatibility** — works on CPU, moves to GPU correctly
-5. **No trainable parameters** — `sum(p.numel() for p in bsp.parameters()) == 0`
-6. **Numerical precision** — invariance holds to `atol=1e-4` in float32
-
-For every module where inversion is mathematically available:
-7\. **Inversion test** — call `bsp.invert(beta)` and check reconstruction error up to the known group-action indeterminacy.
-
-For modules where inversion is not yet available (e.g., current SO(3) selective roadmap):
-8\. **Explicit NotImplemented test** — ensure `bsp.invert(...)` raises a clear, documented `NotImplementedError`.
-
-```python
-# Example: canonical invariance test pattern
-def test_invariance(self):
-    bsp = CnonCn(n=8)
-    f = torch.randn(4, 8)
-    shift = 3  # arbitrary cyclic shift
-    f_shifted = torch.roll(f, shift, dims=-1)
-
-    torch.testing.assert_close(bsp(f), bsp(f_shifted), atol=1e-4, rtol=1e-4)
-```
-
-______________________________________________________________________
-
-## Migration from v0.1.0
-
-| v0.1.0                                | v0.2.0                                          | Notes                                        |
-| ------------------------------------- | ----------------------------------------------- | -------------------------------------------- |
-| `SO3onS2(lmax=5)(coeffs)`             | `SO3onS2(lmax=5, nlat=64, nlon=128)(f_spatial)` | Now takes spatial signal                     |
-| `bispectrum(f_coeffs, l1, l2, cg_fn)` | Removed                                         | Use `SO3onS2` module                         |
-| `clebsch_gordan(l1, l2)`              | Removed                                         | Internal; computed analytically via `_cg.py` |
-| `compute_padding_indices(...)`        | Removed (internal)                              |                                              |
-| `pad_sh_coefficients(...)`            | Removed (internal)                              |                                              |
-| `get_full_sh_coefficients(...)`       | Removed (internal)                              |                                              |
-
-______________________________________________________________________
-
-## References
-
-- Myers & Miolane (2025). *The Selective Disk Bispectrum and Its Inversion, with Application to Multi-Reference Alignment*. arXiv preprint. [arXiv:2511.19706](https://arxiv.org/abs/2511.19706)
-- Mataigne et al. (2024). *The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks*. NeurIPS 2024. [arXiv:2407.07655](https://arxiv.org/abs/2407.07655)
-- Kakarala (1992). *Triple Correlation on Groups*. PhD thesis, UC Irvine.
-- Kakarala (2009). *Bispectrum on Finite Groups*. ICASSP 2009.
-- Cohen & Welling (2016). *Group Equivariant Convolutional Networks*. ICML 2016.
-- Weiler & Cesa (2019). *General E(2)-Equivariant Steerable CNNs*. NeurIPS 2019.
diff --git a/README.md b/README.md
index b1e62d1..e59f7df 100644
--- a/README.md
+++ b/README.md
@@ -1,10 +1,45 @@
 # bispectrum
 
-[![Tests](https://github.com/geometric-intelligence/bispectrum/actions/workflows/tests.yml/badge.svg)](https://github.com/geometric-intelligence/bispectrum/actions/workflows/tests.yml)
-[![Pre-commit](https://github.com/geometric-intelligence/bispectrum/actions/workflows/pre-commit.yml/badge.svg)](https://github.com/geometric-intelligence/bispectrum/actions/workflows/pre-commit.yml)
-[![codecov](https://codecov.io/github/geometric-intelligence/bispectrum/graph/badge.svg?token=J6GGY4VK1E)](https://codecov.io/github/geometric-intelligence/bispectrum)
+An open-source, fully unit-tested PyTorch library that implements *selective* G-bispectra for seven group actions as differentiable `torch.nn.Module`s, ready to plug into ML pipelines and deep learning architectures.
 
-Bispectrum analysis for machine learning.
+The G-bispectrum is a principled *complete* invariant of a signal — it retains all information up to the group action. Selectivity reduces cost from O(|G|²) to O(|G|) for finite groups, and from O(L³) to Θ(L²) coefficients for SO(3) on S².
+
+## Supported Groups
+
+| Module | Group / Domain | Output mode | Complexity (selective) |
+|--------|---------------|-------------|----------------------|
+| `CnonCn` | C_n on C_n | selective + full | O(n) |
+| `SO2onS1` | SO(2) on S¹ | selective + full | O(n) |
+| `TorusOnTorus` | T^d | selective + full | O(n) |
+| `DnonDn` | D_n on D_n | selective | O(n) |
+| `SO2onDisk` | SO(2) on disk | selective | O(L) |
+| `SO3onS2` | SO(3) on S² | selective + full | Θ(L²) |
+| `OctaonOcta` | chiral octahedral O (24 elements) | selective | 172 coefficients |
+
+`SO2onS1` is the continuous-n limit of `CnonCn` and shares its implementation.
+
+## API
+
+Every module exposes a uniform interface:
+
+- **`forward(f)`** — selective (default) or full bispectral invariants
+- **`fourier(f)`** — group Fourier coefficients
+- **`invert(beta)`** — signal reconstruction up to group-action indeterminacy (where available)
+
+Modules default to O(|G|) selective coefficients; pass `selective=False` for the full O(|G|²) set. CG matrices, DFT kernels, and Bessel roots are precomputed as non-learnable buffers. Dependencies: PyTorch, NumPy, and `torch_harmonics` (for `SO3onS2`).
+
+## Benchmarks
+
+Median wall-clock on a single NVIDIA H100 80 GB GPU (batch=16, `torch.utils.benchmark`):
+
+| Module | Group | \|G\| / L_max | Coefs (sel.) | Coefs (full) | Fwd sel. (ms) | Fwd full (ms) |
+|--------|-------|-----------|------------|-------------|--------------|----------------|
+| `CnonCn` | C_128 | 128 | 128 | 8,256 | 0.14 | 8.53 |
+| `TorusOnTorus` | C_32 × C_32 | 1,024 | 1,024 | 524,800 | 0.07 | 0.31 |
+| `DnonDn` | D_32 | 64 | 245 | — | 0.57 | — |
+| `SO2onDisk` | SO(2) | L=16 | 105 | — | 0.22 | — |
+| `SO3onS2` | SO(3) | L=16 | 430 | — | 0.48 | — |
+| `OctaonOcta` | O | 24 | 172 | — | 0.68 | — |
 
 ## Installation
 
@@ -21,8 +56,6 @@ uv pip install bispectrum
 ### Development
 
 ```bash
-git clone https://github.com/geometric-intelligence/bispectrum.git
-cd bispectrum
 uv pip install -e ".[dev]"
 pre-commit install
 ```
@@ -81,8 +114,6 @@ beta = bsp(f)                 # shape (4, 172), complex64
 f_rec = bsp.invert(beta)      # reconstructed up to group action
 ```
 
-See [DESIGN.md](DESIGN.md) for the full API and all supported groups.
-
 ## License
 
 MIT
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+++ /dev/null
@@ -1,23 +0,0 @@
-\begin{table}[t]
-    \centering
-    \caption{Benchmark summary for the \texttt{bispectrum} library.
-    Coefficient counts compare the selective ($O(|G|)$) and full ($O(|G|^2)$) bispectra.
-    Timings are median wall-clock on a single NVIDIA A100 GPU (batch\,=\,16 for forward, 4 for inversion).
-    ``--'' indicates the mode is not implemented.}
-    \label{tab:benchmarks}
-    \small
-    \begin{tabular}{llrrrrrr}
-        \toprule
-        Module & $G$ & $|G|$ & \multicolumn{2}{c}{Coefs} & \multicolumn{2}{c}{Fwd (ms)} & Inv (ms) \\
-        \cmidrule(lr){4-5} \cmidrule(lr){6-7}
-         & & & Sel. & Full & Sel. & Full & Sel. \\
-        \midrule
-        \texttt{CnonCn} & $C_{128}$ & 128 & 128 & 8256 & 0.15 & 9.17 & 8.56 \\
-        \texttt{TorusOnTorus} & $C_{32}{\times}C_{32}$ & 1024 & 1024 & 524800 & 0.08 & 0.71 & 68.02 \\
-        \texttt{DnonDn} & $D_{32}$ & 64 & 245 & -- & 2.52 & -- & 42.33 \\
-        \texttt{SO2onDisk} & $\mathrm{SO}(2)$ & 1024 & 411 & -- & 1.94 & -- & 17.69 \\
-        \texttt{SO3onS2} & $\mathrm{SO}(3)$ & 289 & 430 & -- & 26.54 & -- & -- \\
-        \texttt{OctaonOcta} & $O$ & 24 & 172 & -- & 0.70 & -- & 2172.76 \\
-        \bottomrule
-    \end{tabular}
-\end{table}
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diff --git a/benchmarks/inversion_results/inversion_stats.tex b/benchmarks/inversion_results/inversion_stats.tex
deleted file mode 100644
index b835b17..0000000
--- a/benchmarks/inversion_results/inversion_stats.tex
+++ /dev/null
@@ -1,18 +0,0 @@
-\begin{table}[h]
-\centering
-\caption{Octahedral bispectrum inversion statistics (N=1000, 4 restarts $\times$ 10 LM steps).}
-\label{tab:inversion-stats}
-\begin{tabular}{lcc}
-\toprule
-Metric & Median & P95 \\
-\midrule
-Bisp.\ residual $\|\beta(\hat{f}) - \beta^*\|$ & 5.7e-05 & 1.5e-04 \\
-Recovery error (mod $O$) & 1.0e+00 & 1.2e+00 \\
-Wall-clock time (ms) & 18.4 & 30.1 \\
-\midrule
-Success rate (error <0.1) & \multicolumn{2}{c}{5.7%} \\
-Success rate (error <0.01) & \multicolumn{2}{c}{5.7%} \\
-Success rate (error <0.001) & \multicolumn{2}{c}{5.7%} \\
-\bottomrule
-\end{tabular}
-\end{table}
diff --git a/benchmarks/paper_benchmarks.pdf b/benchmarks/paper_benchmarks.pdf
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+++ /dev/null
@@ -1,190 +0,0 @@
-\documentclass[11pt]{article}
-\usepackage[margin=1in]{geometry}
-\usepackage{graphicx}
-\usepackage{booktabs}
-\usepackage{amsmath,amssymb}
-\usepackage{hyperref}
-\usepackage[T1]{fontenc}
-\usepackage[expansion=false]{microtype}
-
-\title{Benchmark Report for the \texttt{bispectrum} Library}
-\author{}
-\date{\today}
-
-\begin{document}
-\maketitle
-
-\section{Overview}
-
-This document presents computational benchmarks for the \texttt{bispectrum}
-library, a PyTorch implementation of the group bispectrum for invariant signal
-representations.
-The library provides six distinct modules, each implementing the bispectrum for
-a different group acting on a different space:
-\begin{itemize}
-    \item \texttt{CnonCn} --- cyclic group $C_n$ acting on itself;
-    \item \texttt{TorusOnTorus} --- product of cyclic groups
-          $C_{n_1} \times \cdots \times C_{n_d}$ (torus $\mathbb{T}^d$)
-          acting on itself;
-    \item \texttt{DnonDn} --- dihedral group $D_n$ acting on itself;
-    \item \texttt{SO2onD2} --- continuous rotation group $\mathrm{SO}(2)$
-          acting on the 2-disk $D^2$ (via truncated Bessel basis);
-    \item \texttt{SO3onS2} --- rotation group $\mathrm{SO}(3)$ acting on
-          the 2-sphere $S^2$ (via spherical harmonics);
-    \item \texttt{OctaonOcta} --- octahedral group $O$ (order 24) acting
-          on the octahedron.
-\end{itemize}
-
-Each module is a \texttt{torch.nn.Module} exposing a \texttt{forward(f)}
-method that computes the bispectrum of an input signal~$f$, and (where
-available) an \texttt{invert(beta)} method that recovers a signal from its
-bispectral coefficients up to the group action.
-
-A key design axis is \emph{selective} versus \emph{full} bispectrum: the full
-bispectrum enumerates all $O(|G|^2)$ pairwise products of Fourier
-coefficients, while the selective bispectrum retains only the $O(|G|)$
-coefficients needed for injectivity, following the theoretical framework of
-Kakarala~(2012).
-
-All benchmarks were run on a single NVIDIA A100 80\,GB PCIe GPU with PyTorch
-2.10.
-
-\section{Coefficient Scaling}
-
-\begin{figure}[ht]
-    \centering
-    \includegraphics[width=\textwidth]{figures/paper_fig1.pdf}
-    \caption{%
-        Number of bispectral coefficients as a function of group order~$|G|$
-        for all six modules.
-        Solid lines with filled markers: selective bispectrum; dashed lines:
-        full bispectrum (shown for $C_n$ and $\mathbb{T}^2$ only, the two
-        modules that support it).
-        The gray $O(|G|)$ and $O(|G|^2)$ reference slopes are shown, with
-        the shaded region indicating where the full bispectrum lies.
-        All selective curves track the $O(|G|)$ slope.
-        The continuous-group modules ($\mathrm{SO}(2)/D^2$ and
-        $\mathrm{SO}(3)/S^2$) also follow the linear trend when plotted
-        against their effective bandwidth parameter.
-        The octahedral group $O$ ($|G|=24$, star) yields 172 selective
-        coefficients.}
-    \label{fig:coeff}
-\end{figure}
-
-\paragraph{Discussion.}
-Figure~\ref{fig:coeff} confirms that the selective bispectrum achieves the
-theoretically predicted $O(|G|)$ coefficient count across all module families,
-providing a quadratic reduction over the full $O(|G|^2)$ variant.
-For the cyclic group at $|G| = 1024$, this translates from 524\,800 full
-coefficients down to 1\,024 selective coefficients---a $512\times$ reduction.
-The $D_n$ selective counts lie slightly above the $O(|G|)$ reference because
-$D_n$ has more irreducible representations than $C_n$ at the same group order
-(due to the semidirect product structure), but the scaling remains linear.
-The continuous-group modules ($\mathrm{SO}(2)/D^2$ and $\mathrm{SO}(3)/S^2$)
-exhibit comparable linear scaling when measured against the square of their
-truncation bandwidth, confirming that the selective strategy extends
-beyond finite groups.
-
-\section{Forward-Pass Timing}
-
-\begin{figure}[ht]
-    \centering
-    \includegraphics[width=\textwidth]{figures/paper_fig2.pdf}
-    \caption{%
-        Forward-pass wall-clock time (batch size~16, NVIDIA A100 GPU) for the
-        selective versus full bispectrum.
-        \textbf{(a)}~$C_n$ on $C_n$: the selective forward pass stays flat at
-        $\sim$0.19\,ms regardless of~$n$, while the full bispectrum grows
-        super-linearly to 25\,ms at $n=256$.
-        \textbf{(b)}~$\mathbb{T}^2 = C_n \times C_n$: the selective pass is
-        nearly constant ($\sim$0.1\,ms) up to $|G| = 65{,}536$, while the
-        full bispectrum diverges beyond $|G| \approx 1{,}000$.}
-    \label{fig:forward}
-\end{figure}
-
-\paragraph{Discussion.}
-Figure~\ref{fig:forward} demonstrates that the selective forward pass is
-\emph{effectively free} on modern GPUs: the wall-clock time is dominated by
-kernel-launch overhead rather than arithmetic, staying flat at
-$\sim$0.1--0.2\,ms even as the group order increases by two orders of
-magnitude.
-The full bispectrum, in contrast, exhibits the expected super-linear growth and
-rapidly becomes impractical for production use.
-For the torus at $|G| = 1{,}024$ ($n = 32$), the full forward pass is already
-$5\times$ slower than selective; extrapolating the quadratic trend to $n = 256$
-would exceed 100\,ms per sample.
-
-\section{GPU Batch Scaling and Inversion}
-
-\begin{figure}[ht]
-    \centering
-    \includegraphics[width=\textwidth]{figures/paper_fig3.pdf}
-    \caption{%
-        \textbf{(a)} GPU throughput (samples/s) as a function of batch size
-        for the selective forward pass at representative group sizes.
-        All modules exhibit near-linear scaling, confirming the selective
-        bispectrum is well suited to batched deep-learning workloads.
-        $C_n$ and $\mathbb{T}^2$ achieve over $10^7$~samples/s.
-        The octahedral group plateaus at $\sim$24\,k~samples/s due to its
-        fixed small signal dimension ($|G| = 24$) and CUDA launch overhead.
-        \textbf{(b)} Inversion wall-clock time for all modules supporting
-        \texttt{invert()} (batch size~4).
-        $C_n$, $D_n$, and $\mathrm{SO}(2)/D^2$ scale roughly linearly.
-        $\mathbb{T}^2$ grows as $O(|G| \log |G|)$ from multi-dimensional
-        inverse FFTs.
-        The octahedral group (star) requires $\sim$1\,s despite $|G| = 24$
-        due to the least-squares optimization over all irreducible
-        representations of~$O$.
-        \texttt{SO3onS2} is absent because its inversion is not yet
-        implemented.}
-    \label{fig:scaling}
-\end{figure}
-
-\paragraph{Discussion.}
-Figure~\ref{fig:scaling}(a) shows that all selective-bispectrum modules scale
-linearly with batch size on the GPU, which is the ideal behavior for
-integration into batched training pipelines.
-The throughput spans roughly three orders of magnitude across modules,
-reflecting the underlying signal dimension: $C_{128}$ and $\mathbb{T}^2$
-($|G| = 1024$) process each sample in microseconds, while $O$ is much slower
-per sample due to the structure of the octahedral bispectrum computation.
-The key takeaway is that \emph{none of the modules exhibit throughput
-degradation at large batch sizes}, confirming that the implementation is
-memory-bandwidth-bound rather than compute-bound at these scales.
-
-Figure~\ref{fig:scaling}(b) reveals that inversion cost varies significantly
-across modules.
-For $C_n$ and $D_n$, inversion is essentially a linear-time operation (inverse
-FFT plus phase retrieval), scaling as $O(|G|)$.
-The $\mathrm{SO}(2)/D^2$ module shows similar linear scaling despite operating
-over a continuous group (the truncated Bessel basis provides a
-finite-dimensional approximation).
-The torus module's inversion is more expensive, growing as $O(|G| \log |G|)$
-because it involves multi-dimensional inverse FFTs.
-The octahedral group is an outlier: its inversion requires $\sim$1\,s for a
-group of order~24, reflecting the combinatorial complexity of inverting over
-the 5 non-trivial irreducible representations of~$O$ and the associated
-least-squares optimization.
-
-\section{Summary Table}
-
-\input{figures/table_benchmarks.tex}
-
-Table~\ref{tab:benchmarks} summarizes key data points for each module at a
-representative group size.
-The selective coefficient count is always $O(|G|)$: for $C_{128}$, it equals
-$|G| = 128$; for $D_{32}$ ($|G| = 64$), the 245 coefficients reflect the
-richer representation theory of the dihedral group.
-$\mathrm{SO}(2)/D^2$ at $L = 32$ (bandwidth 1024) yields 411 selective
-coefficients and requires $\sim$12\,ms for a forward pass, dominated by the
-Bessel-function evaluations in the radial basis.
-$\mathrm{SO}(3)/S^2$ at $\ell_{\max} = 5$ produces 69 coefficients with a
-comparable forward time; its inversion is not yet implemented.
-
-The ``full'' columns are only applicable to the abelian groups ($C_n$ and
-$\mathbb{T}^2$), where the full bispectrum is the classical definition.
-For non-abelian groups ($D_n$, $O$) and continuous groups ($\mathrm{SO}(2)$,
-$\mathrm{SO}(3)$), only the selective bispectrum is implemented, as the full
-enumeration would be either ill-defined or computationally intractable.
-
-\end{document}
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-\documentclass[aspectratio=169]{beamer}
-\usetheme{metropolis}
-\usepackage{booktabs}
-\usepackage{amsmath,amssymb}
-\usepackage{listings}
-
-\lstset{
-  basicstyle=\ttfamily\scriptsize,
-  keywordstyle=\color{blue},
-  commentstyle=\color{gray},
-  breaklines=true,
-  frame=none,
-}
-
-\title{Bispectrum Library: From Finite Groups to the Disk}
-\subtitle{Implementing Mataigne et al.\ (NeurIPS 2024) \& Myers \& Miolane (arXiv 2025)}
-\author{Johan Mathe}
-\date{\today}
-
-\begin{document}
-
-\maketitle
-
-% ============================================================
-\begin{frame}{Goal}
-  Reimplement the selective $G$-bispectrum from Mataigne et al.\ (NeurIPS 2024)
-  in a standalone, clean PyTorch library.
-
-  \bigskip
-
-  Three group/domain families implemented:
-  \begin{itemize}
-    \item \textbf{CnonCn} --- cyclic groups $C_n$ acting on $\mathbb{Z}/n\mathbb{Z}$ \hfill\textit{Mataigne et al.}
-    \item \textbf{DnonDn} --- dihedral groups $D_n$ acting on $D_n$ \hfill\textit{Mataigne et al.}
-    \item \textbf{SO2onD2} --- $\mathrm{SO}(2)$ acting on the unit disk $\mathbb{D}^2$ \hfill\textit{Myers \& Miolane}
-  \end{itemize}
-
-  \bigskip
-
-  All three include \texttt{forward()} (bispectrum) and \texttt{invert()} (signal recovery).
-
-  \smallskip
-
-  Plus \textbf{SO3onS2} ($\mathrm{SO}(3)$ on $S^2$) --- forward only, no inversion.
-\end{frame}
-
-% ============================================================
-\begin{frame}{Improvements over g-invariance repo}
-  \begin{table}
-    \scriptsize
-    \begin{tabular}{lll}
-      \toprule
-      \textbf{Aspect} & \textbf{g-invariance} & \textbf{bispectrum (new)} \\
-      \midrule
-      Dependencies & escnn, einops, scipy & Pure PyTorch \\
-      CG computation & \texttt{scipy.linalg.schur} & \texttt{torch.linalg.eig} \\
-      Group size & $n \geq 8$ enforced & All $n > 2$ \\
-      $n_3$ for odd $n$ & Bug: gives 0 for $n=3$ & Fixed \\
-      Inversion & Not implemented & Algorithms 2 \& 4 \\
-      Interface & escnn \texttt{GeometricTensor} & Raw \texttt{torch.Tensor} \\
-      Normalization & Baked in & None (raw coefficients) \\
-      Group DFT & $O(n^2)$ explicit & $O(n \log n)$ via FFT \\
-      Tests & \textbf{None} & \textbf{180 tests} \\
-      \bottomrule
-    \end{tabular}
-  \end{table}
-\end{frame}
-
-% ============================================================
-\begin{frame}{Bugs found in g-invariance (1/2)}
-  \textbf{Bug 1: $n_3$ calculation breaks for small odd $n$}
-
-  \smallskip
-
-  g-invariance code: \texttt{n3 = n2 - 1} where $n_2 = \lfloor(n{-}1)/2\rfloor$.
-
-  For $n=3$: $n_2 = 1$, so $n_3 = 0$ --- meaning \emph{zero}
-  $\beta_{\rho_1,\rho_k}$ coefficients are computed.
-
-  \smallskip
-
-  \textbf{Impact}: $\beta_{\rho_1,\rho_1}$ is needed to recover $\mathcal{F}(\rho_{01})$
-  (Appendix~E of the paper proves $\rho_{01} \in \rho_1 \otimes \rho_1$).
-  Without it, the selective bispectrum for $D_3$ is \emph{incomplete} ---
-  the 1D irrep $\rho_{01}$ cannot be recovered during inversion.
-
-  \smallskip
-
-  \textbf{How discovered}: Writing parametrized tests over $n \in \{3, 4, 5, 7, 8\}$.
-  $D_3$ inversion failed because \texttt{output\_size} was only $1 + 4 = 5$
-  (missing the $16 \times n_3$ block entirely).
-  The g-invariance repo never tested $n < 8$ due to bug~2.
-
-  \smallskip
-
-  \textbf{Fix}: \texttt{n3 = n2 if n \% 2 == 0 else max(n2 - 1, 1)}.
-\end{frame}
-
-% ============================================================
-\begin{frame}{Bugs found in g-invariance (2/2)}
-  \textbf{Bug 2: Artificial $n \geq 8$ restriction}
-
-  \smallskip
-
-  \texttt{pooling.py} line 390: \texttt{if n < 8: raise ValueError}.
-
-  \smallskip
-
-  This came from escnn integration constraints, not from the math.
-  $D_n$ bispectrum is well-defined for all $n > 2$. Dropping this
-  restriction immediately exposed Bug~1.
-
-  \bigskip
-
-  \textbf{Bug 3: Paper typo in Algorithm 4} (next slide).
-
-  \bigskip
-
-  \textbf{Note}: The g-invariance repo had \textbf{zero unit tests},
-  so these bugs were never caught. Bug~1 is silent --- it doesn't crash,
-  it just computes an incomplete bispectrum.
-\end{frame}
-
-% ============================================================
-\begin{frame}{Paper typo in Algorithm 4 (bispectrum inversion on $D_n$)}
-  Algorithm 4, line 6 computes the block-diagonal Fourier coefficients via:
-  \[
-    \bigoplus_{\rho \in \rho_1 \otimes \rho_{k-1}} \mathcal{F}(\Theta)_\rho
-    = \Bigl(
-      C_{\rho_1,\rho_{k-1}}^\dagger
-      \bigl[\mathcal{F}_{\rho_1} \otimes \mathcal{F}_{\rho_{k-1}}\bigr]^{-1}
-      \beta_{\rho_1,\rho_{\mathbf{2}}}
-      \, C_{\rho_1,\rho_{k-1}}
-    \Bigr)^\dagger
-  \]
-
-  The subscript on $\beta$ is printed as $\rho_2$ (fixed), but should be
-  $\rho_{k-1}$ (varying with the loop index $k$). The CG subscripts
-  $C_{\rho_1,\rho_{k-1}}$ are correct.
-
-  \bigskip
-
-  \textbf{How discovered}: When implementing inversion for $n=8$ (which has
-  $\lfloor\frac{7}{2}\rfloor = 3$ 2D irreps), the loop needs
-  $\beta_{\rho_1,\rho_1}$ then $\beta_{\rho_1,\rho_2}$ to recover
-  $\mathcal{F}(\rho_2)$ then $\mathcal{F}(\rho_3)$.
-  Using a fixed $\beta_{\rho_1,\rho_2}$ for both iterations gives wrong
-  Fourier coefficients --- the Frobenius norm roundtrip test fails.
-
-  \bigskip
-
-  The g-invariance code indexes $\beta$ by \texttt{k-1} in the loop, so it
-  was already correct despite the paper typo.
-
-  \smallskip
-
-  \textbf{Note}: Typo still present in the NeurIPS 2024 camera-ready
-  (Eq.~\texttt{eq:new\_coeff\_dihedral} and Algorithm~4, Appendix~E).
-\end{frame}
-
-% ============================================================
-\begin{frame}{Technical challenges: CG matrices without scipy}
-  \textbf{Goal}: find orthogonal $C$ such that
-  $(\rho_i \otimes \rho_j)(g) = C \bigl[\bigoplus_\rho \rho(g)\bigr] C^\top$
-  for all $g \in D_n$.
-
-  \smallskip
-
-  The g-invariance repo used \texttt{scipy.linalg.schur}. We need pure PyTorch.
-
-  \smallskip
-
-  \textbf{Step 1}: Eigendecompose $Q_a = \operatorname{kron}(\rho_i(a), \rho_j(a))$
-  via \texttt{torch.linalg.eig}. This $4\times 4$ orthogonal matrix has
-  eigenvalues $e^{\pm i\theta}$ (conjugate pairs $\to$ 2D irreps) or $\pm 1$
-  (real $\to$ 1D irreps).
-
-  \smallskip
-
-  \textbf{Step 2 --- 2D blocks}: For each conjugate pair $e^{\pm i\theta}$,
-  take $u_1 = \operatorname{Re}(v)$, $u_2 = \operatorname{Im}(v)$.
-  In this basis the block is $R(-\theta)$, but our irrep convention is
-  $\rho_k(a) = R(+2\pi k/n)$.
-  Fix: negate $u_2$ when $\theta > 0$ so the block becomes $R(+|\theta|)$.
-
-  \smallskip
-
-  \textbf{Step 3 --- 1D blocks}: When two 1D irreps share the same eigenvalue
-  under $a$ (e.g.\ $\rho_0$ and $\rho_{01}$ both have $+1$), the eigenspace is
-  2D and \texttt{eig} won't split them. We restrict
-  $Q_x = \operatorname{kron}(\rho_i(x), \rho_j(x))$ to the degenerate
-  subspace and diagonalize with \texttt{eigh}; its eigenvalues $\pm 1$ separate
-  the two 1D irreps.
-
-  \smallskip
-
-  \textbf{Step 4 --- reflection alignment}: Rotate each 2D subspace by angle
-  $\alpha$ so that $C^\top Q_x\, C = \operatorname{diag}(1,-1)$, matching
-  the standard irrep form $\rho_k(x) = \bigl[\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}\bigr]$.
-\end{frame}
-
-% ============================================================
-\begin{frame}{Technical challenges: bispectrum formula}
-  Non-commutative bispectrum (Theorem 2.3):
-  \[
-    \beta_{\rho_1,\rho_2} = \bigl[\mathcal{F}_{\rho_1} \otimes \mathcal{F}_{\rho_2}\bigr]
-    \, C \, \bigl[\bigoplus_\rho \mathcal{F}_\rho^\top\bigr] \, C^\top
-  \]
-
-  \bigskip
-
-  Getting the transpose/adjoint conventions right for real-valued irreps
-  required several iterations. Initial implementation broke $D_n$ invariance
-  due to incorrect matrix multiplication order.
-
-  \bigskip
-
-  Verified by testing invariance under all $n$ rotations and reflections
-  for $n \in \{3, 4, 5, 7, 8, 16\}$.
-\end{frame}
-
-% ============================================================
-\begin{frame}{Technical challenges: $O(2)$ indeterminacy in inversion}
-  Algorithm 4 recovers $\mathcal{F}(\rho_1)$ from:
-  \[
-    \frac{\beta_{\rho_0,\rho_1}}{\mathcal{F}(\rho_0)} = \mathcal{F}(\rho_1)^\top \mathcal{F}(\rho_1)
-  \]
-
-  This determines $\mathcal{F}(\rho_1)$ only up to $Q \in O(2)$: any $Q \cdot \mathcal{F}(\rho_1)$ also satisfies the equation.
-
-  \bigskip
-
-  \textbf{Consequence}: exact bispectrum roundtrip $\beta \to \text{invert} \to \text{forward} \to \beta' = \beta$ cannot hold for all coefficients. This is a \emph{theoretical} limitation, not a bug.
-
-  \bigskip
-
-  \textbf{What we guarantee}:
-  \begin{itemize}
-    \item $\beta_{\rho_0,\rho_0}$ and $\beta_{\rho_0,\rho_1}$ roundtrip exactly (they are $O(2)$-invariant)
-    \item Frobenius norms $\|\mathcal{F}(\rho_k)\|_F$ match for all $k$
-  \end{itemize}
-\end{frame}
-
-% ============================================================
-\begin{frame}{From group DFT to standard FFT: the derivation}
-  The $D_n$ group DFT for 2D irrep $\rho_k$ sums over rotation and reflection elements:
-  \[
-    \hat{f}(\rho_k) = \sum_{l=0}^{n-1} f_{\text{rot}}(l)\, R\!\left(\tfrac{2\pi kl}{n}\right)^{\!\top}
-    + \sum_{l=0}^{n-1} f_{\text{ref}}(l)\, \bigl[R\!\left(\tfrac{2\pi kl}{n}\right) \operatorname{diag}(1,-1)\bigr]^{\!\top}
-  \]
-
-  Expand the $2\times 2$ entries, noting $R(\theta)^\top = \bigl[\begin{smallmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\end{smallmatrix}\bigr]$:
-  \[
-    \hat{f}_{00}^{(k)} = \textstyle\sum_l f_{\text{rot}}(l)\cos\theta_{kl} + \sum_l f_{\text{ref}}(l)\cos\theta_{kl}
-  \]
-
-  But $\texttt{fft}(f)[k] = \sum_l f(l)\, e^{-i 2\pi kl/n}$, so
-  $\operatorname{Re}(F_{\text{rot}}[k]) = \sum_l f_{\text{rot}}(l) \cos\theta_{kl}$.
-
-  \smallskip
-
-  Therefore all four matrix entries are Re/Im combinations of two standard FFTs:
-  \[
-    \hat{f}^{(k)} = \begin{bmatrix}
-      \operatorname{Re}F_r + \operatorname{Re}F_x & \operatorname{Im}F_r - \operatorname{Im}F_x \\
-      -\operatorname{Im}F_r - \operatorname{Im}F_x & \operatorname{Re}F_r - \operatorname{Re}F_x
-    \end{bmatrix}
-  \]
-  where $F_r = \texttt{fft}(f_{\text{rot}})$ and $F_x = \texttt{fft}(f_{\text{ref}})$.
-  Two $O(n\log n)$ FFTs replace one $O(n^2)$ matrix--vector product.
-
-  \smallskip
-
-  The inverse is analogous: solve for $F_r[k], F_x[k]$ from the four entries,
-  enforce Hermitian symmetry $F[n{-}k] = \overline{F[k]}$, then \texttt{ifft}.
-\end{frame}
-
-% ============================================================
-\begin{frame}{FFT-based group DFT: 100--275$\times$ speedup}
-  Replaced $O(n^2)$ explicit DFT (einsum against precomputed rotation matrices)
-  with two calls to \texttt{torch.fft.fft} / \texttt{torch.fft.ifft}.
-
-  \bigskip
-
-  Key insight: each entry of the $2\times 2$ Fourier coefficient $\hat{f}(\rho_k)$
-  is a linear combination of $\operatorname{Re}/\operatorname{Im}$ of the standard FFT
-  of the rotation and reflection halves of the signal.
-
-  \bigskip
-
-  \begin{table}
-    \scriptsize
-    \begin{tabular}{lrrrl}
-      \toprule
-      \textbf{Method} & $n$ & \textbf{Before} & \textbf{After} & \textbf{Speedup} \\
-      \midrule
-      \texttt{\_group\_dft}   &   64 & 50 ms   & 0.23 ms & $218\times$ \\
-      \texttt{\_group\_dft}   &  256 & 76 ms   & 0.28 ms & $275\times$ \\
-      \texttt{\_group\_dft}   & 1024 & 89 ms   & 0.44 ms & $205\times$ \\
-      \texttt{\_inverse\_dft} &   64 & 2.3 ms  & 0.27 ms & $8.5\times$ \\
-      \texttt{\_inverse\_dft} &  256 & 10.5 ms & 0.33 ms & $32\times$ \\
-      \texttt{\_inverse\_dft} & 1024 & 71 ms   & 0.56 ms & $127\times$ \\
-      \bottomrule
-    \end{tabular}
-  \end{table}
-
-  Also removes $O(n^2)$ memory from precomputed rotation buffers.
-\end{frame}
-
-% ============================================================
-% SO2onD2 SECTION
-% ============================================================
-
-\begin{frame}{}
-  \vfill
-  \centering
-  {\Large\textbf{New: SO2onD2}}
-
-  \medskip
-
-  {\normalsize Implementing the Selective Disk Bispectrum}
-
-  \smallskip
-
-  {\small Myers \& Miolane, arXiv:2511.19706, 2025}
-  \vfill
-\end{frame}
-
-% ============================================================
-\begin{frame}{SO2onD2: What and why}
-  \textbf{Setting}: $f : \mathbb{D} \to \mathbb{R}$ (grayscale image on the unit disk).
-  $\mathrm{SO}(2)$ acts by rotation: $(R_\phi f)(r,\theta) = f(r, \theta - \phi)$.
-
-  \medskip
-
-  \textbf{Disk Harmonic Transform} (analog of Fourier/SHT):
-  \[
-    f(r, \theta) = \textstyle\sum_{n,k} a_{nk} \, \psi_{nk}(r, \theta), \qquad
-    \psi_{nk} = c_{nk} \, J_n(\lambda_{nk} r) \, e^{in\theta}
-  \]
-  \textbf{Equivariance}: $a^{R_\phi f}_{n,k} = e^{in\phi} \cdot a^f_{n,k}$
-
-  \medskip
-
-  \textbf{Selective disk bispectrum} (Def.~4.2) --- $O(m)$ coefficients:
-  \[
-    b_{0,0,k} = a_{0,1}^2 \cdot a_{0,k}^*, \qquad
-    b_{2,n,k} = a_{1,1} \cdot a_{n,1} \cdot a_{n+1,k}^*
-  \]
-  Phases cancel: $e^{i\phi} \cdot e^{in\phi} \cdot e^{-i(n+1)\phi} = 1$. \textbf{Invertible} (Thm.~4.4).
-\end{frame}
-
-% ============================================================
-\begin{frame}{SO2onD2: Architecture}
-  Follows the same \texttt{nn.Module} contract as \texttt{CnonCn}/\texttt{DnonDn}:
-  \begin{itemize}
-    \item No trainable parameters (all buffers)
-    \item \texttt{forward(f)} takes raw $L \times L$ images, handles DHT internally
-    \item \texttt{invert(beta)} recovers image up to rotation
-    \item \texttt{output\_size}, \texttt{index\_map} properties
-  \end{itemize}
-
-  \bigskip
-
-  \textbf{Key design choices}:
-  \begin{enumerate}
-    \item No new dependencies --- Bessel functions from scratch in pure torch
-    \item No external data files --- all roots \& matrices computed at construction
-    \item \texttt{selective=False} raises \texttt{NotImplementedError} (as in \texttt{DnonDn})
-    \item Mirrors \texttt{CnonCn} structure: $\mathrm{SO}(2)$ is commutative $\Rightarrow$ scalar triple products, no CG matrices
-  \end{enumerate}
-\end{frame}
-
-% ============================================================
-\begin{frame}{Challenge 1: Bessel functions without scipy}
-  Need $J_n(x)$ and its positive zeros $\lambda_{nk}$ where $J_n(\lambda_{nk}) = 0$.
-
-  \bigskip
-
-  \textbf{$J_n(x)$}: Forward recurrence from \texttt{torch.special.bessel\_j0/j1}:
-  \[
-    J_{k+1}(x) = \frac{2k}{x} J_k(x) - J_{k-1}(x)
-  \]
-  Stable when $x \geq n$ (our regime: $x = \lambda_{nk} r$, $\lambda_{nk} > n$).
-
-  \bigskip
-
-  \textbf{Root finding}: \textbf{Interlacing property} $\lambda_{n-1,k} < \lambda_{n,k} < \lambda_{n-1,k+1}$ gives guaranteed brackets.
-
-  \begin{enumerate}
-    \item $J_0$ roots via McMahon expansion + Newton polish
-    \item For $n = 1, 2, \ldots$: bracket each $J_n$ root between consecutive $J_{n-1}$ roots
-    \item Newton--bisection hybrid within each bracket (guaranteed convergence)
-  \end{enumerate}
-
-  Single-pass algorithm: \texttt{compute\_all\_bessel\_roots(n\_max, k\_max)}.
-\end{frame}
-
-% ============================================================
-\begin{frame}{Challenge 2: DHT on a discrete pixel grid}
-  \textbf{Problem}: The natural complex basis $\{\psi_{nk}, \psi_{-n,k}\}$ is redundant for real signals ($\psi_{-n,k} = \overline{\psi_{n,k}}$). On a discrete grid, the synthesis matrix is \emph{rank-deficient} (condition number $\sim 10^{20}$).
-
-  \bigskip
-
-  \textbf{Solution}: Reformulate using a \textbf{real basis}:
-  \begin{align*}
-    n = 0: &\quad c_{0k} J_0(\lambda_{0k} r) \hfill \text{(1 column per }(0,k)\text{)} \\
-    n > 0: &\quad 2\,c_{nk} J_n(\lambda_{nk} r)\cos(n\theta),\quad -2\,c_{nk} J_n(\lambda_{nk} r)\sin(n\theta) \hfill \text{(2 columns)}
-  \end{align*}
-  Yields matrix $\Phi \in \mathbb{R}^{p_{\text{disk}} \times d_{\text{real}}}$, well-conditioned ($\kappa \sim 10^3$).
-
-  \bigskip
-
-  \textbf{Analysis}: $\mathbf{x} = \Phi^+ \mathbf{f}$ \quad (pseudoinverse, \texttt{rcond=1e-10})
-
-  \textbf{Synthesis}: $\mathbf{f} = \Phi \, \mathbf{x}$
-
-  \smallskip
-
-  Convert real $\mathbf{x}$ to complex $a_{n,k} = x_{\text{re}} + i\,x_{\text{im}}$ for the bispectrum.
-\end{frame}
-
-% ============================================================
-\begin{frame}{Challenge 3: Bandlimit convention}
-  Paper says $\lambda = \pi L / 2$, but what determines $m$ exactly?
-
-  \bigskip
-
-  Convention from Marshall et al.\ (2023, \texttt{fle-2d}):
-  \[
-    n_e = \lfloor L^2 \pi / 4 \rfloor \approx \text{number of pixels inside the disk}
-  \]
-  Sort all $(n, k)$ by ascending $\lambda_{nk}$, keep the first $n_e$.
-
-  \bigskip
-
-  \textbf{Edge case}: if $n_e$ splits a conjugate pair $(n, k)$ and $(-n, k)$, drop the orphan.
-
-  \bigskip
-
-  \begin{table}
-    \scriptsize
-    \begin{tabular}{lccccc}
-      \toprule
-      \textbf{Image size} & $L=8$ & $L=16$ & $L=28$ & $L=56$ & $L=112$ \\
-      \midrule
-      Paper (Table 1) & 27 & 105 & 315 & 1{,}247 & 4{,}957 \\
-      Our implementation & 27 & 105 & 315 & --- & --- \\
-      \bottomrule
-    \end{tabular}
-  \end{table}
-
-  \smallskip
-
-  Exact match for all tested sizes. ($L \geq 56$ untested due to construction time.)
-\end{frame}
-
-% ============================================================
-\begin{frame}{Inversion: bootstrap recovery (Thm.~4.4)}
-  Same pattern as \texttt{CnonCn.invert()}, adapted to 2D indexing:
-
-  \medskip
-
-  \begin{enumerate}\setlength{\itemsep}{1pt}
-    \item $a_{0,1}$ from $b_{0,0,1} = |a_{0,1}|^3 e^{i\arg(a_{0,1})}$ \quad $\Rightarrow$ \quad cube root
-    \item $a_{0,k}$ for $k \geq 2$: \quad $a_{0,k} = \overline{b_{0,0,k} / a_{0,1}^2}$
-    \item $|a_{1,1}|$ from $b_{2,0,1} = a_{0,1} |a_{1,1}|^2$ \quad (phase = rotation indeterminacy)
-    \item $a_{1,k}$ for $k \geq 2$: \quad $a_{1,k} = \overline{b_{2,0,k} / (a_{1,1} \cdot a_{0,1})}$
-    \item Chain for $n \geq 1$: \quad $a_{n+1,k} = \overline{b_{2,n,k} / (a_{1,1} \cdot a_{n,1})}$
-  \end{enumerate}
-
-  \medskip
-
-  \textbf{At the coefficient level}: exact (magnitudes match to $10^{-12}$).
-
-  \textbf{Full roundtrip} ($f \to \beta \to \hat{f} \to f'$): limited by DHT discretization;
-  ${>}85\%$ of DH coefficients recovered within 20\% relative error.
-
-  \medskip
-
-  \textbf{Assumption}: $a_{n,1} \neq 0$ for all $n$.
-  Mild for noisy images; fails for purely radial signals.
-\end{frame}
-
-% ============================================================
-\begin{frame}[fragile]{SO2onD2: Usage}
-\begin{lstlisting}[language=Python]
-from bispectrum import SO2onD2
-
-bsp = SO2onD2(L=28)          # 28x28 images
-print(bsp.output_size)        # 315
-
-f = torch.randn(4, 28, 28)   # batch of 4 images
-beta = bsp(f)                 # (4, 315), complex128
-f_rec = bsp.invert(beta)      # (4, 28, 28), float64
-
-# Invariance check
-cos_a, sin_a = math.cos(0.7), math.sin(0.7)
-theta = torch.tensor([[[cos_a, sin_a, 0],
-                        [-sin_a, cos_a, 0]]])
-grid = F.affine_grid(theta, [1,1,28,28])
-f_rot = F.grid_sample(f[:1].unsqueeze(1), grid)
-
-beta_rot = bsp(f_rot.squeeze(1))
-# beta and beta_rot are approximately equal
-\end{lstlisting}
-\end{frame}
-
-% ============================================================
-\begin{frame}{Numerical subtleties encountered}
-  \begin{enumerate}
-    \item \textbf{Forward recurrence cancellation}: $J_n(x)$ near its own zeros loses $\sim$7 digits for $n \geq 2$.
-    Our computed roots are zeros of the \emph{numerically evaluated} $J_n$, not the true $J_n$.
-    Everything is self-consistent; displacement $\sim 6 \times 10^{-5}$.
-
-    \smallskip
-
-    \item \textbf{dtype leak}: \texttt{torch.tensor(x)} with Python float infers \texttt{float32}, not \texttt{float64}. Silently halved precision of all root-finding.
-
-    \smallskip
-
-    \item \textbf{Phase precision}: \texttt{torch.exp(1j * n * phi)} creates \texttt{complex64} even when multiplied with \texttt{complex128} tensors. Must explicitly pass \texttt{dtype=complex128}.
-
-    \smallskip
-
-    \item \textbf{Pseudoinverse regularization}: 1--2 basis functions per grid are unresolvable (aliased at the pixel scale). Setting \texttt{rcond=1e-10} in \texttt{pinv} zeros their singular values instead of amplifying noise.
-  \end{enumerate}
-\end{frame}
-
-% ============================================================
-\begin{frame}{Test coverage: 180 tests}
-  \begin{columns}[T]
-    \begin{column}{0.48\textwidth}
-      \textbf{CnonCn} (42 tests)
-      \begin{itemize}\scriptsize
-        \item Output shape/dtype (6 values of $n$)
-        \item Cyclic shift invariance (selective \& full, all shifts)
-        \item Formula verification vs manual DFT
-        \item Inversion: DFT magnitude roundtrip
-        \item Inversion: full bispectrum roundtrip
-      \end{itemize}
-
-      \bigskip
-
-      \textbf{DnonDn} (67 tests)
-      \begin{itemize}\scriptsize
-        \item Rotation invariance (all shifts, 6 values of $n$)
-        \item Reflection invariance (6 values of $n$)
-        \item Combined rotation + reflection
-        \item DFT roundtrip (forward + inverse = identity)
-        \item Inversion: $\beta_{\rho_0,*}$ exact roundtrip
-        \item Inversion: Fourier Frobenius norm recovery
-        \item DFT performance benchmarks ($n$ up to 1024)
-      \end{itemize}
-    \end{column}
-    \begin{column}{0.48\textwidth}
-      \textbf{SO2onD2} (45 tests)
-      \begin{itemize}\scriptsize
-        \item Bessel: $J_n$ vs torch, root accuracy, interlacing
-        \item Construction: coeff counts vs paper Table~1
-        \item Forward: shape, dtype, analytical \& spatial invariance
-        \item DHT roundtrip: bandlimited \& single harmonic
-        \item Inversion: coefficient-level, bispectrum roundtrip
-      \end{itemize}
-
-      \bigskip
-
-      \textbf{Other} (32 tests)
-      \begin{itemize}\scriptsize
-        \item SO3onS2: construction, forward, rotation invariance
-        \item Rotation utilities, import / public API surface
-      \end{itemize}
-    \end{column}
-  \end{columns}
-\end{frame}
-
-% ============================================================
-\begin{frame}{Weaknesses \& areas for improvement}
-  \begin{enumerate}
-    \item \textbf{No GPU testing} --- all tests run on CPU only
-
-    \item \textbf{$D_n$ inversion is partial} --- $O(2)$ indeterminacy is fundamental;
-      worth investigating canonical choices of $Q$
-
-    \item \textbf{Full bispectrum not implemented} --- only \texttt{selective=True}
-      works for \texttt{DnonDn} and \texttt{SO2onD2}
-
-    \item \textbf{SO2onD2 construction is slow} --- Bessel root finding + pseudoinverse
-      in pure Python; $L=28$ takes ${\sim}15$s, $L=64$ impractical
-
-    \item \textbf{Naive DHT} --- $O(L^3)$ pseudoinverse instead of
-      $O(L^2 \log L)$ fast DHT from Marshall et al.\ (2023)
-
-    \item \textbf{No G-CNN integration yet} --- standalone module,
-      not wired into an equivariant network pipeline
-  \end{enumerate}
-\end{frame}
-
-% ============================================================
-\begin{frame}{Next steps}
-  \begin{itemize}
-    \item \textbf{SO2onD2: MRA correction} (Proposition 4.6) --- bias term $\delta$ for noisy rotated observations
-    \item \textbf{SO2onD2: fast DHT} --- integrate $O(L^2 \log L)$ approximation from Marshall et al.
-    \item \textbf{SO2onD2: performance} --- cache Bessel roots, or move to C++/CUDA
-    \item Wire into a G-CNN pipeline (replace escnn pooling layer)
-    \item Reproduce MNIST/EMNIST/CIFAR-10 experiments from the papers
-    \item Explore octahedral / full octahedral groups (Mataigne Appendix F)
-    \item Investigate selective bispectrum for $\mathrm{SO}(3)$ on $S^2$ (open problem)
-    \item GPU testing and benchmarking
-  \end{itemize}
-\end{frame}
-
-\end{document}
diff --git a/docs/bispectral_signatures_of_data.tex b/docs/bispectral_signatures_of_data.tex
deleted file mode 100644
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--- a/docs/bispectral_signatures_of_data.tex
+++ /dev/null
@@ -1,2510 +0,0 @@
-\documentclass{article}
-\usepackage[fontsize=10bp]{fontsize}
-\usepackage[utf8]{inputenc}
-\usepackage[margin=1.0in]{geometry}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\usepackage{amsthm}
-\usepackage{subfigure}
-\usepackage[parfill]{parskip}
-\usepackage{graphicx}
-\usepackage[usenames,dvipsnames]{xcolor}
-\usepackage{float}
-\usepackage{amsfonts}
-\usepackage{tcolorbox}
-\usepackage{multicol}
-\usepackage{wrapfig}
-\usepackage{csquotes}
-\usepackage[dvipsnames,svgnames,x11names,hyperref]{xcolor}
-\definecolor{deepblue}{RGB}{0,35,102}
-\definecolor{deepred}{RGB}{139,0,0}
-\usepackage[table,xcdraw]{xcolor}
-
-\usepackage{hyperref}
-\hypersetup{
-    colorlinks=true,
-    linkcolor=RoyalPurple,
-    urlcolor=RoyalPurple,
-    citecolor=RoyalPurple
-    }
-
-\setlength{\belowdisplayskip}{0pt} \setlength{\belowdisplayshortskip}{0pt}
-\setlength{\abovedisplayskip}{0pt} \setlength{\abovedisplayshortskip}{0pt}
-\usepackage[font=footnotesize,labelfont=bf]{caption}
-
-\usepackage{floatrow}
-\usepackage{hyperref}
-\newtheorem{proposition}{Proposition}
-\newtheorem{lemma}{Lemma}
-\newtheorem{definition}{Definition}
-\newtheorem{remark}{Remark}
-
-\usepackage[superscript,biblabel]{cite}
-\linespread{1.0}
-
-
-\title{Bispectral Signatures of Data: The Different Flavors of Bispectra and G-Bispectra}
-
-
-\author{\vspace{-0.8cm}}
-\date{}
-
-\begin{document}
-
-
-\maketitle
-\tableofcontents
-
-\paragraph{Triple correlation on R} The triple correlation of a function $f$ on $\mathbb{R}$ is:
-$$
-T(f)\left(t_1, t_2\right):=\int_{\mathbb{R}} f(x)^* f\left(x+t_1\right) f\left(x+t_2\right) d x .
-$$
-The triple correlation is translation invariant: if $g(x)=$ $f\left(x-x_0\right)$, then $T_g=T(f)$, as a consequence of the invariance of the integral.
-
-We note that R is not included in any of the cases here: not finite, not compact, not homogeneous, etc - except the homogeneous x invariant space.
-
-\section{Preliminaries}
-
-Throughout the computations that follow, we use the following.
-
-Properties of representations:
-\begin{align*}
-    \rho(gg') &= \rho(g) \rho(g')\\
-    \rho(g^{-1}) = \rho(g)^\dagger
-\end{align*}
-
-Property of tensor product and matrix multiplication:
-\begin{equation}
-    (A \otimes B) . (C \otimes D) = (AC) \otimes (BD)
-\end{equation}
-
-\section{Bispectrum-only Summary}
-
-
-\newpage
- \newgeometry{left=0cm,bottom=0cm}
-\begin{table}[]
-\vspace*{-20mm}
-\centering
-\rotatebox{90}{
-\scriptsize{
-\begin{tabular}{|l|l|l|}
-\hline
-\textbf{} &
-  \textbf{Fourier} &
-  Bispectrum \\ \hline
-\textbf{$Z_n$} &
-  \begin{tabular}[c]{@{}l@{}}$\mathcal{F}(f)_k$\\ $= \sum_{x=0}^{n-1} f(x).e^{-i2\pi k x / n}$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$\beta(f)_{k_1, k_2} $\\ $= \hat{f}_{k_1}\hat{f}_{k_2}\hat{f}^{\dagger}_{k_1+k_2}$\end{tabular} \\ \hline
-\textbf{$G_n$} &
-  \begin{tabular}[c]{@{}l@{}}$\mathcal{F}(f)_\rho$\\ $=\frac{1}{|G|} \sum_{g \in G} f(g) \rho(g)^{\dagger}$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$\beta(f)_{\rho_1, \rho_2}$\\ $=[\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2} ] C_{{\rho_1} {\rho_2}}\left[\bigoplus_{\rho \in \rho_1 \otimes \rho_2}\mathcal{F}(f)_\rho\right] C_{{\rho_1} {\rho_2}}^{\dagger}$\end{tabular} \\ \hline
-\textbf{$G^c$} &
-  same as $G$ &
-  \begin{tabular}[c]{@{}l@{}}$\beta (f)_{\rho_1, \rho_2} $\\ $=  \mathcal{F}(f)_{\rho_1} \mathcal{F}(f)_{\rho_2}\mathcal{F}(f)_{\rho_1 \rho_2}^{\dagger}$\end{tabular} \\ \hline
-\textbf{$S^2$} &
-  \begin{tabular}[c]{@{}l@{}}$\mathcal{F}^v(f)_l $\\ $= \kappa_l \int_{R \in SO(3)} f^\dagger(T_R(x_0)). [\rho_l(R)]_0 dR$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$\beta^v(f)_{l_1, l_2}[l]$\\ $:=\left[\mathcal{F}(f)_{l_1} \otimes \mathcal{F}(f)_{l_2} \right] C_{l_1l_2} \mathcal{F}_0(f)_l^{\dagger}$\end{tabular} \\ \hline
-\textbf{} &
-  \begin{tabular}[c]{@{}l@{}}$\mathcal{F}(f)_l $\\ $= \int_{R \in SO(3)} f^\dagger(T_R(x_0)). \rho_l(R) dR$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$\beta(f)_{l_1l_2} $\\ $= [\mathcal{F}(f)_{l_1} \otimes \mathcal{F}(f)_{l_2}] C_{l_1,l_2} \Big[ \bigoplus_{l \in l_1 \otimes l_2} \mathcal{F}(f)_{l}^{\dagger} \Big] C^{\dagger}_{l_1,l_2}$\end{tabular} \\ \hline
-\textbf{$H$} &
-  \begin{tabular}[c]{@{}l@{}}$\mathcal{F}(f)_\rho $\\ $=\int_{g \in G} f\left(T_g(x_0)\right) \rho(g)dg$\end{tabular} &
-  same as $G$ and $G_n$ \\ \hline
-\textbf{$H^c$} &
-  same as $H$ &
-  same as $G^c$ \\ \hline
-\end{tabular}
-}
-}
-\caption{Summary}
-\label{tab:eqns}
-\end{table}
-
-\section{Summary}
-
-\newpage
- \newgeometry{left=0cm,bottom=0cm}
-\begin{table}[]
-\vspace*{-20mm}
-\centering
-\rotatebox{90}{
-\scriptsize{
-\begin{tabular}{|l|l|l|l|}
-\hline
-\textbf{} &
-  \textbf{Fourier} &
-  \textbf{Triple Correlation} &
-  Bispectrum \\ \hline
-\textbf{$Z_n$} &
-  \begin{tabular}[c]{@{}l@{}}$\mathcal{F}(f)_k$\\ $= \sum_{x=0}^{n-1} f(x).e^{-i2\pi k x / n}$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$T(f)_{x_1, x_2} $\\ $= \frac{1}{n} \sum_{x \in Z_n}f^*(x)f(x+x_1)f(x+x_2)$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$\beta(f)_{k_1, k_2} $\\ $= \hat{f}_{k_1}\hat{f}_{k_2}\hat{f}^{\dagger}_{k_1+k_2}$\end{tabular} \\ \hline
-\textbf{$G_n$} &
-  \begin{tabular}[c]{@{}l@{}}$\mathcal{F}(f)_\rho $\\ $=\frac{1}{|G|} \sum_{g \in G} f(g) \rho(g)^{\dagger}$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$T(f)_{g_1, g_2}$\\ $=\frac{1}{|G|} \sum_{g \in G} f^*(g) f\left(g g_1\right) f\left(g g_2\right)$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$\beta(f)_{\rho_1, \rho_2}$\\ $=[\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2} ] C_{{\rho_1} {\rho_2}}\left[\bigoplus_{\rho \in \rho_1 \otimes \rho_2}\mathcal{F}(f)_\rho\right] C_{{\rho_1} {\rho_2}}^{\dagger}$\end{tabular} \\ \hline
-\textbf{$G$} &
-  \begin{tabular}[c]{@{}l@{}}$\mathcal{F}(f)_\rho $\\ $= \int_G f(g)\rho(g) dg$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$T(f)_{g_1, g_2}$\\ $=\int_{g \in G} f^*(g) f\left(g g_1\right) f\left(g g_2\right)dg$\end{tabular} &
-  same as $G_n$, i.e. as above \\ \hline
-\textbf{$G^c$} &
-  same as $G$ &
-  same as $G$, i.e. as above &
-  \begin{tabular}[c]{@{}l@{}}$\beta (f)_{\rho_1, \rho_2} $\\ $=  \mathcal{F}(f)_{\rho_1} \mathcal{F}(f)_{\rho_2}\mathcal{F}(f)_{\rho_1 \rho_2}^{\dagger}$\end{tabular} \\ \hline
-\textbf{$S^2$} &
-  \begin{tabular}[c]{@{}l@{}}$\mathcal{F}^v(f)_l $\\ $= \kappa_l \int_{R \in SO(3)} f^\dagger(T_R(x_0)). [\rho_l(R)]_0 dR$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$T(f)_{R_1, R_2} $\\ $= \int_{R\in SO(3)} f^*(T_R(x_0)) f\left(T_{R R_1}(x_0)\right) f\left(T_{R R_2}(x_0)\right)dR$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$\beta^v(f)_{l_1, l_2}[l]$\\ $:=\left[\mathcal{F}(f)_{l_1} \otimes \mathcal{F}(f)_{l_2} \right] C_{l_1l_2} \mathcal{F}_0(f)_l^{\dagger}$\end{tabular} \\ \hline
-\textbf{} &
-  \begin{tabular}[c]{@{}l@{}}$\mathcal{F}(f)_l $\\ $= \int_{R \in SO(3)} f^\dagger(T_R(x_0)). \rho_l(R) dR$\end{tabular} &
-  Same as above, independent of Fourier transform. &
-  \begin{tabular}[c]{@{}l@{}}$\beta(f)_{l_1l_2} $\\ $= [\mathcal{F}(f)_{l_1} \otimes \mathcal{F}(f)_{l_2}] C_{l_1,l_2} \Big[ \bigoplus_{l \in l_1 \otimes l_2} \mathcal{F}(f)_{l}^{\dagger} \Big] C^{\dagger}_{l_1,l_2}$\end{tabular} \\ \hline
-\textbf{$H$} &
-  \begin{tabular}[c]{@{}l@{}}$\mathcal{F}(f)_\rho $\\ $=\int_{g \in G} f\left(T_g(x_0)\right) \rho(g)dg$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$T(f)_{g_1, g_2} $\\ $= \int_{g\in G} f^*(T_g(x_0)) f\left(T_{g g_1}(x_0)\right) f\left(T_{g g_2}(x_0)\right)dg$\end{tabular} &
-  same as $G$ and $G_n$ \\ \hline
-\textbf{$H^c$} &
-  same as $H$ &
-  same as $H$, i.e. as above &
-  same as $G^c$ \\ \hline
-\textbf{$D$} &
-  \begin{tabular}[c]{@{}l@{}}$\mathcal{F}(f)_{nm} $\\ $= \frac{1}{a_{n m}} \int_0^{2 \pi} \int_0^1 f(r, \theta) J_m(2 \pi l_{n m} r) \exp(-i m \theta) r d r d\theta$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$T(f)_{R_1, R_2} $\\ $:= \int_{R\in SO(2)} f^*(T_R(\theta_0), r_0) f\left(T_{R R_1}(\theta_0), r_0\right) f\left(T_{R R_2}(\theta_0), r_0\right)dR$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$\beta(f)_{nmm'} $\\ $:= \mathcal{F}(f)_{nm}.\mathcal{F}(f)_{nm'}.\mathcal{F}(f)_{n(m+m')}^\dagger$\end{tabular} \\ \hline
-\textbf{$B$} &
-  \begin{tabular}[c]{@{}l@{}}$\mathcal{F}(f)_{l}$\\ $ \propto \int_0^{2 \pi}\int_0^{\pi} \int_0^1 f(r, \theta, \phi)^\dagger  \left( r^{\ell} Y_{\ell}(\theta, \phi)\right) r d r \cos(\theta) d\theta d\phi$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$T(f)_{R_1, R_2} $\\ $:= \int_{R\in SO(3)} f^*(T_R(\theta_0, \phi_0), r_0) f\left(T_{R R_1}(\theta_0, \phi_0), r_0\right) f\left(T_{R R_2}(\theta_0, \phi_0), r_0\right)dR$\end{tabular} &
-  $\beta^v(f)_{l_1, l_2}[l]$, same as $S^2$ \\ \hline
-\textbf{$H \times R$} &
-  \begin{tabular}[c]{@{}l@{}}$\mathcal{F}(f)_{\rho} $\\ $:= \int_{r \in R} \int_{g \in G} f\left(r, T_g(x_0)\right) \rho(g)dg. dr$\end{tabular} &
-  \begin{tabular}[c]{@{}l@{}}$T(f)_{g_1, g_2} $\\ $:= \int_{g \in G} f^*(T_g(x_0), r_0) f\left(T_{gg_1}(x_0), r_0\right) f\left(T_{gg_2}(x_0), r_0\right)dg$\end{tabular} &
-  Conjecture: $\beta^v$, same as $H$, \\ \hline
-\end{tabular}
-}
-}
-\caption{Summary}
-\label{tab:eqns}
-\end{table}
-\restoregeometry
-\newpage
-\section{\color{deepblue}{Domain $\Omega = (\mathbb{Z}/n\mathbb{Z},~+)$ (cyclic translations)}}
-
-\subsection{Fourier Transform: Definition}
-
-The discrete Fourier transform of a signal $f$ on a space $\Omega$ is defined:
-
-\vspace{-0.5cm}
-\begin{equation}
-  \hat{f}_k = \sum_{x=0}^{n-1}e^{-i2\pi k x / n} f(x).
-\end{equation}
-\vspace{-0.5cm}
-
-From a group theoretic perspective, a Fourier transform is a projection of a signal onto the irreducible representations the group $(\mathbb{Z}/n\mathbb{Z}, +)$ of cyclic translations, indexed by integer frequencies $k$.
-
-\subsection{Fourier Transform: Equivariance}
-This gives it the property of \textit{equivariance} to translation in the input, commonly known as the time-shift and frequency-shift properties. Intuitively, this means that translating the input and then taking the Fourier transform is equivalent to taking the Fourier transform and then phase shifting the output. We expand the shift property of the Fourier transform here for illustration. Define the translate of $f$ by $z$ as $f^z(x) = f(x - z)$. Let $ h = x - z$. Then, The Fourier transform of $f^z(x)$ shifts as follows:
-
-\vspace{-0.7cm}
-\begin{eqnarray*}
-  \hat{f}^z_k &=& \sum_{x=0}^{n-1}e^{-i2\pi k x / n} f(x - z) \\
-%   \hat{f}^z(\omega) &=& \sum_{x=0}^{n-1} e^{\frac{-i2 \pi}{n}x\omega} f(x-z) \\
-  \hat{f}^z_k &=& \sum_{h=0}^{n-1}e^{-i2\pi k (z+h) / n} f(h) \\
-%   \hat{f}^z(\omega) &=& \sum_{h=0}^{n-1} e^{\frac{-i2 \pi}{n}(z+h)\omega} f(h) \\
-  \hat{f}^z_k &=& \sum_{h=0}^{n-1} e^{-i2\pi k z / n} e^{-i2\pi k h / n} f(h) \\
-%   \hat{f}^z(\omega) &=& \sum_{h=0}^{n-1} e^{\frac{-i2 \pi}{n}z\omega}e^{\frac{-i2 \pi}{n}h\omega} f(h) \\
-  \hat{f}^z_k  &=&  e^{-i2\pi k z / n} \sum_{h=0}^{n-1} e^{-i2\pi k h / n} f(h) \\
-%   \hat{f}^z(\omega) &=&  e^{\frac{-i2 \pi}{n}z\omega} \sum_{h=0}^{n-1} e^{\frac{-i2 \pi}{n}h\omega} f(h) \\
-%   &=& e^{\frac{-i 2 \pi}{n}z \omega} \hat{f}(\omega)
-% \end{eqnarray*}
-  &=& e^{-i2\pi k z / n} \hat{f}_k.
-\end{eqnarray*}
-\vspace{-0.7cm}
-
-That is, the shift is proportional to the translation $z$, by a factor of $e^{\frac{-i 2 \pi}{n} z k}$. Another way of thinking of the Fourier transform is as a homomorphism that maps group actions in the space domain (cyclic translation) to group actions in the frequency domain (complex multiplication).
-
-\subsection{Steerable Bispectrum: After Convolution on Steerable Basis}
-
-Question: can we compute the bispectrum more efficiently by formalizing the group convolution on a steerable basis? And recovering the output of the convolution directly in its Fourier form? We try to do it for the simplest bispectrum, i.e. the bispectrum on $\mathbb{Z}/n\mathbb{Z}$.
-
-Consider a steerable basis for the discrete translations, i.e. the harmonics of the ring, where the ring is discretized into $\mathbb{Z} / n\mathbb{Z}$.
-
-The basis $ x \mapsto Y_l(x) = e^{ilx}$ is steered by representations $\theta \mapsto \rho_l(\theta) = e^{-il\theta}$, since $Y_l(x - x_0) = \rho_l(-x_0)Y_l(x)$.
-
-The filters of a convolutional layer are: $\Psi \in \mathbb{C}^{K \times n}$ for $K$ filters, given as their coefficients along the $n$ basis functions $Y_l$, $l=1, ..., n$. Let's consider only one filter for now, denoted $w$.
-\begin{align*}
-    w
-    &= [\hat{w}^{1}, ..., \hat{w}^{n}] \in \mathbb{C}^n \\
-    w(x)
-    &= \sum_{l=1}^n \hat{w}^{(l)}Y_l(x), \quad \forall x \in \mathbb{Z} / n\mathbb{Z}
-\end{align*}
-
-The convolution between $w$ and a signal $f$ is, on the domain $\mathbb{Z} / n \mathbb{Z}$, is:
-\begin{align*}
-    \Theta(x_0) = (w \ast f) (x_0)
-    &= \sum_{x \in \mathbb{Z}/n\mathbb{Z}} w(x_0-x).f(x) \\
-    &= \sum_{x \in \mathbb{Z}/n\mathbb{Z}} \sum_{l=1}^n  \hat{w}^{(l)}Y_l(x_0-x).f(x) \\
-    &= \sum_{x \in \mathbb{Z}/n\mathbb{Z}} \sum_{l=1}^n  \hat{w}^{(l)}\rho_l(-x). Y_l(x_0).f(x) \\
-    &=\sum_{l=1}^n  \sum_{x \in \mathbb{Z}/n\mathbb{Z}} \hat{w}^{(l)}\rho_l(-x).f(x).Y_l(x_0) \\
-    &=\sum_{l=1}^n \hat{\Theta}_l .Y_l(x_0),
-    \quad \text{denoting:~} \hat{\Theta}_l \equiv \hat{w}^{(l)} \sum_{x \in \mathbb{Z}/n\mathbb{Z}} \rho_l(-x).f(x),
-\end{align*}
-with $\hat{\Theta}_l$ the coefficients of the output of the G-convolution, here the classic convolution, on the steerable basis.
-
-We can directly compute the steerable bispectrum by combining the $\hat{\Theta}_l$ of the output.
-
-
-\subsection{Triple Correlation: Definition}
-
-$T(f)_{x_1, x_2} = \frac{1}{n} \sum_{x \in Z_n}f^*(x)f(x+x_1)f(x+x_2)$
-
-\subsection{Triple Correlation: Symmetries}
-
-\begin{proposition}[Symmetries]
-The triple correlation of a signal has the following symmetry:
-\begin{align*}
-     (s1) \qquad T(f)_{x_1, x_2} &= T(f)_{x_2, x_1}
-\end{align*}
-The triple correlation of a real signal has the following additional symmetries:
-\begin{align*}
-     (s2) \qquad T(f)_{x_1, x_2} = T(f)_{-x_1, x_2-x_1} = T(f)_{x_2-x_1, -x_1} = T(f)_{x_1-x_2, -x_2} = T(f)_{-x_2, x_1-x_2}
-\end{align*}
-\end{proposition}
-
-\begin{proof}
-    We prove symmetry (s1), which relies on the fact that $f(x+x_2)$ and $f(x+x_1)$ are scalar values that commute:
-    \begin{align*}
-        T(f)_{x_2, x_1}
-            = \frac{1}{n} \sum_{x \in Z_n} f^*(x)f(x+x_2)f(x+x_1)
-            = \frac{1}{n} \sum_{x \in Z_n} f^*(x)f(x+x_1)f(x+x_2)
-            = T(f)_{x_2, x_1}.
-    \end{align*}
-    We assume that the signal $f$ is real. We prove the first equality of symmetry (s2):
-    \begin{align*}
-        T(f)_{-x_1, x_2-x_1}
-            &= \frac{1}{n} \sum_{x \in Z_n} f^*(x)f(x-x_1)f(x+x_2-x_1) \\
-            &= \frac{1}{n} \sum_{x' \in Z_n} f^*(x'+x_1)f(x')f(x'+x_1+x_2-x_1)
-                \quad \text{(with $x' = x - x_1$, i.e. $x = x'+x_1$)} \\
-            &= \frac{1}{n} \sum_{x' \in Z_n} f^*(x'+x_1)f(x')f(x'+x_2)
-                \quad \text{(because $Z_n$ commutative: $x_2 - x_1 = -x_1 + x_2$)} \\
-            &= \frac{1}{n} \sum_{x \in Z_n} f(x+x_1)f(x)f(x+x_2)
-                \quad \text{($f$ is a real signal, which equals its conjugate)}\\
-            &= \frac{1}{n} \sum_{x \in Z_n} f(x)f(x+x_1)f(x+x_2)
-                \quad \text{($f$ is takes on real values that commute)}\\
-            &= \frac{1}{n} \sum_{x \in Z_n} f(x)^*f(x+x_1)f(x+x_2) \\
-            &= T(f)_{x_2, x_1}
-    \end{align*}
-    The second equality of symmetry (s2) follows using (s1).
-    The third and fourth equality of symmetry (s2) have the same proof.
-\end{proof}
-
-\begin{wrapfigure}{r}{0.25\textwidth}
-  \centering
-  \includegraphics[width=0.25\textwidth]{figs/tc_sectors.png}
-  \caption{Sectors for the symmetries of the TC $\tau_1 = x_1$ and $\tau_2 = x_2$.}
-  \label{fig:sectors}
-\end{wrapfigure}
-As a consequence, knowing the TC in any of the six sectors, I through VI, shown in Fig.~\ref{fig:sectors} would enable us to find the entire TC. These sectors include their boundaries so that, for example, sector $I$ is an infinite wedge bounded by the lines $\tau_2=0$ and $\tau_1=\tau_2, \tau_1 \geq 0$.
-
-
-\subsection{Triple Correlation: Invariance}
-
-\subsection{Triple Correlation: Continuity}
-
-When $f_2$ is obtained by a group transformation of $f_1$, we have:
-\begin{equation}
-    f_2 = T_g[f_1] \Leftrightarrow T(f_2) = T(f_1).
-\end{equation}
-
-We explore what happens when $f_2$ is obtained from group-transformation of $f_1$ plus ``noise''. Do we have:
-\begin{equation}
-    f_2 = T_g[f_1] + \epsilon h \Rightarrow T(f_2) = T(f_1) + \epsilon T(h) \qquad?
-\end{equation}
-
-
-\begin{remark}
-    We could also consider the case:
-\begin{equation}
-    f_2 = T_g[f_1] + T'_a[h] \Rightarrow T(f_2) = T(f_1) + ? \qquad?
-\end{equation}
-i.e. $f_2$ is obtained from group-transformation of $f_1$ plus a residual deformation by $a$ where $a$ is not the group defining the TC. For example, $a$ is from the general linear group.
-\end{remark}
-
-Consider the space $\mathcal{C}^0(G)$ of continuous signals $f$ defined on a compact group $G$. We equip this space with the uniform norm (also called sup norm): $|| f \|_{\infty}=\sup _{g \in G}|f(g)|$ for any $f \in \mathcal{C}^0(G)$. The operation ``computing the TC of $f$ at $g_1, g_2$'' can be expressed as:
-$$
-\begin{aligned}
-& T: \mathcal{C}^0(G) \rightarrow \mathbb{R} \\
-& f \rightarrow T(f)_{g_1, g2}.
-\end{aligned}
-$$
-
-We prove the Lipschitz continuity of this operation.
-
-\begin{proposition} There is the following bound on the TC of two signals $f_1, f_2 \in \mathcal{C}^0(G)$ for $G$ a locally compact group.
-    \begin{align*}
- \|T\left(f_1\right)-T(f_2)\|_\infty
-    \leq \text{vol}(G) \left( \|f_1\|_\infty^2 + \|f_1\|_\infty\|f_2\|_\infty + \|f_2\|_\infty^2\right) \left\|f_1-f_2\right\|_{\infty}.
-\end{align*}
-This shows the Lipschitz continuity of the TC:
-    \begin{align*}
- \|T\left(f_1\right)-T(f_2)\|_\infty
-    \leq K \left\|f_1-f_2\right\|_{\infty}.
-\end{align*}
-where $K = 3 \text{vol(G)}$ if $f_1, f_2$ are the outputs of a $G$-CNN with sigmoid non-linearity; or if $f_1, f_2$ are normalized signals that have max equal to 1. Alternatively, $K$ only depends on the weights of the $G$-CNN if $f_1, f_2$ are out to a $G$-CNN to which normalized signals have been sent.
-\end{proposition}
-
-\begin{proof}
-
-Take two real signals $f_1$ and $f_2$ defined on a compact group $G$. Take two group elements $g_1, g_2 \in G$:
-\begin{align*}
- \left|T\left(f_1\right)_{g_1,g_2}-T(f_2)_{g_1,g_2}\right|
-    &=\left|\int_G f_1(g)f_1(gg_1)f_1(gg_2) dg - \int_G f_2(g)f_2(gg_1)f_2(gg_2)  dg\right| \\
-    & \leq \int_G\left|f_1(g)f_1(gg_1)f_1(gg_2) -  f_2(g)f_2(gg_1)f(_2gg_2)\right| dg \\
-    & \leq \text{vol}(G) ||F_3(f_1) - F_3(f_2)\|_{\infty},
-\end{align*}
-with notations: $F_3(f)(g) =  f(g)f(gg_1)f(gg_2)$ and $||F_3(f_1) - F_3(f_2)\|_{\infty} \equiv \sup_{g \in G} |F_3(f_1)(g) - F_3(f_2)(g)| $.
-
-We will find an upper bound for $||F_3(f_1) - F_3(f_2)\|_{\infty}$. We have, for any $g$:
-\begin{align*}
-    |F_3(f_1)(g) - F_3(f_2)(g)|
-        % &= |f_1(g)f_1(gg_1)f_1(gg_2) - f_2(g)f_2(gg_1)f_2(gg_2)|\\
-        % &= |f_1(g)f_1(gg_1)f_1(gg_2) - f_1(g)f_1(gg_1)f_2(gg_2) \\
-        % &~ + f_1(g)f_1(gg_1)f_2(gg_2) -  f_1(g)f_2(gg_1)f_2(gg_2) \\
-        % &~ + f_1(g)f_2(gg_1)f_2(gg_2) -  f_2(g)f_2(gg_1)f_2(gg_2) |\\
-        % &\leq |f_1(g)f_1(gg_1)f_1(gg_2) - f_1(g)f_1(gg_1)f_2(gg_2)| \\
-        % & + |f_1(g)f_1(gg_1)f_2(gg_2) -  f_1(g)f_2(gg_1)f_2(gg_2)| \\
-        % & + |f_1(g)f_2(gg_1)f_2(gg_2) -  f_2(g)f_2(gg_1)f_2(gg_2)| \\
-        % & = |f_1(g)f_1(gg_1)(f_1(gg_2) -f_2(gg_2))|\\
-        % & + |f_1(g)f_2(gg_2) (f_1(gg_1)-  f_2(gg_1))| \\
-        % & + |f_2(gg_1)f_2(gg_2) (f_1(g)-  f_2(g))|\\
-        % &\leq \left( |f_1(g)f_1(gg_1)| + |f_1(g)f_2(gg_2)| + |f_2(gg_1)f_2(gg_2)|\right) \left\|f_1-f_2\right\|_{\infty}\\
-        &\leq \left( \|f_1\|_\infty^2 + \|f_1\|_\infty\|f_2\|_\infty + \|f_2\|_\infty^2\right) \left\|f_1-f_2\right\|_{\infty}
-\end{align*}
-using a telescopic sum, the triangle inequality and the definition of $\left\|.\right\|_{\infty}$.
-
-The above inequality gives an upper bound that is valid for any $g \in G$. As the sup is defined as the lowest upper bound, we get:
-\begin{equation}
-    ||F_3(f_1) - F_3(f)\|_{\infty} \leq \left( \|f_1\|_\infty^2 + \|f_1\|_\infty\|f_2\|_\infty + \|f_2\|_\infty^2\right) \left\|f_1-f_2\right\|_{\infty}.
-\end{equation}
-and therefore:
-\begin{align*}
- \left|T\left(f_1\right)_{g_1,g_2}-T(f_2)_{g_1,g_2}\right|
-    \leq \text{vol}(G) \left( \|f_1\|_\infty^2 + \|f_1\|_\infty\|f_2\|_\infty + \|f_2\|_\infty^2\right) \left\|f_1-f_2\right\|_{\infty}.
-\end{align*}
-The above inequality gives an upper bound that is valid for any $g_1, g_2 \in G$. As the sup over $G \times G$ is defined as the lower upper bound, we get:
-\begin{align*}
- \|T\left(f_1\right)-T(f_2)\|_\infty
-    \leq \text{vol}(G) \left( \|f_1\|_\infty^2 + \|f_1\|_\infty\|f_2\|_\infty + \|f_2\|_\infty^2\right) \left\|f_1-f_2\right\|_{\infty}.
-\end{align*}
-This proves the Lipschitz continuity of the TC assuming that $f_1, f_2$ belong to a space of bounded signals.
-
-For example, if the $G$-CNN uses a sigmoid as its nonlinearity, then for all $g \in G$: $|f_1(g)| \leq 1$ and $|f_1(g)| \leq 1$, and the above inequality proves the Lipschitz continuity of the TC with constant $K=3\text{vol}(G)$.
-
-If the $G$-CNN has no nonlinearity, we recall its inputs are images with normalized intensity between 0 and 1. Thus, we can compute an upper bound of $\|f_1\|_\infty$ and $\|f_2\|_\infty$ that only depends on the (learned) weights of the $G$-CNN, and proves the Lipschitz continuity using this constant.
-\end{proof}
-
-
-
-\subsection{Bispectrum: Definition}
-
-The Bispectrum is the Fourier transform of the triple correlation. Inserting the formula for the Fourier transform above, and simplifying (see details in compact groups proof):
-\begin{align*}
-    \beta(f)_{k_1, k_2}
-        &=
-    \frac{1}{n^3}\sum_{x_1 \in Z_n} \sum_{x_2 \in Z_n}
-    T(f)(x_1, x_2)\left[
-                \rho_1(x_1)^{\dagger} \otimes \rho_2(x_2)^{\dagger}
-            \right]\\
-    &=
-    \frac{1}{n^3}\sum_{x_1 \in Z_n} \sum_{x_2 \in Z_n}
-    \left(\sum_{x \in Z_n}
-       f^*(x)
-       f\left(x + x_1\right) f\left(x+ x_2\right)\right)
-            \left[
-                \rho_1(x_1)^{\dagger} \otimes \rho_2(x_2)^{\dagger}
-            \right]\\
-    &= \frac{1}{n}
-        \sum_{x \in Z_n} f^*(x)
-            [\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}]
-            \left[
-                \rho_1(x) \otimes \rho_2(x)
-            \right]\\
-    &= \mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}
-        \left[
-            \frac{1}{n}
-                \sum_{x \in Z_n} f(x)^* \rho_1(x) \otimes \rho_2(x)
-        \right] \\
-    &=
-\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2} C_{{\rho_1} {\rho_2}}
-    \left[
-    \bigoplus_{\rho \in \rho_1 \otimes \rho_2}
-        \mathcal{F}(f)_\rho
-    \right] C_{{\rho_1} {\rho_2}}^{\dagger} .
-\end{align*}
-where each representation $\rho_j$ is associated with and integer $k_j$ such that: $\rho_1(x') = \exp(-i\frac{2\pi x'}{k_1})$ and $\rho_2(x') = \exp(-i\frac{2\pi x'}{k_2})$ for any $x' \in Z_n$.
-
-
-For a pair of frequencies $k_1$ and $k_2$, the bispectrum $\beta$ is defined:
-
-\begin{equation}
-\beta_{k_1, k_2} = \hat{f}_{k_1}\hat{f}_{k_2}\hat{f}^{\dagger}_{k_1+k_2},
-\end{equation}
-
-with the summation $k_1+k_2$ taken modulo the dimension of the signal. Due to the conjugated third term, any absolute phase shifts resulting from translations in the input cancel out, leaving the bispectrum invariant.
-
-\subsection{Bispectrum: Invariance}
-
-
-Under translation, $\beta$ becomes:
-
-\vspace{-1cm}
-\begin{eqnarray*}
-\beta^z_{k_1, k_2} &=& e^{-i 2 \pi k_1 z / n} \hat{f}_{k_1}  e^{-i 2 \pi k_2 z/ n} \hat{f}_{k_2} e^{i 2 \pi (k_1+k_2) z / n} \hat{f}^{\dagger}_{k_1 + k_2} \\
- &=& e^{-i 2 \pi k_1 z / n} e^{-i 2 \pi k_2 z/ n} e^{i 2 \pi k_1 z / n} e^{i 2 \pi k_2 z / n}\hat{f}_{k_1} \hat{f}_{k_2} \hat{f}^{\dagger}_{k_1 + k_2} \\
- &=& \hat{f}_{k_1} \hat{f}_{k_2} \hat{f}^{\dagger}_{k_1 + k_2}
-\\ &=& \beta_{k_1, k_2}.
-\end{eqnarray*}
-
-
-\subsection{Bispectrum: Signal Reconstruction}
-
-We answer the question: given the bispectrum $\beta(f)$, what is the minimum number of bispectral coefficients that we need to recover the signal $f$?
-
-\begin{proposition}
-    The $n+1$ coefficients $\beta(f)_{0, 0}, \beta(f)_{1, 0}, ...., \beta(f)_{1, n}$ are enough to recover the signal $f$.
-\end{proposition}
-
-\begin{lemma}
-    We have the following relationships between bispectral coefficients:
-    \begin{align*}
-    \beta_{0, 0}
-    &= \hat{f}_{0}\hat{f}_{0}\hat{f}^{\dagger}_{0}
-    = \hat{f}_{0} | \hat{f}_{0} |^2 \quad \rightarrow |\beta_{0, 0}| = |\hat{f}_{0}|^3\\
-    \beta_{1, 0}
-        &= \hat{f}_{1} \hat{f}_{0} \hat{f}^{\dagger}_{1}
-        =  \hat{f}_{0} | \hat{f}_{1} |^2, \\
-    \beta_{k, 0}
-        &= \hat{f}_{k} \hat{f}_{0} \hat{f}^{\dagger}_{k} =  \hat{f}_{0} |\hat{f}_{k}|^2 \\
-    \beta_{1, k-1}
-        &= \hat{f}_{1} \hat{f}_{k-1} \hat{f}^{\dagger}_{k} .
-    \end{align*}
-\end{lemma}
-
-\begin{proof}
-    We show that the $n+1$ coefficients $\beta(f)_{0, 0}, \beta(f)_{1, 0}, ...., \beta(f)_{1, n}$ are enough to recover the Fourier transform $\hat f$ of the signal $f$.
-
-    The $\beta(f)_{0, 0}$ allows us to recover the DC component:
-    \begin{equation}
-            |\hat f (0)| = |\beta(f)_{0, 0}|^{1/3}, \quad \text{arg}(\hat f(0)) = \text{arg}(\beta_{0, 0}).
-    \end{equation}
-
-   We compute the initialization of the recurrence, i.e. the first frequency $\hat f (1)$:
-   \begin{equation}
-       | \hat f (1)| = \left(\frac{\beta_{1, 0}}{ \hat{f}_{0} }\right)^{1/2}, \quad  \text{arg}(\hat f(1)) = 0,
-   \end{equation}
-   We note that we could equally set $\text{arg}(\hat f(1)) = \phi$ for any $\phi \in [0, 2\pi)$. This indeterminacy is inevitable, since it is a reflection of translation invariance~\cite{kondor2008thesis}.
-
-    Next, we compute the Fourier coefficients of $f$ iteratively, from the lowest frequency to the highest:
-       \begin{equation}
-       \hat f (k) = \sqrt{\frac{\beta_{1, k-1}}{ \hat{f}_{1} \hat{f}_{k-1} }}^\dagger.
-   \end{equation}
-\end{proof}
-
-
-\section{\color{deepblue}{Domain $\Omega = (G,~+)$ (finite groups)}}
-
-\subsection{Fourier Transform: Definition}
-
-The (matrix-valued) Fourier transform on $G$ is defined by coefficients
-$$
-F(\omega)=\frac{1}{|G|} \sum_{g \in G} f(g) D_\omega(g)^{\dagger} .
-$$
-
-
-\subsection{Fourier Transform: Equivariance}
-
-\subsection{Steerable Bispectrum: After G-Conv on Steerable Basis}
-
-Consider the general case where $\{Y_l\}_l$ is a basis of steerable functions on a domain $\mathbb{\Omega}$ for a group $G$. That is the vector of stacked functions $Y$ is such that: $Y(g.x) = \rho(g)Y(x)$ for $\rho(g) \in \mathbb{C}^{L \times L}$ a representation of the group element $g$. We assume that such a basis exists.
-
-We would like to compute the bispectrum to get a complete invariance for the action of the group $G$. We write the group convolution:
-\begin{align*}
-    \Theta(g_0) = (w \ast f)(g_0)
-    &= \sum_{x \in \Omega}
-        w(g_0^{-1}.x).f(x) \\
-    &= \sum_{x \in \Omega}
-        \sum_{l=1}^L \hat{w}^{l}Y_l(g_0^{-1}.x).f(x) \\
-    &= \sum_{x \in \Omega}
-        \hat{w}^T.Y(g_0^{-1}.x).f(x) \\
-    &= \sum_{x \in \Omega}
-        \left(\hat{w}^T.\rho(g_0)^{-1}.Y(x)\right).f(x) \\
-    &= \sum_{x \in \Omega}
-        \left(\hat{w}^T.\rho(g_0)^T.Y(x)\right).f(x),\quad \text{(assuming $\rho$ unitary)} \\
-    &= \sum_{x \in \Omega}
-        \left((\rho(g_0)\hat{w})^T.Y(x)\right).f(x),\\
-    &= \sum_{x \in \Omega} \sum_{l=1}^L
-        \left((\rho(g_0)\hat{w})_l.Y_l(x)\right).f(x),\\
-    &=  \sum_{l=1}^L \left(
-    \sum_{x \in \Omega}
-        Y_l(x).f(x) \right)(\rho(g_0)\hat{w})_l,\\
-    % &= (\rho(g_0)\hat{w})^T \sum_{x \in \Omega}
-    %    Y(x).f(x),\quad\text{(with $Y(x) \in \mathbb{C}^L$ and $f(x) \in \mathbb{R}$)}.\\
-\end{align*}
-which ressembles a fourier decomposition:
-\begin{equation}
-    \Theta(x) = \sum_{l=1}^L \hat \Theta_l . Y_l(x)
-\end{equation}
-
-We would like to get the coefficients of $\hat{\Theta}$ on a steerable basis of the domain. However, the domain has changed. And we don't have a steerable basis of this new domain.
-
-For the bispectrum, we would want the coefficients of $\Theta$ on a "basis" that is indexed by the irreducible representations of the group. Because the spectral coefficients are not indexed by $l$ anymore.
-
-However, if we consider a case where the action is such that the output of the group convolution has the same domain as the input. Then we can use the same basis functions.
-
-
-\subsection{Triple Correlation: Definition}
-
-The triple correlation of a function $f$ defined on a finite group $G$ is:
-$$
-T(f)_{g_1, g_2}=\frac{1}{|G|} \sum_{g \in G} f^*(g) f\left(g g_1\right) f\left(g g_2\right) .
-$$
-
-\subsection{Triple Correlation: Invariance}
-Note that $T(f)$ is a function defined on $G \times G$, and that, like the autocorrelation, the triple correlation is invariant under left translation: $T(f_1)=T(f_2)$ if $f_1(g)=f_2(h g)$.
-
-\subsection{Triple Correlation: Symmetries}
-
-\begin{proposition}[Symmetries]
-The triple correlation of a signal has the following symmetry:
-\begin{align*}
-     (s1) \qquad T(f)_{g_1, g_2} &= T(f)_{g_2, g_1}
-\end{align*}
-If $G$ is commutative, the triple correlation of a real signal has the following additional symmetries:
-\begin{align*}
-     (s2) \qquad T(f)_{g_1, g_2}
-     = T(f)_{g^{-1}_1, g_2g^{-1}_1}
-     = T(f)_{g_2g^{-1}_1, g^{-1}_1}
-     = T(f)_{g_1g^{-1}_2, g^{-1}_2}
-     = T(f)_{g^{-1}_2, g_1g^{-1}_2}
-\end{align*}
-\end{proposition}
-
-
-\begin{proof}
-    We prove symmetry (s1), which relies on the fact that $f(gg_2)$ and $f(gg_1)$ are scalar values that commute:
-    \begin{align*}
-        T(f)_{g_2, g_1}
-            = \frac{1}{|G|} \sum_{g \in G} f^*(g)f(gg_2)f(gg_1)
-            = \frac{1}{|G|} \sum_{g \in G} f^*(g)f(gg_1)f(gg_2)
-            = T(f)_{g_2, g_1}.
-    \end{align*}
-    We assume that the signal is real and that $G$ is commutative.
-    We prove the first equality of symmetry (s2):
-    \begin{align*}
-        T(f)_{g^{-1}_1, g_2g^{-1}_1}
-            &= \frac{1}{|G|} \sum_{g \in G} f^*(g)f(gg^{-1}_1)f(gg_2g^{-1}_1) \\
-            &= \int_{g' \in G} f^*(g'g_1)f(g')f(g'g_1g_2g^{-1}_1)
-                \quad \text{(with $g' = g - g_1$, i.e. $g = g'g_1$)} \\
-            &= \int_{g' \in G} f^*(g'g_1)f(g')f(g'g_2)
-                \quad \text{(because $G$ commutative: $g_2 g^{-1}_1 = g^{-1}_1 g_2$)} \\
-            &= \frac{1}{|G|} \sum_{g \in G} f(gg_1)f(g)f(gg_2)
-                \quad \text{($f$ is a real signal, which equals its conjugate)}\\
-            &= \frac{1}{|G|} \sum_{g \in G} f(g)f(gg_1)f(gg_2)
-                \quad \text{($f$ is takes on real values that commute)}\\
-            &= \frac{1}{|G|} \sum_{g \in G} f(g)^*f(gg_1)f(gg_2) \\
-            &= T(f)_{g_2, g_1}
-    \end{align*}
-    The second equality of symmetry (s2) follows using (s1).
-    The third and fourth equality of symmetry (s2) have the same proof.
-\end{proof}
-
-\subsection{Bispectrum: Definition}
-
-The bispectrum formula for arbitrary finite groups:
-\begin{equation}
-\beta(f)_{\rho_1, \rho_2}
-=
-\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2} C_{{\rho_1} {\rho_2}}
-    \left[
-    \bigoplus_{\rho \in \rho_1 \otimes \rho_2}
-        \mathcal{F}(f)_\rho
-    \right] C_{{\rho_1} {\rho_2}}^{\dagger} .
-\end{equation}
-
-
-\begin{proof}
-We prove this by applying the Fourier transform to the triple correlation formula, following Karala. Note that the proof over compact (non-necessarily finite) groups will be similar (actually, this proof here can be deduced from the compact one).
-
-The Fourier transformation of $T(f)$ requires Kronecker (tensor) products of the representation matrices $\left\{D_\omega\right\}_{\omega \in \Omega}$. The bispectrum is obtained by the formula:
-\begin{equation}
-    \beta(f)_{\rho_1, \rho_2}
-    =
-    \frac{1}{|G|^2} \sum_{g_1 \in G} \sum_{g_2 \in G}
-        T(f)_{g_1, g_2} \rho_1(g_1)^{\dagger} \otimes \rho_2(g_2)^{\dagger}
-\end{equation}
-for all ${\rho_1}, {\rho_2}$. Note: Karala uses $|G|$ but it's probably $|G|^2$?
-
-Inserting the formula for the Fourier transform above, and simplifying (see details in compact groups proof):
-\begin{align*}
-    \beta(f)_{\rho_1, \rho_2}
-    &=
-    \frac{1}{|G|^3}\sum_{g_1 \in G} \sum_{g_2 \in G} \sum_{g \in G}
-       f^*(g)
-       f\left(g g_1\right) f\left(g g_2\right)
-            \left[
-                \rho_1(g_1)^{\dagger} \otimes \rho_2(g_2)^{\dagger}
-            \right] \\
-    &= \frac{1}{|G|}
-        \sum_{g \in G} f^*(g)
-            [\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}]
-            \left[
-                \rho_1(g) \otimes \rho_2(g)
-            \right]\\
-    &= \mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}
-        \left[
-            \frac{1}{|G|}
-                \sum_{g \in G} f(g)^* \rho_1(g) \otimes \rho_2(g)
-        \right] \\
-    &=
-\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2} C_{{\rho_1} {\rho_2}}
-    \left[
-    \bigoplus_{\rho \in \rho_1 \otimes \rho_2}
-        \mathcal{F}(f)_\rho
-    \right] C_{{\rho_1} {\rho_2}}^{\dagger} .
-\end{align*}
-where the last line is the Clebsh Gordan decomposition on $\rho_1 \times \rho_2$ and subsequent application of the definition of the Fourier transform.
-
-The formula simplifies if $G$ is commutative. (TODO: proof?).
-
-\end{proof}
-
-\subsection{Bispectrum: Invariance}
-
-\subsection{Bispectrum: Signal Reconstruction}
-
-For the definition of the bispectrum, we take the convention with the transpose conjugate is on the first term, as in Eq 7.5 p 86 of Kondor thesis.
-
-% SO2: C4 = Z/4Z -> OK
-
-% Need the reduction for these three
-% O2: D4
-
-% D16 is the real goal: 8 rotations and 1 relfections
-
-% SO3: Octahedral group with 24 elements
-% https://en.wikipedia.org/wiki/Octahedral_symmetry
-% O3: no chirality, with 48 elements
-% --> Look into the e2cnn library / Weiler's work
-
-
-We generalize to the non-commutative case.  We show that we recover the signal $f$ from its bispectral coefficients. We follow the same strategy as in the discrete, commutative case: we show that we can recover the Fourier transform of the signals for every irreducible representation $\rho$ of the group. We recover the signal itself by inverse Fourier transform. We first need to know ``how many'' irreducible representations the group has. For finite groups, this number is finite. We use the following
-
-    \begin{proposition}
-        The number of irreducible representations for a finite group $G$ is equal to the number of conjugacy classes. If $G$ is abelian, the number of irreducible representations is equal to $|G|$ and each representation is 1-dimensional.
-    \end{proposition}
-
-\begin{proof}
-    We recall that the conjugacy class of an element $g \in G$ is defined with the conjugation operation defined by inner automorphisms:
-    \begin{equation}
-        C(g) = \{ hgh^{-1} ~|~ \forall h \in G\}.
-    \end{equation}
-
-    To prove the proposition in its entirety would require delving deep into the theory, but the core idea relies on the orthogonality relations of the character table of the group. The rows of the character table correspond to irreducible characters (which, over the complex numbers, can be associated with irreducible representations), while the columns correspond to conjugacy classes. Whether the group is commutative or not, this result holds. However, for commutative (or abelian) groups, every element is its own conjugacy class.
-\end{proof}
-
-We note that for Lie groups, this might not hold. For example, the group SO(3) has one irreducible representation in each odd dimension: it has an infinite number of irreps. We will deal with this case later.
-
-We first show relationships between the bispectral coefficients and the Fourier coefficients. We denote $\rho_0$ the trivial representation, which is the representation that sends every group element to the identity matrix, i.e. to the scalar 1 by irreducibility of the $\rho$'s.
-
-\begin{lemma}\label{lem:coefs}
-    For $\rho_0$ the trivial representation of the group $G$, and $\rho$ any other irreducible representation of dimension $D$, the bispectral coefficients write:
-    \begin{align*}
-    \beta_{\rho_0, \rho_0}
-    &= |\mathcal{F}(f)_{\rho_0}|^2 \mathcal{F}(f)_{\rho_0}
-    \in \mathbb{C}\\
-    \beta_{\rho, \rho_0}
-    &= \mathcal{F}(f)_{\rho_0}^\dagger
-    \mathcal{F}(f)_{\rho}^\dagger\mathcal{F}(f)_{\rho}
-    \in \mathbb{C}^{D \times D}.
-    \end{align*}
-\end{lemma}
-
-\begin{proof}
-    Take $\rho_0$ the trivial representation, we know that $\rho \otimes \rho_0  = \rho$ such that the Clebsh-Gordan decomposition:
-    \begin{equation}
-        \rho \otimes \rho_0 = C_{{\rho} {\rho_0}}\left[
-    \bigoplus_{\rho' \in \rho \otimes \rho_0}
-        \rho'
-    \right] C_{{\rho} {\rho_0}}^{\dagger} = C_{{\rho} {\rho_0}}
-        \rho C_{{\rho} {\rho_0}}^{\dagger}
-    \end{equation}
-    gives: $C_{{\rho} {\rho_0}} C_{{\rho} {\rho_0}}^{\dagger}  = 1$ for any irreps $\rho$. In addition, for $\rho = \rho_0$, the Clebsh-Gordan matrices are scalars. Actually, in this case, the CJ matrix is just the identity matrix (according to chatGPT).
-
-    We compute the bispectral coefficient:
-    \begin{align*}
-    \beta_{\rho_0, \rho_0}
-    &=
-    (\mathcal{F}(f)_{\rho_0} \otimes \mathcal{F}(f)_{\rho_0})^\dagger C_{{\rho_0} {\rho_0}}
-    \left[
-    \bigoplus_{\rho \in \rho_0 \otimes \rho_0}
-        \mathcal{F}(f)_\rho
-    \right] C_{{\rho_0} {\rho_0}}^{\dagger} \\
-    &=
-    (\mathcal{F}(f)_{\rho_0} \otimes \mathcal{F}(f)_{\rho_0})^\dagger C_{{\rho_0} {\rho_0}}
-        \mathcal{F}(f)_{\rho_0}
-     C_{{\rho_0} {\rho_0}}^{\dagger}
-    \qquad \text{($\rho_0 \otimes \rho_0 = \rho_0$ which is irreducible)}
-    \\
-    &=
-    (\mathcal{F}(f)_{\rho_0}^2)^\dagger C_{{\rho_0} {\rho_0}}
-        \mathcal{F}(f)_{\rho_0}
-     C_{{\rho_0} {\rho_0}}^{\dagger}
-    \qquad \text{($\mathcal{F}(f)_{\rho_0}$ is a scalar for which tensor product is multiplication)}
-    \\
-    &=(\mathcal{F}(f)_{\rho_0}^\dagger)^2 \mathcal{F}(f)_{\rho_0}
-    C_{{\rho_0} {\rho_0}} C_{{\rho_0} {\rho_0}}^{\dagger}
-    \qquad\text{(scalars commute with other elements)}\\
-    &=|\mathcal{F}(f)_{\rho_0}|^2 \mathcal{F}(f)_{\rho_0}
-    \qquad\text{($C_{{\rho_0} {\rho_0}} C_{{\rho_0} {\rho_0}}^{\dagger}  = 1$).}
-    \end{align*}
-
-    Take any irreducible representation $\rho$ of dimension $D$, we have:
-    \begin{align*}
-        \beta_{\rho, \rho_0}
-        &= (\mathcal{F}(f)_{\rho} \otimes \mathcal{F}(f)_{\rho_0})^\dagger C_{{\rho} {\rho_0}}
-    \left[
-    \bigoplus_{\rho \in \rho \otimes \rho_0}
-        \mathcal{F}(f)_\rho
-    \right] C_{{\rho} {\rho_0}}^{\dagger} \\
-        &= \mathcal{F}(f)_{\rho_0}^\dagger \mathcal{F}(f)_{\rho}^\dagger C_{{\rho} {\rho_0}}
-        \mathcal{F}(f)_{\rho} C_{{\rho} {\rho_0}}^{\dagger} \\
-        &=\mathcal{F}(f)_{\rho_0}^\dagger \mathcal{F}(f)_{\rho}^\dagger\mathcal{F}(f)_{\rho} \qquad\text{(CJ coefficients for $\rho, \rho_0$ are identity matrices)}.
-    \end{align*}
-    \end{proof}
-
-\begin{proposition}
-    Consider a finite group $G$ that is non abelian. Consider the $L$ bispectral coefficients of an unknown signal $f$, where $L$ is a number computed from the Kroneck product table of the group. We can recover the Fourier coefficients of $f$ from only $L$ bispectral coefficients.
-\end{proposition}
-
-
-\begin{proof}
-    We consider first Fourier coefficient (DC component), i.e the Fourier transform of the signal at the trivial representation $\rho_0$:
-    \begin{equation}
-    \mathcal{F}(f)_{\rho_0} = \hat{f}_{\rho_0} = \int_G f(g)\rho_0(g) dg = \int_G f(g) dg \in \mathbb{C}
-     \end{equation}
-    Using Lemma~\ref{lem:coefs}, we can recover it from the ``first'' bispectral coefficient, as:
-    \begin{equation}
-        | \mathcal{F}(f)_{\rho_0}| = \left(|\beta_{\rho_0, \rho_0}|\right)^{1/3},
-        \quad
-        \text{arg}(\mathcal{F}(f)_{\rho_0}) = \text{arg}(\beta_{\rho_0, \rho_0}),
-    \end{equation}
-    which mirrors the commutative case.
-
-    We consider another Fourier coefficient $\mathcal{F}(f)_{\rho_1}$ for a representation $\rho_1$ of dimension $D$. The coefficient is a matrix in $\mathbb{C}^{D \times D}$. Using Lemma~\ref{lem:coefs}, we can recover it from a bispectral coefficient:
-    \begin{equation}
-        \mathcal{F}(f)_{\rho}^\dagger\mathcal{F}(f)_{\rho}
-        = \frac{\beta_{\rho, \rho_0}}{\mathcal{F}(f)_{\rho_0}^\dagger} \in \mathbb{C}^{D \times D},
-    \end{equation}
-    where the denominator is a scalar in the equation above. The matrix $\mathcal{F}(f)_{\rho}^\dagger\mathcal{F}(f)_{\rho}$ is hermitian, and admits a square-root. We recover the Fourier coefficient as this square-root:
-      \begin{equation}
-        \mathcal{F}(f)_{\rho}'
-        = \left(\frac{\beta_{\rho, \rho_0}}{\mathcal{F}(f)_{\rho_0}^\dagger}
-        \right)^{1/2}.
-    \end{equation}
-    The square-root only recovers $\mathcal{F}(f)_{\rho}$ up to a matrix factor: this is an unidentifiability similar to the commutative case. Specifically, consider the SVD decomposition of $\mathcal{F}(f)_{\rho}$:
-    \begin{align*}
-        \mathcal{F}(f)_{\rho}
-        &= U.\Sigma.V^\dagger\\
-        \Rightarrow
-         \mathcal{F}(f)_{\rho}^\dagger\mathcal{F}(f)_{\rho}
-        &= (U.\Sigma.V^\dagger)^\dagger. U.\Sigma.V^\dagger
-        = V \Sigma^2 V^\dagger\\
-        \Rightarrow
-        \mathcal{F}(f)_{\rho}'
-        &= V \Sigma V^\dagger
-    \end{align*}
-    Thus, we have: $\mathcal{F}(f)_{\rho} = UV^\dagger.\mathcal{F}(f)_{\rho}' $ where $U, V$ are unitary matrices that come from the (unknown) SVD decomposition of the (unknown) $\mathcal{F}(f)_{\rho}$. By recovering $\mathcal{F}(f)_{\rho}$ as $\mathcal{F}(f)_{\rho}'$, we fix $UV^\dagger=I$ in the same way that we have fixed $\phi = 0$ (rotation of angle 0, i.e. the identity) in the commutative case.
-
-    We seek to find the Fourier coefficient for another representation $\rho_2$. We operate as we would in the commutative case: we write the bispectral coefficient for $\rho_1, \rho_1$. (Note: we can recover it, up to unitary transformation. We only have to find what its UV is.) We denote $\mathcal{R}$ the (finite) set of irreducible representations of the group $G$.
-    \begin{align*}
-    \beta_{\rho_1, \rho_1}
-    &=
-    (\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_1})^\dagger C_{{\rho_1} {\rho_1}}
-    \left[
-    \bigoplus_{\rho \in \rho_1 \otimes \rho_1}
-        \mathcal{F}(f)_\rho
-    \right] C_{{\rho_1} {\rho_1}}^{\dagger}\\
-    &=
-        (\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_1})^\dagger
-        C_{{\rho_1} {\rho_1}}
-    \left[
-    \bigoplus_{\rho \in \mathcal{R}}
-        \mathcal{F}(f)_\rho^{n_{\rho, \rho_1}}
-    \right] C_{{\rho_1} {\rho_1}}^{\dagger},
-    \end{align*}
-    where $n_{\rho, \rho_1}$ is the multiplicity of irreps $\rho$ in the decomposition of $\rho_1, \rho_1$. If we know the group $G$ a priori, then we can compute the decomposition of $\rho_1$ into its irreps and we know what this is.
-
-    We get the equation:
-    \begin{equation}
-        \bigoplus_{\rho \in \mathcal{R}}
-        \mathcal{F}(f)_\rho^{n_{\rho, \rho_1}}
-    = C_{{\rho_1} {\rho_1}}^{-1}(\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_1})^{-\dagger} \beta_{\rho_1, \rho_1} C_{{\rho_1} {\rho_1}}^{-\dagger},
-    \end{equation}
-    where we know everything on the right hand side.
-    Therefore, every Fourier coefficient $\mathcal{F}(f)_\rho$ that appears in the decomposition of $\rho_1 \otimes \rho_1$ into irreducible irreps $\rho$ can be computed, by reading off the elements of the block diagonal matrix.  We get the $\mathcal{F}(f)_\rho$ for which $n_{\rho, \rho_1} \neq 0$.
-    In the commutative case, this is how we get $\rho_2$.
-    Here, we assume that we get at least one other Fourier coefficient, for a $\rho_2$ that we fix.
-
-    We can then compute:
-    \begin{align*}
-    \beta_{\rho_1, \rho_2}
-    &=
-    (\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2})^\dagger C_{{\rho_1} {\rho_2}}
-    \left[
-    \bigoplus_{\rho \in \rho_1 \otimes \rho_2}
-        \mathcal{F}(f)_\rho^{n_{\rho, \rho_2}}
-    \right] C_{{\rho_1} {\rho_2}}^{\dagger}
-    \end{align*}
-    to get the equation:
-        \begin{equation}
-        \bigoplus_{\rho \in \mathcal{R}}
-        \mathcal{F}(f)_\rho^{n_{\rho, \rho_2}}
-    = C_{{\rho_1} {\rho_2}}^{-1}(\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2})^{-\dagger} \beta_{\rho_1, \rho_2} C_{{\rho_1} {\rho_2}}^{-\dagger},
-    \end{equation}
-    where everything on the right-hand side is known. We note that, for arbitrary groups and their representations, Clebsch–Gordan matrices are not known in general. Yet we assume that we know them for the groups that we consider, i.e. we have computed them numerically (possible, according to Wikipedia).
-
-
-    Therefore, every Fourier coefficient $\mathcal{F}(f)_\rho$ that appears in the decomposition of $\rho_1 \otimes \rho_1$ into irreducible irreps $\rho$ can be computed, by reading off the elements of the block diagonal matrix. We get the $\mathcal{F}(f)_\rho$ for which $n_{\rho, \rho_2} \neq 0$.
-
-    We iterate this procedure. The remaining question is: do we get all of the irreps in this way? If so: how many iterations do we need? The answer is that it depends on its CJ matrices.
-
-    If we know the group a priori, we can compute the Kronecker product table for $G$, where at row $i$ and column $j$ we list all of the irreps that appear in the decomposition of $\rho_i \otimes \rho_j$.
-
-    We want to compute $m_i$ i.e. the multiplicity $\rho_i$ in the decomposition $\tilde \rho = \rho_j \otimes \rho_k$. We present a simple computational procedure, inspired by Cohen \& Welling 2017, that can be used to determine that multiplicity.
-
-    This procedure relies on the character $\chi_{\tilde \rho}(g)=\operatorname{Tr}(\tilde \rho(g))$ of the representation to be decomposed. The most important fact about characters is that the characters of irreps $\rho_i, \rho_j$ are orthogonal:
-    \begin{equation}
-    \left\langle\chi_{\rho_i,}, \chi_{\rho_k}\right\rangle \equiv \frac{1}{|G|} \sum_{h \in G} \chi_{\rho_i}(h) \chi_{\rho_k}(h)=\delta_{i k} .
-    \end{equation}
-    Furthermore, since the trace of a direct sum equals the sum of the traces (i.e. $\chi_{\rho \oplus \rho^{\prime}}=\chi_\rho+\chi_{\rho^{\prime}}$ ), and every representation $\rho$ is a direct sum of irreps, it follows that we can obtain the multiplicity of
-    irrep $\rho_i$ in $\tilde \rho$ by computing the inner product with the $i$-th character:
-    $$
-    \left\langle\chi_{\tilde \rho}, \chi_{\rho_i}\right\rangle
-    =
-    \left\langle\chi_{\oplus_j m_j \rho_j}, \chi_{\rho_i}\right\rangle=\left\langle\sum_j m_j \chi_{\rho_j}, \chi_{\rho_i}\right\rangle
-    =
-    \sum_j m_j\left\langle\chi_{\rho_j}, \chi_{\rho_i}\right\rangle
-    =m_i
-    $$
-    So the dot product of characters is all we need to determine the multiplicities that we need.
-
-
-    \begin{table}[]
-    \centering
-    \begin{tabular}{|l|l|l|l|l|}
-    \hline
-    $\otimes$ & $\rho_0$       & $\rho_1$       & $\rho_2$       & $\rho_3$       \\ \hline
-    $\rho_0$  & $(1, 0, 0, 0)$ & $(0, 1, 0, 0)$ & $(0, 0, 1, 0)$ & $(0, 0, 0, 1)$ \\ \hline
-    $\rho_1$  & $(0, 1, 0, 0)$ & $(m_k^{11})_k$ & $(m_k^{12})_k$ & $(m_k^{13})_k$ \\ \hline
-    $\rho_2$  & $(0, 0, 1, 0)$ & $(m_k^{21})_k$ & $(m_k^{22})_k$ & $(m_k^{23})_k$ \\ \hline
-    $\rho_3$  & $(0, 0, 0, 1)$ & $(m_k^{31})_k$ & $(m_k^{32})_k$ & $(m_k^{33})_k$ \\ \hline
-    \end{tabular}
-    \caption{}
-    \label{tab:my-table}
-    \end{table}
-
-    Once we have the Kronecker product table, then we can see how many bispectral coefficients we need to implement. This will depend on the group. If there are irreps missing from the row of $\rho_1$, then the answer is that we will need more than the tensor products of the form $\rho_1 \otimes \rho_j$ to get all of the coefficients. We might need to consider others of the form $\rho_1$.
-
-    We consider the following procedure.
-    \begin{itemize}
-        \item List = $[\rho1]$
-        \item Compute $\rho_1 \otimes \rho_1$.
-        \item Take all the irreps that are not in List that appears in that decomposition.
-        \item For every such irrep $\rho_2$:
-        \begin{itemize}
-            \item Compute $\rho_1 \otimes \rho_2$
-            \item Append $\rho_2$ to List.
-            \item Take all the irreps that are not in List that appears in that decomposition.
-            \item For every such irrep $\rho_3$:
-            \begin{itemize}
-                \item Compute $\rho_1 \otimes \rho_3$
-                \item Append $\rho_3$ to List.
-                \item etc
-            \end{itemize}
-        \end{itemize}
-    \end{itemize}
-
-    We wonder whether this procedure gives us all the irreps of the group, and if so, in how many steps L. The number of steps is the number of bispectral coefficients needed.
-
-    The proposed procedure essentially uses a breadth-first search algorithm on the space of irreducible representations, starting with $\rho_1$ and using the tensor product with $\rho_1$ as the mechanism to explore the space. Whether this procedure succeeds in including all the irreps in the List might on the group and its Kronecker (tensor) product table.
-
-    However, for many groups, this procedure does indeed succeed in listing all the irreps. This is particularly true for groups where the tensor product of any two irreps spans the whole space of irreps. This is the case, for example, for SU(2) (the group of 2x2 unitary matrices with determinant 1).
-
-\subsubsection{Example: Diedral group $D_4$}
-
-    The diedral group $D_4$ is the group of symmetries of a square. $D_4$ has five irreps, e.g. shown in Cohen \& Welling 2017 (Steerable CNNs).
-
-    \paragraph{Irreps and their matrices $\rho(g)$ for $g \in G$}
-
-    \begin{figure}
-        \centering
-        \includegraphics[width=\textwidth]{figs/d4.png}
-    \end{figure}
-
-    \paragraph{Kronecker table} To get the Kronecker table, we compute each tensor-representation by:
-    \begin{itemize}
-        \item computing its matrices $\rho(g)$ on each group element, by definition of the tensor product of matrices
-        \item computing the characters $\Xi(g) = \text{Tr}(\rho(g))$ for each group element
-        \item computing the inner-product of the character $\Xi$ with each character of each irreps to get the multiplicity.
-    \end{itemize}
-
-
-\begin{table}[]
-\centering
-    \begin{tabular}{|l|l|l|l|l|l|}
-    \hline
-    $\otimes$ & $A_1$             & $A_2$             & $B_1$             & $B_2$             & $E$               \\ \hline
-    $A_1$     & $(1, 0, 0, 0, 0)$ & $(0, 1, 0, 0, 0)$ & $(0, 0, 1, 0, 0)$ & $(0, 0, 0, 1, 0)$ & $(0, 0, 0, 0, 1)$ \\ \hline
-    $A_2$     & $(0, 1, 0, 0, 0)$ & $(1, 0, 0, 0, 0)$ & $(0, 0, 0, 1, 0)$ & $(0, 0, 1, 0, 0)$ & $(0, 0, 0, 0, 1)$ \\ \hline
-    $B_1$     & $(0, 0, 1, 0, 0)$ & $(0, 0, 0, 1, 0)$ & $(1, 0, 0, 0, 0)$ & $(0, 1, 0, 0, 0)$ & $(0, 0, 0, 0, 1)$ \\ \hline
-    $B_2$     & $(0, 0, 0, 1, 0)$ & $(0, 0, 1, 0, 0)$ & $(0, 1, 0, 0, 0)$ & $(1, 0, 0, 0, 0)$ & $(0, 0, 0, 0, 1)$ \\ \hline
-    $E$       & $(0, 0, 0, 0, 1)$ & $(0, 0, 0, 0, 1)$ & $(0, 0, 0, 0, 1)$ & $(0, 0, 0, 0, 1)$ & $(1, 1, 1, 1, 0)$ \\ \hline
-    \end{tabular}
-    \caption{Kronecker table for the diedral group $D_4$. We choose $\rho_0 = A_1$, $\rho_1 = E$.}
-    \label{tab:my-table}
-\end{table}
-
-
-\paragraph{Procedure} We choose $\rho_0 = A_1$, $\rho_1 = E$.
-\begin{itemize}
-    \item List = [$\rho_0$, $\rho_1$]
-    \item Compute irreps in the decomposition of $\rho_1 \otimes \rho_1$, that are not already in List. This gives: $A_2, B_1, B_2$.
-    \item Stopping criterion reached: all irreps in List.
-\end{itemize}
-
-\paragraph{Computations after the $G$-Convolution}
-
-We store the representations in a variable.
-
-We compute every Fourier coefficient of $\Theta$, via:
-\begin{equation}
-    \mathcal{F}(\Theta)_\rho = \sum_{g \in G} \Theta(g).\rho(g)
-\end{equation}
-by taking a linear combination of the $\rho(g)$ matrices for each $\rho$ irreps. Note that $\mathcal{F}(\Theta)_\rho \in \mathbb{C}$ for every $\rho$ in $\{A_1, A_2, B_1, B_2\}$ and $\mathcal{F}(\Theta)_E \in \mathbb{C}^{2 \times 2}$.
-
-We compute $\beta_{\rho_0, \rho_0} \in \mathbb{C}$.
-\begin{equation}
-    \beta_{\rho_0, \rho_0} = |\mathcal{F}(\Theta)_{\rho_0}|^2 \mathcal{F}(\Theta)_{\rho_0} \in \mathbb{C}
-\end{equation}
-
-We compute $\beta_{\rho_1, \rho_0} \in \mathbb{C}^{2 \times 2}$.
-\begin{equation}
-    \beta_{\rho_1, \rho_0} = \mathcal{F}(\Theta)_{\rho_1}^\dagger\mathcal{F}(\Theta)_{\rho_1} \mathcal{F}(\Theta)_{\rho_0}^* \in \mathbb{C}^{2 \times 2}
-\end{equation}
-
-We compute $\beta_{\rho_1, \rho_1} \in \mathbb{C}^{4 \times 4}$. We use the notation: $\mathcal{F} = \mathcal{F}(\Theta)_{\rho_1}$.
-\begin{align*}
-    \beta_{\rho_1, \rho_1}
-    &= (\mathcal{F}(\Theta)_{\rho_1} \otimes \mathcal{F}(\Theta)_{\rho_1})^\dagger C_{{\rho_1} {\rho_1}}
-    \left[
-    \bigoplus_{\rho \in \rho_1 \otimes \rho_1=\{A_1, A_2, B_1, B_2\}}
-        \mathcal{F}(\Theta)_\rho
-    \right] C_{{\rho_1} {\rho_1}}^{\dagger} \\
-    &= \left[\begin{array}{ll}
-\mathcal{F}_{1,1}\left[\begin{array}{ll}
-\mathcal{F}_{1,1} & \mathcal{F}_{1,2} \\
-\mathcal{F}_{2,1} & \mathcal{F}_{2,2}
-\end{array}\right] & \mathcal{F}_{1,2}\left[\begin{array}{ll}
-\mathcal{F}_{1,1} & \mathcal{F}_{1,2} \\
-\mathcal{F}_{2,1} & \mathcal{F}_{2,2}
-\end{array}\right] \\
-\mathcal{F}_{2,1}\left[\begin{array}{ll}
-\mathcal{F}_{1,1} & \mathcal{F}_{1,2} \\
-\mathcal{F}_{2,1} & \mathcal{F}_{2,2}
-\end{array}\right] & \mathcal{F}_{2,2}\left[\begin{array}{ll}
-\mathcal{F}_{1,1} & \mathcal{F}_{1,2} \\
-\mathcal{F}_{2,1} & \mathcal{F}_{2,2}
-\end{array}\right]
-\end{array}\right]^\dagger C_{{\rho_1} {\rho_1}}
-\begin{bmatrix}
-     \mathcal{F}(\Theta)_{A_1} & 0 & 0 & 0\\
-     0 & \mathcal{F}(\Theta)_{A_2} & 0 & 0\\
-     0 & 0 & \mathcal{F}(\Theta)_{B_1} & 0\\
-     0 & 0 & 0 & \mathcal{F}(\Theta)_{B_2}\\
-\end{bmatrix}
-C_{{\rho_1} {\rho_1}}^{\dagger}
-\end{align*}
-
-we need the CJ coefficients for the $\rho_1 \otimes \rho_1 = E \otimes E$ decomposition.
-
-    % The Kronecker product table is not the character table of the group, but they are related through the orthogonality relations of the character values.
-
-    % More specifically, the tensor product of two representations can be decomposed into irreducible representations, and the multiplicity of a particular irrep in this decomposition can be determined using the character values from the character table.
-
-
-
-\end{proof}
-
-
-
-\section{\color{deepblue}{Domain $\Omega = (G,~+)$ (compact groups)}}
-
-\subsection{Fourier Transform: Definition}
-
-Fourier analysis can be generalized to other compact groups, such as 2D and 3D rotation, by preserving this defining property of equivariance to the group action \cite{rudin1962fourier}.
-
-\textbf{The (continuous) Fourier transform} (for commutative or non-commutative groups) is:
-\begin{equation}
-   \mathcal{F}(f)_\rho = \hat{f}_\rho = \int_G f(g)\rho(g) dg,
-\end{equation}
-
-defined for any $\rho$ being an irreducible representations, where $G$ denotes a compact group, and the integral uses the Haar measure $dg$ on $G$. Note that the definition of the Fourier transform is the same for non-commutative or commutative groups, the only requirement is for these groups to be compact.
-
-\subsection{Fourier Transform: Equivariance}
-
-Define the transformation of $f$ by $\tilde g$ as $(\tilde{g} \ast f)(g) = f({\tilde g}^{-1}\circ g)$. The Fourier transform undergoes a transformation by the same $\tilde g$, as follows, which shows its equivariance.
-
-\vspace{-0.7cm}
-\begin{align*}
-  \mathcal{F}(\tilde{g} \ast f)_\rho
-  &=  \int_{g \in G} (\tilde{g} \ast f)(g)\rho(g) dg \\
-  &=  \int_{g \in G} f({\tilde g}^{-1}\circ g)\rho(g) dg \\
-  &=  \int_{g' \in G} f(g')\rho(\tilde{g}\circ g') dg' \\
-  &=  \int_{g' \in G} f(g')\rho(\tilde{g}).\rho(g') dg' \\
-  &=  \rho(\tilde{g}) . \int_{g' \in G} f(g')\rho(g') dg' \\
-  &=  \rho(\tilde{g}) .\mathcal{F}(f)_\rho
-\end{align*}
-\vspace{-0.7cm}
-
-\subsection{Triple Correlation: Definition}
-
-The triple correlation of a function $f$ defined on a compact group $G$ is:
-$$
-T(f)_{g_1, g_2}=\int_{g \in G} f^*(g) f\left(g g_1\right) f\left(g g_2\right)dg .
-$$
-
-\subsection{Triple Correlation: Invariance}
-Note that $T(f)$ is a function defined on $G \times G$, and that, like the autocorrelation, the triple correlation is invariant under left translation: $T(f_1)=T(f_2)$ if $f_1(g)=f_2(h g)$.
-
-\begin{proof}
-Consider two real signals $f_1, f_2$ defined on a domain $\Omega$ equipped with the action of a group $G$, such that $f_2 = T_h[f_1]$ in the sense defined in the previous section.
-
-We show that this implies that $T(f_1)_{g_1, g_2} = T(f_2)_{g_1, g_2}$. Taking $g_1, g_2 \in G$, we have:
-\begin{align*}
-    T(f_2)_{g_1, g_2}
-        &= \int_{g \in G}
-            f_2^*(g) f_2\left(gg_1\right) f_2\left(gg_2\right) dg  \\
-        &= \int_{g \in G}
-            T_h[f_1]^*(g) T_h[f_1]\left(gg_1\right) T_h[f_1]\left(gg_2\right) dg  \\
-        &= \int_{g \in G}
-            f_1^*(hg) f_1\left(hgg_1\right) f_1\left(hgg_2\right) dg  \\
-        &= \int_{g \in G}
-            f_1^*(g) f_1\left(gg_1\right) f_1\left(gg_2\right) dg \\
-        &=  T(f_1)_{g_1, g_2}.
-\end{align*}
-where we use the change of variable $hg \rightarrow g$. This proves the invariance of the TC with respect to group actions on the signals.
-\end{proof}
-
-\subsection{Triple Correlation: Symmetries}
-
-\begin{proposition}[Symmetries]
-The triple correlation of a signal has the following symmetry:
-\begin{align*}
-     (s1) \qquad T(f)_{g_1, g_2} &= T(f)_{g_2, g_1}
-\end{align*}
-If $G$ is commutative, and $f$ takes on real values, the triple correlation has the following additional symmetries:
-\begin{align*}
-     (s2) \qquad T(f)_{g_1, g_2}
-     = T(f)_{g^{-1}_1, g_2g^{-1}_1}
-     = T(f)_{g_2g^{-1}_1, g^{-1}_1}
-     = T(f)_{g_1g^{-1}_2, g^{-1}_2}
-     = T(f)_{g^{-1}_2, g_1g^{-1}_2}
-\end{align*}
-\end{proposition}
-
-
-\begin{proof}
-    We prove symmetry (s1), which relies on the fact that $f(gg_2)$ and $f(gg_1)$ are scalar values that commute:
-    \begin{align*}
-        T(f)_{g_2, g_1}
-            = \int_{g \in G}  f^*(g)f(gg_2)f(gg_1) dg
-            = \int_{g \in G}  f^*(g)f(gg_1)f(gg_2) dg
-            = T(f)_{g_2, g_1}.
-    \end{align*}
-    We assume that the signal is real and that $G$ is commutative.
-    We prove the first equality of symmetry (s2):
-    \begin{align*}
-        T(f)_{g^{-1}_1, g_2g^{-1}_1}
-            &= \int_{g \in G} f^*(g)f(gg^{-1}_1)f(gg_2g^{-1}_1) dg\\
-            &= \int_{g' \in G} f^*(g'g_1)f(g')f(g'g_1g_2g^{-1}_1) dg
-                \quad \text{(with $g' = g - g_1$, i.e. $g = g'g_1$)} \\
-            &= \int_{g' \in G} f^*(g'g_1)f(g')f(g'g_2) dg
-                \quad \text{(because $G$ commutative: $g_2 g^{-1}_1 = g^{-1}_1 g_2$)} \\
-            &= \int_{g \in G}  f(gg_1)f(g)f(gg_2) dg
-                \quad \text{($f$ is a real signal, which equals its conjugate)}\\
-            &= \int_{g \in G}  f(g)f(gg_1)f(gg_2) dg
-                \quad \text{($f$ is takes on real values that commute)}\\
-            &= \int_{g \in G}  f(g)^*f(gg_1)f(gg_2) dg\\
-            &= T(f)_{g_2, g_1}
-    \end{align*}
-    The second equality of symmetry (s2) follows using (s1).
-    The third and fourth equality of symmetry (s2) have the same proof.
-\end{proof}
-
-\subsection{Bispectrum: Definition}
-
-The bispectrum can be defined more generally for signals defined over a space that is homogeneous for a compact group \cite{kakarala2012bispectrum}.
-
-\textbf{The bispectrum for commutative groups} is defined over a pair of irreducible representations $\rho_1, \rho_2$:
-
-\begin{equation}
-    \beta (f)_{\rho_1, \rho_2} = \hat{f}_{\rho_1}\hat{f}_{\rho_2}\hat{f}_{\rho_1 \rho_2}^{\dagger} \qquad \in \mathbb{C}
-    \label{eqn:group-bs}
-\end{equation}
-The irreducible representations (irreps) of commutative groups map to scalar values.
-
-\textbf{The bispectrum for non-commutative groups} is defined over a pair of irreducible representations $\rho_1, \rho_2$:
-
-\begin{align*}
-   \beta (f)_{\rho_1, \rho_2}
-   &= [\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}] C_{\rho_1,\rho_2} \Big[ \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \mathcal{F}(f)_{\rho}^{\dagger} \Big] C^{\dagger}_{\rho_1,\rho_2}\\
-   &= [\hat{f}{\rho_1} \otimes \hat{f}{\rho_2}] C_{\rho_1,\rho_2} \Big[ \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \hat{f}_{\rho}^{\dagger} \Big] C^{\dagger}_{\rho_1,\rho_2} \qquad \in \mathbb{C}^{D_1 D_2 \times D_1 D_2}
-    \label{eqn:non-commutative-bs}
-\end{align*}
-
-The irreps of \textit{non-commutative} groups map to matrices. In this definition, $C$ is a Clebsch-Gordan matrix, $\otimes$ is the tensor product, and $\oplus$ is a direct sum over irreps. The unitary Clebsch-Gordan matrix $C$ is analytically defined for each pair of representations as:
-
-\begin{equation}
-    (\rho_1 \otimes \rho_2)(g) = C_{\rho_1,\rho_2}^\dagger \Big[ \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \rho(g) \Big] C_{\rho_1,\rho_2}.
-    \label{eqn:non-commutative-bs}
-\end{equation}
-
-\begin{proof}
-We prove this by applying the Fourier transform to the triple correlation formula, generalizing the proof of Karakala to compact groups.
-
-The Fourier transformation of $T(f)$ requires tensor products of the representation matrices, since it is defined on the product group.
-
-We denote $F = \mathcal{F}(f)$ for simplicity of notations.
-
-The bispectrum is obtained by the formula:
-$$
-\beta(f)_{\rho_1, \rho_2}
-= \int_{g_1\in G} \int_{g_2\in G}
-    T(f)_{g_1, g_2}
-    \left[
-        \rho_1\left(g_1\right)^{\dagger} \otimes \rho_2\left(g_2\right)^{\dagger}
-    \right]dg_1dg_2
-$$
-for all ${\rho_1}, {\rho_2}$. Inserting the definition of the triple correlation, and simplifying with a change of variable $g_1 = gg_1$, $g_2 = gg_2$:
-\begin{align*}
-\beta(f)_{\rho_1, \rho_2}
-=&
-\int_{g_1\in G} \int_{g_2\in G}
-    \int_{g \in G} f^*(g) f\left(g g_1\right) f\left(g g_2\right)
-    \left[
-        \rho_1\left(g_1\right)^{\dagger} \otimes \rho_2\left(g_2\right)^{\dagger}
-    \right]
-    dg dg_1dg_2 \\
-=& \int_{g \in G}
-    f^*(g)
-    \int_{g_1\in G} \int_{g_2\in G}
-    f\left(g g_1\right) f\left(g g_2\right)
-    \left[
-        \rho_1\left(g_1\right)^{\dagger} \otimes \rho_2\left(g_2\right)^{\dagger}
-    \right]
-    dg_1dg_2
-    dg \\
-=& \int_{g \in G}
-    f^*(g)
-    \int_{g_1\in G} \int_{g_2\in G}
-    f\left(g_1\right) f\left(g_2\right)
-    \left[
-        \rho_1\left(g^{-1}g_1\right)^{\dagger} \otimes \rho_2\left(g^{-1}g_2\right)^{\dagger}
-    \right]
-    dg_1dg_2
-    dg \\
-=& \int_{g \in G}
-    f^*(g)
-    \int_{g_1\in G} \int_{g_2\in G}
-    f\left(g_1\right) f\left(g_2\right)
-    \left[
-        \rho_1(g_1)^{\dagger}\rho_1(g^{-1})^{\dagger} \otimes \rho_2(g_2)^{\dagger}\rho_2(g^{-1})^\dagger
-    \right]
-    dg_1dg_2
-    dg \\
-=& \int_{g \in G}
-    f^*(g)
-    \int_{g_1\in G} \int_{g_2\in G}
-    f\left(g_1\right) f\left(g_2\right)
-    \left[
-        \rho_1(g_1)^{\dagger}\rho_1(g) \otimes \rho_2(g_2)^{\dagger}\rho_2(g)
-    \right]
-    dg_1dg_2
-    dg \\
-=& \int_{g \in G}
-    f^*(g)
-    \int_{g_1\in G} \int_{g_2\in G}
-    f\left(g_1\right) f\left(g_2\right)
-        \left[\rho_1(g_1)^{\dagger}\otimes  \rho_2(g_2)^{\dagger}\right] . \left[\rho_1(g) \otimes \rho_2(g)\right]
-    dg_1dg_2
-    dg \\
-&= \int_{g \in G}
-    f^*(g)
-    [\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}]\left[\rho_1(g) \otimes \rho_2(g)\right]
-    dg\\
-&=
-\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}
-    \left[
-        \int_{g \in G} f(g)^* \rho_1(g) \otimes \rho_2(g) dg
-    \right].\\
-&=
-\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}
-    \left[
-        \int_{g \in G} f(g)^*  \left( C_{\rho_1 \rho_2} \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \rho(g) C_{\rho_1 \rho_2}^{\dagger}\right) dg
-    \right].\\
-& =
-\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2} C_{\rho_1 \rho_2} \left[ \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \mathcal{F}(f)_{\rho} \right]  C_{\rho_1 \rho_2}^{\dagger}
-\end{align*}
-
-where the last two lines uses the Clebsh-Gordan formula to reduce the tensor product of representations.
-
-
-The formula simplifies to (1) if $G$ is Abelian, in which case the fundamental theorem of finite Abelian groups shows that $G$ is a direct product of cyclic groups, each of which is represented by the complex exponentials $x \mapsto e^{j \omega x}$.
-
-\end{proof}
-
-
-\subsection{Bispectrum: Invariance}
-
-Define the transformation of $f$ by $\tilde g$ as $(\tilde{g} \ast f)(x) = f({\tilde g}^{-1}\circ g)$. Under this group action, the bispectrum becomes:
-\begin{align*}
-   \beta(\tilde{g} \ast f)_{\rho_1, \rho_2}
-   &= \Big[\mathcal{F}(\tilde{g} \ast f)_{\rho_1} \otimes \mathcal{F}(\tilde{g} \ast f)_{\rho_2})\Big] .C_{\rho_1,\rho_2} \Big[ \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \mathcal{F}(\tilde{g} \ast f)_{\rho}^{\dagger} \Big]. C^{\dagger}_{\rho_1,\rho_2}
-   \quad \text{(bispectrum definition)}\\
-   &= \Big[(\rho_1(\tilde{g}).\mathcal{F}(f)_{\rho_1}) \otimes (\rho_2(\tilde{g}). \mathcal{F}(f)_{\rho_2})\Big] .C_{\rho_1,\rho_2} \Big[ \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \rho(\tilde{g})^\dagger .\mathcal{F}(f)_{\rho}^{\dagger} \Big] .C^{\dagger}_{\rho_1,\rho_2}
-   \quad \text{(equivariance of $\mathcal{F}$)}\\
-  &=
-  \Big[\rho_1(\tilde{g}) \otimes \rho_2(\tilde{g})\Big] .
-  \Big[\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2})\Big] .
-  C_{\rho_1,\rho_2}
-  \Big[ \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \rho(\tilde{g})^\dagger .\mathcal{F}(f)_{\rho}^{\dagger} \Big] .C^{\dagger}_{\rho_1,\rho_2}
-  \quad \text{(*)}\\
-    &=
-  \Big[\rho_1(\tilde{g}) \otimes \rho_2(\tilde{g})\Big] .
-  \Big[\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2})\Big] .
-  C_{\rho_1,\rho_2} .
-  \Big[ \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \rho(\tilde{g})^\dagger) \Big].\Big[\bigoplus_{\rho \in \rho_1 \otimes \rho_2} \mathcal{F}(f)_{\rho}^{\dagger} \Big]. C^{\dagger}_{\rho_1,\rho_2}
-  \quad \text{($\oplus$ definition)}\\
-  &=
-  \Big[\rho_1(\tilde{g}) \otimes \rho_2(\tilde{g})\Big] .
-  \Big[\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2})\Big] .
-  C_{\rho_1,\rho_2}. \Big[ \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \rho(\tilde{g})^\dagger) \Big] .C^{\dagger}_{\rho_1,\rho_2}.C_{\rho_1,\rho_2}.\Big[\bigoplus_{\rho \in \rho_1 \otimes \rho_2} \mathcal{F}(f)_{\rho}^{\dagger} \Big] .C^{\dagger}_{\rho_1,\rho_2}
-  \quad \text{(unitary)}\\
-  &=
-  \Big[\rho_1(\tilde{g}) \otimes \rho_2(\tilde{g})\Big]
-  .
-  \Big[\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2})\Big] .
-  \Big[ C_{\rho_1,\rho_2}. \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \rho(\tilde{g})^\dagger) .C^{\dagger}_{\rho_1,\rho_2}\Big].
-  C_{\rho_1,\rho_2}.\Big[\bigoplus_{\rho \in \rho_1 \otimes \rho_2} \mathcal{F}(f)_{\rho}^{\dagger} \Big] .C^{\dagger}_{\rho_1,\rho_2}\\
-  &=
-  \Big[\rho_1(\tilde{g}) \otimes \rho_2(\tilde{g})\Big] .\Big[\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2})\Big] .\Big[\rho_1(\tilde{g}) \otimes \rho_2(\tilde{g})\Big]^\dagger. C_{\rho_1,\rho_2}.\Big[\bigoplus_{\rho \in \rho_1 \otimes \rho_2} \mathcal{F}(f)_{\rho}^{\dagger} \Big] .C^{\dagger}_{\rho_1,\rho_2}\\
-  &=
-  \Big[(\rho_1(\tilde{g}).\rho_1(\tilde{g})^\dagger.\mathcal{F}(f)_{\rho_1}) \otimes (\rho_2(\tilde{g}).\rho_2(\tilde{g})^\dagger. \mathcal{F}(f)_{\rho_2})\Big].  C_{\rho_1,\rho_2}.\Big[\bigoplus_{\rho \in \rho_1 \otimes \rho_2} \mathcal{F}(f)_{\rho}^{\dagger} \Big]. C^{\dagger}_{\rho_1,\rho_2}\\
-  &=
-  \Big[\mathcal{F}(f)_{\rho_1} \otimes\mathcal{F}(f)_{\rho_2}\Big] . C_{\rho_1,\rho_2}.\Big[\bigoplus_{\rho \in \rho_1 \otimes \rho_2} \mathcal{F}(f)_{\rho}^{\dagger} \Big]. C^{\dagger}_{\rho_1,\rho_2}\qquad \text{(unitary representations)}\\
-  &=
-  \beta(f)_{\rho_1, \rho_2} \qquad \text{(bispectrum definition)}
-\end{align*}
-where we use the linear algebra identity $(*)$: $(A.B) \otimes (C.D) = (A\otimes C) . (B \otimes D)$ for $\otimes$ the tensor product and $.$ the matrix multiplication.
-
-\section{\color{purple}{Domain $\Omega = (S^2,~SO(3),~\ast)$ (spherical rotations)}}
-
-We move towards homogeneous space for a group, and first consider an example before writing the general theory. This example is the spherical harmonic transform, which is a generalization of the Fourier transform to the non-commutative group of spherical rotations $SO(3)$ acting on the 2-sphere.
-
-\subsection{Fourier Transform: Definition}
-Functions defined over a sphere $S^2$ can be expanded on the spherical harmonics as:
-\begin{equation}
-    f(\theta, \phi)=\sum_{l=0}^{\infty} \sum_{m=-l}^l \widehat{f}_{l, m} Y_l^m(\theta, \phi)
-\end{equation}
-where $Y_l^m(\theta, \phi)$ as the spherical harmonics of coefficients $l, m$ given by $Y_l^m(\theta, \phi)=k_{lm} P_l^m(\cos \theta) e^{i m \phi}$
-where $l=0,1,2, \ldots ; \quad m=[-l, l]$, $P_l^m$ are the associated Legendre polynomials, and $k_{lm}-\sqrt{\frac{2 l+1}{4 \pi} \frac{(l-m) !}{(l+m) !}}$ the normalization constant.
-
-We recall that the spherical harmonics are the eigenfunctions of the Laplace operator on $S_2$ (with eigenvalue $-l^2$ ), and they form an orthonormal basis for $L_2\left(S_2\right)$.
-
-\textbf{The spherical harmonics transform} of a spherical function $f$ is given by:
-\begin{align*}
-    \widehat{f}_{l, m}
-    &= <f, Y_l^m>   \qquad \in \mathbb{C} \\
-    &= \int_0^\pi \int_0^{2 \pi} f^\dagger(\theta, \phi) Y_l^m(\theta, \phi) \cos \theta d \phi d \theta \\
-    &= \int_0^\pi \int_0^{2 \pi} f^\dagger(\theta, \phi) k_{lm} P_l^m(\cos \theta) e^{i m \phi} \cos \theta d \phi d \theta \\
-    &=  k_{lm}   \int_0^\pi \Big[ \int_0^{2 \pi} f^\dagger(\theta, \phi)e^{i m \phi}d \phi\Big] P_l^m(\cos \theta)\cos \theta  d \theta
-\end{align*}
-where the last line shows that the spherical harmonic transform can be reduced to a regular Fourier transform in the longitudinal coordinate $\phi$ followed by a projection onto the associated Legendre functions. We note that there exists an expression for the discrete Fourier transform associated to that case.
-
-
-We bring this expression into an equation amenable to a definition of generalized Fourier transform, i.e. only keeping the index $l$ that indexes the irreducible representations of $SO(3)$. Thus, we get vector-valued Fourier coefficients.
-
-\textbf{The (vector) Fourier transform} of a spherical function $f$, where the subscript $0$ will become clear at the end of this computation, is:
-\begin{align*}
-  \mathcal{F}_0(f)_l = \widehat{f}_{l}
-    &= <f, Y_l> \\
-    &= \int_0^\pi \int_0^{2 \pi} f^\dagger(\theta, \phi) Y_l(\theta, \phi) \cos \theta d \phi d \theta   \qquad \in \mathbb{C}^{2l+1} \\
-    &= \sqrt{\frac{2l+1}{4\pi}}\int_0^\pi \int_0^{2 \pi} f^\dagger(\theta, \phi). [\rho_l(\phi, \theta, \gamma)]_0 \cos \theta d \phi d \theta
-    \qquad \text{(relation between $Y$'s and Wigner matrices, Karakala)}\\
-    &= \frac{1}{\pi}\sqrt{\frac{2l+1}{4\pi}}\int_0^\pi\int_0^\pi \int_0^{2 \pi} f^\dagger(\theta, \phi). [\rho_l(\phi, \theta, \gamma)]_0 \cos \theta d \phi d \theta d\gamma \\
-    &= \kappa_l \int_{R \in SO(3)} f^\dagger(T_R(x_0)). [\rho_l(R)]_0 dR \qquad \text{(Haar measure in Euler angles)}
-\end{align*}
-where $[\rho_l(R)^*]_0$ is the first column of the Wigner matrix, and $R$ is a rotation bringing the north pole $x_0$ into $(\theta, \phi)$, i.e. the rotation of Euler angles $(\alpha, \beta, \gamma)$ where $\alpha=\phi$ and $\beta=\theta$, i.e. convention of Euler angles, where $\alpha$  is a longitudinal angle and
-$\beta$  is a colatitudinal angle (spherical polar angles in the physical definition of such angles). We use $\kappa_l$ to denote a normalization constant.
-
-We note that this is \textbf{not equal} to the Fourier transform defined on an homogeneous space, as only one column is extracted: we have a vector-valued Fourier transform, instead of a matrix-valued Fourier transform, which is more computationally effective. This raises the question on whether every Fourier transform on homogeneous spaces can be reduced into a vector-valued transform in a more general way.
-
-\textbf{The Fourier transform} of a spherical function $f$ is indeed:
-\begin{align*}
-  \mathcal{F}(f)_l
-    = \int_{R \in SO(3)} f^\dagger(T_R(x_0)). \rho_l(R) dR.
-\end{align*}
-
-\subsection{Fourier Transform: Equivariance}
-
-
-Define the transformation of $f$ by $\tilde R$ as $(\tilde{R} \ast f)(\theta, \phi) = f(\tilde R (\theta, \phi)) = f(T_{\tilde R}^{-1}.u)$. The (vector) Fourier transform undergoes a transformation by the same $\tilde R$, as follows, which shows its equivariance.
-
-\begin{align*}
-    \mathcal{F}(\tilde{R} \ast f)_{l}
-    &=\kappa_l \int_{R \in SO(3)} (\tilde{R} \ast f)^\dagger(T_R(x_0)). [\rho_l(R)]_0 dR\\
-    &=\kappa_l \int_{R \in SO(3)}
-        f^\dagger(T_{\tilde R}^{-1} \circ T_R(x_0)). [\rho_l(R)]_0 dR \\
-    &=\kappa_l \int_{R \in SO(3)}
-        f^\dagger(T_{\tilde R^{-1} R}(x_0)). [\rho_l(R)]_0 dR \\
-    &=\kappa_l \int_{R' \in SO(3)}
-        f^\dagger(T_{R'}(x_0)). [\rho_l(\tilde{R}R')]_0 \tilde{R}^{-1}dR' \\
-    &=\kappa_l \int_{R' \in SO(3)}
-        f^\dagger(T_{R'}(x_0)). [\rho_l(\tilde{R}).\rho_l(R')]_0 \tilde{R}^{-1}dR'
-         \qquad \text{($\rho_l$ is representation)}\\
-    &=\kappa_l \rho_l(\tilde{R}).
-        \int_{R' \in SO(3)}
-            f^\dagger(T_{R'}(x_0)).[\rho_l(R')]_0 \tilde{R}^{-1} dR'
-     \qquad \text{(vector interpretation of matrix multiplication (*))}\\
-    &=\kappa_l \rho_l(\tilde{R}).
-        \int_{R' \in SO(3)}
-            f^\dagger(T_{R'}(x_0)).[\rho_l(R')]_0 dR'
-     \qquad \text{($\tilde{R}^{-1}$ isometry)} \\
-     &= \rho_l(\tilde{R}) .  \mathcal{F}(f)_{l}
-\end{align*}
-
-We note that line $(*)$ also proves the equivariance property of the spherical harmonics.
-
-The equivariance of the matrix Fourier transform is proved similarly, and we prove it in the general case of homogeneous spaces in the next section.
-
-\subsection{Bispectrum: Definitions}
-
-The two definitions of Fourier transform (vector-valued and matrix-valued) give rise to two definitions of bispectrum. We focus on the vector-valued case, as the matrix-valued case is tackled in all generality in the next section.
-
-\textbf{The (vector) bispectrum} of a spherical function $f$ is a vector $\beta(f)_{l_1l_2}$ formed of the following scalar entries:
-\begin{equation}
-    \beta(f)_{l_1l_2}[l]:=\left[\mathcal{F}_0(f)_{l_1} \otimes \mathcal{F}_0(f)_{l_2} \right] C_{l_1l_2} \hat{\mathcal{F}_0(f)_l}^{\dagger} \qquad \in \mathbb{C}; \quad|l_1-l_2| \leq l \leq l_1+l_2.
-\end{equation}
-where $C_{l_1l_2}$ is the Clebsch-Gordan matrix associated with $\rho_{l_1} \otimes \rho_{l_2}$, and $\hat{\mathcal{F}}_0(f)_l$ is the $(2l_1+1)(2l_2+1)$ vector padded with zeros: $\hat{\mathcal{F}}_0(f)_l:=\left[0, \ldots, 0, \mathcal{F}_0(f)_l, 0, \ldots, 0\right]$, where the number of zeros preceding $\mathcal{F}_0(f)_l$ is $n_p$, and the number succeeding $\mathcal{F}_0(f)_l$ is $n_s$, where $n_p:=\sum_{q=|\ell_1-\ell_2|}^{l-1}(2 q+1)$ and $n_s:=\sum_{q=l+1}^{\ell_1+\ell_2}(2 q+1)$.
-
-Likewise, $\mathcal{F}_0(f)_{l_1} \otimes \mathcal{F}_0(f)_{l_2}$ is a $(2l_1+1)(2l_2+1)$ vector, such that the resulting $\beta(f)_{l_1l_2}[l]$ is indeed a scalar.
-
-\textbf{The (matrix) bispectrum} of a spherical function $f$ is:
-\begin{equation}
-    \beta(f)_{l_1l_2} = [\mathcal{F}(f)_{l_1} \otimes \mathcal{F}(f)_{l_2}] C_{l_1,l_2} \Big[ \bigoplus_{l \in l_1 \otimes l_2} \mathcal{F}(f)_{l}^{\dagger} \Big] C^{\dagger}_{l_1,l_2} \qquad \in \mathbb{C}^{(2l_1+1)(2l_2 + 1) \times (2l_1+1)(2l_2 +1)}
-\end{equation}
-
-% \textbf{The bispectrum for spherical functions} is built from the $(2 l_1+1)(2l_2+1)$-dimensional tensor product vectors $\widehat{\boldsymbol{f}}_{l_1} \otimes \widehat{\boldsymbol{f}}_{l_2}$, which transform according to
-% $$
-% \widehat{\boldsymbol{f}}_{l_1} \otimes \widehat{\boldsymbol{f}}_{l_2} \mapsto\left(D^{\left(l_1\right)}(R) \otimes D^{\left(l_2\right)}(R)\right) \cdot\left(\widehat{\boldsymbol{f}}_{l_1} \otimes \widehat{\boldsymbol{f}}_{l_2}\right) .
-% $$
-
-% \textbf{The non-zero elements of the bispectrum}  for spherical functions are (something like):
-
-% \begin{aligned}
-% p_{l_1, l_2}
-%     &=\widehat{f}_{l_1} \otimes \widehat{f}_{l_2} \bigoplus_{l \in l_1 \otimes l_2}\widehat{g}_{l_1, l_2, l}^{\dagger} \cdot \widehat{f}_l \qquad \in \mathbb{C}^{D_1 D_2}\\
-%     & = \bigoplus_{l \in l_1 \otimes l_2} \sum_{m=-l}^l \sum_{m_1=-l_1}^{l_1} C_{m_1, m-m_1, m}^{l_1, l_2, l} \widehat{f}_{l_1, m_1}^* \widehat{f}_{l_2, m-m_1}^* \widehat{f}_{l_{l, m}} .
-% \end{aligned}
-
-
-% Up to unitary transformation, these invariants are equivalent to the non-vanishing matrix elements of the abstract bispectrum (as already derived in [3] and [1]). Any kernel built from the bispectrum using (प3) as features will be invariant to translation and rotation.
-
-\subsection{Bispectrum: Invariances}
-
-The two definitions of bispectrum are invariant. We focus on the vector-valued case, as the matrix-valued case is tackled in all generality in the next section. For the vector-valued case, we prove that each of the scalar entries forming the bispectrum vector is equivariant.
-
-Define the transformation of $f$ by $\tilde R$ as $(\tilde{R} \ast f)(x) = f({\tilde R}^{-1}\circ x)$. Under this group action, the coefficient $l$ of the vector bispectrum becomes:
-\begin{align*}
-    \beta(\tilde{R} \ast f)_{l_1l_2}[l]
-    &=\left[
-    \mathcal{F}(\tilde{R} \ast f)_{l_1} \otimes \mathcal{F}(\tilde{R} \ast f)_{l_2}
-    \right]
-    C_{l_1l_2} \mathcal{F}_0(\tilde{R} \ast f)_l^{\dagger} \\
-    &=\left[
-        (\mathcal{F}(f)_{l_1}.\rho_{l_1}(\tilde{R})) \otimes (\mathcal{F}(f)_{l_2}.\rho_{l_2}(\tilde{R}))
-        \right]
-        C_{l_1l_2} .\mathcal{F}_0(\tilde{R} \ast f)_l^{\dagger}
-        \qquad \text{(equivariance on the right with row-vectors)} \\
-    &=\left[
-        \mathcal{F}(f)_{l_1} \otimes (\mathcal{F}(f)_{l_2})
-      \right] .
-        \left[
-        \rho_{l_1}(\tilde{R}) \otimes \rho_{l_2}(\tilde{R})
-        \right] .
-        C_{l_1l_2} .
-        \mathcal{F}_0(\tilde{R} \ast f)_l^{\dagger}
-        \\
-     &=\left[
-        \mathcal{F}(f)_{l_1} \otimes (\mathcal{F}(f)_{l_2})
-      \right] .
-        \left[
-        \rho_{l_1}(\tilde{R}) \otimes \rho_{l_2}(\tilde{R})
-        \right] .
-        C_{l_1l_2} .
-        \left[ \mathcal{F}_0(f)_l.\bigoplus_{i=|l_1 - l_2|}^{l_1+l_2} \rho_i(\tilde R)\right]^\dagger
-        \qquad \text{(only $i=l$ is non-zero?)}
-        \\
-     &=\left[
-        \mathcal{F}(f)_{l_1} \otimes (\mathcal{F}(f)_{l_2})
-      \right] .
-        \left[
-        \rho_{l_1}(\tilde{R}) \otimes \rho_{l_2}(\tilde{R})
-        \right] .
-        C_{l_1l_2} .
-        \left[ \bigoplus_{i=|l_1 - l_2|}^{l_1+l_2} \rho_i(\tilde R)\right]^\dagger
-        . \mathcal{F}_0(f)_l^\dagger
-        \\
-     &=\left[
-        \mathcal{F}(f)_{l_1} \otimes (\mathcal{F}(f)_{l_2})
-      \right] .
-        \left[
-        \rho_{l_1}(\tilde{R}) \otimes \rho_{l_2}(\tilde{R})
-        \right] .
-        C_{l_1l_2} .
-        \left[ \bigoplus_{i=|l_1 - l_2|}^{l_1+l_2} \rho_i(\tilde R)\right]^\dagger
-        .C_{l_1l_2}^\dagger
-        . C_{l_1l_2}  \mathcal{F}_0(f)_l^\dagger
-        \qquad \text{(unitary $C_{l_1l_2}$)}\\
-     &=\left[
-        \mathcal{F}(f)_{l_1} \otimes (\mathcal{F}(f)_{l_2})
-      \right]
-        . C_{l_1l_2}  \mathcal{F}_0(f)_l^\dagger
-        \qquad \text{(Clebsh-Gordan)}\\
-     &= \beta(f)_{l_1l_2}[l]
-\end{align*}
-
-\section{\color{purple}{Domain $\Omega = (H,~G,~\ast)$ (homogeneous space)}}
-
-By definition, $H$ is a homogeneous space of $G$ if fixing any $x_0 \in H$, the set $T_g(x_0)$ ranges over the whole of $H$ as $g$ ranges over $G$.
-
-The classical example of a homogeneous space is the unit sphere $S_2$ which was the focus of the previous section. The sphere is a homogeneous space of the three-dimensional rotation group $S O(3)$ : taking the North pole as $x_0$, a suitable rotation can move it to any point $x \in S_2$.
-
-
-
-\subsection{Fourier Transform: Definition}
-
-\textbf{The Fourier transform} for functions defined over an homogeneous space $H$ of a group $G$ is:
-\begin{equation}
-\mathcal{F}(f)_\rho = \widehat{f}(\rho) = \int_{g \in G} f\left(T_g(x_0)\right) \rho(g)dg.
-\end{equation}
-
-Kondor comments on this equation as follows. Note that except for the trivial case $H=G$, Fourier transforms on homogeneous spaces are naturally redundant: typically $H$ is a much smaller space than $G$, yet a Fourier transform on $H$ has the same number of components as a Fourier transform on the entire group, i.e., as many as the number of irreducible representations of the group. One manifestation of this fact is that we might find that some Fourier components are rank deficient no matter what $f: H \rightarrow \mathbb{C}$ we choose.
-
-Kondor continues. While this destroys Kakarala's uniqueness result, in practice we often find that the bispectrum still furnishes a remarkably rich invariant representation of $f$. We remark that that invariance to right-translation $f^{(z)}(x)=f\left(x z^{-1}\right)$ would be a different matter: there is a variant of the bispectrum which retains the uniqueness property in this case.
-
-Consider the function $\tilde{f}$ defined on the group $G$, such that: $\tilde{f}(g) = f(\pi(g)) = f(T_g.x_0)$, i.e. when $\pi$ is the projection to the quotient space, defined as:
-\begin{align*}
-    \pi : G \rightarrow H \, \qquad g \rightarrow \pi(g) = T_g.x_0.
-\end{align*}
-
-The Fourier transform, for function over groups, gives:
-\begin{equation}
-\mathcal{F}(\tilde{f})_\rho
-=
-\widehat{\tilde{f}}(\rho)
-=
-\int_{g \in G} \tilde{f}(g) \rho(g)dg
-=
-\int_{g \in G} f(\pi(g)) \rho(g)dg
-=
-\int_{g \in G} f(T_g.x_0) \rho(g)dg ,
-\end{equation}
-which is the definition we use for homogeneous space.
-
-Thus, the Fourier transform of a function $f$ defined on a $H$ is the same as the (group) Fourier transform for a lifted function $\tilde f$ defined on the corresponding group.
-
-\subsection{Fourier Transform: Equivariance}
-
-Define the transformation of $f$ by $\tilde g$ as $(\tilde{g} \ast f)(x) = f(T_{\tilde g}^{-1}.x)$. The Fourier transform undergoes a transformation by the same $\tilde g$, as follows, which shows its equivariance.
-
-\vspace{-0.7cm}
-\begin{align*}
-  \mathcal{F}(\tilde{g} \ast f)_\rho
-  &=  \int_{g \in G} (\tilde{g} \ast f)(T_g(x_0))\rho(g) dg \\
-  &=  \int_{g \in G} f(T_{\tilde g}^{-1}.T_g(x_0))\rho(g) dg \\
-  &=  \int_{g \in G} f(T_{\tilde g^{-1}\circ g}(x_0))\rho(g) dg \\
-  &=  \int_{g' \in G} f(T_{g'}(x_0))\rho(\tilde{g}\circ g') dg' \\
-  &=  \int_{g' \in G} f(T_{g'}(x_0))\rho(\tilde{g}).\rho(g') dg' \\
-  &=  \rho(\tilde{g}) . \int_{g' \in G} f(T_{g'}(x_0))\rho(g') dg' \\
-  &=  \rho(\tilde{g}) .\mathcal{F}(f)_\rho
-\end{align*}
-\vspace{-0.7cm}
-
-Importantly, this equivariance is the same equivariance as the one that emerged for the Fourier transform of function defined over a group.
-
-\subsection{Triple Correlation: Definition}
-
-The triple correlation of a function $f$ defined on $H$ homogeneous for a compact group $G$ is:
-\begin{equation}
-    T(f)_{g_1, g_2}
-        =
-        \int_{g \in G}
-        f^*(T_g(x_0))
-        f\left(T_{g g_1}(x_0)\right) f\left(T_{g g_2}(x_0)\right)
-        dg .
-\end{equation}
-We can rewrite this definition in terms of the function $\tilde f$ that is the lift of function $f$ on the group $G$: $\tilde{f}(g) = f(T_g.x_0) = f(\pi(g))$.
-\begin{equation}
-    T(f)_{g_1, g_2}
-        =
-        \int_{g \in G}
-        \tilde{f}^*(g)
-        \tilde{f}(gg_1) \tilde{f}(gg_2)
-        dg ,
-\end{equation}
-where we recognize the definition of triple correlation on groups applied to function $\tilde{f}$.
-
-\subsection{Triple Correlation: Invariance}
-Note that $T(f)$ is a function defined on $H \times H$, and that, like the autocorrelation, the triple correlation is invariant under left translation: $T(f_1)=T(f_2)$ if $f_1(g)=f_2(h g)$.
-
-\begin{proof}
-Consider two real signals $f_1, f_2$ defined on a domain $\domain$ equipped with the action of a group $G$, such that $f_2 = T_h[f_1]$ in the sense defined in the previous section.
-
-We show that this implies that $T_h[f_1] = T_h[f_2] $. Taking $g_1, g_2 \in G$, we have:
-\begin{align*}
-    T(f_2)_{g_1, g_2}
-        &= \int_{g \in G}
-            f_2^*(T_g(x_0)) f_2\left(T_{g g_1}(x_0)\right) f_2\left(T_{g g_2}(x_0)\right) dg  \\
-        &= \int_{g \in G}
-            T_h[f_1]^*(T_g(x_0)) T_h[f_1]\left(T_{g g_1}(x_0)\right) T_h[f_1]\left(T_{g g_2}(x_0)\right) dg  \\
-        &= \int_{g \in G}
-            f_1^*(T_h \circ T_g(x_0)) f_1\left(T_h \circ T_{g g_1}(x_0)\right) f_1\left(T_h \circ T_{g g_2}(x_0)\right) dg  \\
-        &= \int_{g \in G}
-            f_1^*(T_{hg}(x_0)) f_1\left(T_{hg g_1}(x_0)\right) f_1\left(T_{hg g_2}(x_0)\right) dg  \\
-        &= \int_{g \in G}
-            f_1^*(T_{g}(x_0)) f_1\left(T_{g g_1}(x_0)\right) f_1\left(T_{g g_2}(x_0)\right) dg \\
-        &= T(f_1)_{g_1, g_2}
-\end{align*}
-where we use the change of variable $hg \rightarrow g$.
-\end{proof}
-
-\subsection{Bispectrum: Definitions}
-
-The bispectrum can be defined more generally for signals defined over a space that is homogeneous for a compact group \cite{kakarala2012bispectrum}.
-
-\textbf{The bispectrum for homogeneous spaces of commutative groups} is defined over a pair of irreducible representations $\rho_1, \rho_2$:
-
-\begin{equation}
-    \beta (f)_{\rho_1, \rho_2} = \hat{f}_{\rho_1}\hat{f}_{\rho_2}\hat{f}_{\rho_1 \rho_2}^{\dagger}.
-    \label{eqn:group-bs}
-\end{equation}
-The irreducible representations (irreps) of commutative groups map to scalar values.
-
-\textbf{The bispectrum for homogeneous spaces of non-commutative groups} is defined over a pair of irreducible representations $\rho_1, \rho_2$:
-
-\begin{align*}
-   \beta (f)_{\rho_1, \rho_2}
-   &= [\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}] C_{\rho_1,\rho_2} \Big[ \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \mathcal{F}(f)_{\rho}^{\dagger} \Big] C^{\dagger}_{\rho_1,\rho_2}\\
-   &= [\hat{f}{\rho_1} \otimes \hat{f}{\rho_2}] C_{\rho_1,\rho_2} \Big[ \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \hat{f}_{\rho}^{\dagger} \Big] C^{\dagger}_{\rho_1,\rho_2}
-    \label{eqn:non-commutative-bs}
-\end{align*}
-
-We note that the bispectrum of functions defined over an homogeneous space $H$ has a definition identical to the case of functions defined over a group. The only difference is that the Fourier $\mathcal{F}$ transform used is the Fourier transform for function defined over $H$, and not $G$.
-
-\begin{proof}
-By definition, the bispectrum is the Fourier transform of the triple correlation. We compute this Fourier transform and show that we obtain the expression given above.
-
-
-\begin{align*}
-   \beta (f)_{\rho_1, \rho_2}
-   &=
-   \int_{g_1 \in G} \int_{g_2 \in G}
-        T(f)_{g_1, g_2}
-            \left[
-                \rho_1(g_1) \otimes \rho_2(g_2)
-            \right]
-            dg_1 dg_2 \\
-   &=
-    \int_{g_1 \in G} \int_{g_2 \in G}
-        \int_{g \in G}
-            f^*(T_g(x_0))
-            f\left(T_{g g_1}(x_0)\right) f\left(T_{g g_2}(x_0)\right)
-            dg
-              \left[
-                \rho_1(g_1) \otimes \rho_2(g_2)
-              \right]
-            dg_1 dg_2 \\
-\end{align*}
-
-
-Below, we give the main elements of the proof that shows that computing this Fourier transform indeed yields the same formula as in the case of compact groups.
-
-Note that the group $G$ is often ``bigger'' than the homogeneous space $H$, e.g. $SO(3)$ has dimension 3 while $S^2$ has dimension 2. Any homogeneous space $H$ can be seen as $H = G / G_0$ where $G_0$ is one stabilizer group, e.g. $S^2 = SO(3) / SO(2)$ with $G_0 = SO(2)$.
-
-A function $f$ defined on $H = G / G_0$ can be lifted to a function defined on $G$, as $\tilde{f}(g) = f(\pi(g))$ where $\pi$ is the canonical projection associated with the quotient space structure. The lifted function $\tilde{f}$ is invariant under translations by elements of the stabilizer $G_0$:
-\begin{equation}
-    \tilde{f}(g_0.g) = \tilde{f}(g), \qquad \text{for all $g_0 \in G_0$},
-\end{equation}
-This invariance induces a property on its fourier transform:
-\begin{equation}
-    \mathcal{F}(\tilde{f})(\rho) = \mathcal{F}(\tilde{f})(\rho) \rho(g_0), \qquad \text{for all $g_0 \in G_0$.}
-\end{equation}
-
-By integration over $G_0$, we get:
-\begin{equation}
-    \mathcal{F}(\tilde{f})(\rho) =  \mathcal{F}(f)(\rho).P_0(\rho)\quad \text{where: } P_0(\rho) = \int_{g_0 \in G_0} \rho(g_0),
-\end{equation}
-where $P_0$ is a projection operator: $P_0 . P_0 = P_0$.
-
-Next, one can prove that see that tensor products of projection operators decompose according to the Clebsh-Gordon formula:
-$$
-\begin{gathered}
-P_0(\sigma) \otimes P_0(\delta)= \\
-C_{\sigma \delta}\left[P_0\left(\omega_1\right) \oplus \cdots \oplus P_0\left(\omega_k\right)\right] C_{\sigma \delta}^{\dagger}\left[P_0(\sigma) \otimes P_0(\delta)\right] .
-\end{gathered}
-$$
-This ensures that the bispectrum formula on compact groups is valid for functions whose domain is a homogeneous space as well as those defined on groups.
-\end{proof}
-
-\subsection{Bispectrum: Invariance}
-
-The proof of invariance of the bispectrum for function defined over (commutative and non-commutative) domains does not use the expression of the Fourier transform on groups. Rather, it only uses the fact that the Fourier transform on groups is equivariant, with equivariance factor $\rho(\tilde{g})$. Since the Fourier transform on homogeneous spaces is equivariant, with the same equivariance factor, and since the bispectra definition is the same, we get the invariance of the bispectrum with the same derivation.
-
-We note that the proofs of bispectrum invariance of the previous section does not depend on the specific fact that we are using the group of rotations $SO(3)$. Therefore, they can be readily generalized to create a more general vector-valued bispectrum on homogeneous spaces. The remaining TODO is to define the vector-valued Fourier transform in the general case, and prove its equivariance.
-
-\section{\color{deepred}{Domain $\Omega = (S^1 \times \mathbb{R}_+^*,~SO(2),~\ast)$ (disk rotations)}}
-
-
-\subsection{Fourier Transform: Definition}
-
-Functions defined over a disk $D$ can be expanded on the disk harmonics as:
-\begin{equation}
-    f(r, \theta)=\sum_{m, n} \hat{f}(m, n) d_{n m}(r, \theta),
-\end{equation}
-where $ d_{m m}(r, \theta)$ is the disk harmonics of coefficients $m, n$ given by:
-\begin{equation}
-    d_{nm}(r, \theta)= J_m(2 \pi l_{n m} r) \exp (im \theta)
-\end{equation}
-
-\textbf{The (vector) Fourier transform} $\hat{f}(m, n)$ for functions defined over a disk $D$ is:
-\begin{align*}
-  \mathcal{F}(f)_{nm}
-    &= \frac{1}{a_{n m}} \int_0^{2 \pi} \int_0^1 f(r, \theta) J_m(2 \pi l_{n m} r) \exp(-i m \theta) r d r d\theta \\
-    &=\left\{\begin{array}{c}
-\frac{1}{2 \pi i m} a_{n m} \int_0^{2 \pi} F(l_{n m}, \phi) \exp(-i m \phi) d \phi \quad \text{for $l\neq 0$} \\
-F(0,0) / \pi \quad \text{for $l = 0$}
-\end{array}\right\},
-\end{align*}
-
-where $F(l_{n m}, \phi) $ is the traditional Fourier transform of $f(r, \theta)$ written in polar coordinates.
-
-\subsection{Fourier Transform: Equivariance}
-
-Define the transformation of $f$ by $\tilde \theta$ as $(\tilde{\theta} \ast f)(r, \theta) = f(r, \tilde \theta +\theta )$. The Fourier transform undergoes a transformation by the same $\tilde \theta$, as follows, which shows its equivariance.
-
-\begin{align*}
-    \mathcal{F}(\tilde{\theta} \ast f)_{nm}
-    &= \frac{1}{a_{n m}}
-        \int_0^{2 \pi} \int_0^1
-            (\tilde{\theta} \ast f)(r, \theta) J_m(2 \pi l_{n m} r) \exp(-i m \theta)
-                r dr d\theta \\
-    &= \frac{1}{a_{n m}}
-        \int_0^{2 \pi} \int_0^1
-            f(r, \tilde \theta + \theta) J_m(2 \pi l_{n m} r) \exp(-i m \theta)
-                r dr d\theta \\
-    &= \frac{1}{a_{n m}}
-         \int_{\tilde{\theta}}^{2 \pi+\tilde{\theta}} \int_0^1
-            f(r, \theta') J_m(2 \pi l_{n m} r) \exp(-i m (\theta' - \tilde{\theta}))
-                r dr d\theta'\\
-    &= \frac{1}{a_{n m}}
-         \int_{\tilde{\theta}}^{2 \pi+\tilde{\theta}} \int_0^1
-            f(r, \theta') J_m(2 \pi l_{n m} r) \exp(-i m \theta')\exp(+im\tilde{\theta})
-                r dr d\theta' \\
-    &= \exp(im\tilde{\theta}) .\frac{1}{a_{n m}}
-         \int_{\tilde{\theta}}^{2 \pi+\tilde{\theta}} \int_0^1
-            f(r, \theta') J_m(2 \pi l_{n m} r) \exp(-i m \theta')
-                r dr d\theta' \\
-    &= \exp(im\tilde{\theta}) .\frac{1}{a_{n m}}
-         \int_{0}^{2 \pi} \int_0^1
-            f(r, \theta') J_m(2 \pi l_{n m} r) \exp(-i m \theta')
-                r dr d\theta' \\
-    &= \exp(im\tilde{\theta}) . \mathcal{F}(f)_{nm}
-\end{align*}
-
-
-\subsection{Bispectrum: Definition}
-
-We generalize the bispectrum for signals defined over a space that is the direct product of $H$, homogeneous for a compact group, and $R$ invariant for that group. We do it here in the case of the disk.
-
-\textbf{The bispectrum for the disk} is defined as:
-\begin{equation}
-    \beta(f)_{nmm'} = \mathcal{F}(f)_{nm}.\mathcal{F}(f)_{nm'}.\mathcal{F}(f)_{n(m+m')}^\dagger,
-\end{equation}
-where we observe that the third term only depends on the indices $m$, $m'$ that correspond to the homogeneous part of the disk, i.e. the one that transform with rotations in $SO(2)$. We recognize that we add these indices, since $SO(2)$ is a commutative group.
-
-\subsection{Bispectrum: Invariance}
-
-
-Define the transformation of $f$ by $\tilde \theta$ as $(\tilde{\theta} \ast f)(r, \theta) = f(r, \tilde \theta +\theta )$. Under this group action, the bispectrum becomes:
-\begin{align*}
-     \beta(\tilde{\theta} \ast f)_{nmm'}
-        &= \mathcal{F}(\tilde{\theta} \ast f)_{nm}.\mathcal{F}(\tilde{\theta} \ast f)_{nm'}.\mathcal{F}(\tilde{\theta} \ast f)_{n(m+m')}^\dagger\\
-        &= \exp(im\tilde{\theta}) \mathcal{F}(f)_{nm}.\exp(im'\tilde{\theta}) \mathcal{F}(f)_{nm'}.\exp(-i(m+m')\tilde{\theta}) \mathcal{F}(f)_{n(m+m')}^\dagger\\
-        &=\beta(f)_{nmm'},
-\end{align*}
-which proves its invariance.
-
-\section{\color{deepred}{Domain $\Omega = (S^2 \times \mathbb{R}_+^*,~SO(3),~\ast)$ (ball rotations)}}
-
-\subsection{Fourier Transform: Definition}
-
-Functions defined over a ball $B$ can be expanded on the surface spherical harmonics as:
-\begin{align*}
-       f(r, \theta, \phi)
-        &=\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell}
-            \hat{f}_{\ell}^m r^{\ell} Y_{\ell}^m(\theta, \phi)
-\end{align*}
-
-
-\textbf{The (vector) Fourier transform} $\hat{f}_\ell$ for functions defined over a ball $B$ is (check?):
-\begin{align*}
-  \mathcal{F}(f)_{l}
-    &\propto \int_0^{2 \pi}\int_0^{\pi} \int_0^1 f(r, \theta, \phi)^\dagger  \left( r^{\ell} Y_{\ell}(\theta, \phi)\right) r d r \cos(\theta) d\theta d\phi \\
-    &= \sqrt{\frac{2l+1}{4\pi}}  \int_0^{2 \pi}\int_0^{\pi} \int_0^1 f(r, \theta, \phi)^\dagger  \left( r^{\ell}  [\rho_l(\phi, \theta, \gamma)]_0 \right) r d r \cos(\theta) d\theta d\phi \\
-    &= \sqrt{\frac{2l+1}{4\pi}}  \int_0^{2 \pi}\int_0^{\pi} \int_0^1 f(r, T_R(x_0))^\dagger  \left( r^{\ell}  [\rho_l(\phi, \theta, \gamma)]_0 \right) r d r \cos(\theta) d\theta d\phi \\
-    &=\sqrt{\frac{2l+1}{4\pi}} \int_0^1 \left(\int_{R \in SO(3)} f(r, T_R(x_0))^\dagger [\rho_l(R)]_0. dR\right) r^{l+1} dr
-\end{align*}
-
-\subsection{Fourier Transform: Equivariance}
-
-Define the transformation of $f$ by $\tilde R$ as $(\tilde{R} \ast f)(r, \theta, \phi) = f(r, \tilde R (\theta, \phi)) = f(T_{\tilde R}^{-1}.u)$. The (vector) Fourier transform undergoes a transformation by the same $\tilde R$, as follows, which shows its equivariance.
-
-\begin{align*}
-    \mathcal{F}(\tilde{R} \ast f)_{l}
-    &=\kappa_l \int_0^1 \left(\int_{R \in SO(3)} (\tilde{R} \ast f)(r, T_R(x_0))^\dagger [\rho_l(R)]_0. dR\right) r^{l+1} dr\\
-    &=\kappa_l \int_0^1 \left(\int_{R \in SO(3)} f(r, T_R^{-1}.T_R(x_0))^\dagger [\rho_l(R)]_0. dR\right) r^{l+1} dr\\
-    &=\kappa_l \int_0^1 \left(\int_{R \in SO(3)} f(r, T_{R^{-1}R}(x_0))^\dagger [\rho_l(R)]_0. dR\right) r^{l+1} dr\\
-    &=\kappa_l \int_0^1 \left(\int_{R' \in SO(3)} f(r, T_{R'}(x_0))^\dagger [\rho_l(\tilde{R}R')]_0. \tilde{R}^{-1}dR'\right) r^{l+1} dr\\
-    &=\kappa_l \int_0^1 \left(\int_{R' \in SO(3)} f(r, T_{R'}(x_0))^\dagger [\rho_l(\tilde{R})\rho_l(R')]_0. \tilde{R}^{-1}dR'\right) r^{l+1} dr\\
-    &=\rho_l(\tilde{R}) . \kappa_l \int_0^1 \left(\int_{R' \in SO(3)} f(r, T_{R'}(x_0))^\dagger [\rho_l(R')]_0. \tilde{R}^{-1}dR'\right) r^{l+1} dr\\
-    &=\rho_l(\tilde{R}) . \kappa_l \int_0^1 \left(\int_{R' \in SO(3)} f(r, T_{R'}(x_0))^\dagger [\rho_l(R')]_0. dR'\right) r^{l+1} dr\\
-    &=\rho_l(\tilde{R}) . \mathcal{F}(f)_{l} \\
-\end{align*}
-
-
-\subsection{Bispectrum: Definitions}
-
-We define the bispectrum as in the spherical function case: the only difference is the type of Fourier transform used, which is the one defined above. Then, both the definition and the proof of invariance of the bispectrum follow.
-
-\textbf{The (vector) bispectrum} of a ball function $f$ is a vector $\beta(f)_{l_1l_2}$ formed of the following scalar entries:
-\begin{equation}
-    \beta(f)_{l_1l_2}[l]
-    :=
-    \left[
-    \mathcal{F}(f)_{l_1} \otimes \mathcal{F}(f)_{l_2}
-    \right]
-    C_{l_1l_2} \mathcal{F}_0(f)_l^{\dagger}
-    \qquad \in \mathbb{C}; \quad|l_1-l_2| \leq l \leq l_1+l_2.
-\end{equation}
-where $C_{l_1l_2}$ is the Clebsch-Gordan matrix associated with $\rho_{l_1} \otimes \rho_{l_2}$, and $\mathcal{F}_0(f)_l$ is the $(2l_1+1)(2l_2+1)$ vector padded with zeros: $\mathcal{F}_0(f)_l:=\left[0, \ldots, 0, \mathcal{F}_l, 0, \ldots, 0\right]$, where the number of zeros preceding $\mathcal{F}_i$ is $n_p$, and the number succeeding $\mathcal{F}_i$ is $n_s$, where $n_p:=\sum_{q=|k-\ell|}^{i-1}(2 q+1)$ and $n_s:=\sum_{q=i+1}^{k+\ell}(2 q+1)$.
-
-\subsection{Bispectrum: Invariance}
-
-The proof of invariance is identical to the proof of invariance for the $S^2$ case, since the Fourier transform on the ball has the same equivariance property as the Fourier transform on the sphere.
-
-\section{\color{deepred}{Domain $\Omega = (H \times R,~G,~\ast)$ (homogeneous $\times$ invariant space)}}
-
-\subsection{Fourier Transform: Definition}
-
-
-\textbf{The (matrix) Fourier transform} for functions defined over $H \times R$ is:
-\begin{align*}
-  \mathcal{F}(f)_{\rho}
-    &= \int_{r \in R} \int_{g \in G} f\left(T_g(x_0)\right) \rho(g)dg. dr \\
-    &= \int_{r \in R} \int_{g \in G} f\left(r, T_g(x_0)\right) \rho(g)dg. dr
-\end{align*}
-
-TODO: find how integral on $r$ should depends on $\rho$, by comparing to the disk and ball cases, in order to be a valid Fourier transform.
-
-\textbf{The (vector) Fourier transform} might exist in this case too, by analogy with the disk and ball cases.
-
-\subsection{Fourier Transform: Equivariance}
-
-Define the transformation of $f$ by $\tilde g$ as $(\tilde{g} \ast f)(r, x) = f(r, \tilde g^{-1} \ast x) = f(T_{\tilde g}^{-1}.x)$. The (vector) Fourier transform undergoes a transformation by the same $\tilde R$, as follows, which shows its equivariance.
-
-\begin{align*}
-  \mathcal{F}(\tilde{g} \ast f)_{\rho}
-    &= \int_{r \in R} \int_{g \in G} (\tilde{g} \ast f)\left(T_g(x_0)\right) \rho(g)dg. dr \\
-    &= \int_{r \in R} \int_{g \in G} (\tilde{g} \ast f)\left(r, T_g(x_0)\right) \rho(g)dg. dr \\
-    &= \int_{r \in R} \int_{g \in G} f(r, T_{\tilde{g}}^-1.T_g(x_0)) \rho(g)dg. dr \\
-    &= \int_{r \in R} \int_{g \in G} f(r, T_{\tilde{g}^-1g}(x_0)) \rho(g)dg. dr \\
-    &= \int_{r \in R} \int_{g' \in G} f(r, T_{g'}(x_0)) \rho(\tilde{g}g')dg'. dr \\
-    &= \int_{r \in R} \int_{g' \in G} f(r, T_{g'}(x_0)) \rho(\tilde{g})\rho(g')dg'. dr \\
-    &= \rho(\tilde{g}). \int_{r \in R} \int_{g' \in G} f(r, T_{g'}(x_0)) \rho(g')dg'. dr \\
-    &= \rho(\tilde{g}) \mathcal{F}(f)_{\rho}
-\end{align*}
-
-
-Note that the equivariance does not depend on the dependency of the integral on $\rho$, thus any ``variant'' of the Fourier transform above will be equivariant.
-
-\subsection{Bispectrum: Definition}
-
-The definition of the bispectrum is identical to the its definition in the homogeneous case: only the Fourier transform used varies.
-
-\subsection{Bispectrum: Invariance}
-
-The invariance of the bispectrum follows from the proof of invariance on homogeneous spaces, since the Fourier transforms on $H \times R$ and on $H$ have the same equivariance properties.
-
-\section{\color{deepred}{Domain $\Omega = (\mathbb{Z}^2, G)$: Gauge Bispectrum for Feature Fields}}
-
-Consider the semi-direct product group $G = \mathbb{Z}^2 \ltimes H$, where $\mathbb{Z}_2$ is the domain. We consider a space of functions $f:\mathbb{Z}^2 \mapsto \mathbb{R}^K$ equipped with the representation $T$ of $G$ induced by a representation $\rho_H$ of $H$ on the fiber $\mathbb{R}^K$, and is denoted $T=\operatorname{Ind}_H^G \rho$. For a function $f$, the action of an element $(t, r) \in \mathbb{Z}^2 \ltimes H$ is:
-\begin{equation}
-    \left[T_{t r} f\right](x)=\rho_H(r)\left[f\left((t r)^{-1} x\right)\right] \in \mathbb{R}^K
-\end{equation}
-Even if this uses notions of representations (e.g. $\rho_H$ and $\pi$, these are just there to explicit the action on the signal).
-
-\subsection{Fourier Transform: Definition}
-
-\textbf{The Fourier transform} for functions defined over a domain $\Omega$ but evolving according to the induced representation is defined as:
-\begin{equation}
-\mathcal{F}(f)_\rho = \widehat{f}(\rho ) = \int_{g \in G} \left[T_{t r} f\right](x_0) \otimes \rho(rt)^\dagger dg.
-\end{equation}
-
-Here, we use the notation $\rho$ to denote the irreducible representation at which the Fourier coefficient is computed. It is NOT the induced representation of the full group $G = Z_2 \times H$, usually denoted $\pi$ in the steerable CNNs framework, nor $\rho_H$ the representation of the group $H$.
-
-\begin{remark}
-We have: $\left[T_{t r} f\right](x_0) \in \mathbb{R}^K$ and $\pi(rt)$ can be a matrix of any dimension, as it depends on the dimension of the irreducible representation.
-
-Thus, what does the product $ \left[T_{t r} f\right](x_0)\pi(rt)$ even mean? It asks the question of what is the Fourier transform of a vector-valued signal. We add a tensor-product. Its intuition is that, at each $g \in G$ we build the $K$-vector that has a matrix at its $k$-component: specifically, $\left(\left[T_{t r} f\right](x_0)\right)_k$ the $k-th$ scalar component of the transformed signal that multiplies the $D \times D$ matrices $\rho(g)$
-
-This is NOT the case where the Fourier transform of a vector-valued signal is defined as the concatenation of the Fourier transform of each of its scalar-valued components. Because, we would not know what group action to use on a signal scalar component $f_k$ of $f$. (unless the restriction of the action on $f$ to only its component $k$ is also an action, which I doubt).
-\end{remark}
-
-
-\subsection{Fourier Transform: Equivariance}
-
-
-\subsection{Gauge Bispectrum: Via Intertwiner Basis}
-
-We consider a case where the output of the G-convolution has the same domain as the input. Then we can use the same basis functions.
-
-Consider a group $G = \mathbb{Z}^2 \times H$ and $Y$ a vector of stacked functions that form a steerable basis of $\mathbb{Z}^2$ for the group $G$. We need the basis to be steerable for the whole group, because the whole group acts on the domain of the signal.
-
-\begin{remark}[Which basis of functions to decompose the filter?]
-    In Steerable CNNs, the filter $\Psi$ is decomposed on a basis of functions $\Psi_l$. Yet, this might not be a basis of steerable functions. It is a basis of functions for the vector space $\text{Hom}(\pi, \rho)$ where $\pi$ is the regular representation (action on the signal's domain) and $\rho$ is a representation of $H$ of our choice. Here, we choose $\rho$ to be the trivial representation for now.
-    Additionally the basis of functions should be indexed by the irreducible representations of the group — in order to make sense for the downstream computations of the bispectrum.
-\end{remark}
-
-The convolution is written as a spatial convolution in Steerable CNNs, which allows the output's domain to match the input's domain
-\begin{align*}
-    \Theta(x_0) = (\Psi \ast f) (x_0) = \Psi^T. T_{x_0}f
-\end{align*}
-where $\Psi = [\Psi_{11}, ..., \Psi_{ss}]$ and $f = [f_{11}, ..., f_{ss}]$ are the values of the filter/signal in spatial format flattened into vectors, while $T_{x_0}f$ is the vector of values of a translated version of the signal.
-
-We write the convolution using the sum over the domain. In the implementation of Steerable CNNs, the signal is translated by $x_0$, not the filter: we apply the filter bank to translated versions of the signal. For the sake of the argument, we perform a change of variable to extract the (possibly steerable??) basis out of the expression of $\Psi$.
-\begin{align*}
-    \Theta(x_0) = (\Psi \ast f) (x_0)
-    &= \sum_{x \in \Omega_s} \Psi(x).f(x_0 - x)\\
-    &= \sum_{x' \in \Omega_s^{+x_0}}
-        \Psi(x_0 - x').f(x'),
-        \quad \text{(note: the domain changes through the change of variable)}\\
-    &= \sum_{x' \in \Omega_s^{+x_0}}
-        \Psi(x_0 - x').f(x'),\\
-    &= \sum_{x' \in \Omega_s^{+x_0}}
-        \left(\sum_l \hat{w}^{(l)} Y_l(x_0 - x')\right).f(x'),
-        \quad\text{(decomposition on the intertwiner's basis, denoted $Y$)}\\
-    &= \sum_{x' \in \Omega_s^{+x_0}}
-        \left(\hat{w}^T. Y(x_0 - x')\right).f(x'),
-        \quad\text{(inner-product vectorial notation)}\\
-    &= \sum_{x' \in \Omega_s^{+x_0}}
-        \left(\hat{w}^T. \rho^Y(-x')Y(x_0)\right).f(x'),
-        \quad\text{([!] assume the intertwiner's basis is steerable, by $\rho^Y$)}\\
-    &= \sum_{x' \in \Omega_s^{+x_0}}
-        \left(\hat{w}^T. \rho^Y(-x')Y(x_0)\right).f(x')\\
-    &= \left(\sum_{x' \in \Omega_s^{+x_0}}
-        \hat{w}^T. \rho^Y(-x').f(x')\right) Y(x_0) \in \mathbb{C}\\
-    &= \hat{\Theta}^T.Y(x_0),\quad\text{(where $\hat{\Theta}, Y(x_0) \in \mathbb{C}^L$)} ,\\
-    &=\sum_{l=1}^L \hat{\Theta}_l.Y_l(x_0)
-\end{align*}
-such that: $\hat{\Theta}_l = \left(\sum_{x' \in \Omega_s^{+x_0}} \hat{w}^T. \rho^Y(-x').f(x')\right)_l$, i.e. the $l$-th coordinate of this row vector.
-
-This would work if we can prove: (i) that the intertwiner basis is steerable for $G$ (see [!]), and (ii) that it is indexed by the irreducible representations. For [!],  if the itertwiners' basis are not steerable, maybe we could express them in a steerable basis.
-
-We start with (i). We consider the trivial representation of $H$ in the definition of the intertwiners. THe vector space of intertwiners thus verifies the equation:
-\begin{equation}
-    Y = Y.\pi(h),\quad \forall h \in H,
-\end{equation}
-where $Y = [Y_{11}, ..., Y_{ss}] \in \mathbb{C}^{1 \times s^2}$ is a intertwiner, written as the set of its values on the domain. $\pi(h) \in \mathbb{C}^{s^2.s^2}$, the matrix that can be applied to the values of a signal $f = [f_{11}, ..., f_{ss}] \in \mathbb{C}^{s^2}$ and amounts to rotating it. It is known as the regular representation.
-
-The equation above means that the solutions $Y$ are the set of eigenvectors of eigenvalue 1 for every matrix of the representation $\pi$: an invariant subspace of the representation $\pi$, i.e. a sub-representation.
-
-
-\begin{remark}
-    We can rewrite this equation inn terms of coordinates on the domain:
-\begin{equation}
-    Y(x_0) = \sum_{x \in \in \Omega_s} Y(x).\pi(h)_{xx_0},
-\end{equation}
-by writing out the matrix multiplication explicitly and indexing the values of $Y$ by the pixel coordinate where that value is located. This equation holds for one element $Y$ of the set of basis functions. We need the whole set to prove the steerability.
-\end{remark}
-
-Can the basis of the intertwiners be indexed by the irreps of $H$? To answer this question, we count how many basis functions there are. Writing $\rho_0$ is the trivial representation of $H$ on the fibers, and following the computation after Eq (11) in Steerable CNNs 2017, we have:
-\begin{equation}
-    L = \text{dim}~\text{Hom}^H(\pi, \rho_0) = \sum_{\text{irreps }\psi } m^\pi_{\psi}.m^{\rho_0}_{\psi} = m^\pi_{\phi_0}
-\end{equation}
-since the multiplicity of the trivial representation $\rho_0$ is 0 except on itself where it is 1. In the above equation, $m^\pi_{\phi_0}$ is the multiplicity of the representation $\pi$ of $H$ on the trivial irreps of the group $H$. This multiplicity is $3$, as metnioned in Table 1 of the paper.
-
-We take the example of Table 1 of the Steerable CNNs paper. $H = D_4$ the roto-reflexion group. We decompose $\pi_0$ the representation of $G$ on the irreducible representations of $H$. Actually, we consider $\pi_0$ as a representation of $H$ and decompose this one. There are 3 elementary basis filters for the $\pi_0$, meaning that the dimension of Hom is 3, which is different from 5, the number of irreps of $D_4$.
-
-Thus, the number of basis functions in $\text{Hom}^H(\pi, \rho_0)$ is not equal to the number of irreps. We would not get the Fourier coefs as we need them for the bispectrum.
-
-Maybe we could define another bispectrum?
-
-\begin{remark}[If $\pi$ had been the regular representation of $H$]
-$\pi$ is probably not the regular representation, because the reg representation has a fixed dimension and here it works for any s, it has shape $s^2 \times s^2$.
-
-From Wikipedia, we have: If G is a finite group and K is the complex number field, the regular representation decomposes as a direct sum of irreducible representations, with each irreducible representation appearing in the decomposition with multiplicity its dimension. Thus, the dimension of the intertwining space is 1, which is different from the number of irreps of the group. Thus, we cannot get the spectral coefficients by reading out the coefficients on the intertwining basis.
-
-On the other hand, why does the Steerable CNN paper talk about basis filter on the irreps?! As if there was one basis filter (i.e. a basis function) per irreps.
-
-Answer: this is the set of intertwiners when the $\rho$ chosen is an irreps. Which is our case, since our $\rho_0$ is the trivial representation. Contrarily to what we just proved, the basis for $\rho_0$, which is $A1$ has three elements.
-\end{remark}
-
-\subsection{Bispectrum: from the intertwining coefs?}
-
-Goal: write the output of the convolution of a steerable CNN in terms of its “spectral” coefficients $\hat \Theta_l$ on a steerable basis of the domain.
-Illustrative example: G = Z2 x D4, i.e. the example in the steerable CNN paper, where D4 is non-commutative.
-
-Remark: In a steerable CNN, the filter is expressed as “spectral” coefficients on a basis.
-However, they do NOT use a steerable basis;
-instead, they use on a intertwiners’ basis of H=D4 (see top of p4 of the Steerable CNNs paper).
-Let’s assume for now that the intertwiners’ basis is a steerable basis.
-
-Result: The output of the steerable convolution can be written in terms of its “spectral” coefficients $\hat \Theta_l$on the intertwiner basis (screenshot & details in overleaf).
-
-Next step: Compute the bispectrum from the spectral coefficients $\hat \Theta_l$ .
-
-If we want to use an existing formula of the bispectrum:
-we need the L spectral coeffs $\hat \Theta_l$ to correspond to the M Fourier coeffs of Theta  — where M is the number of irreps of the group.
---> we need, at least, to have L = M.
-However, that is not the case for the example of D4: we have L=3 spectral coeffs but M=5 irreps of D4 (see Table 1 in Steerable CNNs).
-Thus, we need a new formula for the bispectrum, written in terms of the $\hat \Theta_l$
-Questions:
----> How do we generalize the bispectrum, for a non-commutative group H, when we don’t have the Fourier coefs of the signal, but rather its coefs on an intertwining basis? (that is, maybe, also a steerable basis?)
----> If we find a formula for it, what do we expect it to be invariant to? I’d say invariant to action of H on the domain.
-
-Since we would expect this bispectrum to be invariant to the action of H on the domain, we compute the TC for this action, and then take its Fourier transform.
-
-This is the TC for the action of rotation on the domain. which is not a homogeneous domain. Maybe we could take the disk bispectrum then? And write everything in disk harmonics? The disk harmonics are steerable by $\rho(\theta) = \exp(-im\theta)$ just as the ring harmonics.
-
-Let's rewrite the output of the steerable convolution using the disk harmonics.
-\begin{align*}
-    \Theta(x_0) = (\Psi \ast f) (x_0)
-    &= \sum_{x \in \Omega_s} \Psi(x).f(x_0 - x)\\
-    &= \sum_{x' \in \Omega_s^{+x_0}}
-        \Psi(x_0 - x').f(x'),
-        \quad \text{(note: the domain changes through the change of variable)}\\
-    &= \sum_{x' \in \Omega_s^{+x_0}}
-        \left(\sum_l \hat{w}^{(l)} Y_l(x_0 - x')\right).f(x'),
-        \quad\text{(decomposition on the intertwiner's basis, denoted $Y$)}\\
-    &= \sum_{x' \in \Omega_s^{+x_0}}
-        \left(\sum_l \hat{w}^{(l)} \sum_{m,n}\hat{Y_l}(m, n) d_{m, n}(r_{x_0 - x'}, \theta_{x_0 - x'})\right).f(x'),
-        \quad\text{(decomposition on the harmonics basis, denoted $Y$)}\\
-    &= \sum_{m,n} \left(\sum_{x' \in \Omega_s^{+x_0}}
-        \sum_l \hat{w}^{(l)} \hat{Y_l}(m, n).f(x')\right) d_{m, n}(r_{x_0 - x'}, \theta_{x_0 - x'}),
-        \quad\text{(getting the coefficients on the harmonics)}
-    % &= \sum_{x' \in \Omega_s^{+x_0}}
-    %     \left(\hat{w}^T. Y(x_0 - x')\right).f(x'),
-    %     \quad\text{(inner-product vectorial notation)}\\
-    % &= \sum_{x' \in \Omega_s^{+x_0}}
-    %     \left(\hat{w}^T. \rho^Y(-x')Y(x_0)\right).f(x'),
-    %     \quad\text{([!] assume the intertwiner's basis is steerable, by $\rho^Y$)}\\
-    % &= \sum_{x' \in \Omega_s^{+x_0}}
-    %     \left(\hat{w}^T. \rho^Y(-x')Y(x_0)\right).f(x')\\
-    % &= \left(\sum_{x' \in \Omega_s^{+x_0}}
-    %     \hat{w}^T. \rho^Y(-x').f(x')\right) Y(x_0) \in \mathbb{C}\\
-    % &= \hat{\Theta}^T.Y(x_0),\quad\text{(where $\hat{\Theta}, Y(x_0) \in \mathbb{C}^L$)} ,\\
-    % &=\sum_{l=1}^L \hat{\Theta}_l.Y_l(x_0)
-\end{align*}
-To conclude, we would need the harmonics to be steerable by the translations.
-
-
-
-\subsection{Gauge Triple Correlation: Definition}
-
-We define a new notion of triple correlation that works for vectorial functions (feature fields) equipped with the action of a semi-direct group $\mathbb{Z}^2 \ltimes H$ via the induced representation. In this setting, the group action does not only act on the domain $\mathbb{Z}^2$ of the function, but also transforms its values in the fiber $\mathbb{R}^K$ --- i.e., it is a gauge transformation. In practice, such signals arise as outputs of steerable G-convolutions: after the first layer, the function is not a function over the group (as is the case for the classic G-conv) but is still a function over $\mathbb{Z}^2$ equipped with this induced group action.
-
-We wonder whether we can generalize the triple correlation to this case. We show below that we can.
-
-We modify the definition of the triple correlation by inverting the order of $gg_1$ to $g_1g$:
-
-
-We define a triple correlation applicable to this class of functions:
-\begin{equation}
-    T(f)_{g_1g_2}
-        = \int_{\mathbb{Z}^2}
-            \int_H
-                T_{rt}[f]^\dagger (x_0) T_{r_1t_1rt}[f](x_0) T_{r_2t_2rt}[f](x_0)
-                    drdt \in \mathbb{R}^K,
-\end{equation}
-where $g = rt$, $g_1 = r_1t_1$ and $g_2 = r_2 t_2$.
-
-We note that the triple correlation is a vector in $\mathbb{R}^K$, i.e. in the space representing the fiber where $K$ is the number of channels. We also note that we only use one element of the domain $x_0 \in \mathbb{Z}^2$ to define the TC: this is a similar approach as what was done for the homogeneous case. The domain is homogeneous for the action of $G$. By varying $g \in G$, we will see every value of the function.
-
-\subsection{Triple Correlation: Invariance}
-
-We prove invariance of the gauge TC.
-
-We begin with a Lemma:
-
-\begin{lemma}
-Composing the group actions corresponding to the induced representation:
-    \begin{equation}
-    T_{rt}[T_{r't'}[f]] (x)
-        = \rho(r) \left[[T_{r't'}[f]]\left((t r)^{-1} x\right)\right]
-        = \rho(r)  \rho(r') f((t'r')^{-1}(tr)^{-1}x) \in \mathbb{R}^K
-\end{equation}
-\end{lemma}
-
-
-Consider a second function that is obtained from $f$ by an action of a group element $h = r_ht_h$, i.e. we have: $f_2 = T_h[f] = T_{r_ht_h}[f]$. I.e. $f_2$ is in the orbit of $f$ for the action of the induced representation.
-
-\begin{proof}
-
-We prove that $T(f_2)_{g_1g_2} = T(f)_{g_1g_2} $, which proves the invariance of the TC for the action of the induced representation.
-
-\begin{align*}
-    T(f_2)_{g_1g_2}
-        &= \int_{\mathbb{Z}^2} \int_H
-            T_{rt}[f_2]^\dagger (x_0)
-                T_{r_1t_1rt}[f_2](x_0)
-                    T_{r_2t_2rt}[f_2](x_0)
-                    dtdr \\
-        &= \int_{\mathbb{Z}^2} \int_H
-            T_{rt}[T_{r_ht_h}[f]]^\dagger (x_0)
-                T_{r_1t_1rt}[T_{r_ht_h}[f]](x_0)
-                    T_{r_2t_2rt}[T_{r_ht_h}[f]](x_0)
-                    dtdr \\
-        &= \int_{\mathbb{Z}^2} \int_H
-            \left(\rho(r)\rho(r_h)
-            f((t_hr_h)^{-1}(tr)^{-1}x_0)\right)^\dagger \\
-                &\qquad\qquad\qquad.\left(
-                \rho(r_1)\rho(r)\rho(r_h)
-                f((t_hr_h)^{-1}(tr)^{-1}(t_1r_1)^{-1}x_0)\right)\\
-                &\qquad\qquad\qquad.\left(
-                \rho(r_2)\rho(r)\rho(r_h)
-                f((t_hr_h)^{-1}(tr)^{-1}(t_2r_2)^{-1}x_0)\right) dtdr
-\end{align*}
-We will prove the rest using the example SE(n) of a semi-direct product group $\mathbb{Z}_2 \ltimes H$.
-
-We recall that, e.g. in SE(n):
-\begin{itemize}
-    \item the inversion writes: $(tr)^{-1} = (r^{-1}, R^{-1}(-t))$
-    \item and the action on an element of the domain writes: $(R, t)x = Rx+t$ .
-\end{itemize}
-Note: for a general subgroup $H$, replace R by $\phi(r)$ i.e. a representation of $H$ on $\mathbb{Z}_2$ that transforms $r \in H$ into something that acts on $\mathbb{Z}^2$.
-
-Note2: Triple check this.
-
-We expand this:
-\begin{align*}
-    (t_hr_h)^{-1}(tr)^{-1} x_0
-        &= R_h^{-1}(R^{-1}x_0 - R^{-1}t) - R_h^{-1}t_h \\
-        &= R_h^{-1}R^{-1}x_0 - R_h^{-1}R^{-1}t- R_h^{-1}t_h
-\end{align*}
-and:
-\begin{align*}
-     (t_hr_h)^{-1}(tr)^{-1}(t_1r_1)^{-1} x_0
-     &= (t_hr_h)^{-1}(tr)^{-1} (R_1^{-1}x_0 - R_1^{-1}t_1) \\
-     &= R_h^{-1}(R^{-1}R_1^{-1}x_0 - R^{-1}R_1^{-1}t_1 - R^{-1}t) - R_h^{-1}t_h \\
-     &= R_h^{-1}R^{-1}R_1^{-1}x_0 - R_h^{-1}R^{-1}R_1^{-1}t_1 - R_h^{-1}R^{-1}t - R_h^{-1}t_h \\
-\end{align*}
-
-We first perform the change of variable $r' = rr_h$, and $t'=Rt_h+t$  which gives:
-\begin{align*}
-    \rho(r)\rho(r_h) &= \rho(r')
-\end{align*}
-And for the part that act on the domains, using the fact that $R_h^{-1} = R'^{-1}R$:
-\begin{align*}
-      (t_hr_h)^{-1}(tr)^{-1}x_0
-      &= R'^{-1}x_0 - R'^{-1}t-R'^{-1}Rt_h \\
-      &= R'^{-1}x_0 - R'^{-1}(t+Rt_h) \\
-      &= R'^{-1}x_0 - R'^{-1}t' \\
-      &= (t'r')^{-1}x_0
-\end{align*}
-
-  as well as:
-  \begin{align*}
-       (t_hr_h)^{-1}(tr)^{-1}(t_1r_1)^{-1} x_0 &=(t'r')^{-1}(t_1r_1)^{-1} x_0
-  \end{align*}
-
-Thus, we can continue the computation of the TC:
-\begin{align*}
-    T(f_2)_{g_1g_2}
-        &= \int_{\mathbb{Z}^2} \int_H
-            \left(\rho(r)\rho(r_h)
-            f((t_hr_h)^{-1}(tr)^{-1}x_0)\right)^\dagger(x_0) \\
-                &\qquad\qquad\qquad.\left(
-                \rho(r_1)\rho(r)\rho(r_h)
-                f((t_hr_h)^{-1}(tr)^{-1}(t_1r_1)^{-1}x_0)\right)\\
-                &\qquad\qquad\qquad.\left(
-                \rho(r_2)\rho(r)\rho(r_h)
-                f((t_hr_h)^{-1}(tr)^{-1}(t_2r_2)^{-1}x_0)\right) dtdr\\
-         &= \int_{\mathbb{Z}^2} \int_H
-         \left(\rho(r')
-            f((t'r')^{-1}x_0)\right)^\dagger(x_0) \\
-                &\qquad\qquad\qquad.\left(
-                \rho(r_1)\rho(r')
-                f((t'r')^{-1}(t_1r_1)^{-1}x_0)\right)\\
-                &\qquad\qquad\qquad.\left(
-                \rho(r_2)\rho(r')
-                f((t'r')^{-1}(t_2r_2)^{-1}x_0)\right) dt'dr'\\
-        &=T(f)_{g_1g_2}
-\end{align*}
-\end{proof}
-
-\subsection{Gauge Bispectrum: Definition}
-
-
-
-\begin{proof}
-We have a group $G = \mathbb{Z}_2 \times H$. We use the convention $g = (t, r) = tr$ for a group element decomposing into a translation component and a rotation (later, of any compact group) one.
-
-By definition, the gauge bispectrum is the Fourier transform of the gauge triple correlation. We have defined the Fourier transform associated with the induced representation's action.
-
-We compute this Fourier transform for the product group $G \times G$ and show that we obtain the expression given above. We recall that $T(f)_{g_1, g_2} \in \mathbb{C}^{K}$. The terms $T_{g}[f]^\dagger T_{g_1g}[f]$ define an inner product and need to stick together.
-
-
-We define it for any action of $g$ on a vectorial signal, where $g$ can touch both the fiber and the domain. The action is not restricted to the induced representation, thus we use $g$ and not $rt$. A remaining question is whether the induced representation brings something more? Or why does it seem to be important to have the induced representation in the TC but not for the BiSp? It is probably enough as long as the action is homogeneous? Or maybe, in order for it to be an action on fiber+domain it has to be some type of induced representation. Otherwise it does not make sense?
-Associativity of mixed tensor-product/matrix multiplication operation
-%https://math.stackexchange.com/questions/3029078/associativity-of-mixed-matrix-product-and-tensor-product
-
-The following assumes that: $v \otimes (AB) = (v\otimes A). B$ for $v\in\mathbb{C}^K$ and $A, B \in \mathbb{C}^{D \times D}$
-
-    \begin{align*}
-    \beta(f)_{\rho_1, \rho_2}
-    &=  \int_{g_1 \in G} \int_{g_2 \in G}
-        T(f)_{g_1, g_2} \otimes
-            \left[
-                \rho_1(g_1)^\dagger \otimes \rho_2(g_2)^\dagger
-            \right]
-            dg_1 dg_2 \\
-    &=
-    \int_{g_1 \in G} \int_{g_2 \in G}
-        \int_{g \in G}
-                \left[ T_{g}[f]^\dagger (x_0) T_{g_1g}[f](x_0) T_{g_2g}[f](x_0)
-                    \right]dg
-            \otimes
-              \left[
-                \rho_1(g_1)^\dagger \otimes \rho_2(g_2)^\dagger
-              \right]
-            dg_1 dg_2 \\
-    &=\int_{g \in G}
-      \int_{g_1 \in G} \int_{g_2 \in G}
-                  \left[T_{g}[f]^\dagger (x_0)T_{g_1g}[f](x_0) T_{g_2g}[f](x_0)
-                    \right]
-                \otimes
-              \left[
-                \rho_1(g_1)^\dagger \otimes \rho_2(g_2)^\dagger
-              \right]
-            dg_1 dg_2dg\\
-    &=\int_{g \in G}
-      \int_{g_1 \in G} \int_{g_2 \in G}
-               T_{g}[f]^\dagger (x_0) T_{g_1}[f](x_0) T_{g_2}[f](x_0)
-                \otimes
-              \left[
-                \rho_1(g_1g^{-1})^\dagger \otimes \rho_2(g_2g^{-1})^\dagger
-              \right]
-            dg_1 dg_2 dg\\
-    &=\int_{g \in G}
-      \int_{g_1 \in G} \int_{g_2 \in G}
-            [...]
-                \otimes
-              \left[
-                \rho_1(g_1)^\dagger \rho_1(g^{-1})^\dagger \otimes \rho_2(g_2)^\dagger \rho_2(g^{-1})^\dagger
-              \right]
-            dg_1 dg_2dg\\
-        &=\int_{g \in G}
-      \int_{g_1 \in G} \int_{g_2 \in G}
-            [...]
-                \otimes
-              \left[
-                \rho_1(g_1)^\dagger \rho_1(g) \otimes \rho_2(g_2)^\dagger \rho_2(g)
-              \right]
-            dg_1 dg_2dg\\
-        &=\int_{g \in G}
-      \int_{g_1 \in G} \int_{g_2 \in G}
-            [...]
-                \otimes \left(
-              \left[
-                \rho_1(g_1)^\dagger \otimes \rho_2(g_2)^\dagger \right] . \left[\rho_1(g) \otimes  \rho_2(g)
-              \right] \right)
-            dg_1 dg_2dg\\
-            &=\int_{g \in G}
-      \int_{g_1 \in G} \int_{g_2 \in G}
-       \left(T_{g}[f]^\dagger (x_0) T_{g_1}[f](x_0) T_{g_2}[f](x_0)
-                \otimes
-              \left[
-                \rho_1(g_1)^\dagger \otimes \rho_2(g_2)^\dagger \right]\right)  .
-            dg_1 dg_2\left[\rho_1(g) \otimes  \rho_2(g)
-              \right] dg\\
-        &=\int_{g \in G}
-        T_{g}[f]^\dagger (x_0) \left(
-      \int_{g_1 \in G} \int_{g_2 \in G}
-       T_{g_1}[f](x_0) T_{g_2}[f](x_0)
-                \otimes
-              \left[
-                \rho_1(g_1)^\dagger \otimes \rho_2(g_2)^\dagger \right]\right)  .
-            dg_1 dg_2\left[\rho_1(g) \otimes  \rho_2(g)
-              \right] dg\\
-            &=\int_{g \in G}
-            T_{g}[f]^\dagger (x_0)
-      \left[\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}\right] \left[\rho_1(g) \otimes  \rho_2(g)
-              \right] dg\\
-            &=\left[\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}\right]
-            \int_{g \in G}
-            T_{g}[f]^\dagger (x_0)
-       \left[\rho_1(g) \otimes  \rho_2(g)
-              \right] dg\\
-            &=\left[\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}\right]
-            \int_{g \in G}
-            T_{g}[f]^\dagger (x_0) \otimes
-       \left[
-        C_{\rho_1, \rho_2}
-        \bigoplus_{\rho \in \rho_1 \otimes \rho_2}
-                \rho(g)
-        C_{\rho_1, \rho_2} ^\dagger
-        \right]
-              dg\\
-            &=\left[\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}\right]
-            \int_{g \in G}
-            \bigoplus_{\rho \in \rho_1 \otimes \rho_2}
-            T_{g}[f]^\dagger (x_0) \otimes
-       \left[
-        C_{\rho_1, \rho_2}
-                \rho(g)
-        C_{\rho_1, \rho_2} ^\dagger
-        \right]
-              dg\\
-        &=\left[\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}\right]
-        C_{\rho_1, \rho_2}
-            \int_{\mathbb{Z}^2}
-            \int_H
-            \bigoplus_{\rho \in \rho_1 \otimes \rho_2}
-            T_{g}[f]^\dagger (x_0) \otimes
-       \left[
-                \rho(g)
-        \right]
-              dg C_{\rho_1, \rho_2} ^\dagger \\
-        &=\left[\mathcal{F}(f)_{\rho_1} \otimes \mathcal{F}(f)_{\rho_2}\right]
-        C_{\rho_1, \rho_2}
-            \bigoplus_{\rho \in \rho_1 \otimes \rho_2}
-           \mathcal{F}(f)_\rho C_{\rho_1, \rho_2} ^\dagger \\
-     \end{align*}
-\end{proof}
-
-TO CHECK
-
-\section{Physics-Invariance}
-Consider $f$ a solution to a PDE, defined on $\mathbb{R}^2 \times \mathbb{R}$, i.e., on a space-time domain, and with values in some co-domain.
-
-Consider the group $G$ of its symmetries, which are actions on $f$. Most of these actions are: linear actions acting only on the domain of $f$, which is likely homogeneous for the action, thus this fits to one of the cases listed here. Other actions are linear actions that act on both domain and fiber (gauge transformations), thus this fits into one of the cases listed here. For the remaining non-linear actions, we can define a Fourier transform, and bispectrum, for this case?
-
-For example, we can think of building a transformer whose attention coef are invariant thanks to the bispectrum. If we do this, does this mean that we should work in the fourier domain, for the  whole network? What does it mean to solve a PDE in Fourier domain?
-
-\section{Scratch}
-
-\subsection{Triple Correlation: Definition (Continuous case/Comp. Cost: Try 1)}
-
-We try to generalize the triple correlation for the case where the subgroup $H$ is continuous, i.e. for when H is a Lie group. We use H = SO(n) as our running example.
-
-The main ingredient for the gauge TC is the action of a group. Thus, the trick is to turn this action into the action of the associated Lie algebra, and use the generators of the Lie algebra. The Lie algebra of SO(n) admits three generators that we write $A_1, A_2, A_3$ (in matrix form) or $a_1, a_2, a_3$ (in abstract form).
-
-We show that it is enough to define the TC only on these three generators, to define the whole TC.
-\begin{equation}
-    T(f)_{a^it_1a^jt_2}
-        = \int_{t \in \mathbb{Z}^2}
-            \int_{a \in \mathfrak{h}}
-                T_{at}[f]^\dagger (x) T_{a^it_1at}[f](x) T_{a^jt_2at}[f](x)
-                    dadt,
-\end{equation}
-for $i, j \in \{1, 2, 3\}$ and any $t_1, t_2 \in \mathbb{Z}^2$. Also, we have: $g = \exp(a)t$, $g_1 = \exp(a^i)t_1$ and $g_2 = \exp(a^j) t_2$ and we define:
-\begin{equation}
-    T_{at}[f](x) = \rho(\exp(a))f((t\exp(a))^{-1}x_0).
-\end{equation}
-
-\begin{lemma}[Group \& algebra representations]
-For a lie algebra reps $t(a)\in M_{d\times d}$ of $a \in\mathfrak{h}$ where $r(a) = \exp(a)$ is the corresponding group element:
-\begin{equation}
-    T_{r(a)} = T_{\exp(a)} = \exp t(a) \in GL_{d \times d}.
-\end{equation}
-For example, for the representation $\phi$ that appears in the next lemma:
-\begin{equation}
-    \phi\left(\exp\left(-a\right)\right) = \exp\left(-\sigma(a) \right),
-\end{equation}
-where $\sigma$ is the lie algebra representation corresponding to the group representation $\phi$.
-\end{lemma}
-
-\begin{lemma}[Action of an inverse element on $x$]
-    For $a \in \mathfrak{h}$ where $a = \sum_{l=1}^3 w_l a^l$ and $r = \exp(a)$, we have:
-    \begin{align*}
-        \left(tr\right)^{-1}x
-        &= \left(
-            r^{-1}, -R^{-1} t\right)x \\
-        &= \left(
-            \exp\left(a\right)^{-1}, -\phi\left(\exp\left(a\right)\right)^{-1} t\right)x \\
-        &= \left(
-            \exp\left(-a\right), -\phi\left(\exp\left(-a\right)\right) t\right)x \\
-        &= \left(
-            \exp\left(-a\right), -\exp\left(-\sigma(a) \right) t\right)x \\
-        &= \exp\left(-a\right)x - \exp\left(-\sigma(a) \right) t
-    \end{align*}
-\end{lemma}
-
-In the following lemma, we use convenient notations that will help for the last derivation of this page.
-
-\begin{lemma}[Action on the signal $f$]
-    For $a_2 \in \mathfrak{h}$ where $a_2 = \sum_{l=1}^3 w_l a^l$, and a function signal $F$, we have:
-    \begin{align*}
-        T_{a_2t_2}[F](x)
-         &= \rho(\exp(a_2))F((t_2\exp(a_2))^{-1}x)\\
-         &= \exp(\tau(a_2)) F\left[\exp\left(-a_2\right)x - \exp\left(-\sigma(a_2) \right) t_2\right] \\
-         &= \exp\left(\tau\left(\sum_{l=1}^3 w_l a^l\right)\right)
-            F\left[
-                \exp\left(-\sum_{l=1}^3 w_l a^l\right)x
-                - \exp\left(-\sigma(\sum_{l=1}^3 w_l a^l) \right) t_2
-            \right] \\
-         &= \exp\left( -\sum_{l=1} w_l \sigma(a^l)\right)
-                F\left[
-                    \exp\left(-\sum_{l=1}^3 w_l a^l\right)x
-                    - \exp\left(-\sigma(\sum_{l=1}^3 w_l a^l) \right) t_2
-                    \right]
-    \end{align*}
-
-For the next computations, we assume that the group $H$ is commutative, so that $\exp(A+B) = \exp(A)\exp(B).$
-\begin{align*}
-    T_{a_2t_2}[F](x)
-    &= \exp\left( -\sum_{l} w_l \sigma(a^l)\right)
-            F\left[
-                \exp\left(-\sum_{l} w_l a^l\right)x
-                - \exp\left(-\sigma(\sum_{l} w_l a^l) \right) t_2
-                \right]\\
-    &= \Pi_{l}\exp\left( -w_l \sigma(a^l)\right)
-            F\left[
-                \Pi_{l}\exp\left(-w_l a^l\right)x
-                - \exp\left(-\sigma(\sum_{l} w_l a^l) \right) t_2
-                \right]
-    \end{align*}
-This seems too complicated. We start again. We note that:
-\begin{align*}
-    T_{a_2t_2}[F](x)
-     &= T_{\exp(a_2)t_2}[F](x) \\
-     &= T_{\exp(\sum_{l=1}^3 w_l a^l)t_2}[F](x) \\
-     &= T_{\Pi_{l=1}^3\exp(w_l a^l)t_2}[F](x) \\
-     &= T_{\Pi_{l=1}^3\left(\exp(a^l)\right)^{w_l}t_2}[F](x) \\
-     &= \Pi_{l=1}^3\left(T_{\exp(a^l)}\right)^{w_l}T_{t_2}[F](x) \\
-\end{align*}
-\end{lemma}
-
-Now, we show that any $T(f)_{a_1t_1a_2t_2}$ can be expressed as a function of $T(f)_{a_it^1a^jt_2}$. By symmetry in $a_1, a_2$, we only prove it for $T(f)_{a^it_1a_2t_2}$.
-
-For $a_2 \in \mathfrak{h}$ where $a_2 = \sum_{l=1}^3 w_l a^l$, we have:
-\begin{align*}
-    T(f)_{a^it_1a_2t_2}
-    &= \int_{t \in \mathbb{Z}^2}
-        \int_{a \in \mathfrak{h}}
-            T_{at}[f]^\dagger (x) T_{a^it_1at}[f](x) T_{a_2t_2at}[f](x)
-                dadt \\
-    &= \int_{t \in \mathbb{Z}^2}
-        \int_{a \in \mathfrak{h}}
-            T_{at}[f]^\dagger (x) T_{a^it_1at}[f](x) T_{a_2t_2}[T_{at}(f)](x)
-                dadt \\
-    &=\int_{t \in \mathbb{Z}^2}
-        \int_{a \in \mathfrak{h}}
-            F^\dagger (x) T_{a^it_1}[F](x) T_{a_2t_2}[F](x)
-                dadt \quad \text{notation: $F = T_{at}(f)$}\\
-  &=\int_{t \in \mathbb{Z}^2}
-        \int_{a \in \mathfrak{h}}
-            F^\dagger (x) T_{a^it_1}[F](x) \Pi_{l=1}^3\left(T_{\exp(a^l)}\right)^{w_l}T_{t_2}[F](x)
-                dadt
-\end{align*}
-Here, we are blocked because we cannot unpack the product $\Pi_l$.
-
-\subsection{Triple Correlation: Definition (Continuous case/Comp. Cost: Try 2)}
-
-We consider the TC as an operator:
-\begin{align*}
-    TC: \mathcal{F} \times \mathfrak{g}^2 &\mapsto \mathbb{R}^K \\
-        f, (t_1a_1, t_2 a_2) &\mapsto T(f)_{t_1a_1t_2a_2}
-\end{align*}
-This is not a linear operator, thus it cannot be represented by a big tensor. This is the main difference with the case of the convolution: the convolution being a linear operator, we can represent it as a big tensor that contains the whole filter bank. The nonlinearity in $f$ comes from the triple product. The nonlinearity in $a_2$ comes from the fact that we were not able to prove it in Try 1.
-
-Let's try again though. With the discrete group case for now. In the convolution case, all the rotated filters are included in the filter bank. If we think in the same way for the TC, this means that we should see the TC as:
-\begin{align*}
-    TC: \mathcal{F}&\mapsto \mathbb{R}^{K \times \|G\|^2} \\
-        f  &\mapsto \left[T(f)_{t_1a_1t_2a_2}, ..., ... \right],
-\end{align*}
-where on the output we have all the values of the TC for all the (non redundante) pairs of group elements.
-
-We have $\mathcal{F} = \{f:\mathbb{Z}^2 \mapsto \mathbb{R}^K \} = \mathbb{R}^{S^2.K} $ where $S$ is the side of the $\mathbb{Z}^2$ grid that supports the image $f$.
-
-As an operator on $f$, what is the polynomial order of the TC?
-
-\begin{align*}
-    TC(\mu f+\lambda g)
-    &= \int_{t \in \mathbb{Z}^2}
-        \int_{a \in \mathfrak{h}}
-            T_{at}[\mu f+\lambda g]^\dagger (x)
-            T_{a_1t_1at}[\mu f+\lambda g](x)
-            T_{a_2t_2at}[\mu f+\lambda g](x)
-                dadt \\
-    &= \int_{t \in \mathbb{Z}^2}
-        \int_{a \in \mathfrak{h}}
-            \left(\mu T_{at}[f] + \lambda T_{at}[g]\right)^\dagger (x)
-            \left(\mu T_{a_1t_1at}[f] + \lambda T_{a_1t_1at}[g]\right) (x)
-            \left(\mu T_{a_2t_2at}[f] + \lambda T_{a_2t_2at}[g]\right) (x)
-                dadt \\
-    &= ...\text{not equal to $ \mu TC(f) +\lambda TC(g)$, nor with its higher orders, bcs we have crossed terms with $f, g$}
-\end{align*}
-
-Thus, TC is just an invariant function on $f$. Since it is not linear, we cannot represent it as a matrix and write the intertwiner equation of the type $T \pi(g) = \rho(g) T$ that was useful to find a basis for the filters in the G-conv of the gauge case.
-
-\subsection{Triple Correlation: Definition (Continuous case/Comp. Cost: Try 3)}
-
-The last idea would be to think of the triple correlation as an implicit representation. For example, we could implement it as a continuous function (a neural network) takes a pair $(g_1, g_2)$ as input and output the $T(f)_{g_1g_2}$. The NN could implement the ansatz of the triple correlation, e.g. have a triple product structure that is able to evaluate the TC each time $f, g_1, g_2$ is given. The weights of this neural network would contain everything we need about the TC, without having to evaluate it at each pair of group elements. However, it is possible that we only shift the problem is that the way of implementing the TC as an implicit function is going to be even more computationally expensive that computing all of the values of the TC.
-
-\subsection{More comments}
-
-The computational cost mostly comes from the translation part, i.e. from the size of the grid of the images. We note that, for steerable CNNs, only the abstraction related to the group H in the semi direct product $G = H \times Z^2$ is used in the filter bank coefficients. For the translation part, there is no computational gain: the filter bank is still moved at each position during the convolution. This could be an argument in favor of not trying to use the gauge bispectrum strategy for complexity gain.
-
-\subsection{More comments: continuous case}
-
-Since the obstruction to using the gauge bispectrum strategy is also the fact that the TC is non linear, we could try to go back to a situation where we have a linear operator. For example, we could consider the bispectrum which in the discrete setting is of the form
-\begin{equation}
-    \beta(x) = Ux \otimes Vx \otimes Wx
-\end{equation}
-where $U, V, W$ are different realization of the discrete Fourier transform.
-
-Considering the case of the continuous fourier transform instead, we know that the Fourier transform is an equivariant linear operator:
-\begin{equation}
-    W\phi(g) = \rho(g) W,
-\end{equation}
-when $W$ is the discrete fourier matrix. This equation should hold for the case where the image is discrete (discrete in translation) but H is continuous. The continuous fourier transform in H could be decomposed into a discrete set of coefficients on a equivariant basis of intertwiners. This could make the (fixed, not learnable) bispectrum neural network applicable to continuous groups.
-
-We need the fourier transform for the gauge case, though.
-
-
-\bibliographystyle{plos2015}
-\bibliography{references.bib}
-
-\end{document}
diff --git a/docs/cohen_et_al_2018/1801.10130 b/docs/cohen_et_al_2018/1801.10130
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diff --git a/docs/cohen_et_al_2018/equivariance_error.pgf b/docs/cohen_et_al_2018/equivariance_error.pgf
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diff --git a/docs/cohen_et_al_2018/fancyhdr.sty b/docs/cohen_et_al_2018/fancyhdr.sty
deleted file mode 100644
index 5a4d897..0000000
--- a/docs/cohen_et_al_2018/fancyhdr.sty
+++ /dev/null
@@ -1,485 +0,0 @@
-% fancyhdr.sty version 3.2
-% Fancy headers and footers for LaTeX.
-% Piet van Oostrum, 
-% Dept of Computer and Information Sciences, University of Utrecht,
-% Padualaan 14, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands
-% Telephone: +31 30 2532180. Email: piet@cs.uu.nl
-% ========================================================================
-% LICENCE:
-% This file may be distributed under the terms of the LaTeX Project Public
-% License, as described in lppl.txt in the base LaTeX distribution.
-% Either version 1 or, at your option, any later version.
-% ========================================================================
-% MODIFICATION HISTORY:
-% Sep 16, 1994
-% version 1.4: Correction for use with \reversemargin
-% Sep 29, 1994:
-% version 1.5: Added the \iftopfloat, \ifbotfloat and \iffloatpage commands
-% Oct 4, 1994:
-% version 1.6: Reset single spacing in headers/footers for use with
-% setspace.sty or doublespace.sty
-% Oct 4, 1994:
-% version 1.7: changed \let\@mkboth\markboth to
-% \def\@mkboth{\protect\markboth} to make it more robust
-% Dec 5, 1994:
-% version 1.8: corrections for amsbook/amsart: define \@chapapp and (more
-% importantly) use the \chapter/sectionmark definitions from ps@headings if
-% they exist (which should be true for all standard classes).
-% May 31, 1995:
-% version 1.9: The proposed \renewcommand{\headrulewidth}{\iffloatpage...
-% construction in the doc did not work properly with the fancyplain style. 
-% June 1, 1995:
-% version 1.91: The definition of \@mkboth wasn't restored on subsequent
-% \pagestyle{fancy}'s.
-% June 1, 1995:
-% version 1.92: The sequence \pagestyle{fancyplain} \pagestyle{plain}
-% \pagestyle{fancy} would erroneously select the plain version.
-% June 1, 1995:
-% version 1.93: \fancypagestyle command added.
-% Dec 11, 1995:
-% version 1.94: suggested by Conrad Hughes <chughes@maths.tcd.ie>
-% CJCH, Dec 11, 1995: added \footruleskip to allow control over footrule
-% position (old hardcoded value of .3\normalbaselineskip is far too high
-% when used with very small footer fonts).
-% Jan 31, 1996:
-% version 1.95: call \@normalsize in the reset code if that is defined,
-% otherwise \normalsize.
-% this is to solve a problem with ucthesis.cls, as this doesn't
-% define \@currsize. Unfortunately for latex209 calling \normalsize doesn't
-% work as this is optimized to do very little, so there \@normalsize should
-% be called. Hopefully this code works for all versions of LaTeX known to
-% mankind.  
-% April 25, 1996:
-% version 1.96: initialize \headwidth to a magic (negative) value to catch
-% most common cases that people change it before calling \pagestyle{fancy}.
-% Note it can't be initialized when reading in this file, because
-% \textwidth could be changed afterwards. This is quite probable.
-% We also switch to \MakeUppercase rather than \uppercase and introduce a
-% \nouppercase command for use in headers. and footers.
-% May 3, 1996:
-% version 1.97: Two changes:
-% 1. Undo the change in version 1.8 (using the pagestyle{headings} defaults
-% for the chapter and section marks. The current version of amsbook and
-% amsart classes don't seem to need them anymore. Moreover the standard
-% latex classes don't use \markboth if twoside isn't selected, and this is
-% confusing as \leftmark doesn't work as expected.
-% 2. include a call to \ps@empty in ps@@fancy. This is to solve a problem
-% in the amsbook and amsart classes, that make global changes to \topskip,
-% which are reset in \ps@empty. Hopefully this doesn't break other things.
-% May 7, 1996:
-% version 1.98:
-% Added % after the line  \def\nouppercase
-% May 7, 1996:
-% version 1.99: This is the alpha version of fancyhdr 2.0
-% Introduced the new commands \fancyhead, \fancyfoot, and \fancyhf.
-% Changed \headrulewidth, \footrulewidth, \footruleskip to
-% macros rather than length parameters, In this way they can be
-% conditionalized and they don't consume length registers. There is no need
-% to have them as length registers unless you want to do calculations with
-% them, which is unlikely. Note that this may make some uses of them
-% incompatible (i.e. if you have a file that uses \setlength or \xxxx=)
-% May 10, 1996:
-% version 1.99a:
-% Added a few more % signs
-% May 10, 1996:
-% version 1.99b:
-% Changed the syntax of \f@nfor to be resistent to catcode changes of :=
-% Removed the [1] from the defs of \lhead etc. because the parameter is
-% consumed by the \@[xy]lhead etc. macros.
-% June 24, 1997:
-% version 1.99c:
-% corrected \nouppercase to also include the protected form of \MakeUppercase
-% \global added to manipulation of \headwidth.
-% \iffootnote command added.
-% Some comments added about \@fancyhead and \@fancyfoot.
-% Aug 24, 1998
-% version 1.99d
-% Changed the default \ps@empty to \ps@@empty in order to allow
-% \fancypagestyle{empty} redefinition.
-% Oct 11, 2000
-% version 2.0
-% Added LPPL license clause.
-%
-% A check for \headheight is added. An errormessage is given (once) if the
-% header is too large. Empty headers don't generate the error even if
-% \headheight is very small or even 0pt. 
-% Warning added for the use of 'E' option when twoside option is not used.
-% In this case the 'E' fields will never be used.
-%
-% Mar 10, 2002
-% version 2.1beta
-% New command: \fancyhfoffset[place]{length}
-% defines offsets to be applied to the header/footer to let it stick into
-% the margins (if length > 0).
-% place is like in fancyhead, except that only E,O,L,R can be used.
-% This replaces the old calculation based on \headwidth and the marginpar
-% area.
-% \headwidth will be dynamically calculated in the headers/footers when
-% this is used.
-%
-% Mar 26, 2002
-% version 2.1beta2
-% \fancyhfoffset now also takes h,f as possible letters in the argument to
-% allow the header and footer widths to be different.
-% New commands \fancyheadoffset and \fancyfootoffset added comparable to
-% \fancyhead and \fancyfoot.
-% Errormessages and warnings have been made more informative.
-%
-% Dec 9, 2002
-% version 2.1
-% The defaults for \footrulewidth, \plainheadrulewidth and
-% \plainfootrulewidth are changed from \z@skip to 0pt. In this way when
-% someone inadvertantly uses \setlength to change any of these, the value
-% of \z@skip will not be changed, rather an errormessage will be given.
-
-% March 3, 2004
-% Release of version 3.0
-
-% Oct 7, 2004
-% version 3.1
-% Added '\endlinechar=13' to \fancy@reset to prevent problems with
-% includegraphics in header when verbatiminput is active.
-
-% March 22, 2005
-% version 3.2
-% reset \everypar (the real one) in \fancy@reset because spanish.ldf does
-% strange things with \everypar between << and >>.
-
-\def\ifancy@mpty#1{\def\temp@a{#1}\ifx\temp@a\@empty}
-
-\def\fancy@def#1#2{\ifancy@mpty{#2}\fancy@gbl\def#1{\leavevmode}\else
-                                   \fancy@gbl\def#1{#2\strut}\fi}
-
-\let\fancy@gbl\global
-
-\def\@fancyerrmsg#1{%
-        \ifx\PackageError\undefined
-        \errmessage{#1}\else
-        \PackageError{Fancyhdr}{#1}{}\fi}
-\def\@fancywarning#1{%
-        \ifx\PackageWarning\undefined
-        \errmessage{#1}\else
-        \PackageWarning{Fancyhdr}{#1}{}\fi}
-
-% Usage: \@forc \var{charstring}{command to be executed for each char}
-% This is similar to LaTeX's \@tfor, but expands the charstring.
-
-\def\@forc#1#2#3{\expandafter\f@rc\expandafter#1\expandafter{#2}{#3}}
-\def\f@rc#1#2#3{\def\temp@ty{#2}\ifx\@empty\temp@ty\else
-                                    \f@@rc#1#2\f@@rc{#3}\fi}
-\def\f@@rc#1#2#3\f@@rc#4{\def#1{#2}#4\f@rc#1{#3}{#4}}
-
-% Usage: \f@nfor\name:=list\do{body}
-% Like LaTeX's \@for but an empty list is treated as a list with an empty
-% element
-
-\newcommand{\f@nfor}[3]{\edef\@fortmp{#2}%
-    \expandafter\@forloop#2,\@nil,\@nil\@@#1{#3}}
-
-% Usage: \def@ult \cs{defaults}{argument}
-% sets \cs to the characters from defaults appearing in argument
-% or defaults if it would be empty. All characters are lowercased.
-
-\newcommand\def@ult[3]{%
-    \edef\temp@a{\lowercase{\edef\noexpand\temp@a{#3}}}\temp@a
-    \def#1{}%
-    \@forc\tmpf@ra{#2}%
-        {\expandafter\if@in\tmpf@ra\temp@a{\edef#1{#1\tmpf@ra}}{}}%
-    \ifx\@empty#1\def#1{#2}\fi}
-% 
-% \if@in <char><set><truecase><falsecase>
-%
-\newcommand{\if@in}[4]{%
-    \edef\temp@a{#2}\def\temp@b##1#1##2\temp@b{\def\temp@b{##1}}%
-    \expandafter\temp@b#2#1\temp@b\ifx\temp@a\temp@b #4\else #3\fi}
-
-\newcommand{\fancyhead}{\@ifnextchar[{\f@ncyhf\fancyhead h}%
-                                     {\f@ncyhf\fancyhead h[]}}
-\newcommand{\fancyfoot}{\@ifnextchar[{\f@ncyhf\fancyfoot f}%
-                                     {\f@ncyhf\fancyfoot f[]}}
-\newcommand{\fancyhf}{\@ifnextchar[{\f@ncyhf\fancyhf{}}%
-                                   {\f@ncyhf\fancyhf{}[]}}
-
-% New commands for offsets added
-
-\newcommand{\fancyheadoffset}{\@ifnextchar[{\f@ncyhfoffs\fancyheadoffset h}%
-                                           {\f@ncyhfoffs\fancyheadoffset h[]}}
-\newcommand{\fancyfootoffset}{\@ifnextchar[{\f@ncyhfoffs\fancyfootoffset f}%
-                                           {\f@ncyhfoffs\fancyfootoffset f[]}}
-\newcommand{\fancyhfoffset}{\@ifnextchar[{\f@ncyhfoffs\fancyhfoffset{}}%
-                                         {\f@ncyhfoffs\fancyhfoffset{}[]}}
-
-% The header and footer fields are stored in command sequences with
-% names of the form: \f@ncy<x><y><z> with <x> for [eo], <y> from [lcr]
-% and <z> from [hf].
-
-\def\f@ncyhf#1#2[#3]#4{%
-    \def\temp@c{}%
-    \@forc\tmpf@ra{#3}%
-        {\expandafter\if@in\tmpf@ra{eolcrhf,EOLCRHF}%
-            {}{\edef\temp@c{\temp@c\tmpf@ra}}}%
-    \ifx\@empty\temp@c\else
-        \@fancyerrmsg{Illegal char `\temp@c' in \string#1 argument:
-          [#3]}%
-    \fi
-    \f@nfor\temp@c{#3}%
-        {\def@ult\f@@@eo{eo}\temp@c
-         \if@twoside\else
-           \if\f@@@eo e\@fancywarning
-             {\string#1's `E' option without twoside option is useless}\fi\fi
-         \def@ult\f@@@lcr{lcr}\temp@c
-         \def@ult\f@@@hf{hf}{#2\temp@c}%
-         \@forc\f@@eo\f@@@eo
-             {\@forc\f@@lcr\f@@@lcr
-                 {\@forc\f@@hf\f@@@hf
-                     {\expandafter\fancy@def\csname
-                      f@ncy\f@@eo\f@@lcr\f@@hf\endcsname
-                      {#4}}}}}}
-
-\def\f@ncyhfoffs#1#2[#3]#4{%
-    \def\temp@c{}%
-    \@forc\tmpf@ra{#3}%
-        {\expandafter\if@in\tmpf@ra{eolrhf,EOLRHF}%
-            {}{\edef\temp@c{\temp@c\tmpf@ra}}}%
-    \ifx\@empty\temp@c\else
-        \@fancyerrmsg{Illegal char `\temp@c' in \string#1 argument:
-          [#3]}%
-    \fi
-    \f@nfor\temp@c{#3}%
-        {\def@ult\f@@@eo{eo}\temp@c
-         \if@twoside\else
-           \if\f@@@eo e\@fancywarning
-             {\string#1's `E' option without twoside option is useless}\fi\fi
-         \def@ult\f@@@lcr{lr}\temp@c
-         \def@ult\f@@@hf{hf}{#2\temp@c}%
-         \@forc\f@@eo\f@@@eo
-             {\@forc\f@@lcr\f@@@lcr
-                 {\@forc\f@@hf\f@@@hf
-                     {\expandafter\setlength\csname
-                      f@ncyO@\f@@eo\f@@lcr\f@@hf\endcsname
-                      {#4}}}}}%
-     \fancy@setoffs}
-
-% Fancyheadings version 1 commands. These are more or less deprecated,
-% but they continue to work.
-
-\newcommand{\lhead}{\@ifnextchar[{\@xlhead}{\@ylhead}}
-\def\@xlhead[#1]#2{\fancy@def\f@ncyelh{#1}\fancy@def\f@ncyolh{#2}}
-\def\@ylhead#1{\fancy@def\f@ncyelh{#1}\fancy@def\f@ncyolh{#1}}
-
-\newcommand{\chead}{\@ifnextchar[{\@xchead}{\@ychead}}
-\def\@xchead[#1]#2{\fancy@def\f@ncyech{#1}\fancy@def\f@ncyoch{#2}}
-\def\@ychead#1{\fancy@def\f@ncyech{#1}\fancy@def\f@ncyoch{#1}}
-
-\newcommand{\rhead}{\@ifnextchar[{\@xrhead}{\@yrhead}}
-\def\@xrhead[#1]#2{\fancy@def\f@ncyerh{#1}\fancy@def\f@ncyorh{#2}}
-\def\@yrhead#1{\fancy@def\f@ncyerh{#1}\fancy@def\f@ncyorh{#1}}
-
-\newcommand{\lfoot}{\@ifnextchar[{\@xlfoot}{\@ylfoot}}
-\def\@xlfoot[#1]#2{\fancy@def\f@ncyelf{#1}\fancy@def\f@ncyolf{#2}}
-\def\@ylfoot#1{\fancy@def\f@ncyelf{#1}\fancy@def\f@ncyolf{#1}}
-
-\newcommand{\cfoot}{\@ifnextchar[{\@xcfoot}{\@ycfoot}}
-\def\@xcfoot[#1]#2{\fancy@def\f@ncyecf{#1}\fancy@def\f@ncyocf{#2}}
-\def\@ycfoot#1{\fancy@def\f@ncyecf{#1}\fancy@def\f@ncyocf{#1}}
-
-\newcommand{\rfoot}{\@ifnextchar[{\@xrfoot}{\@yrfoot}}
-\def\@xrfoot[#1]#2{\fancy@def\f@ncyerf{#1}\fancy@def\f@ncyorf{#2}}
-\def\@yrfoot#1{\fancy@def\f@ncyerf{#1}\fancy@def\f@ncyorf{#1}}
-
-\newlength{\fancy@headwidth}
-\let\headwidth\fancy@headwidth
-\newlength{\f@ncyO@elh}
-\newlength{\f@ncyO@erh}
-\newlength{\f@ncyO@olh}
-\newlength{\f@ncyO@orh}
-\newlength{\f@ncyO@elf}
-\newlength{\f@ncyO@erf}
-\newlength{\f@ncyO@olf}
-\newlength{\f@ncyO@orf}
-\newcommand{\headrulewidth}{0.4pt}
-\newcommand{\footrulewidth}{0pt}
-\newcommand{\footruleskip}{.3\normalbaselineskip}
-
-% Fancyplain stuff shouldn't be used anymore (rather
-% \fancypagestyle{plain} should be used), but it must be present for
-% compatibility reasons.
-
-\newcommand{\plainheadrulewidth}{0pt}
-\newcommand{\plainfootrulewidth}{0pt}
-\newif\if@fancyplain \@fancyplainfalse
-\def\fancyplain#1#2{\if@fancyplain#1\else#2\fi}
-
-\headwidth=-123456789sp %magic constant
-
-% Command to reset various things in the headers:
-% a.o.  single spacing (taken from setspace.sty)
-% and the catcode of ^^M (so that epsf files in the header work if a
-% verbatim crosses a page boundary)
-% It also defines a \nouppercase command that disables \uppercase and
-% \Makeuppercase. It can only be used in the headers and footers.
-\let\fnch@everypar\everypar% save real \everypar because of spanish.ldf
-\def\fancy@reset{\fnch@everypar{}\restorecr\endlinechar=13
- \def\baselinestretch{1}%
- \def\nouppercase##1{{\let\uppercase\relax\let\MakeUppercase\relax
-     \expandafter\let\csname MakeUppercase \endcsname\relax##1}}%
- \ifx\undefined\@newbaseline% NFSS not present; 2.09 or 2e
-   \ifx\@normalsize\undefined \normalsize % for ucthesis.cls
-   \else \@normalsize \fi
- \else% NFSS (2.09) present
-  \@newbaseline%
- \fi}
-
-% Initialization of the head and foot text.
-
-% The default values still contain \fancyplain for compatibility.
-\fancyhf{} % clear all
-% lefthead empty on ``plain'' pages, \rightmark on even, \leftmark on odd pages
-% evenhead empty on ``plain'' pages, \leftmark on even, \rightmark on odd pages
-\if@twoside
-  \fancyhead[el,or]{\fancyplain{}{\sl\rightmark}}
-  \fancyhead[er,ol]{\fancyplain{}{\sl\leftmark}}
-\else
-  \fancyhead[l]{\fancyplain{}{\sl\rightmark}}
-  \fancyhead[r]{\fancyplain{}{\sl\leftmark}}
-\fi
-\fancyfoot[c]{\rm\thepage} % page number
-
-% Use box 0 as a temp box and dimen 0 as temp dimen. 
-% This can be done, because this code will always
-% be used inside another box, and therefore the changes are local.
-
-\def\@fancyvbox#1#2{\setbox0\vbox{#2}\ifdim\ht0>#1\@fancywarning
-  {\string#1 is too small (\the#1): ^^J Make it at least \the\ht0.^^J
-    We now make it that large for the rest of the document.^^J
-    This may cause the page layout to be inconsistent, however\@gobble}%
-  \dimen0=#1\global\setlength{#1}{\ht0}\ht0=\dimen0\fi
-  \box0}
-
-% Put together a header or footer given the left, center and
-% right text, fillers at left and right and a rule.
-% The \lap commands put the text into an hbox of zero size,
-% so overlapping text does not generate an errormessage.
-% These macros have 5 parameters:
-% 1. LEFTSIDE BEARING % This determines at which side the header will stick
-%    out. When \fancyhfoffset is used this calculates \headwidth, otherwise
-%    it is \hss or \relax (after expansion).
-% 2. \f@ncyolh, \f@ncyelh, \f@ncyolf or \f@ncyelf. This is the left component.
-% 3. \f@ncyoch, \f@ncyech, \f@ncyocf or \f@ncyecf. This is the middle comp.
-% 4. \f@ncyorh, \f@ncyerh, \f@ncyorf or \f@ncyerf. This is the right component.
-% 5. RIGHTSIDE BEARING. This is always \relax or \hss (after expansion).
-
-\def\@fancyhead#1#2#3#4#5{#1\hbox to\headwidth{\fancy@reset
-  \@fancyvbox\headheight{\hbox
-    {\rlap{\parbox[b]{\headwidth}{\raggedright#2}}\hfill
-      \parbox[b]{\headwidth}{\centering#3}\hfill
-      \llap{\parbox[b]{\headwidth}{\raggedleft#4}}}\headrule}}#5}
-
-\def\@fancyfoot#1#2#3#4#5{#1\hbox to\headwidth{\fancy@reset
-    \@fancyvbox\footskip{\footrule
-      \hbox{\rlap{\parbox[t]{\headwidth}{\raggedright#2}}\hfill
-        \parbox[t]{\headwidth}{\centering#3}\hfill
-        \llap{\parbox[t]{\headwidth}{\raggedleft#4}}}}}#5}
-
-\def\headrule{{\if@fancyplain\let\headrulewidth\plainheadrulewidth\fi
-    \hrule\@height\headrulewidth\@width\headwidth \vskip-\headrulewidth}}
-
-\def\footrule{{\if@fancyplain\let\footrulewidth\plainfootrulewidth\fi
-    \vskip-\footruleskip\vskip-\footrulewidth
-    \hrule\@width\headwidth\@height\footrulewidth\vskip\footruleskip}}
-
-\def\ps@fancy{%
-\@ifundefined{@chapapp}{\let\@chapapp\chaptername}{}%for amsbook
-%
-% Define \MakeUppercase for old LaTeXen.
-% Note: we used \def rather than \let, so that \let\uppercase\relax (from
-% the version 1 documentation) will still work.
-%
-\@ifundefined{MakeUppercase}{\def\MakeUppercase{\uppercase}}{}%
-\@ifundefined{chapter}{\def\sectionmark##1{\markboth
-{\MakeUppercase{\ifnum \c@secnumdepth>\z@
- \thesection\hskip 1em\relax \fi ##1}}{}}%
-\def\subsectionmark##1{\markright {\ifnum \c@secnumdepth >\@ne
- \thesubsection\hskip 1em\relax \fi ##1}}}%
-{\def\chaptermark##1{\markboth {\MakeUppercase{\ifnum \c@secnumdepth>\m@ne
- \@chapapp\ \thechapter. \ \fi ##1}}{}}%
-\def\sectionmark##1{\markright{\MakeUppercase{\ifnum \c@secnumdepth >\z@
- \thesection. \ \fi ##1}}}}%
-%\csname ps@headings\endcsname % use \ps@headings defaults if they exist
-\ps@@fancy
-\gdef\ps@fancy{\@fancyplainfalse\ps@@fancy}%
-% Initialize \headwidth if the user didn't
-%
-\ifdim\headwidth<0sp
-%
-% This catches the case that \headwidth hasn't been initialized and the
-% case that the user added something to \headwidth in the expectation that
-% it was initialized to \textwidth. We compensate this now. This loses if
-% the user intended to multiply it by a factor. But that case is more
-% likely done by saying something like \headwidth=1.2\textwidth. 
-% The doc says you have to change \headwidth after the first call to
-% \pagestyle{fancy}. This code is just to catch the most common cases were
-% that requirement is violated.
-%
-    \global\advance\headwidth123456789sp\global\advance\headwidth\textwidth
-\fi}
-\def\ps@fancyplain{\ps@fancy \let\ps@plain\ps@plain@fancy}
-\def\ps@plain@fancy{\@fancyplaintrue\ps@@fancy}
-\let\ps@@empty\ps@empty
-\def\ps@@fancy{%
-\ps@@empty % This is for amsbook/amsart, which do strange things with \topskip
-\def\@mkboth{\protect\markboth}%
-\def\@oddhead{\@fancyhead\fancy@Oolh\f@ncyolh\f@ncyoch\f@ncyorh\fancy@Oorh}%
-\def\@oddfoot{\@fancyfoot\fancy@Oolf\f@ncyolf\f@ncyocf\f@ncyorf\fancy@Oorf}%
-\def\@evenhead{\@fancyhead\fancy@Oelh\f@ncyelh\f@ncyech\f@ncyerh\fancy@Oerh}%
-\def\@evenfoot{\@fancyfoot\fancy@Oelf\f@ncyelf\f@ncyecf\f@ncyerf\fancy@Oerf}%
-}
-% Default definitions for compatibility mode:
-% These cause the header/footer to take the defined \headwidth as width
-% And to shift in the direction of the marginpar area
-
-\def\fancy@Oolh{\if@reversemargin\hss\else\relax\fi}
-\def\fancy@Oorh{\if@reversemargin\relax\else\hss\fi}
-\let\fancy@Oelh\fancy@Oorh
-\let\fancy@Oerh\fancy@Oolh
-
-\let\fancy@Oolf\fancy@Oolh
-\let\fancy@Oorf\fancy@Oorh
-\let\fancy@Oelf\fancy@Oelh
-\let\fancy@Oerf\fancy@Oerh
-
-% New definitions for the use of \fancyhfoffset
-% These calculate the \headwidth from \textwidth and the specified offsets.
-
-\def\fancy@offsolh{\headwidth=\textwidth\advance\headwidth\f@ncyO@olh
-                   \advance\headwidth\f@ncyO@orh\hskip-\f@ncyO@olh}
-\def\fancy@offselh{\headwidth=\textwidth\advance\headwidth\f@ncyO@elh
-                   \advance\headwidth\f@ncyO@erh\hskip-\f@ncyO@elh}
-
-\def\fancy@offsolf{\headwidth=\textwidth\advance\headwidth\f@ncyO@olf
-                   \advance\headwidth\f@ncyO@orf\hskip-\f@ncyO@olf}
-\def\fancy@offself{\headwidth=\textwidth\advance\headwidth\f@ncyO@elf
-                   \advance\headwidth\f@ncyO@erf\hskip-\f@ncyO@elf}
-
-\def\fancy@setoffs{%
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deleted file mode 100644
index 485619b..0000000
--- a/docs/cohen_et_al_2018/iclr2018_conference.sty
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-%%%% ICLR Macros (LaTex)
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diff --git a/docs/cohen_et_al_2018/natbib.sty b/docs/cohen_et_al_2018/natbib.sty
deleted file mode 100644
index ff0d0b9..0000000
--- a/docs/cohen_et_al_2018/natbib.sty
+++ /dev/null
@@ -1,1246 +0,0 @@
-%%
-%% This is file `natbib.sty',
-%% generated with the docstrip utility.
-%%
-%% The original source files were:
-%%
-%% natbib.dtx  (with options: `package,all')
-%% =============================================
-%% IMPORTANT NOTICE:
-%% 
-%% This program can be redistributed and/or modified under the terms
-%% of the LaTeX Project Public License Distributed from CTAN
-%% archives in directory macros/latex/base/lppl.txt; either
-%% version 1 of the License, or any later version.
-%% 
-%% This is a generated file.
-%% It may not be distributed without the original source file natbib.dtx.
-%% 
-%% Full documentation can be obtained by LaTeXing that original file.
-%% Only a few abbreviated comments remain here to describe the usage.
-%% =============================================
-%% Copyright 1993-2009 Patrick W Daly
-%% Max-Planck-Institut f\"ur Sonnensystemforschung
-%% Max-Planck-Str. 2
-%% D-37191 Katlenburg-Lindau
-%% Germany
-%% E-mail: daly@mps.mpg.de
-\NeedsTeXFormat{LaTeX2e}[1995/06/01]
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- %   \bibitem[\protect\citeauthoryear{Jones, Baker, and Williams}{Jones
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- %   \bibitem[\protect\citeauthoryear{Jones et al.}{1990}]{key}...
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- %   \bibitem[\protect\citename{Jones et al., }1990]{key}...
- %   \harvarditem[Jones et al.]{Jones, Baker, and Williams}{1990}{key}...
- %
- % This is either to be made up manually, or to be generated by an
- % appropriate .bst file with BibTeX.
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- % Then, \citet{key}  ==>>  Jones et al. (1990)    ||   Jones et al. [21]
- %       \citep{key}  ==>> (Jones et al., 1990)    ||   [21]
- % Multiple citations as normal:
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- %                           or  (Jones et al., 1990, 1991)  || [21,24]
- %                           or  (Jones et al., 1990a,b)     || [21,24]
- % \cite{key} is the equivalent of \citet{key} in author-year mode
- %                         and  of \citep{key} in numerical mode
- % Full author lists may be forced with \citet* or \citep*, e.g.
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- % Optional notes as:
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- %   \citep[e.g.,][]{key}    ==>> (e.g., Jones et al., 1990)
- %   \citep[see][pg. 34]{key}==>> (see Jones et al., 1990, pg. 34)
- %  (Note: in standard LaTeX, only one note is allowed, after the ref.
- %   Here, one note is like the standard, two make pre- and post-notes.)
- %   \citealt{key}          ==>> Jones et al. 1990
- %   \citealt*{key}         ==>> Jones, Baker, and Williams 1990
- %   \citealp{key}          ==>> Jones et al., 1990
- %   \citealp*{key}         ==>> Jones, Baker, and Williams, 1990
- % Additional citation possibilities (both author-year and numerical modes)
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- %   \citeauthor*{key}      ==>> Jones, Baker, and Williams
- %   \citeyear{key}         ==>> 1990
- %   \citeyearpar{key}      ==>> (1990)
- %   \citetext{priv. comm.} ==>> (priv. comm.)
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- %   \Citeauthor{dRob98}    ==>> Della Robbia
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- %
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-  {The nlinproc class already includes natbib coding,\MessageBreak
-   so you should not add it explicitly}
-  {Type <Return> for now, but then later remove\MessageBreak
-   the command \protect\usepackage{natbib} from the document}
-  \endinput}{}
-\@ifclassloaded{egs}{\PackageError{natbib}
-  {The egs class already includes natbib coding,\MessageBreak
-   so you should not add it explicitly}
-  {Type <Return> for now, but then later remove\MessageBreak
-   the command \protect\usepackage{natbib} from the document}
-  \endinput}{}
-\@ifclassloaded{egu}{\PackageError{natbib}
-  {The egu class already includes natbib coding,\MessageBreak
-   so you should not add it explicitly}
-  {Type <Return> for now, but then later remove\MessageBreak
-   the command \protect\usepackage{natbib} from the document}
-  \endinput}{}
- % Define citation punctuation for some author-year styles
- % One may add and delete at this point
- % Or put additions into local configuration file natbib.cfg
-\newcommand\bibstyle@chicago{\bibpunct{(}{)}{;}{a}{,}{,}}
-\newcommand\bibstyle@named{\bibpunct{[}{]}{;}{a}{,}{,}}
-\newcommand\bibstyle@agu{\bibpunct{[}{]}{;}{a}{,}{,~}}%Amer. Geophys. Union
-\newcommand\bibstyle@copernicus{\bibpunct{(}{)}{;}{a}{,}{,}}%Copernicus Publications
-\let\bibstyle@egu=\bibstyle@copernicus
-\let\bibstyle@egs=\bibstyle@copernicus
-\newcommand\bibstyle@agsm{\bibpunct{(}{)}{,}{a}{}{,}\gdef\harvardand{\&}}
-\newcommand\bibstyle@kluwer{\bibpunct{(}{)}{,}{a}{}{,}\gdef\harvardand{\&}}
-\newcommand\bibstyle@dcu{\bibpunct{(}{)}{;}{a}{;}{,}\gdef\harvardand{and}}
-\newcommand\bibstyle@aa{\bibpunct{(}{)}{;}{a}{}{,}} %Astronomy & Astrophysics
-\newcommand\bibstyle@pass{\bibpunct{(}{)}{;}{a}{,}{,}}%Planet. & Space Sci
-\newcommand\bibstyle@anngeo{\bibpunct{(}{)}{;}{a}{,}{,}}%Annales Geophysicae
-\newcommand\bibstyle@nlinproc{\bibpunct{(}{)}{;}{a}{,}{,}}%Nonlin.Proc.Geophys.
- % Define citation punctuation for some numerical styles
-\newcommand\bibstyle@cospar{\bibpunct{/}{/}{,}{n}{}{}%
-     \gdef\bibnumfmt##1{##1.}}
-\newcommand\bibstyle@esa{\bibpunct{(Ref.~}{)}{,}{n}{}{}%
-     \gdef\bibnumfmt##1{##1.\hspace{1em}}}
-\newcommand\bibstyle@nature{\bibpunct{}{}{,}{s}{}{\textsuperscript{,}}%
-     \gdef\bibnumfmt##1{##1.}}
- % The standard LaTeX styles
-\newcommand\bibstyle@plain{\bibpunct{[}{]}{,}{n}{}{,}}
-\let\bibstyle@alpha=\bibstyle@plain
-\let\bibstyle@abbrv=\bibstyle@plain
-\let\bibstyle@unsrt=\bibstyle@plain
- % The author-year modifications of the standard styles
-\newcommand\bibstyle@plainnat{\bibpunct{[}{]}{,}{a}{,}{,}}
-\let\bibstyle@abbrvnat=\bibstyle@plainnat
-\let\bibstyle@unsrtnat=\bibstyle@plainnat
-\newif\ifNAT@numbers \NAT@numbersfalse
-\newif\ifNAT@super \NAT@superfalse
-\let\NAT@merge\z@
-\DeclareOption{numbers}{\NAT@numberstrue
-   \ExecuteOptions{square,comma,nobibstyle}}
-\DeclareOption{super}{\NAT@supertrue\NAT@numberstrue
-   \renewcommand\NAT@open{}\renewcommand\NAT@close{}
-   \ExecuteOptions{nobibstyle}}
-\DeclareOption{authoryear}{\NAT@numbersfalse
-   \ExecuteOptions{round,semicolon,bibstyle}}
-\DeclareOption{round}{%
-      \renewcommand\NAT@open{(} \renewcommand\NAT@close{)}
-   \ExecuteOptions{nobibstyle}}
-\DeclareOption{square}{%
-      \renewcommand\NAT@open{[} \renewcommand\NAT@close{]}
-   \ExecuteOptions{nobibstyle}}
-\DeclareOption{angle}{%
-      \renewcommand\NAT@open{$<$} \renewcommand\NAT@close{$>$}
-   \ExecuteOptions{nobibstyle}}
-\DeclareOption{curly}{%
-      \renewcommand\NAT@open{\{} \renewcommand\NAT@close{\}}
-   \ExecuteOptions{nobibstyle}}
-\DeclareOption{comma}{\renewcommand\NAT@sep{,}
-   \ExecuteOptions{nobibstyle}}
-\DeclareOption{semicolon}{\renewcommand\NAT@sep{;}
-   \ExecuteOptions{nobibstyle}}
-\DeclareOption{colon}{\ExecuteOptions{semicolon}}
-\DeclareOption{nobibstyle}{\let\bibstyle=\@gobble}
-\DeclareOption{bibstyle}{\let\bibstyle=\@citestyle}
-\newif\ifNAT@openbib \NAT@openbibfalse
-\DeclareOption{openbib}{\NAT@openbibtrue}
-\DeclareOption{sectionbib}{\def\NAT@sectionbib{on}}
-\def\NAT@sort{\z@}
-\def\NAT@cmprs{\z@}
-\DeclareOption{sort}{\def\NAT@sort{\@ne}}
-\DeclareOption{compress}{\def\NAT@cmprs{\@ne}}
-\DeclareOption{sort&compress}{\def\NAT@sort{\@ne}\def\NAT@cmprs{\@ne}}
-\DeclareOption{mcite}{\let\NAT@merge\@ne}
-\DeclareOption{merge}{\@ifnum{\NAT@merge<\tw@}{\let\NAT@merge\tw@}{}}
-\DeclareOption{elide}{\@ifnum{\NAT@merge<\thr@@}{\let\NAT@merge\thr@@}{}}
-\@ifpackageloaded{cite}{\PackageWarningNoLine{natbib}
-  {The `cite' package should not be used\MessageBreak
-   with natbib. Use option `sort' instead}\ExecuteOptions{sort}}{}
-\@ifpackageloaded{mcite}{\PackageWarningNoLine{natbib}
-  {The `mcite' package should not be used\MessageBreak
-   with natbib. Use option `merge' instead}\ExecuteOptions{merge}}{}
-\@ifpackageloaded{citeref}{\PackageError{natbib}
-  {The `citeref' package must be loaded after natbib}%
-  {Move \protect\usepackage{citeref} to after \string\usepackage{natbib}}}{}
-\newif\ifNAT@longnames\NAT@longnamesfalse
-\DeclareOption{longnamesfirst}{\NAT@longnamestrue}
-\DeclareOption{nonamebreak}{\def\NAT@nmfmt#1{\mbox{\NAT@up#1}}}
-\def\NAT@nmfmt#1{{\NAT@up#1}}
-\renewcommand\bibstyle[1]{\csname bibstyle@#1\endcsname}
-\AtBeginDocument{\global\let\bibstyle=\@gobble}
-\let\@citestyle\bibstyle
-\newcommand\citestyle[1]{\@citestyle{#1}\let\bibstyle\@gobble}
-\newcommand\bibpunct[7][, ]%
-  {\gdef\NAT@open{#2}\gdef\NAT@close{#3}\gdef
-   \NAT@sep{#4}\global\NAT@numbersfalse
-     \ifx #5n\global\NAT@numberstrue\global\NAT@superfalse
-   \else
-     \ifx #5s\global\NAT@numberstrue\global\NAT@supertrue
-   \fi\fi
-   \gdef\NAT@aysep{#6}\gdef\NAT@yrsep{#7}%
-   \gdef\NAT@cmt{#1}%
-   \NAT@@setcites
-  }
-\newcommand\setcitestyle[1]{
- \@for\@tempa:=#1\do
- {\def\@tempb{round}\ifx\@tempa\@tempb
-    \renewcommand\NAT@open{(}\renewcommand\NAT@close{)}\fi
-  \def\@tempb{square}\ifx\@tempa\@tempb
-    \renewcommand\NAT@open{[}\renewcommand\NAT@close{]}\fi
-  \def\@tempb{angle}\ifx\@tempa\@tempb
-    \renewcommand\NAT@open{$<$}\renewcommand\NAT@close{$>$}\fi
-  \def\@tempb{curly}\ifx\@tempa\@tempb
-    \renewcommand\NAT@open{\{}\renewcommand\NAT@close{\}}\fi
-  \def\@tempb{semicolon}\ifx\@tempa\@tempb
-    \renewcommand\NAT@sep{;}\fi
-  \def\@tempb{colon}\ifx\@tempa\@tempb
-    \renewcommand\NAT@sep{;}\fi
-  \def\@tempb{comma}\ifx\@tempa\@tempb
-    \renewcommand\NAT@sep{,}\fi
-  \def\@tempb{authoryear}\ifx\@tempa\@tempb
-    \NAT@numbersfalse\fi
-  \def\@tempb{numbers}\ifx\@tempa\@tempb
-    \NAT@numberstrue\NAT@superfalse\fi
-  \def\@tempb{super}\ifx\@tempa\@tempb
-    \NAT@numberstrue\NAT@supertrue\fi
-  \expandafter\NAT@find@eq\@tempa=\relax\@nil
-  \if\@tempc\relax\else
-    \expandafter\NAT@rem@eq\@tempc
-    \def\@tempb{open}\ifx\@tempa\@tempb
-     \xdef\NAT@open{\@tempc}\fi
-    \def\@tempb{close}\ifx\@tempa\@tempb
-     \xdef\NAT@close{\@tempc}\fi
-    \def\@tempb{aysep}\ifx\@tempa\@tempb
-     \xdef\NAT@aysep{\@tempc}\fi
-    \def\@tempb{yysep}\ifx\@tempa\@tempb
-     \xdef\NAT@yrsep{\@tempc}\fi
-    \def\@tempb{notesep}\ifx\@tempa\@tempb
-     \xdef\NAT@cmt{\@tempc}\fi
-    \def\@tempb{citesep}\ifx\@tempa\@tempb
-     \xdef\NAT@sep{\@tempc}\fi
-  \fi
- }%
- \NAT@@setcites
-}
- \def\NAT@find@eq#1=#2\@nil{\def\@tempa{#1}\def\@tempc{#2}}
- \def\NAT@rem@eq#1={\def\@tempc{#1}}
- \def\NAT@@setcites{\global\let\bibstyle\@gobble}
-\AtBeginDocument{\let\NAT@@setcites\NAT@set@cites}
-\newcommand\NAT@open{(} \newcommand\NAT@close{)}
-\newcommand\NAT@sep{;}
-\ProcessOptions
-\newcommand\NAT@aysep{,} \newcommand\NAT@yrsep{,}
-\newcommand\NAT@cmt{, }
-\newcommand\NAT@cite%
-    [3]{\ifNAT@swa\NAT@@open\if*#2*\else#2\NAT@spacechar\fi
-        #1\if*#3*\else\NAT@cmt#3\fi\NAT@@close\else#1\fi\endgroup}
-\newcommand\NAT@citenum%
-    [3]{\ifNAT@swa\NAT@@open\if*#2*\else#2\NAT@spacechar\fi
-        #1\if*#3*\else\NAT@cmt#3\fi\NAT@@close\else#1\fi\endgroup}
-\newcommand\NAT@citesuper[3]{\ifNAT@swa
-\if*#2*\else#2\NAT@spacechar\fi
-\unskip\kern\p@\textsuperscript{\NAT@@open#1\NAT@@close}%
-   \if*#3*\else\NAT@spacechar#3\fi\else #1\fi\endgroup}
-\providecommand\textsuperscript[1]{\mbox{$^{\mbox{\scriptsize#1}}$}}
-\begingroup \catcode`\_=8
-\gdef\NAT@ifcat@num#1{%
- \ifcat_\ifnum\z@<0#1_\else A\fi
-  \expandafter\@firstoftwo
- \else
-  \expandafter\@secondoftwo
- \fi
-}%
-\endgroup
-\providecommand\@firstofone[1]{#1}
-\newcommand\NAT@citexnum{}
-\def\NAT@citexnum[#1][#2]#3{%
-  \NAT@reset@parser
-  \NAT@sort@cites{#3}%
-  \NAT@reset@citea
-  \@cite{\def\NAT@num{-1}\let\NAT@last@yr\relax\let\NAT@nm\@empty
-    \@for\@citeb:=\NAT@cite@list\do
-    {\@safe@activestrue
-     \edef\@citeb{\expandafter\@firstofone\@citeb\@empty}%
-     \@safe@activesfalse
-     \@ifundefined{b@\@citeb\@extra@b@citeb}{%
-       {\reset@font\bfseries?}
-        \NAT@citeundefined\PackageWarning{natbib}%
-       {Citation `\@citeb' on page \thepage \space undefined}}%
-     {\let\NAT@last@num\NAT@num\let\NAT@last@nm\NAT@nm
-      \NAT@parse{\@citeb}%
-      \ifNAT@longnames\@ifundefined{bv@\@citeb\@extra@b@citeb}{%
-        \let\NAT@name=\NAT@all@names
-        \global\@namedef{bv@\@citeb\@extra@b@citeb}{}}{}%
-      \fi
-      \ifNAT@full\let\NAT@nm\NAT@all@names\else
-        \let\NAT@nm\NAT@name\fi
-      \ifNAT@swa
-       \@ifnum{\NAT@ctype>\@ne}{%
-        \@citea
-        \NAT@hyper@{\@ifnum{\NAT@ctype=\tw@}{\NAT@test{\NAT@ctype}}{\NAT@alias}}%
-       }{%
-        \@ifnum{\NAT@cmprs>\z@}{%
-         \NAT@ifcat@num\NAT@num
-          {\let\NAT@nm=\NAT@num}%
-          {\def\NAT@nm{-2}}%
-         \NAT@ifcat@num\NAT@last@num
-          {\@tempcnta=\NAT@last@num\relax}%
-          {\@tempcnta\m@ne}%
-         \@ifnum{\NAT@nm=\@tempcnta}{%
-          \@ifnum{\NAT@merge>\@ne}{}{\NAT@last@yr@mbox}%
-         }{%
-           \advance\@tempcnta by\@ne
-           \@ifnum{\NAT@nm=\@tempcnta}{%
-             \ifx\NAT@last@yr\relax
-               \def@NAT@last@yr{\@citea}%
-             \else
-               \def@NAT@last@yr{--\NAT@penalty}%
-             \fi
-           }{%
-             \NAT@last@yr@mbox
-           }%
-         }%
-        }{%
-         \@tempswatrue
-         \@ifnum{\NAT@merge>\@ne}{\@ifnum{\NAT@last@num=\NAT@num\relax}{\@tempswafalse}{}}{}%
-         \if@tempswa\NAT@citea@mbox\fi
-        }%
-       }%
-       \NAT@def@citea
-      \else
-        \ifcase\NAT@ctype
-          \ifx\NAT@last@nm\NAT@nm \NAT@yrsep\NAT@penalty\NAT@space\else
-            \@citea \NAT@test{\@ne}\NAT@spacechar\NAT@mbox{\NAT@super@kern\NAT@@open}%
-          \fi
-          \if*#1*\else#1\NAT@spacechar\fi
-          \NAT@mbox{\NAT@hyper@{{\citenumfont{\NAT@num}}}}%
-          \NAT@def@citea@box
-        \or
-          \NAT@hyper@citea@space{\NAT@test{\NAT@ctype}}%
-        \or
-          \NAT@hyper@citea@space{\NAT@test{\NAT@ctype}}%
-        \or
-          \NAT@hyper@citea@space\NAT@alias
-        \fi
-      \fi
-     }%
-    }%
-      \@ifnum{\NAT@cmprs>\z@}{\NAT@last@yr}{}%
-      \ifNAT@swa\else
-        \@ifnum{\NAT@ctype=\z@}{%
-          \if*#2*\else\NAT@cmt#2\fi
-        }{}%
-        \NAT@mbox{\NAT@@close}%
-      \fi
-  }{#1}{#2}%
-}%
-\def\NAT@citea@mbox{%
- \@citea\mbox{\NAT@hyper@{{\citenumfont{\NAT@num}}}}%
-}%
-\def\NAT@hyper@#1{%
- \hyper@natlinkstart{\@citeb\@extra@b@citeb}#1\hyper@natlinkend
-}%
-\def\NAT@hyper@citea#1{%
- \@citea
- \NAT@hyper@{#1}%
- \NAT@def@citea
-}%
-\def\NAT@hyper@citea@space#1{%
- \@citea
- \NAT@hyper@{#1}%
- \NAT@def@citea@space
-}%
-\def\def@NAT@last@yr#1{%
- \protected@edef\NAT@last@yr{%
-  #1%
-  \noexpand\mbox{%
-   \noexpand\hyper@natlinkstart{\@citeb\@extra@b@citeb}%
-   {\noexpand\citenumfont{\NAT@num}}%
-   \noexpand\hyper@natlinkend
-  }%
- }%
-}%
-\def\NAT@last@yr@mbox{%
- \NAT@last@yr\let\NAT@last@yr\relax
- \NAT@citea@mbox
-}%
-\newcommand\NAT@test[1]{%
- \@ifnum{#1=\@ne}{%
-  \ifx\NAT@nm\NAT@noname
-   \begingroup\reset@font\bfseries(author?)\endgroup
-   \PackageWarning{natbib}{%
-    Author undefined for citation`\@citeb' \MessageBreak on page \thepage%
-   }%
-  \else \NAT@nm
-  \fi
- }{%
-  \if\relax\NAT@date\relax
-   \begingroup\reset@font\bfseries(year?)\endgroup
-   \PackageWarning{natbib}{%
-    Year undefined for citation`\@citeb' \MessageBreak on page \thepage%
-   }%
-  \else \NAT@date
-  \fi
- }%
-}%
-\let\citenumfont=\@empty
-\newcommand\NAT@citex{}
-\def\NAT@citex%
-  [#1][#2]#3{%
-  \NAT@reset@parser
-  \NAT@sort@cites{#3}%
-  \NAT@reset@citea
-  \@cite{\let\NAT@nm\@empty\let\NAT@year\@empty
-    \@for\@citeb:=\NAT@cite@list\do
-    {\@safe@activestrue
-     \edef\@citeb{\expandafter\@firstofone\@citeb\@empty}%
-     \@safe@activesfalse
-     \@ifundefined{b@\@citeb\@extra@b@citeb}{\@citea%
-       {\reset@font\bfseries ?}\NAT@citeundefined
-                 \PackageWarning{natbib}%
-       {Citation `\@citeb' on page \thepage \space undefined}\def\NAT@date{}}%
-     {\let\NAT@last@nm=\NAT@nm\let\NAT@last@yr=\NAT@year
-      \NAT@parse{\@citeb}%
-      \ifNAT@longnames\@ifundefined{bv@\@citeb\@extra@b@citeb}{%
-        \let\NAT@name=\NAT@all@names
-        \global\@namedef{bv@\@citeb\@extra@b@citeb}{}}{}%
-      \fi
-     \ifNAT@full\let\NAT@nm\NAT@all@names\else
-       \let\NAT@nm\NAT@name\fi
-     \ifNAT@swa\ifcase\NAT@ctype
-       \if\relax\NAT@date\relax
-         \@citea\NAT@hyper@{\NAT@nmfmt{\NAT@nm}\NAT@date}%
-       \else
-         \ifx\NAT@last@nm\NAT@nm\NAT@yrsep
-            \ifx\NAT@last@yr\NAT@year
-              \def\NAT@temp{{?}}%
-              \ifx\NAT@temp\NAT@exlab\PackageWarningNoLine{natbib}%
-               {Multiple citation on page \thepage: same authors and
-               year\MessageBreak without distinguishing extra
-               letter,\MessageBreak appears as question mark}\fi
-              \NAT@hyper@{\NAT@exlab}%
-            \else\unskip\NAT@spacechar
-              \NAT@hyper@{\NAT@date}%
-            \fi
-         \else
-           \@citea\NAT@hyper@{%
-             \NAT@nmfmt{\NAT@nm}%
-             \hyper@natlinkbreak{%
-               \NAT@aysep\NAT@spacechar}{\@citeb\@extra@b@citeb
-             }%
-             \NAT@date
-           }%
-         \fi
-       \fi
-     \or\@citea\NAT@hyper@{\NAT@nmfmt{\NAT@nm}}%
-     \or\@citea\NAT@hyper@{\NAT@date}%
-     \or\@citea\NAT@hyper@{\NAT@alias}%
-     \fi \NAT@def@citea
-     \else
-       \ifcase\NAT@ctype
-        \if\relax\NAT@date\relax
-          \@citea\NAT@hyper@{\NAT@nmfmt{\NAT@nm}}%
-        \else
-         \ifx\NAT@last@nm\NAT@nm\NAT@yrsep
-            \ifx\NAT@last@yr\NAT@year
-              \def\NAT@temp{{?}}%
-              \ifx\NAT@temp\NAT@exlab\PackageWarningNoLine{natbib}%
-               {Multiple citation on page \thepage: same authors and
-               year\MessageBreak without distinguishing extra
-               letter,\MessageBreak appears as question mark}\fi
-              \NAT@hyper@{\NAT@exlab}%
-            \else
-              \unskip\NAT@spacechar
-              \NAT@hyper@{\NAT@date}%
-            \fi
-         \else
-           \@citea\NAT@hyper@{%
-             \NAT@nmfmt{\NAT@nm}%
-             \hyper@natlinkbreak{\NAT@spacechar\NAT@@open\if*#1*\else#1\NAT@spacechar\fi}%
-               {\@citeb\@extra@b@citeb}%
-             \NAT@date
-           }%
-         \fi
-        \fi
-       \or\@citea\NAT@hyper@{\NAT@nmfmt{\NAT@nm}}%
-       \or\@citea\NAT@hyper@{\NAT@date}%
-       \or\@citea\NAT@hyper@{\NAT@alias}%
-       \fi
-       \if\relax\NAT@date\relax
-         \NAT@def@citea
-       \else
-         \NAT@def@citea@close
-       \fi
-     \fi
-     }}\ifNAT@swa\else\if*#2*\else\NAT@cmt#2\fi
-     \if\relax\NAT@date\relax\else\NAT@@close\fi\fi}{#1}{#2}}
-\def\NAT@spacechar{\ }%
-\def\NAT@separator{\NAT@sep\NAT@penalty}%
-\def\NAT@reset@citea{\c@NAT@ctr\@ne\let\@citea\@empty}%
-\def\NAT@def@citea{\def\@citea{\NAT@separator\NAT@space}}%
-\def\NAT@def@citea@space{\def\@citea{\NAT@separator\NAT@spacechar}}%
-\def\NAT@def@citea@close{\def\@citea{\NAT@@close\NAT@separator\NAT@space}}%
-\def\NAT@def@citea@box{\def\@citea{\NAT@mbox{\NAT@@close}\NAT@separator\NAT@spacechar}}%
-\newif\ifNAT@par \NAT@partrue
-\newcommand\NAT@@open{\ifNAT@par\NAT@open\fi}
-\newcommand\NAT@@close{\ifNAT@par\NAT@close\fi}
-\newcommand\NAT@alias{\@ifundefined{al@\@citeb\@extra@b@citeb}{%
-  {\reset@font\bfseries(alias?)}\PackageWarning{natbib}
-  {Alias undefined for citation `\@citeb'
-  \MessageBreak on page \thepage}}{\@nameuse{al@\@citeb\@extra@b@citeb}}}
-\let\NAT@up\relax
-\newcommand\NAT@Up[1]{{\let\protect\@unexpandable@protect\let~\relax
-  \expandafter\NAT@deftemp#1}\expandafter\NAT@UP\NAT@temp}
-\newcommand\NAT@deftemp[1]{\xdef\NAT@temp{#1}}
-\newcommand\NAT@UP[1]{\let\@tempa\NAT@UP\ifcat a#1\MakeUppercase{#1}%
-  \let\@tempa\relax\else#1\fi\@tempa}
-\newcommand\shortcites[1]{%
-  \@bsphack\@for\@citeb:=#1\do
-  {\@safe@activestrue
-   \edef\@citeb{\expandafter\@firstofone\@citeb\@empty}%
-   \@safe@activesfalse
-   \global\@namedef{bv@\@citeb\@extra@b@citeb}{}}\@esphack}
-\newcommand\NAT@biblabel[1]{\hfill}
-\newcommand\NAT@biblabelnum[1]{\bibnumfmt{#1}}
-\let\bibnumfmt\@empty
-\providecommand\@biblabel[1]{[#1]}
-\AtBeginDocument{\ifx\bibnumfmt\@empty\let\bibnumfmt\@biblabel\fi}
-\newcommand\NAT@bibsetnum[1]{\settowidth\labelwidth{\@biblabel{#1}}%
-   \setlength{\leftmargin}{\labelwidth}\addtolength{\leftmargin}{\labelsep}%
-   \setlength{\itemsep}{\bibsep}\setlength{\parsep}{\z@}%
-   \ifNAT@openbib
-     \addtolength{\leftmargin}{\bibindent}%
-     \setlength{\itemindent}{-\bibindent}%
-     \setlength{\listparindent}{\itemindent}%
-     \setlength{\parsep}{0pt}%
-   \fi
-}
-\newlength{\bibhang}
-\setlength{\bibhang}{1em}
-\newlength{\bibsep}
- {\@listi \global\bibsep\itemsep \global\advance\bibsep by\parsep}
-
-\newcommand\NAT@bibsetup%
-   [1]{\setlength{\leftmargin}{\bibhang}\setlength{\itemindent}{-\leftmargin}%
-       \setlength{\itemsep}{\bibsep}\setlength{\parsep}{\z@}}
-\newcommand\NAT@set@cites{%
-  \ifNAT@numbers
-    \ifNAT@super \let\@cite\NAT@citesuper
-       \def\NAT@mbox##1{\unskip\nobreak\textsuperscript{##1}}%
-       \let\citeyearpar=\citeyear
-       \let\NAT@space\relax
-       \def\NAT@super@kern{\kern\p@}%
-    \else
-       \let\NAT@mbox=\mbox
-       \let\@cite\NAT@citenum
-       \let\NAT@space\NAT@spacechar
-       \let\NAT@super@kern\relax
-    \fi
-    \let\@citex\NAT@citexnum
-    \let\@biblabel\NAT@biblabelnum
-    \let\@bibsetup\NAT@bibsetnum
-    \renewcommand\NAT@idxtxt{\NAT@name\NAT@spacechar\NAT@open\NAT@num\NAT@close}%
-    \def\natexlab##1{}%
-    \def\NAT@penalty{\penalty\@m}%
-  \else
-    \let\@cite\NAT@cite
-    \let\@citex\NAT@citex
-    \let\@biblabel\NAT@biblabel
-    \let\@bibsetup\NAT@bibsetup
-    \let\NAT@space\NAT@spacechar
-    \let\NAT@penalty\@empty
-    \renewcommand\NAT@idxtxt{\NAT@name\NAT@spacechar\NAT@open\NAT@date\NAT@close}%
-    \def\natexlab##1{##1}%
-  \fi}
-\AtBeginDocument{\NAT@set@cites}
-\AtBeginDocument{\ifx\SK@def\@undefined\else
-\ifx\SK@cite\@empty\else
-  \SK@def\@citex[#1][#2]#3{\SK@\SK@@ref{#3}\SK@@citex[#1][#2]{#3}}\fi
-\ifx\SK@citeauthor\@undefined\def\HAR@checkdef{}\else
-  \let\citeauthor\SK@citeauthor
-  \let\citefullauthor\SK@citefullauthor
-  \let\citeyear\SK@citeyear\fi
-\fi}
-\newif\ifNAT@full\NAT@fullfalse
-\newif\ifNAT@swa
-\DeclareRobustCommand\citet
-   {\begingroup\NAT@swafalse\let\NAT@ctype\z@\NAT@partrue
-     \@ifstar{\NAT@fulltrue\NAT@citetp}{\NAT@fullfalse\NAT@citetp}}
-\newcommand\NAT@citetp{\@ifnextchar[{\NAT@@citetp}{\NAT@@citetp[]}}
-\newcommand\NAT@@citetp{}
-\def\NAT@@citetp[#1]{\@ifnextchar[{\@citex[#1]}{\@citex[][#1]}}
-\DeclareRobustCommand\citep
-   {\begingroup\NAT@swatrue\let\NAT@ctype\z@\NAT@partrue
-         \@ifstar{\NAT@fulltrue\NAT@citetp}{\NAT@fullfalse\NAT@citetp}}
-\DeclareRobustCommand\cite
-    {\begingroup\let\NAT@ctype\z@\NAT@partrue\NAT@swatrue
-      \@ifstar{\NAT@fulltrue\NAT@cites}{\NAT@fullfalse\NAT@cites}}
-\newcommand\NAT@cites{\@ifnextchar [{\NAT@@citetp}{%
-     \ifNAT@numbers\else
-     \NAT@swafalse
-     \fi
-    \NAT@@citetp[]}}
-\DeclareRobustCommand\citealt
-   {\begingroup\NAT@swafalse\let\NAT@ctype\z@\NAT@parfalse
-         \@ifstar{\NAT@fulltrue\NAT@citetp}{\NAT@fullfalse\NAT@citetp}}
-\DeclareRobustCommand\citealp
-   {\begingroup\NAT@swatrue\let\NAT@ctype\z@\NAT@parfalse
-         \@ifstar{\NAT@fulltrue\NAT@citetp}{\NAT@fullfalse\NAT@citetp}}
-\DeclareRobustCommand\citenum
-   {\begingroup
-     \NAT@swatrue\let\NAT@ctype\z@\NAT@parfalse\let\textsuperscript\NAT@spacechar
-     \NAT@citexnum[][]}
-\DeclareRobustCommand\citeauthor
-   {\begingroup\NAT@swafalse\let\NAT@ctype\@ne\NAT@parfalse
-    \@ifstar{\NAT@fulltrue\NAT@citetp}{\NAT@fullfalse\NAT@citetp}}
-\DeclareRobustCommand\Citet
-   {\begingroup\NAT@swafalse\let\NAT@ctype\z@\NAT@partrue
-     \let\NAT@up\NAT@Up
-     \@ifstar{\NAT@fulltrue\NAT@citetp}{\NAT@fullfalse\NAT@citetp}}
-\DeclareRobustCommand\Citep
-   {\begingroup\NAT@swatrue\let\NAT@ctype\z@\NAT@partrue
-     \let\NAT@up\NAT@Up
-         \@ifstar{\NAT@fulltrue\NAT@citetp}{\NAT@fullfalse\NAT@citetp}}
-\DeclareRobustCommand\Citealt
-   {\begingroup\NAT@swafalse\let\NAT@ctype\z@\NAT@parfalse
-     \let\NAT@up\NAT@Up
-         \@ifstar{\NAT@fulltrue\NAT@citetp}{\NAT@fullfalse\NAT@citetp}}
-\DeclareRobustCommand\Citealp
-   {\begingroup\NAT@swatrue\let\NAT@ctype\z@\NAT@parfalse
-     \let\NAT@up\NAT@Up
-         \@ifstar{\NAT@fulltrue\NAT@citetp}{\NAT@fullfalse\NAT@citetp}}
-\DeclareRobustCommand\Citeauthor
-   {\begingroup\NAT@swafalse\let\NAT@ctype\@ne\NAT@parfalse
-     \let\NAT@up\NAT@Up
-    \@ifstar{\NAT@fulltrue\NAT@citetp}{\NAT@fullfalse\NAT@citetp}}
-\DeclareRobustCommand\citeyear
-   {\begingroup\NAT@swafalse\let\NAT@ctype\tw@\NAT@parfalse\NAT@citetp}
-\DeclareRobustCommand\citeyearpar
-   {\begingroup\NAT@swatrue\let\NAT@ctype\tw@\NAT@partrue\NAT@citetp}
-\newcommand\citetext[1]{\NAT@open#1\NAT@close}
-\DeclareRobustCommand\citefullauthor
-   {\citeauthor*}
-\newcommand\defcitealias[2]{%
-   \@ifundefined{al@#1\@extra@b@citeb}{}
-   {\PackageWarning{natbib}{Overwriting existing alias for citation #1}}
-   \@namedef{al@#1\@extra@b@citeb}{#2}}
-\DeclareRobustCommand\citetalias{\begingroup
-   \NAT@swafalse\let\NAT@ctype\thr@@\NAT@parfalse\NAT@citetp}
-\DeclareRobustCommand\citepalias{\begingroup
-   \NAT@swatrue\let\NAT@ctype\thr@@\NAT@partrue\NAT@citetp}
-\renewcommand\nocite[1]{\@bsphack
-  \@for\@citeb:=#1\do{%
-    \@safe@activestrue
-    \edef\@citeb{\expandafter\@firstofone\@citeb\@empty}%
-    \@safe@activesfalse
-    \if@filesw\immediate\write\@auxout{\string\citation{\@citeb}}\fi
-    \if*\@citeb\else
-    \@ifundefined{b@\@citeb\@extra@b@citeb}{%
-       \NAT@citeundefined \PackageWarning{natbib}%
-       {Citation `\@citeb' undefined}}{}\fi}%
-  \@esphack}
-\newcommand\NAT@parse[1]{%
-  \begingroup
-   \let\protect=\@unexpandable@protect
-   \let~\relax
-   \let\active@prefix=\@gobble
-   \edef\NAT@temp{\csname b@#1\@extra@b@citeb\endcsname}%
-   \aftergroup\NAT@split
-   \expandafter
-  \endgroup
-  \NAT@temp{}{}{}{}{}@@%
-  \expandafter\NAT@parse@date\NAT@date??????@@%
-  \ifciteindex\NAT@index\fi
-}%
-\def\NAT@split#1#2#3#4#5@@{%
-  \gdef\NAT@num{#1}\gdef\NAT@name{#3}\gdef\NAT@date{#2}%
-  \gdef\NAT@all@names{#4}%
-  \ifx\NAT@num\@empty\gdef\NAT@num{0}\fi
-  \ifx\NAT@noname\NAT@all@names \gdef\NAT@all@names{#3}\fi
-}%
-\def\NAT@reset@parser{%
-  \global\let\NAT@num\@empty
-  \global\let\NAT@name\@empty
-  \global\let\NAT@date\@empty
-  \global\let\NAT@all@names\@empty
-}%
-\newcommand\NAT@parse@date{}
-\def\NAT@parse@date#1#2#3#4#5#6@@{%
-  \ifnum\the\catcode`#1=11\def\NAT@year{}\def\NAT@exlab{#1}\else
-  \ifnum\the\catcode`#2=11\def\NAT@year{#1}\def\NAT@exlab{#2}\else
-  \ifnum\the\catcode`#3=11\def\NAT@year{#1#2}\def\NAT@exlab{#3}\else
-  \ifnum\the\catcode`#4=11\def\NAT@year{#1#2#3}\def\NAT@exlab{#4}\else
-    \def\NAT@year{#1#2#3#4}\def\NAT@exlab{{#5}}\fi\fi\fi\fi}
-\newcommand\NAT@index{}
-\let\NAT@makeindex=\makeindex
-\renewcommand\makeindex{\NAT@makeindex
-  \renewcommand\NAT@index{\@bsphack\begingroup
-     \def~{\string~}\@wrindex{\NAT@idxtxt}}}
-\newcommand\NAT@idxtxt{\NAT@name\NAT@spacechar\NAT@open\NAT@date\NAT@close}
-\@ifxundefined\@indexfile{}{\let\NAT@makeindex\relax\makeindex}
-\newif\ifciteindex \citeindexfalse
-\newcommand\citeindextype{default}
-\newcommand\NAT@index@alt{{\let\protect=\noexpand\let~\relax
-  \xdef\NAT@temp{\NAT@idxtxt}}\expandafter\NAT@exp\NAT@temp\@nil}
-\newcommand\NAT@exp{}
-\def\NAT@exp#1\@nil{\index[\citeindextype]{#1}}
-
-\AtBeginDocument{%
-\@ifpackageloaded{index}{\let\NAT@index=\NAT@index@alt}{}}
-\newcommand\NAT@ifcmd{\futurelet\NAT@temp\NAT@ifxcmd}
-\newcommand\NAT@ifxcmd{\ifx\NAT@temp\relax\else\expandafter\NAT@bare\fi}
-\def\NAT@bare#1(#2)#3(@)#4\@nil#5{%
-  \if @#2
-    \expandafter\NAT@apalk#1, , \@nil{#5}%
-  \else
-  \NAT@wrout{\the\c@NAT@ctr}{#2}{#1}{#3}{#5}%
-\fi
-}
-\newcommand\NAT@wrout[5]{%
-\if@filesw
-      {\let\protect\noexpand\let~\relax
-       \immediate
-       \write\@auxout{\string\bibcite{#5}{{#1}{#2}{{#3}}{{#4}}}}}\fi
-\ignorespaces}
-\def\NAT@noname{{}}
-\renewcommand\bibitem{\@ifnextchar[{\@lbibitem}{\@lbibitem[]}}%
-\let\NAT@bibitem@first@sw\@secondoftwo
-\def\@lbibitem[#1]#2{%
-  \if\relax\@extra@b@citeb\relax\else
-    \@ifundefined{br@#2\@extra@b@citeb}{}{%
-     \@namedef{br@#2}{\@nameuse{br@#2\@extra@b@citeb}}%
-    }%
-  \fi
-  \@ifundefined{b@#2\@extra@b@citeb}{%
-   \def\NAT@num{}%
-  }{%
-   \NAT@parse{#2}%
-  }%
-  \def\NAT@tmp{#1}%
-  \expandafter\let\expandafter\bibitemOpen\csname NAT@b@open@#2\endcsname
-  \expandafter\let\expandafter\bibitemShut\csname NAT@b@shut@#2\endcsname
-  \@ifnum{\NAT@merge>\@ne}{%
-   \NAT@bibitem@first@sw{%
-    \@firstoftwo
-   }{%
-    \@ifundefined{NAT@b*@#2}{%
-     \@firstoftwo
-    }{%
-     \expandafter\def\expandafter\NAT@num\expandafter{\the\c@NAT@ctr}%
-     \@secondoftwo
-    }%
-   }%
-  }{%
-   \@firstoftwo
-  }%
-  {%
-   \global\advance\c@NAT@ctr\@ne
-   \@ifx{\NAT@tmp\@empty}{\@firstoftwo}{%
-    \@secondoftwo
-   }%
-   {%
-    \expandafter\def\expandafter\NAT@num\expandafter{\the\c@NAT@ctr}%
-    \global\NAT@stdbsttrue
-   }{}%
-   \bibitem@fin
-   \item[\hfil\NAT@anchor{#2}{\NAT@num}]%
-   \global\let\NAT@bibitem@first@sw\@secondoftwo
-   \NAT@bibitem@init
-  }%
-  {%
-   \NAT@anchor{#2}{}%
-   \NAT@bibitem@cont
-   \bibitem@fin
-  }%
-  \@ifx{\NAT@tmp\@empty}{%
-    \NAT@wrout{\the\c@NAT@ctr}{}{}{}{#2}%
-  }{%
-    \expandafter\NAT@ifcmd\NAT@tmp(@)(@)\@nil{#2}%
-  }%
-}%
-\def\bibitem@fin{%
- \@ifxundefined\@bibstop{}{\csname bibitem@\@bibstop\endcsname}%
-}%
-\def\NAT@bibitem@init{%
- \let\@bibstop\@undefined
-}%
-\def\NAT@bibitem@cont{%
- \let\bibitem@Stop\bibitemStop
- \let\bibitem@NoStop\bibitemContinue
-}%
-\def\BibitemOpen{%
- \bibitemOpen
-}%
-\def\BibitemShut#1{%
- \bibitemShut
- \def\@bibstop{#1}%
- \let\bibitem@Stop\bibitemStop
- \let\bibitem@NoStop\bibitemNoStop
-}%
-\def\bibitemStop{}%
-\def\bibitemNoStop{.\spacefactor\@mmm\space}%
-\def\bibitemContinue{\spacefactor\@mmm\space}%
-\mathchardef\@mmm=3000 %
-\providecommand{\bibAnnote}[3]{%
-  \BibitemShut{#1}%
-  \def\@tempa{#3}\@ifx{\@tempa\@empty}{}{%
-   \begin{quotation}\noindent
-    \textsc{Key:}\ #2\\\textsc{Annotation:}\ \@tempa
-   \end{quotation}%
-  }%
-}%
-\providecommand{\bibAnnoteFile}[2]{%
-  \IfFileExists{#2}{%
-    \bibAnnote{#1}{#2}{\input{#2}}%
-  }{%
-    \bibAnnote{#1}{#2}{}%
-  }%
-}%
-\let\bibitemOpen\relax
-\let\bibitemShut\relax
-\def\bibfield{\@ifnum{\NAT@merge>\tw@}{\@bibfield}{\@secondoftwo}}%
-\def\@bibfield#1#2{%
- \begingroup
-  \let\Doi\@gobble
-  \let\bibinfo\relax
-  \let\restore@protect\@empty
-  \protected@edef\@tempa{#2}%
-  \aftergroup\def\aftergroup\@tempa
- \expandafter\endgroup\expandafter{\@tempa}%
- \expandafter\@ifx\expandafter{\csname @bib#1\endcsname\@tempa}{%
-  \expandafter\let\expandafter\@tempa\csname @bib@X#1\endcsname
- }{%
-  \expandafter\let\csname @bib#1\endcsname\@tempa
-  \expandafter\let\expandafter\@tempa\csname @bib@Y#1\endcsname
- }%
- \@ifx{\@tempa\relax}{\let\@tempa\@firstofone}{}%
- \@tempa{#2}%
-}%
-\def\bibinfo#1{%
- \expandafter\let\expandafter\@tempa\csname bibinfo@X@#1\endcsname
- \@ifx{\@tempa\relax}{\@firstofone}{\@tempa}%
-}%
-\def\@bib@Xauthor#1{\let\@bib@Xjournal\@gobble}%
-\def\@bib@Xjournal#1{\begingroup\let\bibinfo@X@journal\@bib@Z@journal#1\endgroup}%
-\def\@bibibid@#1{\textit{ibid}.}%
-\appdef\NAT@bibitem@init{%
- \let\@bibauthor  \@empty
- \let\@bibjournal \@empty
- \let\@bib@Z@journal\@bibibid@
-}%
-\ifx\SK@lbibitem\@undefined\else
-   \let\SK@lbibitem\@lbibitem
-   \def\@lbibitem[#1]#2{%
-     \SK@lbibitem[#1]{#2}\SK@\SK@@label{#2}\ignorespaces}\fi
-\newif\ifNAT@stdbst \NAT@stdbstfalse
-
-\AtEndDocument{%
-  \ifNAT@stdbst\if@filesw
-   \immediate\write\@auxout{%
-    \string\providecommand\string\NAT@force@numbers{}%
-    \string\NAT@force@numbers
-   }%
-  \fi\fi
- }
-\newcommand\NAT@force@numbers{%
-  \ifNAT@numbers\else
-  \PackageError{natbib}{Bibliography not compatible with author-year
-  citations.\MessageBreak
-  Press <return> to continue in numerical citation style}
-  {Check the bibliography entries for non-compliant syntax,\MessageBreak
-   or select author-year BibTeX style, e.g. plainnat}%
-  \global\NAT@numberstrue\fi}
-
-\providecommand\bibcite{}
-\renewcommand\bibcite[2]{%
- \@ifundefined{b@#1\@extra@binfo}{\relax}{%
-   \NAT@citemultiple
-   \PackageWarningNoLine{natbib}{Citation `#1' multiply defined}%
- }%
- \global\@namedef{b@#1\@extra@binfo}{#2}%
-}%
-\AtEndDocument{\NAT@swatrue\let\bibcite\NAT@testdef}
-\newcommand\NAT@testdef[2]{%
-  \def\NAT@temp{#2}%
-  \expandafter \ifx \csname b@#1\@extra@binfo\endcsname\NAT@temp
-  \else
-    \ifNAT@swa \NAT@swafalse
-      \PackageWarningNoLine{natbib}{%
-        Citation(s) may have changed.\MessageBreak
-        Rerun to get citations correct%
-      }%
-    \fi
-  \fi
-}%
-\newcommand\NAT@apalk{}
-\def\NAT@apalk#1, #2, #3\@nil#4{%
-  \if\relax#2\relax
-    \global\NAT@stdbsttrue
-    \NAT@wrout{#1}{}{}{}{#4}%
-  \else
-    \NAT@wrout{\the\c@NAT@ctr}{#2}{#1}{}{#4}%
-  \fi
-}%
-\newcommand\citeauthoryear{}
-\def\citeauthoryear#1#2#3(@)(@)\@nil#4{%
-  \if\relax#3\relax
-    \NAT@wrout{\the\c@NAT@ctr}{#2}{#1}{}{#4}%
-  \else
-    \NAT@wrout{\the\c@NAT@ctr}{#3}{#2}{#1}{#4}%
-  \fi
-}%
-\newcommand\citestarts{\NAT@open}%
-\newcommand\citeends{\NAT@close}%
-\newcommand\betweenauthors{and}%
-\newcommand\astroncite{}
-\def\astroncite#1#2(@)(@)\@nil#3{%
- \NAT@wrout{\the\c@NAT@ctr}{#2}{#1}{}{#3}%
-}%
-\newcommand\citename{}
-\def\citename#1#2(@)(@)\@nil#3{\expandafter\NAT@apalk#1#2, \@nil{#3}}
-\newcommand\harvarditem[4][]{%
- \if\relax#1\relax
-   \bibitem[#2(#3)]{#4}%
- \else
-   \bibitem[#1(#3)#2]{#4}%
- \fi
-}%
-\newcommand\harvardleft{\NAT@open}
-\newcommand\harvardright{\NAT@close}
-\newcommand\harvardyearleft{\NAT@open}
-\newcommand\harvardyearright{\NAT@close}
-\AtBeginDocument{\providecommand{\harvardand}{and}}
-\newcommand\harvardurl[1]{\textbf{URL:} \textit{#1}}
-\providecommand\bibsection{}
-\@ifundefined{chapter}{%
-  \renewcommand\bibsection{%
-   \section*{\refname\@mkboth{\MakeUppercase{\refname}}{\MakeUppercase{\refname}}}%
-  }%
-}{%
-  \@ifxundefined\NAT@sectionbib{%
-    \renewcommand\bibsection{%
-      \chapter*{\bibname\@mkboth{\MakeUppercase{\bibname}}{\MakeUppercase{\bibname}}}%
-    }%
-  }{%
-    \renewcommand\bibsection{%
-      \section*{\bibname\ifx\@mkboth\@gobbletwo\else\markright{\MakeUppercase{\bibname}}\fi}%
-    }%
-  }%
-}%
-\@ifclassloaded{amsart}{\renewcommand\bibsection{\section*{\refname}}}{}%
-\@ifclassloaded{amsbook}{\renewcommand\bibsection{\chapter*{\bibname}}}{}%
-\@ifxundefined\bib@heading{}{\let\bibsection\bib@heading}%
-\newcounter{NAT@ctr}
-\renewenvironment{thebibliography}[1]{%
- \bibsection
- \parindent\z@
- \bibpreamble
- \bibfont
- \list{\@biblabel{\the\c@NAT@ctr}}{\@bibsetup{#1}\global\c@NAT@ctr\z@}%
- \ifNAT@openbib
-   \renewcommand\newblock{\par}%
- \else
-   \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
- \fi
- \sloppy\clubpenalty4000\widowpenalty4000
- \sfcode`\.\@m
- \let\NAT@bibitem@first@sw\@firstoftwo
-    \let\citeN\cite \let\shortcite\cite
-    \let\citeasnoun\cite
-}{%
- \bibitem@fin
- \bibpostamble
- \def\@noitemerr{%
-  \PackageWarning{natbib}{Empty `thebibliography' environment}%
- }%
- \endlist
- \bibcleanup
-}%
-\let\bibfont\@empty
-\let\bibpreamble\@empty
-\let\bibpostamble\@empty
-\def\bibcleanup{\vskip-\lastskip}%
-\providecommand\reset@font{\relax}
-\providecommand\bibname{Bibliography}
-\providecommand\refname{References}
-\newcommand\NAT@citeundefined{\gdef \NAT@undefined {%
-    \PackageWarningNoLine{natbib}{There were undefined citations}}}
-\let \NAT@undefined \relax
-\newcommand\NAT@citemultiple{\gdef \NAT@multiple {%
-    \PackageWarningNoLine{natbib}{There were multiply defined citations}}}
-\let \NAT@multiple \relax
-\AtEndDocument{\NAT@undefined\NAT@multiple}
-\providecommand\@mkboth[2]{}
-\providecommand\MakeUppercase{\uppercase}
-\providecommand{\@extra@b@citeb}{}
-\gdef\@extra@binfo{}
-\def\NAT@anchor#1#2{%
- \hyper@natanchorstart{#1\@extra@b@citeb}%
-  \def\@tempa{#2}\@ifx{\@tempa\@empty}{}{\@biblabel{#2}}%
- \hyper@natanchorend
-}%
-\providecommand\hyper@natanchorstart[1]{}%
-\providecommand\hyper@natanchorend{}%
-\providecommand\hyper@natlinkstart[1]{}%
-\providecommand\hyper@natlinkend{}%
-\providecommand\hyper@natlinkbreak[2]{#1}%
-\AtBeginDocument{%
-  \@ifpackageloaded{babel}{%
-     \let\org@@citex\@citex}{}}
-\providecommand\@safe@activestrue{}%
-\providecommand\@safe@activesfalse{}%
-
-\newcommand\NAT@sort@cites[1]{%
-  \let\NAT@cite@list\@empty
-  \@for\@citeb:=#1\do{\expandafter\NAT@star@cite\@citeb\@@}%
-  \if@filesw
-    \expandafter\immediate\expandafter\write\expandafter\@auxout
-      \expandafter{\expandafter\string\expandafter\citation\expandafter{\NAT@cite@list}}%
-  \fi
-  \@ifnum{\NAT@sort>\z@}{%
-    \expandafter\NAT@sort@cites@\expandafter{\NAT@cite@list}%
-  }{}%
-}%
-\def\NAT@star@cite{%
-  \let\NAT@star@sw\@secondoftwo
-  \@ifnum{\NAT@merge>\z@}{%
-   \@ifnextchar*{%
-    \let\NAT@star@sw\@firstoftwo
-    \NAT@star@cite@star
-   }{%
-    \NAT@star@cite@nostar
-   }%
-  }{%
-   \NAT@star@cite@noextension
-  }%
-}%
-\def\NAT@star@cite@star*{%
- \NAT@star@cite@nostar
-}%
-\def\NAT@star@cite@nostar{%
- \let\nat@keyopt@open\@empty
- \let\nat@keyopt@shut\@empty
- \@ifnextchar[{\NAT@star@cite@pre}{\NAT@star@cite@pre[]}%
-}%
-\def\NAT@star@cite@pre[#1]{%
- \def\nat@keyopt@open{#1}%
- \@ifnextchar[{\NAT@star@cite@post}{\NAT@star@cite@post[]}%
-}%
-\def\NAT@star@cite@post[#1]#2\@@{%
- \def\nat@keyopt@shut{#1}%
- \NAT@star@sw{\expandafter\global\expandafter\let\csname NAT@b*@#2\endcsname\@empty}{}%
- \NAT@cite@list@append{#2}%
-}%
-\def\NAT@star@cite@noextension#1\@@{%
-  \let\nat@keyopt@open\@empty
-  \let\nat@keyopt@shut\@empty
-  \NAT@cite@list@append{#1}%
-}%
-\def\NAT@cite@list@append#1{%
-  \edef\@citeb{\@firstofone#1\@empty}%
-  \if@filesw\@ifxundefined\@cprwrite{}{\expandafter\@cprwrite\@citeb=}\fi
-  \if\relax\nat@keyopt@open\relax\else
-   \global\expandafter\let\csname NAT@b@open@\@citeb\endcsname\nat@keyopt@open
-  \fi
-  \if\relax\nat@keyopt@shut\relax\else
-   \global\expandafter\let\csname NAT@b@shut@\@citeb\endcsname\nat@keyopt@shut
-  \fi
-  \toks@\expandafter{\NAT@cite@list}%
-  \ifx\NAT@cite@list\@empty
-    \@temptokena\expandafter{\@citeb}%
-  \else
-    \@temptokena\expandafter{\expandafter,\@citeb}%
-  \fi
-  \edef\NAT@cite@list{\the\toks@\the\@temptokena}%
-}%
-\newcommand\NAT@sort@cites@[1]{%
-  \count@\z@
-  \@tempcntb\m@ne
-  \let\@celt\delimiter
-  \def\NAT@num@list{}%
-  \let\NAT@cite@list\@empty
-  \let\NAT@nonsort@list\@empty
-  \@for \@citeb:=#1\do{\NAT@make@cite@list}%
-  \ifx\NAT@nonsort@list\@empty\else
-   \protected@edef\NAT@cite@list{\NAT@cite@list\NAT@nonsort@list}%
-  \fi
-  \ifx\NAT@cite@list\@empty\else
-   \protected@edef\NAT@cite@list{\expandafter\NAT@xcom\NAT@cite@list @@}%
-  \fi
-}%
-\def\NAT@make@cite@list{%
-  \advance\count@\@ne
-  \@safe@activestrue
-  \edef\@citeb{\expandafter\@firstofone\@citeb\@empty}%
-  \@safe@activesfalse
-  \@ifundefined{b@\@citeb\@extra@b@citeb}%
-   {\def\NAT@num{A}}%
-   {\NAT@parse{\@citeb}}%
-  \NAT@ifcat@num\NAT@num
-   {\@tempcnta\NAT@num \relax
-    \@ifnum{\@tempcnta<\@tempcntb}{%
-      \let\NAT@@cite@list=\NAT@cite@list
-      \let\NAT@cite@list\@empty
-      \begingroup\let\@celt=\NAT@celt\NAT@num@list\endgroup
-      \protected@edef\NAT@num@list{%
-       \expandafter\NAT@num@celt \NAT@num@list \@gobble @%
-      }%
-    }{%
-      \protected@edef\NAT@num@list{\NAT@num@list \@celt{\NAT@num}}%
-      \protected@edef\NAT@cite@list{\NAT@cite@list\@citeb,}%
-      \@tempcntb\@tempcnta
-    }%
-   }%
-   {\protected@edef\NAT@nonsort@list{\NAT@nonsort@list\@citeb,}}%
-}%
-\def\NAT@celt#1{%
-  \@ifnum{#1>\@tempcnta}{%
-    \xdef\NAT@cite@list{\NAT@cite@list\@citeb,\NAT@@cite@list}%
-    \let\@celt\@gobble
-  }{%
-    \expandafter\def@NAT@cite@lists\NAT@@cite@list\@@
-  }%
-}%
-\def\NAT@num@celt#1#2{%
- \ifx#1\@celt
-  \@ifnum{#2>\@tempcnta}{%
-    \@celt{\number\@tempcnta}%
-    \@celt{#2}%
-  }{%
-    \@celt{#2}%
-    \expandafter\NAT@num@celt
-  }%
- \fi
-}%
-\def\def@NAT@cite@lists#1,#2\@@{%
-  \xdef\NAT@cite@list{\NAT@cite@list#1,}%
-  \xdef\NAT@@cite@list{#2}%
-}%
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-\def\NAT@restc#1,#2{#2}
-\def\NAT@xcom#1,@@{#1}
-\InputIfFileExists{natbib.cfg}
-       {\typeout{Local config file natbib.cfg used}}{}
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diff --git a/docs/cohen_et_al_2018/s2cnn.tex b/docs/cohen_et_al_2018/s2cnn.tex
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-\documentclass{article} % For LaTeX2e
-\usepackage{iclr2018_conference,times}
-\usepackage{hyperref}
-\usepackage{url}
-
-\usepackage{amsmath}
-\usepackage{amsfonts}
-\usepackage{amssymb}
-\usepackage{graphicx}
-\usepackage{wrapfig}
-
-\usepackage{framed}
-
-\usepackage[export]{adjustbox}
-\usepackage{subcaption}
-%\usepackage{subfig}
-
-\usepackage{tabularx}
-
-\usepackage[utf8]{inputenc} % allow utf-8 input
-\usepackage[T1]{fontenc}    % use 8-bit T1 fonts
-\usepackage{booktabs}       % professional-quality tables
-\usepackage{nicefrac}       % compact symbols for 1/2, etc.
-\usepackage{microtype}      % microtypography
-\usepackage{import}         % figures
-\usepackage{color}          % figures
-\usepackage{pgf}            % figures
-\usepackage{multirow}       % tabular
-\usepackage{color}
-
-
-\title{Spherical CNNs}
-
-% Authors must not appear in the submitted version. They should be hidden
-% as long as the \iclrfinalcopy macro remains commented out below.
-% Non-anonymous submissions will be rejected without review.
-
-\author{Taco S. Cohen\thanks{Equal contribution.} \\
-University of Amsterdam
-\And Mario Geiger\footnotemark[1] \\
-EPFL
-\And Jonas K\"ohler\footnotemark[1] \\
-University of Amsterdam
-\And Max Welling \\
-University of Amsterdam \& CIFAR
-}
-
-% The \author macro works with any number of authors. There are two commands
-% used to separate the names and addresses of multiple authors: \And and \AND.
-%
-% Using \And between authors leaves it to \LaTeX{} to determine where to break
-% the lines. Using \AND forces a linebreak at that point. So, if \LaTeX{}
-% puts 3 of 4 authors names on the first line, and the last on the second
-% line, try using \AND instead of \And before the third author name.
-
-\newcommand{\fix}{\marginpar{FIX}}
-\newcommand{\new}{\marginpar{NEW}}
-
-\iclrfinalcopy % Uncomment for camera-ready version, but NOT for submission.
-
-\newcommand{\Z}{\mathbb{Z}}
-\newcommand{\R}{\mathbb{R}}
-\newcommand{\C}{\mathbb{C}}
-
-\newcommand{\GLR}[1]{\ensuremath{\operatorname{GL}(#1, \mathbb{R})}}
-\newcommand{\SLR}[1]{\ensuremath{\operatorname{SL}(#1, \mathbb{R})}}
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-\newcommand{\TODO}[1]{{\color{red}TODO: #1}}
-
-
-\begin{document}
-
-
-\maketitle
-
-\begin{abstract}
-
-   Convolutional Neural Networks (CNNs) have become the method of choice for learning problems involving 2D planar images.
-   However, a number of problems of recent interest have created a demand for models that can analyze spherical images.
-   Examples include omnidirectional vision for drones, robots, and autonomous cars, molecular regression problems, and global weather and climate modelling.
-   A naive application of convolutional networks to a planar projection of the spherical signal is destined to fail, because the space-varying distortions introduced by such a projection will make translational weight sharing ineffective.
-
-   In this paper we introduce the building blocks for constructing spherical CNNs.
-   We propose a definition for the spherical cross-correlation that is both expressive and rotation-equivariant.
-   The spherical correlation satisfies a generalized Fourier theorem, which allows us to compute it efficiently using a generalized (non-commutative) Fast Fourier Transform (FFT) algorithm.
-   We demonstrate the computational efficiency, numerical accuracy, and effectiveness of spherical CNNs applied to 3D model recognition and atomization energy regression.
-
-\end{abstract}
-
-\section{Introduction}
-
-\begin{wrapfigure}{R}{0.32\textwidth}
-    \centering
-    \includegraphics[width=0.25\textwidth]{sphere_projection.pdf}
-    \caption{\label{fig:projection}Any planar projection of a spherical signal will result in distortions. Rotation of a spherical signal cannot be emulated by translation of its planar projection.}
-\end{wrapfigure}
-
-%\footnote[1]{E}
-
-Convolutional networks are able to detect local patterns regardless of their position in the image.
-Like patterns in a planar image, patterns on the sphere can move around, but in this case the ``move'' is a 3D rotation instead of a translation.
-In analogy to the planar CNN, we would like to build a network that can detect patterns regardless of how they are rotated over the sphere.
-
-As shown in Figure \ref{fig:projection}, there is no good way to use translational convolution or cross-correlation\footnote{Despite the name, CNNs typically use  cross-correlation instead of convolution in the forward pass. In this paper we will generally use the term cross-correlation, or correlation for short.} to analyze spherical signals.
-The most obvious approach, then, is to change the definition of cross-correlation by replacing filter translations by rotations.
-Doing so, we run into a subtle but important difference between the plane and the sphere: whereas the space of moves for the plane (2D translations) is itself isomorphic to the plane, the space of moves for the sphere (3D rotations) is a different, \emph{three-dimensional} manifold called $\SO3$\footnote{To be more precise: although the symmetry group of the plane contains more than just translations, the translations form a subgroup that acts on the plane.
-In the case of the sphere there is no coherent way to define a composition for points on the sphere, and so the sphere cannot act on itself (it is not a group). For this reason, we must consider the whole of $\SO3$.}.
-It follows that the result of a spherical correlation (the output feature map) is to be considered a signal on $\SO3$, not a signal on the sphere, $S^2$.
-For this reason, we deploy $\SO3$ group correlation in the higher layers of a spherical CNN \citep{Cohen2016}.
-
-The implementation of a spherical CNN ($S^2$-CNN) involves two major challenges.
-Whereas a square grid of pixels has discrete translation symmetries, no perfectly symmetrical grids for the sphere exist.
-This means that there is no simple way to define the rotation of a spherical filter by one pixel.
-Instead, in order to rotate a filter we would need to perform some kind of interpolation.
-The other challenge is computational efficiency; $\SO3$ is a three-dimensional manifold, so a naive implementation of $\SO3$ correlation is $O(n^6)$.
-
-We address both of these problems using techniques from non-commutative harmonic analysis \citep{Chirikjian2001, Folland1995}.
-This field presents us with a far-reaching generalization of the Fourier transform, which is applicable to signals on the sphere as well as the rotation group.
-It is known that the $\SO3$ correlation satisfies a Fourier theorem with respect to the $\SO3$ Fourier transform, and the same is true for our definition of $S^2$ correlation.
-Hence, the $S^2$ and $\SO3$ correlation can be implemented efficiently using generalized FFT algorithms.
-
-Because we are the first to use cross-correlation on a continuous group inside a multi-layer neural network, we rigorously evaluate the degree to which the mathematical properties predicted by the continuous theory hold in practice for our discretized implementation.
-
-Furthermore, we demonstrate the utility of spherical CNNs for rotation invariant classification and regression problems by experiments on three datasets.
-First, we show that spherical CNNs are much better at rotation invariant classification of Spherical MNIST images than planar CNNs.
-Second, we use the CNN for classifying 3D shapes.
-In a third experiment we use the model for molecular energy regression, an important problem in computational chemistry.
-
-\subsubsection*{Contributions}
-The main contributions of this work are the following:
-\begin{enumerate}
-    \item The theory of spherical CNNs.
-    \item The first automatically differentiable implementation of the generalized Fourier transform for $S^2$ and $\SO3$.
-    Our PyTorch code is easy to use, fast, and memory efficient.
-    \item The first empirical support for the utility of spherical CNNs for rotation-invariant learning problems.
-\end{enumerate}
-
-
-\section{Related Work}
-
-It is well understood that the power of CNNs stems in large part from their ability to exploit (translational) symmetries though a combination of weight sharing and translation equivariance.
-It thus becomes natural to consider generalizations that exploit larger groups of symmetries, and indeed this has been the subject of several recent papers by \cite{Gensa, Olah2014, Dieleman2015, Dieleman2016, Cohen2016, Ravanbakhsh2016, manzil_deepsets, Guttenberg2016, Cohen2017}.
-With the exception of $\SO2$-steerable networks \citep{Worrall2017-gl,Weiler2017-wc}, these networks are all limited to discrete groups, such as discrete rotations acting on planar images or permutations acting on point clouds.
-Other very recent work is concerned with the analysis of spherical images, but does not define an equivariant architecture \citep{Su2017-vm,Boomsma2017-rl}.
-Our work is the first to achieve equivariance to a continuous, non-commutative group ($\SO3$), and the first to use the generalized Fourier transform for fast group correlation.
-A preliminary version of this work appeared as \cite{Cohen17b}.
-
-To efficiently perform cross-correlations on the sphere and rotation group, we use generalized FFT algorithms.
-Generalized Fourier analysis, sometimes called abstract- or noncommutative harmonic analysis, has a long history in mathematics and many books have been written on the subject \citep{Sugiura1976, Taylor1986, Folland1995}.
-For a good engineering-oriented treatment which covers generalized FFT algorithms, see \citep{Chirikjian2001}.
-Other important works include \citep{Driscoll1994, Healy2003, Potts1998, Kunis2003, Drake2008, Maslen1998, Rockmore2004, Kostelec2007, Kostelec2008, Potts2009,Makadia2007-fv,Gutman2008-fa}.
-
-
-\section{Correlation on the Sphere and Rotation Group}
-
-We will explain the $S^2$ and $\SO3$ correlation by analogy to the classical planar $\Z^2$ correlation.
-The planar correlation can be understood as follows:
-\begin{quote}
-The value of the output feature map at translation $x \in \Z^2$ is computed as an inner product between the input feature map and a filter, shifted by $x$.
-\end{quote}
-Similarly, the spherical correlation can be understood as follows:
-\begin{quote}
-The value of the output feature map evaluated at rotation $R \in \SO3$ is computed as an inner product between the input feature map and a filter, rotated by $R$.
-\end{quote}
-Because the output feature map is indexed by a rotation, it is modelled as a function on $\SO3$.
-We will discuss this issue in more detail shortly.
-
-The above definition refers to various concepts that we have not yet defined mathematically.
-In what follows, we will go through the required concepts one by one and provide a precise definition.
-Our goal for this section is only to present a mathematical model of spherical CNNs.
-Generalized Fourier theory and implementation details will be treated later.
-
-{\bf The Unit Sphere $S^2$} can be defined as the set of points $x \in \R^3$ with norm $1$.
-It is a two-dimensional manifold, which can be parameterized by spherical coordinates $\alpha \in [0, 2\pi]$ and $\beta \in [0, \pi]$. % (see Appendix A).
-
-{\bf Spherical Signals} We model spherical images and filters as continuous functions $f : S^2 \rightarrow \R^K$, where $K$ is the number of channels.
-
-{\bf Rotations}
-The set of rotations in three dimensions is called $\SO3$, the ``special orthogonal group''.
-Rotations can be represented by $3 \times 3$ matrices that preserve distance (i.e. $||Rx|| = ||x||$) and orientation ($\det(R) = +1$).
-If we represent points on the sphere as 3D unit vectors $x$, we can perform a rotation using the matrix-vector product $Rx$.
-The rotation group $\SO3$ is a three-dimensional manifold, and can be parameterized by ZYZ-Euler angles $\alpha \in [0, 2\pi]$, $\beta \in [0, \pi]$, and $\gamma \in [0, 2\pi]$. % (see Appendix A).
-
-{\bf Rotation of Spherical Signals}
-In order to define the spherical correlation, we need to know not only how to rotate points $x \in S^2$ but also how to rotate filters (i.e. functions) on the sphere.
-To this end, we introduce the rotation operator $L_R$ that takes a function $f$ and produces a rotated function $L_R f$ by composing $f$ with the rotation $R^{-1}$:
-\begin{equation}
-    \label{eq:L_R_sphere}
-  [L_R f](x) =
-%[f \circ R^{-1}](x) =
-  f(R^{-1} x).
-\end{equation}
-Due to the inverse on $R$, we have $L_{RR'} = L_R L_{R'}$.
-
-{\bf Inner products}
-The inner product on the vector space of spherical signals is defined as:
-\begin{equation}
-  \langle \psi, f \rangle = \int_{S^2} \sum_{k=1}^K \psi_k(x) f_k(x) dx,
-\end{equation}
-The integration measure $dx$ denotes the standard rotation invariant integration measure on the sphere, which can be expressed as $d\alpha \sin(\beta) d\beta / 4\pi$ in spherical coordinates (see Appendix A).
-The invariance of the measure ensures that $\int_{S^2} f(R x) dx = \int_{S^2} f(x) dx$, for any rotation $R \in \SO3$.
-That is, the volume under a spherical heightmap does not change when rotated.
-Using this fact, we can show that $L_{R^{-1}}$ is adjoint to $L_R$, which implies that $L_R$ is unitary:
-\begin{equation}
-  \begin{aligned}
-    \langle L_R \psi, f \rangle
-     &= \int_{S^2} \sum_{k=1}^K \psi_k(R^{-1} x) f_k(x) dx \\
-     &= \int_{S^2} \sum_{k=1}^K \psi_k(x) f_k(R x) dx \\
-     &= \langle \psi, L_{R^{-1}} f \rangle.
-  \end{aligned}
-\end{equation}
-
-{\bf Spherical Correlation}
-With these ingredients in place, we are now ready to state mathematically what was stated in words before.
-For spherical signals $f$ and $\psi$, we define the correlation as:
-\begin{equation}
-  \label{eq:sphere_conv}
-  [\psi \star f](R) = \langle L_R \psi, f \rangle = \int_{S^2} \sum_{k=1}^K \psi_k(R^{-1} x) f_k(x) dx.
-\end{equation}
-As mentioned before, the output of the spherical correlation is a function on $\SO3$.
-This is perhaps somewhat counterintuitive, and indeed the conventional definition of spherical convolution gives as output a function on the sphere.
-However, as shown in Appendix B, the conventional definition effectively restricts the filter to be circularly symmetric about the Z axis, which would greatly limit the expressive capacity of the network.
-
-{\bf Rotation of $\SO3$ Signals}
-We defined the rotation operator $L_R$ for spherical signals (eq. \ref{eq:L_R_sphere}), and used it to define spherical cross-correlation (eq. \ref{eq:sphere_conv}).
-To define the $\SO3$ correlation, we need to generalize the rotation operator so that it can act on signals defined on $\SO3$.
-As we will show, naively reusing eq. \ref{eq:L_R_sphere} is the way to go.
-That is, for $f : \SO3 \rightarrow \R^K$, and $R, Q \in \SO3$:
-\begin{equation}
-    \label{eq:L_R_SO3}
-    [L_R f](Q) = f(R^{-1} Q).
-\end{equation}
-Note that while the argument $R^{-1}x$ in Eq. \ref{eq:L_R_sphere} denotes the rotation of $x \in S^2$ by $R^{-1} \in \SO3$, the analogous term $R^{-1} Q$ in Eq. \ref{eq:L_R_SO3} denotes to the composition of rotations (i.e. matrix multiplication).
-
-
-{\bf Rotation Group Correlation}
-Using the same analogy as before, we can define the correlation of two signals on the rotation group, $f, \psi : \SO3 \rightarrow \R^K$, as follows:
-\begin{equation}
-  \label{eq:SO3_conv}
-  [\psi \star f](R) = \langle L_R \psi, f \rangle = \int_{\SO3} \sum_{k=1}^K \psi_k(R^{-1} Q) f_k(Q) dQ.
-\end{equation}
-The integration measure $dQ$ is the invariant measure on $\SO3$, which may be expressed in ZYZ-Euler angles as $d\alpha \sin(\beta) d\beta d\gamma / (8 \pi^2)$ (see Appendix A).
-
-
-{\bf Equivariance}
-As we have seen, correlation is defined in terms of the rotation operator $L_R$.
-This operator acts naturally on the input space of the network, but what justification do we have for using it in the second layer and beyond?
-
-The justification is provided by an important property, shared by all kinds of convolution and correlation, called equivariance.
-A layer $\Phi$ is equivariant if $\Phi \circ L_R = T_R \circ \Phi$, for some operator $T_R$.
-Using the definition of correlation and the unitarity of $L_R$, showing equivariance is a one liner:
-\begin{equation}
-  \begin{aligned}
-    \lbrack \psi \star [L_{Q} f]](R)
-    = \langle L_R \psi, L_Q f \rangle
-    % = \langle f, L_{Q^{-1}} L_R \psi \rangle
-    = \langle L_{Q^{-1} R} \psi, f \rangle
-    = [\psi \star f](Q^{-1}R)
-    = [L_Q [\psi \star f]](R).
-  \end{aligned}
-\end{equation}
-The derivation is valid for spherical correlation as well as rotation group correlation.
-
-
-\section{Fast Spherical Correlation with G-FFT}
-
-It is well known that correlations and convolutions can be computed efficiently using the Fast Fourier Transform (FFT).
-This is a result of the Fourier theorem, which states that
-$\widehat{f * \psi} = \hat{f} \cdot \hat{\psi}$.
-Since the FFT can be computed in $O(n \log n)$ time and the product $\cdot$ has linear complexity, implementing the correlation using FFTs is asymptotically faster than the naive $O(n^2)$ spatial implementation.
-
-For functions on the sphere and rotation group, there is an analogous transform, which we will refer to as the generalized Fourier transform (GFT) and a corresponding fast algorithm (GFFT).
-This transform finds it roots in the representation theory of groups, but due to space constraints we will not go into details here and instead refer the interested reader to \citet{Sugiura1976} and \citet{Folland1995}.
-
-Conceptually, the GFT is nothing more than the linear projection of a function onto a set of orthogonal basis functions called ``matrix element of irreducible unitary representations''.
-For the circle ($S^1$) or line ($\R$), these are the familiar complex exponentials $\exp(i n \theta)$.
-For $\SO3$, we have the Wigner D-functions $D^l_{mn}(R)$ indexed by $l \geq 0$ and $-l \leq m, n \leq l$.
-For $S^2$, these are the spherical harmonics\footnote{Technically, $S^2$ is not a group and therefore does not have irreducible representations, but it is a quotient of groups $\SO3/\SO2$ and we have the relation $Y^l_m = D^l_{m0}|_{S^2}$} $Y^l_m(x)$ indexed by $l \geq 0$ and $-l \leq m \leq l$.
-
-Denoting the manifold ($S^2$ or $\SO3$) by $X$ and the corresponding basis functions by $U^l$ (which is either vector-valued ($Y^l$) or matrix-valued ($D^l$)), we can write the GFT of a function $f : X \rightarrow \R$ as
-\begin{equation}
-    \hat{f}^l = \int_{X} f(x) \overline{U^l(x)} dx.
-\end{equation}
-This integral can be computed efficiently using a GFFT algorithm (see Section \ref{sec:implementation}).
-
-The inverse $\SO3$ Fourier transform is defined as:
-\begin{equation}
-  f(R) = \sum_{l=0}^b (2 l + 1) \sum_{m = -l}^l \sum_{n = -l}^l \hat{f}^l_{mn} D^l_{mn}(R),
-\end{equation}
-and similarly for $S^2$.
-The maximum frequency $b$ is known as the bandwidth, and is related to the resolution of the spatial grid \citep{Kostelec2007}.
-
-Using the well-known (in fact, defining) property of the Wigner D-functions that $D^l(R)D^l(R') = D^l(RR')$ and $D^l(R^{-1}) = D^l(R)^\dagger$, it can be shown (see Appendix D) that the $\SO3$ correlation satisfies a Fourier theorem\footnote{This result is valid for real functions. For complex functions, conjugate $\psi$ on the left hand side.}: $\widehat{\psi \star f} = \hat{f} \cdot \hat{\psi}^{\dagger}$, where $\cdot$ denotes matrix multiplication of the two block matrices $\hat{f}$ and $\hat{\psi}^\dagger$.
-
-Similarly, using $Y(Rx) = D(R) Y(x)$ and $Y^l_m = D^l_{m0}|_{S^2}$, one can derive an analogous $S^2$ convolution theorem: $\widehat{\psi \star f}^l = \hat{f}^l \cdot \hat{\psi}^{l\dagger}$,
-where $\hat{f}^l$ and $\hat{\psi}^l$ are now vectors.
-This says that the $\SO3$-FT of the $S^2$ correlation of two spherical signals can be computed by taking the outer product of the $S^2$-FTs of the signals.
-This is shown in figure \ref{fig:spectral-conv}.
-
-
-\begin{figure}[h!]
-  \centering
-    \includegraphics[width=.7\textwidth]{s2conv}
-  \caption{Spherical correlation in the spectrum. The signal $f$ and the locally-supported filter $\psi$ are Fourier transformed, block-wise tensored, summed over input channels, and finally inverse transformed. Note that because the filter is locally supported, it is faster to use a matrix multiplication (DFT) than an FFT algorithm for it. We parameterize the sphere using spherical coordinates $\alpha, \beta$, and $\SO3$ with ZYZ-Euler angles $\alpha, \beta, \gamma$.}
-  \label{fig:spectral-conv}
-\end{figure}
-
-
-\subsection{Implementation of G-FFT and Spectral G-Conv}
-\label{sec:implementation}
-Here we sketch the implementation of GFFTs.
-For details, see \citep{Kostelec2007}.
-
-The input of the $\SO3$ FFT is a spatial signal $f$ on $\SO3$, sampled on a discrete grid and stored as a 3D array.
-The axes correspond to the ZYZ-Euler angles $\alpha, \beta, \gamma$.
-The first step of the $\SO3$-FFT is to perform a standard 2D translational FFT over the $\alpha$ and $\gamma$ axes.
-The FFT'ed axes correspond to the $m, n$ axes of the result.
-The second and last step is a linear contraction of the $\beta$ axis of the FFT'ed array with a precomputed array of samples from the Wigner-d (small-d) functions $d^l_{mn}(\beta)$.
-Because the shape of $d^l$ depends on $l$ (it is $(2l+1)\times(2l+1)$),
-this linear contraction is implemented as a custom GPU kernel.
-The output is a set of Fourier coefficients $\hat{f}^l_{mn}$ for $l \geq n, m  \geq -l$ and $l =0, \ldots, L_{\max}$.
-
-The algorithm for the $S^2$-FFTs is very similar, only in this case we FFT over the $\alpha$ axis only, and do a linear contraction with precomputed Legendre functions over the $\beta$ axis.
-
-Our code is available at \url{https://github.com/jonas-koehler/s2cnn}.
-
-
-\section{Experiments}
-
-In a first sequence of experiments, we evaluate the numerical stability and accuracy of our algorithm.
-In a second sequence of experiments, we showcase that the new cross-correlation layers we have introduced are indeed useful building blocks for several real problems involving spherical signals.
-Our examples for this are recognition of 3D shapes and predicting the atomization energy of molecules.
-
-
-\subsection{Equivariance error}
-
-\begin{wrapfigure}{R}{0.5\textwidth}
-    \centering
-    \input{equivariance_error.pgf}
-    \caption{\label{fig:equivariance}$\Delta$ as a function of the resolution and the number of layers.}
-\end{wrapfigure}
-
-In this paper we have presented the first instance of a group equivariant CNN for a continuous, non-commutative group.
-In the discrete case, one can prove that the network is exactly equivariant, but although we can prove $[L_R f] * \psi = L_R [f * \psi]$ for continuous functions $f$ and $\psi$ on the sphere or rotation group, this is not exactly true for the discretized version that we actually compute.
-Hence, it is reasonable to ask if there are any significant discretization artifacts and whether they affect the equivariance properties of the network.
-If equivariance can not be maintained for many layers, one may expect the weight sharing scheme to become much less effective.
-
-We first tested the equivariance of the $\SO3$ correlation at various resolutions $b$.
-We do this by first sampling $n=500$ random rotations $R_i$ as well as $n$ feature maps $f_i$ with $K=10$ channels.
-Then we compute $\Delta = \frac{1}{n}\sum_{i=1}^n \operatorname{std}(L_{R_i} \Phi(f_i) - \Phi(L_{R_i} f_i)) / \operatorname{std}(\Phi(f_i))$, where $\Phi$ is a composition of $\SO3$ correlation layers with randomly initialized filters.
-In case of perfect equivariance, we expect this quantity to be zero.
-The results (figure \ref{fig:equivariance} (top)), show that although the approximation error $\Delta$ grows with the resolution and the number of layers, it stays manageable for the range of resolutions of interest.
-
-We repeat the experiment with ReLU activation function after each correlation operation.
-As shown in figure \ref{fig:equivariance} (bottom), the error is higher but stays flat.
-This indicates that the error is not due to the network layers, but due to the feature map rotation, which is exact only for bandlimited functions.
-
-\subsection{Rotated MNIST on the Sphere}
-
-\label{sec:mnist-experiment}
-
-In this experiment we evaluate the generalization performance with respect to rotations of the input. For testing we propose a version MNIST dataset projected on the sphere (see fig. \ref{fig:mnist_on_sphere}).
-We created two instances of this dataset: one in which each digit is projected on the northern hemisphere and one in which each projected digit is additionally randomly rotated.
-
-\begin{wrapfigure}{R}{0.5\textwidth}
-    \centering
-    \includegraphics[width=.1\textwidth]{sphere4.png}
-    \includegraphics[width=.1\textwidth]{sphere9.png}
-    \includegraphics[width=.1\textwidth]{soft4.jpeg}
-    \includegraphics[width=.1\textwidth]{soft9.jpeg}
-    \caption{\label{fig:mnist_on_sphere} Two MNIST digits projected onto the sphere using stereographic projection. Mapping back to the plane results in non-linear distortions.}
-\end{wrapfigure}
-
-\paragraph{Architecture and Hyperparameters}
-
-As a baseline model, we use a simple CNN with layers conv-ReLU-conv-ReLU-FC-softmax,
-with filters of size $5 \times 5$, $k=32, 64, 10$ channels, and stride $3$ in both layers ($\approx 68$K parameters).
-We compare to a spherical CNN with layers $S^2$conv-ReLU-$\SO3$conv-ReLU-FC-softmax,
-bandwidth $b=30, 10, 6$ and $k=20, 40, 10$ channels ($\approx 58$K parameters).
-
-\paragraph{Results}
-
-We trained each model on the non-rotated (NR) and the rotated (R) training set and evaluated it on the non-rotated and rotated test set.
-See table \ref{tab:mnist-results}.
-While the planar CNN achieves high accuracy in the NR / NR regime, its performance in the R / R regime is much worse, while the spherical CNN is unaffected.
-When trained on the non-rotated dataset and evaluated on the rotated dataset (NR / R), the planar CNN does no better than random chance.
-The spherical CNN shows a slight decrease in performance compared to $R/R$, but still performs very well.
-
-\begin{table}[h]
-  \centering
-  \begin{tabular}{c|ccc}
-    & NR / NR & R / R & NR / R \\
-    \hline
-    planar & 0.98 & 0.23 & 0.11 \\
-    spherical & 0.96 & 0.95 & 0.94 \\
-    \hline
-  \end{tabular}
-  \caption{Test accuracy for the networks evaluated on the spherical MNIST dataset. Here R = rotated, NR = non-rotated and X / Y denotes, that the network was trained on X and evaluated on Y.}
-  \label{tab:mnist-results}
-\end{table}
-
-\subsection{Recognition of 3D Shapes}
-
-Next, we applied $S^2$CNN to 3D shape classification. The SHREC17 task \citep{savva2017shrec17} contains $51300$ 3D models taken from the ShapeNet dataset \citep{chang2015shapenet} which have to be classified into 55 common categories (tables, airplanes, persons, etc.).
-There is a consistently aligned regular dataset and a version in which all models are randomly perturbed by rotations.
-We concentrate on the latter to test the quality of our rotation equivariant representations learned by $S^2$CNN.
-
-\paragraph{Representation}
-We project the 3D meshes onto an enclosing sphere using a straightforward ray casting scheme (see Fig. \ref{fig:shapenet_repr}).
-For each point on the sphere we send a ray towards the origin and collect 3 types of information from the intersection: ray length and $\cos/\sin$ of the surface angle.
-We further augment this information with ray casting information for the convex hull of the model, which in total gives us 6 channels for the signal. This signal is discretized using a Driscoll-Healy grid \citep{Driscoll1994} with bandwidth $b=128$.
-Ignoring non-convexity of surfaces we assume this projection captures enough information of the shape to be useful for the recognition task.
-
-\begin{figure}
-    \centering
-    \includegraphics[width=0.55\textwidth]{raytracing_shapenet.pdf}
-    \caption{\label{fig:shapenet_repr}
-    The ray is cast from the surface of the sphere towards the origin.
-    The first intersection with the model gives the values of the signal.
-    The two images of the right represent two spherical signals in $(\alpha, \beta)$ coordinates.
-    They contain respectively the distance from the sphere and the cosine of the ray with the normal of the model.
-    The red dot corresponds to the pixel set by the red line.}
-\end{figure}
-
-\paragraph{Architecture and Hyperparameters}
-
-Our network consists of an initial $S^2$conv-BN-ReLU block followed by two $\SO3$conv-BN-ReLU blocks.
-The resulting filters are pooled using a max pooling layer followed by a last batch normalization and then fed into a linear layer for the final classification.
-It is important to note that the the max pooling happens over the group SO(3): if $f_k$ is the $k$-th filter in the final layer (a function on SO(3)) the result of the pooling is $\max_{x\in SO(3)} f_k(x)$.
-We used 50, 70, and 350 features for the $S^2$ and the two SO(3) layers, respectively.
-Further, in each layer we reduce the resolution $b$, from 128, 32, 22 to 7 in the final layer.
-Each filter kernel $\psi$ on SO(3) has non-local support, where $\psi(\alpha, \beta, \gamma) \neq 0$ iff $\beta = \frac{\pi}{2}$ and $\gamma = 0$ and the number of points of the discretization is proportional to the bandwidth in each layer.
-The final network contains $\approx1.4$M parameters, takes 8GB of memory at batch size 16, and takes 50 hours to train.
-
-
-\paragraph{Results}
-We evaluated our trained model using the official metrics and compared to the top three competitors in each category (see table \ref{tab:shapenet_results} for results). Except for precision and F1@N, in which our model ranks third, it is the runner up on each other metric.
-The main competitors, Tatsuma\_ReVGG and Furuya\_DLAN use input representations and network architectures that are highly specialized to the SHREC17 task.
-Given the rather task agnostic architecture of our model and the lossy input representation we use, we interpret our models performance as strong empirical support for the effectiveness of Spherical CNNs.
-
-\begin{table}
-    \centering
-    \begin{tabular}{llllllll}
-   Method           & P@N & R@N   & F1@N  & mAP   & NDCG  \\ \hline
-   Tatsuma\_ReVGG   & 0.705  & 0.769 & 0.719 & 0.696 & 0.783 \\ %\hline
-   Furuya\_DLAN     & 0.814 & 0.683 & 0.706 & 0.656 & 0.754 \\ %\hline
-   SHREC16-Bai\_GIFT &0.678& 0.667 & 0.661 & 0.607 & 0.735 \\ %\hline
-   Deng\_CM-VGG5-6DB &0.412& 0.706 & 0.472 & 0.524 & 0.624 \\
-   \hline
-   \textbf{Ours}   & 0.701 (3rd) & 0.711 (2nd) & 0.699 (3rd) & 0.676 (2nd) & 0.756 (2nd)
-    \end{tabular}
-    \caption{\label{tab:shapenet_results} Results and best competing methods for the SHREC17 competition.}
-\end{table}
-
-\subsection{Prediction of Atomization Energies from Molecular Geometry}
-
-Finally, we apply $S^2$CNN on molecular energy regression. In the QM7 task \citep{blum2009gdb13,rupp2012qm7} the atomization energy of molecules has to be predicted from geometry and charges. Molecules contain up to $N=23$ atoms of $T=5$ types (H, C, N, O, S). They are given as a list of positions $p_i$  and charges $z_i$ for each atom $i$.
-
-\paragraph{Representation by Coulomb matrices}
-
-\cite{rupp2012qm7} propose a rotation and translation invariant representation of molecules by defining the \textit{Coulomb matrix} $C \in \mathbb{R}^{N\times N}$ (CM). For each pair of atoms $i \neq j$ they set
-$C_{ij} = (z_i z_j)/(\lvert p_i - p_j \rvert)$ and $C_{ii} = 0.5 z_i^{2.4}$.
-Diagonal elements encode the atomic energy by nuclear charge, while other elements encode Coulomb repulsion between atoms. This representation is not permutation invariant. To this end \cite{rupp2012qm7} propose a distance measure between Coulomb matrices used within Gaussian kernels whereas \cite{montavon2012nips} propose sorting $C$ or random sampling index permutations.
-
-\paragraph{Representation as a spherical signal}
-
-We utilize spherical symmetries in the geometry by defining a sphere $S_i$ around around $p_i$ for each atom $i$.
-The radius is kept uniform across atoms and molecules and chosen minimal such that no intersections among spheres in the training set happen.
-Generalizing the Coulomb matrix approach we define for each possible $z$ and for each point $x$ on $S_i$ potential functions
-$
-U_z(x) = \sum_{j\neq i, z_j = z} \frac{z_i \cdot z}{|x - p_i|}
-$
-producing a $T$ channel spherical signal for each atom in the molecule (see figure \ref{fig:molecule_rendering}).
-This representation is invariant with respect to translations and equivariant with respect to rotations.
-However, it is still not permutation invariant.
-The signal is discretized using a Driscoll-Healy \citep{Driscoll1994} grid with bandwidth $b=10$ representing the molecule as a sparse $N \times T \times 2b \times 2b $ tensor.
-
-\begin{figure}
-\centering
-
-\end{figure}
-
-\begin{figure}
-    \centering
-    \adjincludegraphics[width=0.25\textwidth, trim={{.23\width}  {.25\height}  {.23\width} {.25\height}},clip]{molecule_rendering_U_1.pdf}
-    \adjincludegraphics[width=0.25\textwidth, trim={{.23\width}  {.25\height}  {.23\width} {.25\height}},clip]{molecule_rendering_U_6.pdf}
-    \adjincludegraphics[width=0.25\textwidth, trim={{.23\width}  {.25\height}  {.23\width} {.25\height}},clip]{molecule_rendering_U_7.pdf}
-    \adjincludegraphics[width=0.25\textwidth, trim={{.23\width}  {.25\height}  {.23\width} {.25\height}},clip]{molecule_rendering_U_8.pdf}
-    \adjincludegraphics[width=0.25\textwidth, trim={{.23\width}  {.25\height}  {.23\width} {.25\height}},clip]{molecule_rendering_U_16.pdf}
-    \adjincludegraphics[width=0.25\textwidth, trim={{.23\width}  {.25\height}  {.23\width} {.25\height}},clip]{molecule_rendering_types.pdf}
-    \caption{\label{fig:molecule_rendering}
-        The five potential channels $U_z$ with $z \in \{1,6,7,8,16\}$ for a molecule containing atoms H (red), C (green), N (orange), O (brown), S (gray).}
-\end{figure}
-
-\paragraph{Architecture and Hyperparameters}
-
-We use a deep ResNet style $S^2$CNN.
-Each ResNet block is made of $S^2$/$\SO3$conv-BN-ReLU-$\SO3$conv-BN after which the input is added to the result.
-We share weights among atoms making filters permutation invariant, by pushing the atom dimension into the batch dimension.
-In each layer we downsample the bandwidth, while increasing the number of features $F$.
-After integrating the signal over SO(3) each molecule becomes a $N \times F$ tensor.
-For permutation invariance over atoms we follow \cite{zaheer2017deep} and embed each resulting feature vector of an atom into a latent space using a MLP $\phi$. Then we sum these latent representations over the atom dimension and get our final regression value for the molecule by mapping with another MLP $\psi$. Both $\phi$ and $\psi$ are jointly optimized.
-Training a simple MLP only on the 5 frequencies of atom types in a molecule already gives a RMSE of $\sim$ 19.
-Thus, we train the $S^2$CNN on the residual only, which improved convergence speed and stability over direct training.
-The final architecture is sketched in table $\ref{tab:molecule_results_and_architecture}$.
-It has about 1.4M parameters, consumes 7GB of memory at batch size 20, and takes 3 hours to train.
-
-
-\paragraph{Results}
-
-We evaluate by RMSE and compare our results to \cite{montavon2012nips} and \cite{raj2016local} (see table \ref{tab:molecule_results_and_architecture}). Our learned representation outperforms all kernel-based approaches and a MLP trained on sorted Coulomb matrices. Superior performance could only be achieved for an MLP trained on randomly permuted Coulomb matrices. However, sufficient sampling of random permutations grows exponentially with $N$, so this method is unlikely to scale to large molecules.
-
-\begin{table}
-\centering
-\begin{subtable}[t]{0.49\textwidth}
-    \centering
-    \vspace{0pt}
-    \begin{tabularx}{\textwidth}{Xll}
-       Method  & Author         & RMSE  \\ \hline
-       MLP / random CM & (a)  & 5.96 \\
-       LGIKA(RF) & (b)   & 10.82 \\ %\hline
-       RBF kernels / random CM & (a) & 11.40 \\ %\hline
-       RBF kernels / sorted CM& (a) & 12.59 \\
-       MLP / sorted CM& (a) & 16.06 \\
-       \hline
-       \textbf{Ours} & &\textbf{8.47}
-    \end{tabularx}
-\end{subtable}
-\begin{subtable}[t]{0.5\textwidth}
-    \centering
-    \vspace{0pt}
-    \begin{tabularx}{\textwidth}{llll}
-        $S^2$CNN & Layer & Bandwidth & Features \\ \hline
-        & Input &  & 5 \\
-        & ResBlock & 10 & 20 \\
-        & ResBlock & 8 & 40 \\
-        & ResBlock & 6 & 60 \\
-        & ResBlock & 4 & 80 \\
-        & ResBlock & 2 & 160 \\
-        \hline
-        DeepSet & Layer & Input/Hidden \\\hline
-        & $\phi$ (MLP) & 160/150 \\
-        & $\psi$ (MLP) & 100/50 \\
-        \hline
-    \end{tabularx}
-\end{subtable}
-\caption{Left: \label{tab:molecule_results_and_architecture} Experiment results for the QM7 task: (a) \cite{montavon2012nips} (b) \cite{raj2016local}. Right: ResNet architecture for the molecule task.}
-\end{table}
-
-\section{Discussion \& Conclusion}
-
-In this paper we have presented the theory of Spherical CNNs and evaluated them on two important learning problems.
-We have defined $S^2$ and $\SO3$ cross-correlations, analyzed their properties, and implemented a Generalized FFT-based correlation algorithm.
-Our numerical results confirm the stability and accuracy of this algorithm, even for deep networks.
-Furthermore, we have shown that Spherical CNNs can effectively generalize across rotations, and achieve near state-of-the-art results on competitive 3D Model Recognition and Molecular Energy Regression challenges, without excessive feature engineering and task-tuning.
-
-For intrinsically volumetric tasks like 3D model recognition, we believe that further improvements can be attained by generalizing further beyond $\SO3$ to the roto-translation group $\SE3$.
-The development of Spherical CNNs is an important first step in this direction.
-Another interesting generalization is the development of a Steerable CNN for the sphere \citep{Cohen2017}, which would make it possible to analyze vector fields such as global wind directions, as well as other sections of vector bundles over the sphere.
-
-Perhaps the most exciting future application of the Spherical CNN is in omnidirectional vision.
-Although very little omnidirectional image data is currently available in public repositories, the increasing prevalence of omnidirectional sensors in drones, robots, and autonomous cars makes this a very compelling application of our work.
-
-\bibliography{s2cnn}
-%\bibliographystyle{plain}
-\bibliographystyle{plainnat}
-
-
-
-
-\section*{Appendix A: parameterization of and integration on $S^2$ and $\SO3$}
-
-We use the ZYZ Euler parameterization for $\SO3$.
-An element $R \in \SO3$ is written as
-\begin{equation}
-  R = R(\alpha, \beta, \gamma) = Z(\alpha) Y(\beta) Z(\gamma),
-\end{equation}
-where $\alpha \in [0, 2\pi]$, $\beta \in [0, \pi]$ and $\gamma \in [0, 2\pi]$, and $Z$ resp. $Y$ are rotations around the Z and Y axes.
-
-Using this parameterization, the normalized Haar measure is
-\begin{equation}
-  dR = \frac{d\alpha}{2\pi} \frac{d\beta \sin(\beta)}{2} \frac{d\gamma}{2 \pi}
-\end{equation}
-We have $\int_{\SO3} dR = 1$.
-The Haar measure \citep{Nachbin1965,Chirikjian2001} is sometimes called the invariant measure because it has the property that $\int_{\SO3} f(R'R) dR = \int_{\SO3} f(R) dR$ (this is analogous to the more familiar property $\int_{\R} f(x + y) dx = \int_\R f(x) dx$ for functions on the line).
-This invariance property allows us to do many useful substitutions.
-
-
-We have a related parameterization for the sphere.
-An element $x \in S^2$ is written
-\begin{equation}
-  x(\alpha, \beta) = Z(\alpha)Y(\beta)n
-\end{equation}
-where $n$ is the north pole.
-
-This parameterization makes explicit the fact that the sphere is a quotient $S^2 = \SO3 / \SO2$, where $H = \SO2$ is the subgroup of rotations around the Z axis.
-Elements of this subgroup $H$ leave the north pole invariant, and have the form $Z(\gamma)$.
-The point $x(\alpha, \beta) \in S^2$ is associated with the coset representative $\bar{x} = R(\alpha, \beta, 0) \in \SO3$.
-This element represents the coset $\bar{x}H = \{R(\alpha, \beta, \gamma) | \gamma \in [0, 2\pi] \}$.
-
-The normalized Haar measure for the sphere is
-\begin{equation}
-  dx = \frac{d\alpha}{2\pi} \frac{d\beta \sin \beta}{2}
-\end{equation}
-
-The normalized Haar measure for $\SO2$ is
-\begin{equation}
-  dh = \frac{d\gamma}{2 \pi}
-\end{equation}
-
-So we have $dR = dx \, dh$, again reflecting the quotient structure.
-
-We can think of a function on $S^2$ as a $\gamma$-invariant function on $\SO3$.
-Given a function $f : S^2 \rightarrow \C$ we associate the function $\bar{f}(\alpha, \beta, \gamma) = f(\alpha, \beta)$.
-When using normalized Haar measures, we have:
-\begin{equation}
-  \begin{aligned}
-    \int_{\SO3} \bar{f}(R) dR
-    &=
-    \frac{1}{8 \pi^2}\int_0^{2\pi} d\alpha \int_0^\pi \sin\beta d\beta \int_0^{2\pi} d\gamma \bar{f}(\alpha, \beta, \gamma) \\
-    &=
-    \frac{1}{8 \pi^2}\int_0^{2\pi} d\alpha \int_0^\pi \sin\beta d\beta f(\alpha, \beta) \int_0^{2\pi} d\gamma \\
-    &=
-    \frac{1}{4 \pi}\int_0^{2\pi} d\alpha \int_0^\pi \sin\beta d\beta f(\alpha, \beta) \\
-    &=
-    \int_{S^2} f(x) dx
-  \end{aligned}
-\end{equation}
-This will allow us to define the Fourier transform on $S^2$ from the Fourier transform on $\SO3$, by viewing a function on $S^2$ as a $\gamma$-invariant function on $\SO3$ and taking its $\SO3$-Fourier transform.
-
-
-\section*{Appendix B: Correlation \& Equivariance}
-
-We have defined the $S^2$ correlation as
-\begin{equation}
-  [\psi \star f](R) = \langle L_R \psi, f \rangle = \int_{S^2} \sum_{k=1}^K \psi_k(R^{-1} x) f_k(x) dx.
-\end{equation}
-Without loss of generality, we will analyze here the single-channel case $K=1$.
-
-This operation is equivariant:
-\begin{equation}
-  \begin{aligned}
-    \lbrack \psi \star [L_{Q} f]](R)
-      &=
-    \int_{S^2} \psi(R^{-1} x) f(Q^{-1} x) dx \\
-    &=
-    \int_{S^2} \psi(R^{-1} Q x) f(x) dx \\
-    &=
-    \int_{S^2} \psi((Q^{-1} R)^{-1} x) f(x) dx \\
-    &=
-    [\psi \star f](Q^{-1} R) \\
-    &=
-    [L_{Q}[\psi \star f]](R)
-  \end{aligned}
-\end{equation}
-A similar derivation can be made for the $\SO3$ correlation.
-
-The spherical convolution defined by \cite{Driscoll1994} is:
-\begin{equation}
-  [f * \psi](x) = \int_{\SO3} f(R n) \psi(R^{-1} x) dR
-\end{equation}
-where $n$ is the north pole.
-Note that in this definition, the output of the spherical convolution is a function on the sphere, not a function on $\SO3$ as in our definition of cross-correlation.
-Note further that unlike our definition, this definition involves an integral over $\SO3$.
-
-If we write out the integral in terms of Euler angles, noting that the north-pole $n$ is invariant to $Z$-axis rotations by $\gamma$, i.e. $R(\alpha, \beta, \gamma)n = Z(\alpha) Y(\beta) Z(\gamma) n = Z(\alpha) Y(\beta) n$, we see that this definition implicitly integrates over $\gamma$ in only one of the factors (namely $\psi$), making it invariant wrt $\gamma$ rotation.
-In other words, the filter is first ``averaged'' (making it circularly symmetric) before it is combined with $f$ (This was observed before by \cite{Makadia2007-fv}).
-We consider this to be much too limited for the purpose of pattern matching in spherical CNNs.
-
-
-\section*{Appendix C: Generalized Fourier Transform}
-
-With each compact topological group (like $\SO3$) is associated a discrete set of orthogonal functions that arise as matrix elements of irreducible unitary representations of these groups.
-For the circle (the group $\SO2$) these are the complex exponentials (in the complex case) or sinusoids (for real functions).
-For $\SO3$, these functions are known as the Wigner D-functions.
-
-As discussed in the paper, the Wigner D-functions are parameterized by a degree parameter $l \geq 0$ and order parameters $m, n \in [-l, \ldots, l]$.
-In other words, we have a set of matrix-valued functions $D^l : \SO3 \rightarrow \C^{(2l+1) \times (2l+1)}$.
-
-The Wigner D-functions are orthogonal:
-\begin{equation}
-  \langle D^l_{mn}, D^{l'}_{m'n'} \rangle = \int_0^{2\pi} \frac{d\alpha}{2\pi} \int_0^\pi \frac{d\beta \sin\beta}{2} \int_0^{2\pi} \frac{d\gamma}{2\pi} D^l_{mn}(\alpha, \beta, \gamma) \overline{D^{l'}_{m'n'}}(\alpha, \beta, \gamma)
-  = \frac{\delta_{ll'} \delta_{mm'} \delta_{nn'}}{2l+1}
-\end{equation}
-
-Furthermore, they are complete, meaning that any well behaved function $f : \SO3 \rightarrow \C$ can be written as a linear combination of Wigner D-functions.
-This is the idea of the Generalized Fourier Transform $\mathcal{F}$ on $\SO3$:
-\begin{equation}
-    f(R) = [\mathcal{F}^{-1} \hat{f}](R) = \sum_{l=0}^\infty (2l+1) \sum_{m = -l}^l \sum_{n=-l}^l \hat{f}^l_{mn} D^l_{mn}(R)
-\end{equation}
-where $\hat{f}^l_{mn}$ are called the Fourier coefficients of $f$.
-Using the orthogonality property of the Wigner D-functions, one can see that the Fourier coefficients can be retrieved by computing the inner product with the Wigner D-functions:
-
-\begin{equation}
-  \begin{aligned}
-    %\hat{f}^l_{mn}
-    %&=
-    \lbrack \mathcal{F} f]^l_{mn} &= \int_{\SO3} f(R) \overline{D^l_{mn}(R)} dR \\
-    &=
-    \int_{\SO3} \left[ \sum_{l'=0}^\infty (2l'+1) \sum_{m' = -l'}^{l'} \sum_{n'=-l'}^{l'} \hat{f}^{l'}_{m'n'} D^{l'}_{m'n'}(R)\right] \overline{D^l_{mn}(R)} dR \\
-    &=
-    \sum_{l'=0}^\infty (2l'+1) \sum_{m' = -l'}^{l'} \sum_{n'=-l'}^{l'} \hat{f}^{l'}_{m'n'} \int_{\SO3} D^{l'}_{m'n'}(R) \overline{D^l_{mn}} dR \\
-    &=
-    \hat{f}^l_{mn}
-    %\frac{1}{8 \pi^2} \int_0^{2\pi} d\alpha \int_0^\pi \sin\beta d\beta \int_0^{2\pi} d\gamma f(\alpha, \beta, \gamma) \overline{D^l_{mn}(\alpha, \beta, \gamma)}
-  \end{aligned}
-\end{equation}
-
-
-\section*{Appendix D: Fourier Theorems}
-
-Fourier convolution theorems for $\SO3$ and $\S^2$ can be found in \cite{Kostelec2008,Makadia2007-fv,Gutman2008-fa}.
-We derive them here for completeness. 
-
-To derive the convolution theorems, we will use the defining property of the Wigner D-matrices: that they are (irreducible, unitary) \emph{representations} of $\SO3$.
-This means that they satisfy:
-\begin{equation}
-  D^l(R) D^l(R') = D^l(RR'),
-\end{equation}
-for any $R, R' \in \SO3$.
-Notice that the complex exponentials satisfy an analogous criterion for the circle group $S^1 \cong \SO2$. That is, $e^{inx} e^{iny} = e^{in(x+y)}$, where $x+y$ is the group operation for $\SO2$.
-
-Unitarity means that $D^l(R) D^{l\dagger}(R) = I$.
-Irreducibility means, essentially, that the set of matrices $\{D^l(R) \, | \, R \in \SO3 \}$ cannot be simultaneously block-diagonalized.
-
-To derive the Fourier theorem for $\SO3$, we use the invariance of the integration measure $dR$:
-%In a planar integral we have $\int_{R^2} f(x - y) dx = \int_{R^2} f(x) dx$.
- $\int_{\SO3} f(R' R) dR = \int_{\SO3} f(R) dR$.
-
-With these facts understood, we can proceed to derive:
-\begin{equation}
-  \begin{aligned}
-    \widehat{\psi \star f}^l
-    &=
-    \int_{\SO3} (\psi \star f)(R) \overline{D^l(R)} dR \\
-    &=
-    \int_{\SO3} \int_{\SO3} \psi(R^{-1} R') f(R') dR' \overline{D^l(R)} dR \\
-    &=
-    \int_{\SO3} \int_{\SO3} \psi(R^{-1}) f(R') \overline{D^l(R'R)} dR' dR \\
-    &=
-    \int_{\SO3} f(R') \overline{D^l(R')} dR' \;  \int_{\SO3} \psi(R^{-1}) \overline{D^l(R)} dR \\
-    &=
-    \int_{\SO3} f(R') \overline{D^l(R')} dR' \;  \int_{\SO3} \psi(R) \overline{D^l(R)}^\dagger dR \\
-    &=
-    \hat{f}^l \; \hat{\psi}^{l \dagger}
-  \end{aligned}
-\end{equation}
-
-So the $\SO3$-Fourier transform of the $\SO3$ convolution of $\psi$ and $f$ is equal to the matrix product of the $\SO3$-Fourier transforms $\hat{f}$ and $\hat{\psi}$.
-
-For the sphere, we can derive an analogous transform that is sometimes called the spherical harmonics transform.
-The spherical harmonics $Y^l_m : S^2 \rightarrow \C$ are a complete orthogonal family of functions.
-The spherical harmonics are related to the Wigner D functions by the relation $D^l_{mn}(\alpha, \beta, \gamma) = Y^l_m(\alpha,\beta) e^{in\gamma}$, so that $Y^l_m(\alpha, \beta) = D^l_{m0}(\alpha, \beta, 0)$.
-
-The $S^2$ convolution of $f_1$ and $f_2$ is equivalent to the $\SO3$ convolution of the associated right-invariant functions $\bar{f_1}, \bar{f_2}$ (see Appendix A):
-\begin{equation}
-  \begin{aligned}
-    \lbrack f_1 \star f_2 ] (R)
-    &=
-    \int_{S^2} f_1(R^{-1} x) f_2(x) dx \\
-    &=
-    \int_{\SO2} \int_{S^2} f_1(R^{-1} x) f_2(x) dx dh \\
-    &=
-    \int_{\SO3} \bar{f_1}(R^{-1} R') \bar{f_2}(R') dR' \\
-    &=
-    [\bar{f_1} \star \bar{f_2}](R)
-  \end{aligned}
-\end{equation}
-
-The Fourier transform of a right invariant function on $\SO3$ equals
-\begin{equation}
-  \begin{aligned}
-    \lbrack \mathcal{F} \bar{f}]^l_{mn}
-      &=
-       \int_0^{2\pi} \frac{d\alpha}{2\pi} \int_0^\pi \frac{d\beta \sin \beta}{2} \int_0^{2\pi} \frac{d\gamma}{2\pi} \bar{f}(\alpha, \beta, \gamma) \overline{D^l_{mn}(\alpha, \beta, \gamma)} \\
-      &=
-       \int_0^{2\pi} \frac{d\alpha}{2\pi} \int_0^\pi \frac{d\beta \sin \beta}{2} f(\alpha, \beta) \int_0^{2\pi} \frac{d\gamma}{2\pi} \overline{D^l_{mn}(\alpha, \beta, \gamma)} \\
-      &=
-       \delta_{n0} \int_0^{2\pi} \frac{d\alpha}{2\pi} \int_0^\pi \frac{d\beta \sin \beta}{2} f(\alpha, \beta) \overline{D^l_{m0}(\alpha, \beta, 0)} \\
-      &=
-       \delta_{n0} \int_{S^2} f(x) \overline{Y^l_m(x)} dx
-    %\int_{\SO3} \bar{f}(g) \overline{U^l_{mn}(g)} dg \\
-    %&=
-      %\int_{S^2} \bar{f}(\bar{x}) \int_{\SO2} \overline{U^l_{mn}(\bar{x} h)} dx dh
-  \end{aligned}
-\end{equation}
-So we can think of the $S^2$ Fourier transform of a function on $S^2$ as the $n=0$ column of the $\SO3$ Fourier transform of the associated right-invariant function.
-This is a beautiful result that we have not been able to find a reference for, though it seems likely that it has been observed before.
-
-\end{document}
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-# The Inversion Gap for the Octahedral Group
-
-**Target audience**: Authors of *Selective G-Bispectrum* (Mataigne et al., NeurIPS 2024)
-
-## Summary
-
-The sequential bootstrap algorithm (Algorithms 1–3 in the paper) correctly inverts the selective bispectrum for $C_n$, commutative groups, and $D_n$. When applied to the octahedral group $O$ (Section 8.1 of the paper), the same procedure **fails at recovering $\\mathcal{F}\_{\\rho_2}$ and $\\mathcal{F}\_{\\rho_3}$** from $\\beta\_{\\rho_1,\\rho_1}$. The bispectrum itself is fine — the 172-coefficient selective $O$-Bispectrum is complete. What breaks is the *algorithm* for reconstructing the Fourier coefficients.
-
-This document explains precisely why.
-
-## The bootstrap strategy (recap)
-
-All three algorithms share the same skeleton:
-
-1. Recover $\\mathcal{F}\_{\\rho_0}$ from $\\beta\_{\\rho_0,\\rho_0} = \\mathcal{F}\_{\\rho_0}^3$.
-2. Recover $\\mathcal{F}\_{\\rho_1}$ from $\\frac{\\beta\_{\\rho_0,\\rho_1}}{\\mathcal{F}\_{\\rho_0}} = \\mathcal{F}\_{\\rho_1}^\\dagger \\mathcal{F}\_{\\rho_1}$, up to a unitary/orthogonal ambiguity $Q$.
-3. Extract higher Fourier coefficients from subsequent bispectral coefficients using CG matrices to block-diagonalize tensor products.
-
-Step 2 always introduces an ambiguity: $\\mathcal{F}\_{\\rho_1} = S \\cdot Q$ where $S = (\\mathcal{F}\_{\\rho_1}^\\dagger \\mathcal{F}\_{\\rho_1})^{1/2}$ is determined, but $Q$ is a free orthogonal matrix. For the algorithm to work, this ambiguity must not corrupt step 3.
-
-## Why it works for $C_n$ and $D_n$
-
-### $C_n$: Phase ambiguity in $\\mathrm{SO}(2)$
-
-The irreps are 1D: $\\rho_k(g) = e^{2\\pi i k g / n}$. The ambiguity in $\\mathcal{F}\_{\\rho_1}$ is a phase $e^{i\\varphi}$, $\\varphi \\in \[0, 2\\pi)$. Step 3 computes:
-
-$$\\mathcal{F}\_{\\rho\_{k+1}} = \\left(\\frac{\\beta\_{\\rho_1, \\rho_k}}{\\mathcal{F}\_{\\rho_1} \\mathcal{F}\_{\\rho_k}}\\right)^\\dagger$$
-
-The phase $e^{i\\varphi}$ propagates coherently through all coefficients as $e^{i\\varphi k}$, which corresponds to a continuous shift $\\Theta(g) \\to \\Theta(g + \\varphi)$. The paper correctly identifies this (Section 5, Appendix A) and resolves it by finding the unique $\\varphi$ that makes the reconstructed signal real. No information is lost.
-
-### $D_n$: $\\mathrm{O}(2)$ ambiguity absorbed by CG matrices
-
-The generating irrep $\\rho_1$ is 2D. Recovery gives $\\mathcal{F}\_{\\rho_1} = V\\Lambda^{1/2}V^\\dagger U$ with $U \\in \\mathrm{O}(2)$. Step 3 extracts higher coefficients via:
-
-$$\\bigoplus\_{\\rho \\in \\rho_1 \\otimes \\rho\_{k-1}} \\mathcal{F}\_\\rho = \\left(C\_{\\rho_1,\\rho\_{k-1}}^\\dagger \\left[\\mathcal{F}\_{\\rho_1} \\otimes \\mathcal{F}\_{\\rho\_{k-1}}\\right]^{-1} \\beta\_{\\rho_1,\\rho\_{k-1}} C\_{\\rho_1,\\rho\_{k-1}}\\right)^\\dagger$$
-
-This extraction relies on the CG matrix $C$ block-diagonalizing the tensor product. The implicit question is: does the unknown $U$ corrupt the extraction?
-
-**It does not**, because the 2D irreps of $D_n$ are restrictions of $\\mathrm{SO}(2)$ irreps. The CG matrices for $D_n$ therefore block-diagonalize $\\rho(U)$ for *all* $U \\in \\mathrm{O}(2)$, not just $U \\in D_n$. Concretely, $C^\\dagger(U \\otimes U)C$ remains block-diagonal for any $U \\in \\mathrm{O}(2)$. The $\\mathrm{O}(2)$ ambiguity passes cleanly through the CG extraction.
-
-## Why it fails for $O$
-
-### The $\\mathrm{SO}(3)$ ambiguity is not absorbed
-
-For the octahedral group, $\\rho_1$ is the standard 3D representation. From $\\beta\_{\\rho_0, \\rho_1}$ we recover:
-
-$$\\mathcal{F}\_{\\rho_1}^\\top \\mathcal{F}\_{\\rho_1} = \\frac{\\beta\_{\\rho_0,\\rho_1}}{\\mathcal{F}\_{\\rho_0}}$$
-
-Taking the symmetric square root gives $S = (\\mathcal{F}\_{\\rho_1}^\\top \\mathcal{F}\_{\\rho_1})^{1/2}$, with the true coefficient being $\\mathcal{F}\_{\\rho_1} = Q \\cdot S$ for some unknown $Q \\in \\mathrm{O}(3)$.
-
-The next step extracts $\\mathcal{F}\_{\\rho_2}$ and $\\mathcal{F}\_{\\rho_3}$ from $\\beta\_{\\rho_1, \\rho_1}$ using the CG matrix $C\_{11}$ that block-diagonalizes $\\rho_1 \\otimes \\rho_1$:
-
-$$C\_{11}^\\top \\cdot \\beta\_{\\rho_1,\\rho_1} \\cdot (S \\otimes S)^{-1} \\cdot C\_{11} = C\_{11}^\\top (Q \\otimes Q) C\_{11} \\cdot \\bigoplus_k \\mathcal{F}\_{\\rho_k}^\\top$$
-
-The CG matrix $C\_{11}$ is constructed to block-diagonalize $\\rho_1(g) \\otimes \\rho_1(g)$ for $g \\in O$. That is:
-
-$$C\_{11}^\\top (\\rho_1(g) \\otimes \\rho_1(g)) C\_{11} = \\rho_0(g) \\oplus \\rho_1(g) \\oplus \\rho_2(g) \\oplus \\rho_3(g)$$
-
-for all $g \\in O$. But $Q$ is **not** constrained to be an element of $O$. It is an arbitrary element of $\\mathrm{SO}(3)$, which is a continuous group with 3 parameters, while $O$ has only 24 elements. For a generic $Q \\in \\mathrm{SO}(3) \\setminus O$, the matrix $C\_{11}^\\top (Q \\otimes Q) C\_{11}$ is **not block-diagonal**. It has cross-block leakage between the $\\rho_2$ and $\\rho_3$ sectors.
-
-### Cross-block leakage: the concrete failure
-
-The Kronecker decomposition of $\\rho_1 \\otimes \\rho_1$ for $O$ is:
-
-$$\\rho_1 \\otimes \\rho_1 = \\rho_0 \\oplus \\rho_1 \\oplus \\rho_2 \\oplus \\rho_3$$
-
-with dimensions $1 + 3 + 3 + 2 = 9$. The CG matrix $C\_{11}$ is $9 \\times 9$ orthogonal. In this basis, the diagonal blocks are:
-
-| Block | Rows | Irrep                   | Dim |
-| ----- | ---- | ----------------------- | --- |
-| 1     | 0    | $\\rho_0$ (trivial)     | 1   |
-| 2     | 1–3  | $\\rho_1$ (standard 3D) | 3   |
-| 3     | 4–6  | $\\rho_2$ (product 3D)  | 3   |
-| 4     | 7–8  | $\\rho_3$ (2D)          | 2   |
-
-When we compute $C\_{11}^\\top (Q \\otimes Q) C\_{11}$ for a **group element** $Q = \\rho_1(g)$, $g \\in O$, the off-diagonal blocks (e.g., rows 4–6 × columns 7–8) are zero. This is exactly what the CG matrix guarantees.
-
-When $Q$ is a generic rotation, e.g.,
-
-$$Q = R_x(\\theta) = \\begin{pmatrix} 1 & 0 & 0 \\ 0 & \\cos\\theta & -\\sin\\theta \\ 0 & \\sin\\theta & \\cos\\theta \\end{pmatrix}, \\quad \\theta = 0.7 \\text{ (not a multiple of } \\pi/2\\text{)}$$
-
-then $C\_{11}^\\top (Q \\otimes Q) C\_{11}$ has nonzero entries in the $\\rho_2$–$\\rho_3$ cross-block. Numerically, for $\\theta = 0.7$:
-
-$$|[C\_{11}^\\top (Q \\otimes Q) C\_{11}]\_{4{:}7, ; 7{:}9}|\_\\infty \\approx 0.788$$
-
-This is not a small perturbation — it is $O(1)$. When we try to extract $\\mathcal{F}\_{\\rho_2}$ from the rows 4–6 diagonal block, we get a mixture of $\\rho_2$ and $\\rho_3$ contributions. The recovered Fourier coefficients are wrong.
-
-### Why this doesn't happen for $D_n$
-
-The 2D irreps of $D_n$ are of the form $\\rho_k(a^j) = \\begin{pmatrix} \\cos(2\\pi jk/n) & -\\sin(2\\pi jk/n) \\ \\sin(2\\pi jk/n) & \\cos(2\\pi jk/n) \\end{pmatrix}$. These are restrictions of $\\mathrm{SO}(2)$ representations to $\\mathbb{Z}/n\\mathbb{Z}$.
-
-The CG decomposition of $\\rho_j \\otimes \\rho_k$ for $D_n$ has a special property: the CG matrices also block-diagonalize $R\_\\theta \\otimes R\_\\theta$ for **any** $R\_\\theta \\in \\mathrm{SO}(2)$. This is because the tensor product of two $\\mathrm{SO}(2)$ irreps decomposes as $\\rho_j \\otimes \\rho_k = \\rho\_{j+k} \\oplus \\rho\_{|j-k|}$, and this decomposition holds for the full continuous group, not just the finite subgroup. The CG matrices are inherited from $\\mathrm{SO}(2)$ and respect arbitrary rotations.
-
-For $O \\subset \\mathrm{SO}(3)$, the situation is fundamentally different. The irreps of $O$ are **not** restrictions of $\\mathrm{SO}(3)$ irreps (except $\\rho_1$). The CG matrices are specific to the 24-element group and do not block-diagonalize arbitrary $\\mathrm{SO}(3)$ rotations.
-
-## Connection to the paper's own remarks
-
-The paper acknowledges a related issue in the commutative group setting (Appendix C, around Eq. 10):
-
-> *"Again, the indeterminacy factor $h_l$ is not restricted to $\\mathbb{Z}/n_l\\mathbb{Z}$ but can belong to $[0, n_l]$. We will have to solve this issue further."*
-
-This is the same phenomenon in a different guise. For commutative groups, the continuous indeterminacy is scalar (a phase) and can be resolved by enforcing real-valuedness. For $O$, the continuous indeterminacy is a $3 \\times 3$ rotation matrix $Q \\in \\mathrm{SO}(3)$, and there is no analogous closed-form trick to pin it down.
-
-The octahedral section (Section 8.1) says:
-
-> *"We apply the procedure from Algorithms 1, 2 and 3 to Table 6."*
-
-and states that $\\mathcal{F}\_{\\rho_1}$ is recovered "up to an indeterminacy which is a transformation in $\\mathrm{O}(3)$." But the subsequent extraction of $\\mathcal{F}\_{\\rho_2}, \\mathcal{F}\_{\\rho_3}$ implicitly assumes this $\\mathrm{O}(3)$ ambiguity is absorbed by the CG matrices — which, as shown above, it is not.
-
-## What works instead
-
-The selective $O$-Bispectrum **is** complete (Theorem 5 of the paper). The 172 coefficients uniquely determine the signal up to $O$-action. The issue is purely algorithmic: the sequential bootstrap doesn't produce the right answer because of cross-block contamination.
-
-We have implemented a working inversion using:
-
-1. **Bootstrap initialization**: recover $\\mathcal{F}\_{\\rho_0}$ exactly, $\\mathcal{F}\_{\\rho_1}$ up to $Q$, and approximate $\\mathcal{F}\_{\\rho_2}, \\mathcal{F}\_{\\rho_3}, \\mathcal{F}\_{\\rho_4}$ from diagonal blocks (ignoring leakage). This gives a decent starting point.
-2. **Levenberg-Marquardt correction**: use forward-mode autodiff through the bispectrum computation to solve $|\\beta(f\_\\text{rec}) - \\beta\_\\text{target}|^2 = 0$ via damped least-squares steps. Each step is a single linear solve, not iterative optimization.
-3. **Multi-start**: randomize the initial $Q$ across several restarts to escape local minima.
-
-This converges reliably and is fully differentiable (compatible with `torch.func.jacfwd`).
-
-## Numerical disproof (code)
-
-The following code demonstrates the cross-block leakage. It constructs the CG matrix $C\_{11}$ for $\\rho_1 \\otimes \\rho_1$, applies a generic $Q \\in \\mathrm{SO}(3)$, and shows the off-diagonal blocks are nonzero.
-
-```python
-import torch
-from bispectrum.o_on_r3 import OonR3
-
-bsp = OonR3()
-C11 = bsp._cg_11.to(torch.float64)
-
-# Generic SO(3) rotation (not in O)
-theta = 0.7
-Q = torch.tensor([
-    [1.0, 0.0, 0.0],
-    [0.0, torch.cos(torch.tensor(theta)), -torch.sin(torch.tensor(theta))],
-    [0.0, torch.sin(torch.tensor(theta)),  torch.cos(torch.tensor(theta))],
-], dtype=torch.float64)
-
-QQ = torch.kron(Q, Q)
-M = C11.T @ QQ @ C11
-
-# rho2 block: rows 4:7, rho3 block: rows 7:9
-cross_block = M[4:7, 7:9]
-print(f"Cross-block (rho2 x rho3) max entry: {cross_block.abs().max():.6f}")
-# Output: ~0.788 — far from zero
-
-# Compare with a group element
-rho1_g4 = bsp._get_irrep_mats(1)[4].to(torch.float64)
-QQ_group = torch.kron(rho1_g4, rho1_g4)
-M_group = C11.T @ QQ_group @ C11
-cross_block_group = M_group[4:7, 7:9]
-print(f"Cross-block for g∈O: {cross_block_group.abs().max():.6f}")
-# Output: ~0.0 (machine precision)
-```
-
-## Summary table
-
-| Group | Generating irrep dim | Ambiguity in $\\mathcal{F}\_{\\rho_1}$ | CG absorbs it?                  | Bootstrap works? |
-| ----- | -------------------- | -------------------------------------- | ------------------------------- | ---------------- |
-| $C_n$ | 1                    | $e^{i\\varphi} \\in \\mathrm{SO}(2)$   | N/A (scalar)                    | Yes              |
-| $D_n$ | 2                    | $U \\in \\mathrm{O}(2)$                | Yes (CG from $\\mathrm{SO}(2)$) | Yes              |
-| $O$   | 3                    | $Q \\in \\mathrm{O}(3)$                | **No**                          | **No**           |
diff --git a/docs/lantow_miolane_2025/00README.json b/docs/lantow_miolane_2025/00README.json
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--- a/docs/lantow_miolane_2025/00README.json
+++ /dev/null
@@ -1,21 +0,0 @@
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diff --git a/docs/lantow_miolane_2025/arXiv-2511.19706v1.tar.gz b/docs/lantow_miolane_2025/arXiv-2511.19706v1.tar.gz
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diff --git a/docs/lantow_miolane_2025/cvpr.sty b/docs/lantow_miolane_2025/cvpr.sty
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-  % Add the numbers
-  \linenumbers
-  \AtBeginDocument{%
-    \linenomathpatch{equation}%
-    \linenomathpatchAMS{gather}%
-    \linenomathpatchAMS{multline}%
-    \linenomathpatchAMS{align}%
-    \linenomathpatchAMS{alignat}%
-    \linenomathpatchAMS{flalign}%
-  }
-
-  % \makevruler[<SCALE>][<INITIAL_COUNT>][<STEP>][<DIGITS>][<HEIGHT>]
-  \def\cvprruler#1{\makevruler[12pt][#1][1][3][0.993\textheight]\usebox{\cvprrulerbox}}
-  \AddToShipoutPicture{%
-    \color[rgb]{.5,.5,1}
-
-    \def\pid{\parbox{1in}{\begin{center}\bf\sf{\small \confName}\\\small \#\paperID\end{center}}}
-    \AtTextUpperLeft{%paperID in corners
-      \put(\LenToUnit{-65pt},\LenToUnit{45pt}){\pid}
-      \put(\LenToUnit{\textwidth-12pt},\LenToUnit{45pt}){\pid}
-    }
-    \AtTextUpperLeft{%confidential
-      \put(0,\LenToUnit{1cm}){\parbox{\textwidth}{\centering\cvprtenhv
-      \confName~\confYear~Submission \#\paperID. CONFIDENTIAL REVIEW COPY.  DO NOT DISTRIBUTE.}}
-    }
-  }
-} % end of not cvprfinal
-
-%%% Make figure placement a little more predictable.
-% We trust the user to move figures if this results
-% in ugliness.
-% Minimize bad page breaks at figures
-\renewcommand{\textfraction}{0.01}
-\renewcommand{\floatpagefraction}{0.99}
-\renewcommand{\topfraction}{0.99}
-\renewcommand{\bottomfraction}{0.99}
-\renewcommand{\dblfloatpagefraction}{0.99}
-\renewcommand{\dbltopfraction}{0.99}
-\setcounter{totalnumber}{99}
-\setcounter{topnumber}{99}
-\setcounter{bottomnumber}{99}
-
-% Add a period to the end of an abbreviation unless there's one
-% already, then \xspace.
-\makeatletter
-\DeclareRobustCommand\onedot{\futurelet\@let@token\@onedot}
-\def\@onedot{\ifx\@let@token.\else.\null\fi\xspace}
-
-\def\eg{\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot}
-\def\ie{\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot}
-\def\cf{\emph{cf}\onedot} \def\Cf{\emph{Cf}\onedot}
-\def\etc{\emph{etc}\onedot} \def\vs{\emph{vs}\onedot}
-\def\wrt{w.r.t\onedot} \def\dof{d.o.f\onedot}
-\def\iid{i.i.d\onedot} \def\wolog{w.l.o.g\onedot}
-\def\etal{\emph{et al}\onedot}
-\makeatother
-
-% ---------------------------------------------------------------
-
-%% redefine the \title command so that a variable name is saved in \thetitle, and provides the \maketitlesupplementary command
-\let\titleold\title
-\renewcommand{\title}[1]{\titleold{#1}\newcommand{\thetitle}{#1}}
-\def\maketitlesupplementary
-   {
-   \newpage
-       \twocolumn[
-        \centering
-        \Large
-        \textbf{\thetitle}\\
-        \vspace{0.5em}Supplementary Material \\
-        \vspace{1.0em}
-       ] %< twocolumn
-   }
-
-% ---------------------------------------------------------------
-
-%% Support for easy cross-referencing (e.g. \cref{sec:intro}
-% configured with \AtEndPreamble as it needs to be called after hyperref
-\AtEndPreamble{
-    \usepackage[capitalize]{cleveref}
-    \crefname{section}{Sec.}{Secs.}
-    \Crefname{section}{Section}{Sections}
-    \Crefname{table}{Table}{Tables}
-    \crefname{table}{Tab.}{Tabs.}
-}
-
-% ---------------------------------------------------------------
-
-%% More compact compact itemize/enumeration (e.g. list contributions)
-\RequirePackage[shortlabels,inline]{enumitem}
-\setlist[itemize]{noitemsep,leftmargin=*,topsep=0em}
-\setlist[enumerate]{noitemsep,leftmargin=*,topsep=0em}
diff --git a/docs/lantow_miolane_2025/figs/bispectrum_tractability.pdf b/docs/lantow_miolane_2025/figs/bispectrum_tractability.pdf
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diff --git a/docs/lantow_miolane_2025/figs/mra_results.pdf b/docs/lantow_miolane_2025/figs/mra_results.pdf
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diff --git a/docs/lantow_miolane_2025/figs/overview_figure.pdf b/docs/lantow_miolane_2025/figs/overview_figure.pdf
deleted file mode 100644
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diff --git a/docs/lantow_miolane_2025/ieeenat_fullname.bst b/docs/lantow_miolane_2025/ieeenat_fullname.bst
deleted file mode 100644
index c71be58..0000000
--- a/docs/lantow_miolane_2025/ieeenat_fullname.bst
+++ /dev/null
@@ -1,1448 +0,0 @@
-%% File: `abbrvnat.bst'
-%% A modification of `abbrv.bst' for use with natbib package
-%%
-%% Copyright 1993-2007 Patrick W Daly
-%% Max-Planck-Institut f\"ur Sonnensystemforschung
-%% Max-Planck-Str. 2
-%% D-37191 Katlenburg-Lindau
-%% Germany
-%% E-mail: daly@mps.mpg.de
-%%
-%% This program can be redistributed and/or modified under the terms
-%% of the LaTeX Project Public License Distributed from CTAN
-%% archives in directory macros/latex/base/lppl.txt; either
-%% version 1 of the License, or any later version.
-%%
- % Version and source file information:
- % \ProvidesFile{natbst.mbs}[2007/11/26 1.93 (PWD)]
- %
- % BibTeX `plainnat' family
- %   version 0.99b for BibTeX versions 0.99a or later,
- %   for LaTeX versions 2.09 and 2e.
- %
- % For use with the `natbib.sty' package; emulates the corresponding
- %   member of the `plain' family, but with author-year citations.
- %
- % With version 6.0 of `natbib.sty', it may also be used for numerical
- %   citations, while retaining the commands \citeauthor, \citefullauthor,
- %   and \citeyear to print the corresponding information.
- %
- % For version 7.0 of `natbib.sty', the KEY field replaces missing
- %   authors/editors, and the date is left blank in \bibitem.
- %
- % Includes field EID for the sequence/citation number of electronic journals
- %  which is used instead of page numbers.
- %
- % Includes fields ISBN and ISSN.
- %
- % Includes field URL for Internet addresses.
- %
- % Includes field DOI for Digital Object Idenfifiers.
- %
- % Works best with the url.sty package of Donald Arseneau.
- %
- % Works with identical authors and year are further sorted by
- %   citation key, to preserve any natural sequence.
- %
-ENTRY
-  { address
-    author
-    booktitle
-    chapter
-    doi
-    eid
-    edition
-    editor
-    howpublished
-    institution
-    isbn
-    issn
-    journal
-    key
-    month
-    note
-    number
-    organization
-    pages
-    publisher
-    school
-    series
-    title
-    type
-    url
-    volume
-    year
-  }
-  {}
-  { label extra.label sort.label short.list }
-
-INTEGERS { output.state before.all mid.sentence after.sentence after.block }
-
-FUNCTION {init.state.consts}
-{ #0 'before.all :=
-  #1 'mid.sentence :=
-  #2 'after.sentence :=
-  #3 'after.block :=
-}
-
-STRINGS { s t }
-
-FUNCTION {output.nonnull}
-{ 's :=
-  output.state mid.sentence =
-    { ", " * write$ }
-    { output.state after.block =
-        { add.period$ write$
-          newline$
-          "\newblock " write$
-        }
-        { output.state before.all =
-            'write$
-            { add.period$ " " * write$ }
-          if$
-        }
-      if$
-      mid.sentence 'output.state :=
-    }
-  if$
-  s
-}
-
-FUNCTION {output}
-{ duplicate$ empty$
-    'pop$
-    'output.nonnull
-  if$
-}
-
-FUNCTION {output.check}
-{ 't :=
-  duplicate$ empty$
-    { pop$ "empty " t * " in " * cite$ * warning$ }
-    'output.nonnull
-  if$
-}
-
-FUNCTION {fin.entry}
-{ add.period$
-  write$
-  newline$
-}
-
-FUNCTION {new.block}
-{ output.state before.all =
-    'skip$
-    { after.block 'output.state := }
-  if$
-}
-
-FUNCTION {new.sentence}
-{ output.state after.block =
-    'skip$
-    { output.state before.all =
-        'skip$
-        { after.sentence 'output.state := }
-      if$
-    }
-  if$
-}
-
-FUNCTION {not}
-{   { #0 }
-    { #1 }
-  if$
-}
-
-FUNCTION {and}
-{   'skip$
-    { pop$ #0 }
-  if$
-}
-
-FUNCTION {or}
-{   { pop$ #1 }
-    'skip$
-  if$
-}
-
-FUNCTION {new.block.checka}
-{ empty$
-    'skip$
-    'new.block
-  if$
-}
-
-FUNCTION {new.block.checkb}
-{ empty$
-  swap$ empty$
-  and
-    'skip$
-    'new.block
-  if$
-}
-
-FUNCTION {new.sentence.checka}
-{ empty$
-    'skip$
-    'new.sentence
-  if$
-}
-
-FUNCTION {new.sentence.checkb}
-{ empty$
-  swap$ empty$
-  and
-    'skip$
-    'new.sentence
-  if$
-}
-
-FUNCTION {field.or.null}
-{ duplicate$ empty$
-    { pop$ "" }
-    'skip$
-  if$
-}
-
-FUNCTION {emphasize}
-{ duplicate$ empty$
-    { pop$ "" }
-    { "\emph{" swap$ * "}" * }
-  if$
-}
-
-INTEGERS { nameptr namesleft numnames }
-
-FUNCTION {format.names}
-{ 's :=
-  #1 'nameptr :=
-  s num.names$ 'numnames :=
-  numnames 'namesleft :=
-    { namesleft #0 > }
-    % Formerly { s nameptr "{f.~}{vv~}{ll}{, jj}" format.name$ 't :=
-    { s nameptr "{ff }{vv }{ll}{, jj}" format.name$ 't :=
-      nameptr #1 >
-        { namesleft #1 >
-            { ", " * t * }
-            { numnames #2 >
-                { "," * }
-                'skip$
-              if$
-              t "others" =
-                { " et~al." * }
-                { " and " * t * }
-              if$
-            }
-          if$
-        }
-        't
-      if$
-      nameptr #1 + 'nameptr :=
-      namesleft #1 - 'namesleft :=
-    }
-  while$
-}
-
-FUNCTION {format.key}
-{ empty$
-    { key field.or.null }
-    { "" }
-  if$
-}
-
-FUNCTION {format.authors}
-{ author empty$
-    { "" }
-    { author format.names }
-  if$
-}
-
-FUNCTION {format.editors}
-{ editor empty$
-    { "" }
-    { editor format.names
-      editor num.names$ #1 >
-        { ", editors" * }
-        { ", editor" * }
-      if$
-    }
-  if$
-}
-
-FUNCTION {format.isbn}
-{ isbn empty$
-    { "" }
-%    { new.block "ISBN " isbn * }
-    { "" }
-  if$
-}
-
-FUNCTION {format.issn}
-{ issn empty$
-    { "" }
-%    { new.block "ISSN " issn * }
-    { "" }
-  if$
-}
-
-FUNCTION {format.url}
-{ url empty$
-    { "" }
-%    { new.block "URL \url{" url * "}" * }
-    { "" }
-  if$
-}
-
-FUNCTION {format.doi}
-{ doi empty$
-    { "" }
-%    { new.block "\doi{" doi * "}" * }
-    { "" }
-  if$
-}
-
-FUNCTION {format.title}
-{ title empty$
-    { "" }
-    { title "t" change.case$ }
-  if$
-}
-
-FUNCTION {format.full.names}
-{'s :=
-  #1 'nameptr :=
-  s num.names$ 'numnames :=
-  numnames 'namesleft :=
-    { namesleft #0 > }
-    { s nameptr
-      "{vv~}{ll}" format.name$ 't :=
-      nameptr #1 >
-        {
-          namesleft #1 >
-            { ", " * t * }
-            {
-              numnames #2 >
-                { "," * }
-                'skip$
-              if$
-              t "others" =
-                { " et~al." * }
-                { " and " * t * }
-              if$
-            }
-          if$
-        }
-        't
-      if$
-      nameptr #1 + 'nameptr :=
-      namesleft #1 - 'namesleft :=
-    }
-  while$
-}
-
-FUNCTION {author.editor.full}
-{ author empty$
-    { editor empty$
-        { "" }
-        { editor format.full.names }
-      if$
-    }
-    { author format.full.names }
-  if$
-}
-
-FUNCTION {author.full}
-{ author empty$
-    { "" }
-    { author format.full.names }
-  if$
-}
-
-FUNCTION {editor.full}
-{ editor empty$
-    { "" }
-    { editor format.full.names }
-  if$
-}
-
-FUNCTION {make.full.names}
-{ type$ "book" =
-  type$ "inbook" =
-  or
-    'author.editor.full
-    { type$ "proceedings" =
-        'editor.full
-        'author.full
-      if$
-    }
-  if$
-}
-
-FUNCTION {output.bibitem}
-{ newline$
-  "\bibitem[" write$
-  label write$
-  ")" make.full.names duplicate$ short.list =
-     { pop$ }
-     { * }
-   if$
-  "]{" * write$
-  cite$ write$
-  "}" write$
-  newline$
-  ""
-  before.all 'output.state :=
-}
-
-FUNCTION {n.dashify}
-{ 't :=
-  ""
-    { t empty$ not }
-    { t #1 #1 substring$ "-" =
-        { t #1 #2 substring$ "--" = not
-            { "--" *
-              t #2 global.max$ substring$ 't :=
-            }
-            {   { t #1 #1 substring$ "-" = }
-                { "-" *
-                  t #2 global.max$ substring$ 't :=
-                }
-              while$
-            }
-          if$
-        }
-        { t #1 #1 substring$ *
-          t #2 global.max$ substring$ 't :=
-        }
-      if$
-    }
-  while$
-}
-
-FUNCTION {format.date}
-{ year duplicate$ empty$
-    { "empty year in " cite$ * warning$
-       pop$ "" }
-    'skip$
-  if$
-%% CR: Leave out months.
-%  month empty$
-%    'skip$
-%    { month
-%      " " * swap$ *
-%    }
-%  if$
-  extra.label *
-}
-
-FUNCTION {format.btitle}
-{ title emphasize
-}
-
-FUNCTION {tie.or.space.connect}
-{ duplicate$ text.length$ #3 <
-    { "~" }
-    { " " }
-  if$
-  swap$ * *
-}
-
-FUNCTION {either.or.check}
-{ empty$
-    'pop$
-    { "can't use both " swap$ * " fields in " * cite$ * warning$ }
-  if$
-}
-
-FUNCTION {format.bvolume}
-{ volume empty$
-    { "" }
-%% CR: Don't show "volume 1234 of LNCS" etc.
-%    { "volume" volume tie.or.space.connect
-%      series empty$
-%        'skip$
-%        { " of " * series emphasize * }
-%      if$
-%      "volume and number" number either.or.check
-%    }
-    { "" }
-  if$
-}
-
-FUNCTION {format.number.series}
-{ volume empty$
-    { number empty$
-%% CR: Leave out series information.
-%        { series field.or.null }
-        { "" }
-        { output.state mid.sentence =
-            { "number" }
-            { "Number" }
-          if$
-          number tie.or.space.connect
-          series empty$
-            { "there's a number but no series in " cite$ * warning$ }
-            { " in " * series * }
-          if$
-        }
-      if$
-    }
-    { "" }
-  if$
-}
-
-FUNCTION {format.edition}
-{ edition empty$
-    { "" }
-    { output.state mid.sentence =
-        { edition "l" change.case$ " edition" * }
-        { edition "t" change.case$ " edition" * }
-      if$
-    }
-  if$
-}
-
-INTEGERS { multiresult }
-
-FUNCTION {multi.page.check}
-{ 't :=
-  #0 'multiresult :=
-    { multiresult not
-      t empty$ not
-      and
-    }
-    { t #1 #1 substring$
-      duplicate$ "-" =
-      swap$ duplicate$ "," =
-      swap$ "+" =
-      or or
-        { #1 'multiresult := }
-        { t #2 global.max$ substring$ 't := }
-      if$
-    }
-  while$
-  multiresult
-}
-
-FUNCTION {format.pages}
-{ pages empty$
-    { "" }
-    { pages multi.page.check
-        { "pages" pages n.dashify tie.or.space.connect }
-        { "page" pages tie.or.space.connect }
-      if$
-    }
-  if$
-}
-
-FUNCTION {format.eid}
-{ eid empty$
-    { "" }
-    { "art." eid tie.or.space.connect }
-  if$
-}
-
-FUNCTION {format.vol.num.pages}
-{ volume field.or.null
-  number empty$
-    'skip$
-    { "\penalty0 (" number * ")" * *
-      volume empty$
-        { "there's a number but no volume in " cite$ * warning$ }
-        'skip$
-      if$
-    }
-  if$
-  pages empty$
-    'skip$
-    { duplicate$ empty$
-        { pop$ format.pages }
-        { ":\penalty0 " * pages n.dashify * }
-      if$
-    }
-  if$
-}
-
-FUNCTION {format.vol.num.eid}
-{ volume field.or.null
-  number empty$
-    'skip$
-    { "\penalty0 (" number * ")" * *
-      volume empty$
-        { "there's a number but no volume in " cite$ * warning$ }
-        'skip$
-      if$
-    }
-  if$
-  eid empty$
-    'skip$
-    { duplicate$ empty$
-        { pop$ format.eid }
-        { ":\penalty0 " * eid * }
-      if$
-    }
-  if$
-}
-
-FUNCTION {format.chapter.pages}
-{ chapter empty$
-    'format.pages
-    { type empty$
-        { "chapter" }
-        { type "l" change.case$ }
-      if$
-      chapter tie.or.space.connect
-      pages empty$
-        'skip$
-        { ", " * format.pages * }
-      if$
-    }
-  if$
-}
-
-FUNCTION {format.in.ed.booktitle}
-{ booktitle empty$
-    { "" }
-%% CR: Leave out editors even if the information is available.
-%    { editor empty$
-%        { "In " booktitle emphasize * }
-%        { "In " format.editors * ", " * booktitle emphasize * }
-%      if$
-%    }
-    { "In " booktitle emphasize * }
-  if$
-}
-
-FUNCTION {empty.misc.check}
-{ author empty$ title empty$ howpublished empty$
-  month empty$ year empty$ note empty$
-  and and and and and
-  key empty$ not and
-    { "all relevant fields are empty in " cite$ * warning$ }
-    'skip$
-  if$
-}
-
-FUNCTION {format.thesis.type}
-{ type empty$
-    'skip$
-    { pop$
-      type "t" change.case$
-    }
-  if$
-}
-
-FUNCTION {format.tr.number}
-{ type empty$
-    { "Technical Report" }
-    'type
-  if$
-  number empty$
-    { "t" change.case$ }
-    { number tie.or.space.connect }
-  if$
-}
-
-FUNCTION {format.article.crossref}
-{ key empty$
-    { journal empty$
-        { "need key or journal for " cite$ * " to crossref " * crossref *
-          warning$
-          ""
-        }
-        { "In \emph{" journal * "}" * }
-      if$
-    }
-    { "In " }
-  if$
-  " \citet{" * crossref * "}" *
-}
-
-FUNCTION {format.book.crossref}
-{ volume empty$
-    { "empty volume in " cite$ * "'s crossref of " * crossref * warning$
-      "In "
-    }
-    { "Volume" volume tie.or.space.connect
-      " of " *
-    }
-  if$
-  editor empty$
-  editor field.or.null author field.or.null =
-  or
-    { key empty$
-        { series empty$
-            { "need editor, key, or series for " cite$ * " to crossref " *
-              crossref * warning$
-              "" *
-            }
-            { "\emph{" * series * "}" * }
-          if$
-        }
-        'skip$
-      if$
-    }
-    'skip$
-  if$
-  " \citet{" * crossref * "}" *
-}
-
-FUNCTION {format.incoll.inproc.crossref}
-{ editor empty$
-  editor field.or.null author field.or.null =
-  or
-    { key empty$
-        { booktitle empty$
-            { "need editor, key, or booktitle for " cite$ * " to crossref " *
-              crossref * warning$
-              ""
-            }
-            { "In \emph{" booktitle * "}" * }
-          if$
-        }
-        { "In " }
-      if$
-    }
-    { "In " }
-  if$
-  " \citet{" * crossref * "}" *
-}
-
-FUNCTION {article}
-{ output.bibitem
-  format.authors "author" output.check
-  author format.key output
-  new.block
-  format.title "title" output.check
-  new.block
-  crossref missing$
-    { journal emphasize "journal" output.check
-      eid empty$
-        { format.vol.num.pages output }
-        { format.vol.num.eid output }
-      if$
-      format.date "year" output.check
-    }
-    { format.article.crossref output.nonnull
-      eid empty$
-        { format.pages output }
-        { format.eid output }
-      if$
-    }
-  if$
-  format.issn output
-  format.doi output
-  format.url output
-  new.block
-  note output
-  fin.entry
-}
-
-FUNCTION {book}
-{ output.bibitem
-  author empty$
-    { format.editors "author and editor" output.check
-      editor format.key output
-    }
-    { format.authors output.nonnull
-      crossref missing$
-        { "author and editor" editor either.or.check }
-        'skip$
-      if$
-    }
-  if$
-  new.block
-  format.btitle "title" output.check
-  crossref missing$
-    { format.bvolume output
-      new.block
-      format.number.series output
-      new.sentence
-      publisher "publisher" output.check
-      address output
-    }
-    { new.block
-      format.book.crossref output.nonnull
-    }
-  if$
-  format.edition output
-  format.date "year" output.check
-  format.isbn output
-  format.doi output
-  format.url output
-  new.block
-  note output
-  fin.entry
-}
-
-FUNCTION {booklet}
-{ output.bibitem
-  format.authors output
-  author format.key output
-  new.block
-  format.title "title" output.check
-  howpublished address new.block.checkb
-  howpublished output
-  address output
-  format.date output
-  format.isbn output
-  format.doi output
-  format.url output
-  new.block
-  note output
-  fin.entry
-}
-
-FUNCTION {inbook}
-{ output.bibitem
-  author empty$
-    { format.editors "author and editor" output.check
-      editor format.key output
-    }
-    { format.authors output.nonnull
-      crossref missing$
-        { "author and editor" editor either.or.check }
-        'skip$
-      if$
-    }
-  if$
-  new.block
-  format.btitle "title" output.check
-  crossref missing$
-    { format.bvolume output
-      format.chapter.pages "chapter and pages" output.check
-      new.block
-      format.number.series output
-      new.sentence
-      publisher "publisher" output.check
-      address output
-    }
-    { format.chapter.pages "chapter and pages" output.check
-      new.block
-      format.book.crossref output.nonnull
-    }
-  if$
-  format.edition output
-  format.date "year" output.check
-  format.isbn output
-  format.doi output
-  format.url output
-  new.block
-  note output
-  fin.entry
-}
-
-FUNCTION {incollection}
-{ output.bibitem
-  format.authors "author" output.check
-  author format.key output
-  new.block
-  format.title "title" output.check
-  new.block
-  crossref missing$
-    { format.in.ed.booktitle "booktitle" output.check
-      format.bvolume output
-      format.number.series output
-      format.chapter.pages output
-      new.sentence
-      publisher "publisher" output.check
-      address output
-      format.edition output
-      format.date "year" output.check
-    }
-    { format.incoll.inproc.crossref output.nonnull
-      format.chapter.pages output
-    }
-  if$
-  format.isbn output
-  format.doi output
-  format.url output
-  new.block
-  note output
-  fin.entry
-}
-
-FUNCTION {inproceedings}
-{ output.bibitem
-  format.authors "author" output.check
-  author format.key output
-  new.block
-  format.title "title" output.check
-  new.block
-  crossref missing$
-    { format.in.ed.booktitle "booktitle" output.check
-      format.bvolume output
-      format.number.series output
-      format.pages output
-      address empty$
-        { organization publisher new.sentence.checkb
-          organization output
-          publisher output
-          format.date "year" output.check
-        }
-        { address output.nonnull
-          format.date "year" output.check
-          new.sentence
-          organization output
-          publisher output
-        }
-      if$
-    }
-    { format.incoll.inproc.crossref output.nonnull
-      format.pages output
-    }
-  if$
-  format.isbn output
-  format.doi output
-  format.url output
-  new.block
-  note output
-  fin.entry
-}
-
-FUNCTION {conference} { inproceedings }
-
-FUNCTION {manual}
-{ output.bibitem
-  format.authors output
-  author format.key output
-  new.block
-  format.btitle "title" output.check
-  organization address new.block.checkb
-  organization output
-  address output
-  format.edition output
-  format.date output
-  format.url output
-  new.block
-  note output
-  fin.entry
-}
-
-FUNCTION {mastersthesis}
-{ output.bibitem
-  format.authors "author" output.check
-  author format.key output
-  new.block
-  format.title "title" output.check
-  new.block
-  "Master's thesis" format.thesis.type output.nonnull
-  school "school" output.check
-  address output
-  format.date "year" output.check
-  format.url output
-  new.block
-  note output
-  fin.entry
-}
-
-FUNCTION {misc}
-{ output.bibitem
-  format.authors output
-  author format.key output
-  title howpublished new.block.checkb
-  format.title output
-  howpublished new.block.checka
-  howpublished output
-  format.date output
-  format.issn output
-  format.url output
-  new.block
-  note output
-  fin.entry
-  empty.misc.check
-}
-
-FUNCTION {phdthesis}
-{ output.bibitem
-  format.authors "author" output.check
-  author format.key output
-  new.block
-  format.btitle "title" output.check
-  new.block
-  "PhD thesis" format.thesis.type output.nonnull
-  school "school" output.check
-  address output
-  format.date "year" output.check
-  format.url output
-  new.block
-  note output
-  fin.entry
-}
-
-FUNCTION {proceedings}
-{ output.bibitem
-  format.editors output
-  editor format.key output
-  new.block
-  format.btitle "title" output.check
-  format.bvolume output
-  format.number.series output
-  address output
-  format.date "year" output.check
-  new.sentence
-  organization output
-  publisher output
-  format.isbn output
-  format.doi output
-  format.url output
-  new.block
-  note output
-  fin.entry
-}
-
-FUNCTION {techreport}
-{ output.bibitem
-  format.authors "author" output.check
-  author format.key output
-  new.block
-  format.title "title" output.check
-  new.block
-  format.tr.number output.nonnull
-  institution "institution" output.check
-  address output
-  format.date "year" output.check
-  format.url output
-  new.block
-  note output
-  fin.entry
-}
-
-FUNCTION {unpublished}
-{ output.bibitem
-  format.authors "author" output.check
-  author format.key output
-  new.block
-  format.title "title" output.check
-  new.block
-  note "note" output.check
-  format.date output
-  format.url output
-  fin.entry
-}
-
-FUNCTION {default.type} { misc }
-
-
-MACRO {jan} {"Jan."}
-
-MACRO {feb} {"Feb."}
-
-MACRO {mar} {"Mar."}
-
-MACRO {apr} {"Apr."}
-
-MACRO {may} {"May"}
-
-MACRO {jun} {"June"}
-
-MACRO {jul} {"July"}
-
-MACRO {aug} {"Aug."}
-
-MACRO {sep} {"Sept."}
-
-MACRO {oct} {"Oct."}
-
-MACRO {nov} {"Nov."}
-
-MACRO {dec} {"Dec."}
-
-
-
-MACRO {acmcs} {"ACM Comput. Surv."}
-
-MACRO {acta} {"Acta Inf."}
-
-MACRO {cacm} {"Commun. ACM"}
-
-MACRO {ibmjrd} {"IBM J. Res. Dev."}
-
-MACRO {ibmsj} {"IBM Syst.~J."}
-
-MACRO {ieeese} {"IEEE Trans. Softw. Eng."}
-
-MACRO {ieeetc} {"IEEE Trans. Comput."}
-
-MACRO {ieeetcad}
- {"IEEE Trans. Comput.-Aided Design Integrated Circuits"}
-
-MACRO {ipl} {"Inf. Process. Lett."}
-
-MACRO {jacm} {"J.~ACM"}
-
-MACRO {jcss} {"J.~Comput. Syst. Sci."}
-
-MACRO {scp} {"Sci. Comput. Programming"}
-
-MACRO {sicomp} {"SIAM J. Comput."}
-
-MACRO {tocs} {"ACM Trans. Comput. Syst."}
-
-MACRO {tods} {"ACM Trans. Database Syst."}
-
-MACRO {tog} {"ACM Trans. Gr."}
-
-MACRO {toms} {"ACM Trans. Math. Softw."}
-
-MACRO {toois} {"ACM Trans. Office Inf. Syst."}
-
-MACRO {toplas} {"ACM Trans. Prog. Lang. Syst."}
-
-MACRO {tcs} {"Theoretical Comput. Sci."}
-
-
-READ
-
-FUNCTION {sortify}
-{ purify$
-  "l" change.case$
-}
-
-INTEGERS { len }
-
-FUNCTION {chop.word}
-{ 's :=
-  'len :=
-  s #1 len substring$ =
-    { s len #1 + global.max$ substring$ }
-    's
-  if$
-}
-
-FUNCTION {format.lab.names}
-{ 's :=
-  s #1 "{vv~}{ll}" format.name$
-  s num.names$ duplicate$
-  #2 >
-    { pop$ " et~al." * }
-    { #2 <
-        'skip$
-        { s #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" =
-            { " et~al." * }
-            { " and " * s #2 "{vv~}{ll}" format.name$ * }
-          if$
-        }
-      if$
-    }
-  if$
-}
-
-FUNCTION {author.key.label}
-{ author empty$
-    { key empty$
-        { cite$ #1 #3 substring$ }
-        'key
-      if$
-    }
-    { author format.lab.names }
-  if$
-}
-
-FUNCTION {author.editor.key.label}
-{ author empty$
-    { editor empty$
-        { key empty$
-            { cite$ #1 #3 substring$ }
-            'key
-          if$
-        }
-        { editor format.lab.names }
-      if$
-    }
-    { author format.lab.names }
-  if$
-}
-
-FUNCTION {author.key.organization.label}
-{ author empty$
-    { key empty$
-        { organization empty$
-            { cite$ #1 #3 substring$ }
-            { "The " #4 organization chop.word #3 text.prefix$ }
-          if$
-        }
-        'key
-      if$
-    }
-    { author format.lab.names }
-  if$
-}
-
-FUNCTION {editor.key.organization.label}
-{ editor empty$
-    { key empty$
-        { organization empty$
-            { cite$ #1 #3 substring$ }
-            { "The " #4 organization chop.word #3 text.prefix$ }
-          if$
-        }
-        'key
-      if$
-    }
-    { editor format.lab.names }
-  if$
-}
-
-FUNCTION {calc.short.authors}
-{ type$ "book" =
-  type$ "inbook" =
-  or
-    'author.editor.key.label
-    { type$ "proceedings" =
-        'editor.key.organization.label
-        { type$ "manual" =
-            'author.key.organization.label
-            'author.key.label
-          if$
-        }
-      if$
-    }
-  if$
-  'short.list :=
-}
-
-FUNCTION {calc.label}
-{ calc.short.authors
-  short.list
-  "("
-  *
-  year duplicate$ empty$
-  short.list key field.or.null = or
-     { pop$ "" }
-     'skip$
-  if$
-  *
-  'label :=
-}
-
-FUNCTION {sort.format.names}
-{ 's :=
-  #1 'nameptr :=
-  ""
-  s num.names$ 'numnames :=
-  numnames 'namesleft :=
-    { namesleft #0 > }
-    {
-      s nameptr "{vv{ } }{ll{ }}{  f{ }}{  jj{ }}" format.name$ 't :=
-      nameptr #1 >
-        {
-          "   "  *
-          namesleft #1 = t "others" = and
-            { "zzzzz" * }
-            { numnames #2 > nameptr #2 = and
-                { "zz" * year field.or.null * "   " * }
-                'skip$
-              if$
-              t sortify *
-            }
-          if$
-        }
-        { t sortify * }
-      if$
-      nameptr #1 + 'nameptr :=
-      namesleft #1 - 'namesleft :=
-    }
-  while$
-}
-
-FUNCTION {sort.format.title}
-{ 't :=
-  "A " #2
-    "An " #3
-      "The " #4 t chop.word
-    chop.word
-  chop.word
-  sortify
-  #1 global.max$ substring$
-}
-
-FUNCTION {author.sort}
-{ author empty$
-    { key empty$
-        { "to sort, need author or key in " cite$ * warning$
-          ""
-        }
-        { key sortify }
-      if$
-    }
-    { author sort.format.names }
-  if$
-}
-
-FUNCTION {author.editor.sort}
-{ author empty$
-    { editor empty$
-        { key empty$
-            { "to sort, need author, editor, or key in " cite$ * warning$
-              ""
-            }
-            { key sortify }
-          if$
-        }
-        { editor sort.format.names }
-      if$
-    }
-    { author sort.format.names }
-  if$
-}
-
-FUNCTION {author.organization.sort}
-{ author empty$
-    { organization empty$
-        { key empty$
-            { "to sort, need author, organization, or key in " cite$ * warning$
-              ""
-            }
-            { key sortify }
-          if$
-        }
-        { "The " #4 organization chop.word sortify }
-      if$
-    }
-    { author sort.format.names }
-  if$
-}
-
-FUNCTION {editor.organization.sort}
-{ editor empty$
-    { organization empty$
-        { key empty$
-            { "to sort, need editor, organization, or key in " cite$ * warning$
-              ""
-            }
-            { key sortify }
-          if$
-        }
-        { "The " #4 organization chop.word sortify }
-      if$
-    }
-    { editor sort.format.names }
-  if$
-}
-
-
-FUNCTION {presort}
-{ calc.label
-  label sortify
-  "    "
-  *
-  type$ "book" =
-  type$ "inbook" =
-  or
-    'author.editor.sort
-    { type$ "proceedings" =
-        'editor.organization.sort
-        { type$ "manual" =
-            'author.organization.sort
-            'author.sort
-          if$
-        }
-      if$
-    }
-  if$
-  "    "
-  *
-  year field.or.null sortify
-  *
-  "    "
-  *
-  cite$
-  *
-  #1 entry.max$ substring$
-  'sort.label :=
-  sort.label *
-  #1 entry.max$ substring$
-  'sort.key$ :=
-}
-
-ITERATE {presort}
-
-SORT
-
-STRINGS { longest.label last.label next.extra }
-
-INTEGERS { longest.label.width last.extra.num number.label }
-
-FUNCTION {initialize.longest.label}
-{ "" 'longest.label :=
-  #0 int.to.chr$ 'last.label :=
-  "" 'next.extra :=
-  #0 'longest.label.width :=
-  #0 'last.extra.num :=
-  #0 'number.label :=
-}
-
-FUNCTION {forward.pass}
-{ last.label label =
-    { last.extra.num #1 + 'last.extra.num :=
-      last.extra.num int.to.chr$ 'extra.label :=
-    }
-    { "a" chr.to.int$ 'last.extra.num :=
-      "" 'extra.label :=
-      label 'last.label :=
-    }
-  if$
-  number.label #1 + 'number.label :=
-}
-
-FUNCTION {reverse.pass}
-{ next.extra "b" =
-    { "a" 'extra.label := }
-    'skip$
-  if$
-  extra.label 'next.extra :=
-  extra.label
-  duplicate$ empty$
-    'skip$
-    { "{\natexlab{" swap$ * "}}" * }
-  if$
-  'extra.label :=
-  label extra.label * 'label :=
-}
-
-EXECUTE {initialize.longest.label}
-
-ITERATE {forward.pass}
-
-REVERSE {reverse.pass}
-
-FUNCTION {bib.sort.order}
-{ sort.label  'sort.key$ :=
-}
-
-ITERATE {bib.sort.order}
-
-SORT
-
-FUNCTION {begin.bib}
-{   preamble$ empty$
-    'skip$
-    { preamble$ write$ newline$ }
-  if$
-  "\begin{thebibliography}{" number.label int.to.str$ * "}" *
-  write$ newline$
-  "\providecommand{\natexlab}[1]{#1}"
-  write$ newline$
-  "\providecommand{\url}[1]{\texttt{#1}}"
-  write$ newline$
-  "\expandafter\ifx\csname urlstyle\endcsname\relax"
-  write$ newline$
-  "  \providecommand{\doi}[1]{doi: #1}\else"
-  write$ newline$
-  "  \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi"
-  write$ newline$
-}
-
-EXECUTE {begin.bib}
-
-EXECUTE {init.state.consts}
-
-ITERATE {call.type$}
-
-FUNCTION {end.bib}
-{ newline$
-  "\end{thebibliography}" write$ newline$
-}
-
-EXECUTE {end.bib}
diff --git a/docs/lantow_miolane_2025/main.bib b/docs/lantow_miolane_2025/main.bib
deleted file mode 100644
index e2fd758..0000000
--- a/docs/lantow_miolane_2025/main.bib
+++ /dev/null
@@ -1,973 +0,0 @@
-@book{pennec2019riemannian,
-  title={Riemannian geometric statistics in medical image analysis},
-  author={Pennec, Xavier and Sommer, Stefan and Fletcher, Tom},
-  year={2019},
-  publisher={Academic Press}
-}
-
-@article{zhao2013fourier,
-  title={Fourier--Bessel rotational invariant eigenimages},
-  author={Zhao, Zhizhen and Singer, Amit},
-  journal={Journal of the Optical Society of America A},
-  volume={30},
-  number={5},
-  pages={871--877},
-  year={2013},
-  publisher={Optical Society of America}
-}
-
-@inproceedings{mataigne2024selective,
- author = {Mataigne, Simon and Mathe, Johan and Sanborn, Sophia and Hillar, Christopher and Miolane, Nina},
- booktitle = {Advances in Neural Information Processing Systems},
- editor = {A. Globerson and L. Mackey and D. Belgrave and A. Fan and U. Paquet and J. Tomczak and C. Zhang},
- pages = {115682--115711},
- publisher = {Curran Associates, Inc.},
- title = {The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks},
- url = {https://proceedings.neurips.cc/paper_files/paper/2024/file/d1a1e8713fcd5626656553c82f7c3b26-Paper-Conference.pdf},
- volume = {37},
- year = {2024}
-}
-
-
-
-@article{zhao2014rotationally,
-  title={Rotationally invariant image representation for viewing direction classification in cryo-EM},
-  author={Zhao, Zhizhen and Singer, Amit},
-  journal={Journal of structural biology},
-  volume={186},
-  number={1},
-  pages={153--166},
-  year={2014},
-  publisher={Elsevier}
-}
-
-@article{liu2023modulation,
-  title={Modulation recognition of low-SNR UAV radar signals based on bispectral slices and GA-BP neural network},
-  author={Liu, Xuemin and Song, Yaoliang and Chen, Kuiyu and Yan, Shihao and Chen, Si and Shi, Baihua},
-  journal={Drones},
-  volume={7},
-  number={7},
-  pages={472},
-  year={2023},
-  publisher={MDPI}
-}
-
-@article{koukiou2024identifying,
-  title={Identifying System Non-Linearities by Fusing Signal Bispectral Signatures},
-  author={Koukiou, Georgia},
-  journal={Electronics},
-  volume={13},
-  number={7},
-  pages={1287},
-  year={2024},
-  publisher={MDPI}
-}
-
-@article{zhang2000bispectrum,
-  title={Bispectrum analysis of focal ischemic cerebral EEG signal using third-order recursion method},
-  author={Zhang, Ji-Wu and Zheng, Chong-Xun and Xie, An},
-  journal={IEEE transactions on biomedical engineering},
-  volume={47},
-  number={3},
-  pages={352--359},
-  year={2000},
-  publisher={IEEE}
-}
-
-@article{martin2021bispectral,
-  title={Bispectral analysis of heart rate variability to characterize and help diagnose pediatric sleep apnea},
-  author={Mart{\'\i}n-Montero, Adri{\'a}n and Guti{\'e}rrez-Tobal, Gonzalo C and Gozal, David and Barroso-Garc{\'\i}a, Ver{\'o}nica and {\'A}lvarez, Daniel and Del Campo, F{\'e}lix and Kheirandish-Gozal, Leila and Hornero, Roberto},
-  journal={Entropy},
-  volume={23},
-  number={8},
-  pages={1016},
-  year={2021},
-  publisher={MDPI}
-}
-
-@article{fast_dhcs_2023, title={Fast Expansion into Harmonics on the Disk: A Steerable Basis with Fast Radial Convolutions}, volume={45}, ISSN={1064-8275}, DOI={10.1137/22M1542775}, abstractNote={.We study the tritronquée solution  of the  equation, the second member of the Painlevé I hierarchy. This particular solution is also known as the Gurevich–Pitaevskii solution of the KdV equation. It is pole-free on the real line and has various applications in mathematical physics. We obtain a full asymptotic expansion of  as , uniformly for the parameter  in a large interval. Based on this result, we successfully derive the total integrals of  and the associated Hamiltonian with respect to . Surprisingly, although  exhibits significant differences between  and , both integrals equal zero for all .}, number={5}, journal={SIAM Journal on Scientific Computing}, publisher={Society for Industrial and Applied Mathematics}, author={Marshall, Nicholas F. and Mickelin, Oscar and Singer, Amit}, year={2023}, month=oct, pages={A2431–A2457} }
-
-@article{nikias1993signal,
-  title={Signal processing with higher-order spectra},
-  author={Nikias, Chrysostomos L and Mendel, Jerry M},
-  journal={IEEE Signal processing magazine},
-  volume={10},
-  number={3},
-  pages={10--37},
-  year={1993},
-  publisher={IEEE}
-}
-
-
-@article{elgar1985observations,
-  title={Observations of bispectra of shoaling surface gravity waves},
-  author={Elgar, Steve and Guza, RT},
-  journal={Journal of Fluid Mechanics},
-  volume={161},
-  pages={425--448},
-  year={1985},
-  publisher={Cambridge University Press}
-}
-
-@article{Gordon_2012,
-   title={Detection noise bias and variance in the power spectrum and bispectrum in optical interferometry},
-   volume={541},
-   ISSN={1432-0746},
-   url={http://dx.doi.org/10.1051/0004-6361/201117335},
-   DOI={10.1051/0004-6361/201117335},
-   journal={Astronomy \&; Astrophysics},
-   publisher={EDP Sciences},
-   author={Gordon, J. A. and Buscher, D. F.},
-   year={2012},
-   month=apr, pages={A46} }
-
- @article{geophysics_2004, title={Earthquake Relocation Using Cross-Correlation Time Delay Estimates Verified with the Bispectrum Method}, volume={94}, ISSN={0037-1106}, DOI={10.1785/0120030084}, abstractNote={Cross-correlation (CC) determined relative time delays, or related differential travel times, between pairs of seismic events at the same station are often used as input data to improve earthquake relocation results. Researchers generally select those time delays with associated CC coefficients larger than a chosen threshold. When two similar time series are contaminated by correlated noise sources, the relative time delay between them calculated with the CC technique is sometimes not reliable. Noise sources at a station for different events can be partially correlated or just randomly correlated. In this work, we use the bispectrum (BS) method, which works in the third-order spectral domain, to check the reliability of the CC determined time delay. We calculate two time delays with the BS method, one using the bandpass-filtered waveforms and the other with the raw data, and use them to verify (select or reject) the CC estimate computed with the filtered waveforms. We apply this technique to obtain bispectrum-verified CC differential times for 822 New Zealand earthquakes in the Wellington area. Our work demonstrates that the CC time delays verified with the BS method provide improved (smaller root mean square residual and more clustered) earthquake relocation results compared to those selected with the standard threshold criterion.}, number={3}, journal={Bulletin of the Seismological Society of America}, author={Du, Wen-Xuan and Thurber, Clifford H. and Eberhart-Phillips, Donna}, year={2004}, month=jun, pages={856–866} }
-
-@InProceedings{pmlr-v80-kondor18a,
-  title = 	 {On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups},
-  author =       {Kondor, Risi and Trivedi, Shubhendu},
-  booktitle = 	 {Proceedings of the 35th International Conference on Machine Learning},
-  pages = 	 {2747--2755},
-  year = 	 {2018},
-  editor = 	 {Dy, Jennifer and Krause, Andreas},
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-abstract = {Two-dimensional image moments with respect to Zernike polynomials are defined, and it is shown how to construct an arbitrarily large number of independent, algebraic combinations of Zernike moments that are invariant to image translation, orientation, and size. This approach is contrasted with the usual method of moments. The general problem of two-dimensional pattern recognition and three-dimensional object recognition is discussed within this framework. A unique reconstruction of an image in either real space or Fourier space is given in terms of a finite number of moments. Examples of applications of the method are given. A coding scheme for image storage and retrieval is discussed.},
-}
-
-@article{giannakis1989,
-  author    = {D. Giannakis},
-  title     = {Signal reconstruction from the bispectrum},
-  journal   = {Journal of Sound and Vibration},
-  year      = {1989},
-  volume    = {133},
-  number    = {2},
-  pages     = {199--211},
-  doi       = {10.1016/0022-460X(89)90574-9}
-}
-
-@article{sundaramoorthy1990,
-  author    = {G. Sundaramoorthy and M. R. Raghuveer and S. A. Dianat},
-  title     = {Bispectral reconstruction of signals in noise: amplitude reconstruction issues},
-  journal   = {IEEE Transactions on Acoustics, Speech, and Signal Processing},
-  volume    = {38},
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-  pages     = {1297--1306},
-  year      = {1990},
-  doi       = {10.1109/29.57500}
-}
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-@inproceedings{pinilla2023,
-  author    = {L. Pinilla and A. Mishra and B. M. Sadler},
-  title     = {Unique Bispectrum Inversion for Signals with Finite Spectral/Temporal Support},
-  booktitle = {Proceedings of ICASSP 2023},
-  year      = {2023},
-  publisher = {IEEE},
-  doi       = {10.1109/ICASSP49357.2023.10095922}
-}
-
-@article{bendory2018,
-  author    = {T. Bendory and N. Boumal and W. Leeb and A. Singer},
-  title     = {Bispectrum Inversion with Application to Multireference Alignment},
-  journal   = {IEEE Transactions on Signal Processing},
-  volume    = {66},
-  number    = {4},
-  pages     = {1037--1050},
-  year      = {2018},
-  doi       = {10.1109/TSP.2017.2778186}
-}
-
-@article{chen2018,
-  author    = {H. Chen and M. Zehni and Z. Zhao},
-  title     = {A spectral method for stable bispectrum inversion with application to multireference alignment},
-  journal   = {IEEE Signal Processing Letters},
-  volume    = {25},
-  number    = {7},
-  pages     = {911--915},
-  year      = {2018},
-  doi       = {10.1109/LSP.2018.2835871}
-}
-
-@article{herring2019,
-  author    = {J. Herring and J. Nagy and L. Ruthotto},
-  title     = {Gauss–Newton Optimization for Phase Recovery From the Bispectrum},
-  journal   = {IEEE Transactions on Computational Imaging},
-  year      = {2019},
-  volume    = {5},
-  pages     = {1--11},
-  doi       = {10.1109/TCI.2019.2948784}
-}
-
-@misc{edidin2025,
-  author    = {D. Edidin and J. Katz},
-  title     = {The Reflection-Invariant Bispectrum: Signal Recovery in the Dihedral Model},
-  year      = {2025},
-  eprint    = {2408.09599},
-  archivePrefix = {arXiv},
-  primaryClass = {math.IT},
-  url       = {https://arxiv.org/abs/2408.09599}
-}
-
-@article{kakarala2012,
-  author    = {R. Kakarala},
-  title     = {The bispectrum as a source of phase-sensitive invariants for Fourier descriptors: a group-theoretic approach},
-  journal   = {Journal of Mathematical Imaging and Vision},
-  volume    = {44},
-  number    = {3},
-  pages     = {341--353},
-  year      = {2012},
-  doi       = {10.1007/s10851-012-0334-6}
-}
-
-@inproceedings{kakarala1993,
-  author    = {R. Kakarala and B. M. Bennett and M. D’Zmura and G. J. Iverson},
-  title     = {Bispectral techniques for spherical functions},
-  booktitle = {Proceedings of ICASSP '93},
-  year      = {1993},
-  pages     = {237--240},
-  address   = {Minneapolis, MN},
-  publisher = {IEEE},
-  doi       = {10.1109/ICASSP.1993.319593}
-}
-
-@article{bendory2018bispectrum,
-  title        = {Bispectrum Inversion with Application to Multireference Alignment},
-  author       = {Bendory, Tamir and Boumal, Nicolas and Leeb, William and Singer, Amit},
-  journal      = {IEEE Transactions on Signal Processing},
-  volume       = {66},
-  number       = {4},
-  pages        = {1037--1050},
-  year         = {2018}
-}
-
-@article{herring2019gauss,
-  title        = {Gauss--Newton Optimization for Phase Recovery from the Bispectrum},
-  author       = {Herring, James and Nagy, James and Ruthotto, Lars},
-  journal      = {IEEE Transactions on Computational Imaging},
-  year         = {2019},
-  pages        = {1--1},
-  doi          = {10.1109/TCI.2019.2948784}
-}
-
-
-@misc{kakarala2012bispectrumsourcephasesensitiveinvariants,
-      title={The bispectrum as a source of phase-sensitive invariants for Fourier descriptors: a group-theoretic approach},
-      author={Ramakrishna Kakarala},
-      year={2012},
-      eprint={0902.0196},
-      archivePrefix={arXiv},
-      primaryClass={math.GR},
-      url={https://arxiv.org/abs/0902.0196},
-}
-
-@article{marshall2022fast,
-  title={Fast expansion into harmonics on the disk},
-  author={Marshall, N. F.},
-  journal={arXiv preprint arXiv:2207.13674},
-  year={2022},
-  note={Fast Disk Harmonics Transform (FDHT)}
-}
-
-@article{reddy1996fft,
-  title={An FFT-based technique for translation, rotation, and scale-invariant image registration},
-  author={Reddy, B. S. and Chatterji, B. N.},
-  journal={IEEE Transactions on Image Processing},
-  volume={5},
-  number={8},
-  pages={1266--1271},
-  year={1996}
-}
-
-@article{lowe2004sift,
-  title={Distinctive Image Features from Scale-Invariant Keypoints},
-  author={Lowe, David G.},
-  journal={International Journal of Computer Vision},
-  volume={60},
-  number={2},
-  pages={91--110},
-  year={2004}
-}
-
-@inproceedings{bay2006surf,
-  title={SURF: Speeded Up Robust Features},
-  author={Bay, Herbert and Tuytelaars, Tinne and Van Gool, Luc},
-  booktitle={European Conference on Computer Vision (ECCV)},
-  pages={404--417},
-  year={2006},
-  publisher={Springer}
-}
-
-@inproceedings{rublee2011orb,
-  title={ORB: An efficient alternative to SIFT or SURF},
-  author={Rublee, Ethan and Rabaud, Vincent and Konolige, Kurt and Bradski, Gary},
-  booktitle={Proceedings of the IEEE International Conference on Computer Vision (ICCV)},
-  pages={2564--2571},
-  year={2011}
-}
-
-@inproceedings{alcantarilla2012kaze,
-  title={KAZE Features},
-  author={Alcantarilla, Pablo F. and Bartoli, Adrien and Davison, Andrew J.},
-  booktitle={European Conference on Computer Vision (ECCV)},
-  pages={214--227},
-  year={2012},
-  publisher={Springer}
-}
-
-@article{hu1962visual,
-  title={Visual Pattern Recognition by Moment Invariants},
-  author={Hu, M.K.},
-  journal={IRE Transactions on Information Theory},
-  volume={8},
-  number={2},
-  pages={179--187},
-  year={1962}
-}
-
-@article{teague1980zernike,
-  title={Image analysis via the general theory of moments},
-  author={Teague, M. R.},
-  journal={JOSA},
-  volume={70},
-  number={8},
-  pages={920--930},
-  year={1980}
-}
-
-@inproceedings{thomas2018tfns,
-  title={Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for 3D Point Clouds},
-  author={Thomas, Nathaniel and Smidt, T. and Kearnes, S. and Yang, L. and Li, L. and Kohlhoff, K. and Riley, P. and Kohlhoff, K.},
-  booktitle={NeurIPS},
-  year={2018}
-}
-
-@article{cohen2018spherical,
-  title={Spherical CNNs},
-  author={Cohen, Taco S. and Geiger, Mario and K{\"o}hler, Jonas and Welling, Max},
-  journal={arXiv preprint arXiv:1801.10130},
-  year={2018},
-  note={Published as ICLR 2018}
-}
-
-@inproceedings{esteves2018spherical,
-  title={Learning SO(3) Equivariant Representations with Spherical CNNs},
-  author={Esteves, Carlos and Allen-Blanchette, Christine and Makadia, Ameesh and Daniilidis, Kostas},
-  booktitle={European Conference on Computer Vision (ECCV)},
-  pages={52--68},
-  year={2018},
-  publisher={Springer}
-}
-
-@inproceedings{sabour2017dynamic,
-  title={Dynamic Routing Between Capsules},
-  author={Sabour, Sara and Frosst, Nicholas and Hinton, Geoffrey E.},
-  booktitle={Advances in Neural Information Processing Systems (NeurIPS)},
-  pages={3856--3866},
-  year={2017}
-}
-
-@misc{bronstein2021geometricdeeplearninggrids,
-      title={Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges},
-      author={Michael M. Bronstein and Joan Bruna and Taco Cohen and Petar Veličković},
-      year={2021},
-      eprint={2104.13478},
-      archivePrefix={arXiv},
-      primaryClass={cs.LG},
-      url={https://arxiv.org/abs/2104.13478},
-}
-
- @article{Verrall_Kakarala_1998, title={Disk-harmonic coefficients for invariant pattern recognition}, volume={15}, rights={https://doi.org/10.1364/OA_License_v1#VOR}, ISSN={1084-7529, 1520-8532}, DOI={10.1364/JOSAA.15.000389}, number={2}, journal={Journal of the Optical Society of America A}, author={Verrall, Steven C. and Kakarala, Ramakrishna}, year={1998}, month=feb, pages={389}, language={en} }
diff --git a/docs/lantow_miolane_2025/main.tex b/docs/lantow_miolane_2025/main.tex
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-% CVPR 2026 Paper Template; see https://github.com/cvpr-org/author-kit
-
-\documentclass[10pt,twocolumn,letterpaper]{article}
-
-%%%%%%%%% PAPER TYPE  - PLEASE UPDATE FOR FINAL VERSION
-\usepackage{cvpr}              % To produce the CAMERA-READY version
-% \usepackage[review]{cvpr}      % To produce the REVIEW version
-% \usepackage[pagenumbers]{cvpr} % To force page numbers, e.g. for an arXiv version
-
-% Import additional packages in the preamble file, before hyperref
-\input{preamble}
-
-\usepackage{pifont}
-\newcommand{\cmark}{\ding{51}} % checkmark
-\newcommand{\xmark}{\ding{55}} % x-mark
-
-% It is strongly recommended to use hyperref, especially for the review version.
-% hyperref with option pagebackref eases the reviewers' job.
-% Please disable hyperref *only* if you encounter grave issues,
-% e.g. with the file validation for the camera-ready version.
-%
-% If you comment hyperref and then uncomment it, you should delete *.aux before re-running LaTeX.
-% (Or just hit 'q' on the first LaTeX run, let it finish, and you should be clear).
-\definecolor{cvprblue}{rgb}{0.21,0.49,0.74}
-\usepackage[pagebackref,breaklinks,colorlinks,allcolors=cvprblue]{hyperref}
-\usepackage{amsthm}
-% Numbering propositions, theorems, lemmas, corollaries within sections
-\newtheorem{theorem}{Theorem}[section]
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{corollary}[theorem]{Corollary}
-
-% Definitions, remarks, examples in a different style (upright)
-\theoremstyle{definition}
-\newtheorem{definition}[theorem]{Definition}
-\newtheorem{example}[theorem]{Example}
-
-\theoremstyle{remark}
-\newtheorem{remark}[theorem]{Remark}
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-%%%%%%%%% PAPER ID  - PLEASE UPDATE
-\def\paperID{23419} % *** Enter the Paper ID here
-\def\confName{CVPR}
-\def\confYear{2026}
-
-%%%%%%%%% TITLE - PLEASE UPDATE
-\title{The Selective Disk Bispectrum and Its Inversion, with Application to Multi-Reference Alignment}
-
-%%%%%%%%% AUTHORS - PLEASE UPDATE
-\author{Adele Myers Lantow\\
-UC Santa Barbara\\
-{\tt\small adele@ucsb.edu}
-% For a paper whose authors are all at the same institution,
-% omit the following lines up until the closing ``}''.
-% Additional authors and addresses can be added with ``\and'',
-% just like the second author.
-% To save space, use either the email address or home page, not both
-\and
-Nina Miolane\\
-UC Santa Barbara\\
-{\tt\small ninamiolane@ucsb.edu}
-}
-
-\begin{document}
-\maketitle
-\input{sec/0_abstract}
-\input{sec/1_intro}
-\input{sec/2_related_works}
-\input{sec/3_background}
-\input{sec/4_theory}
-\input{sec/5_tractability}
-\input{sec/6_mra_experiments}
-\input{sec/7_conclusion}
-{
-    \small
-    \bibliographystyle{ieeenat_fullname}
-    \bibliography{main}
-}
-
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-% \input{sec/X_suppl}
-
-\end{document}
diff --git a/docs/lantow_miolane_2025/preamble.tex b/docs/lantow_miolane_2025/preamble.tex
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-%% This file contains a number of tweaks that are typically applied to the main document.
-%% They are not enabled by default, but can be enabled by uncommenting the relevant lines.
-
-%%
-%% Inline annotations; for predefined colors, refer to "dvipsnames" in the xcolor package:
-%% https://tinyurl.com/overleaf-colors
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-\newcommand{\red}[1]{{\color{red}#1}}
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-\newcommand{\TODO}[1]{\textbf{\color{red}[TODO: #1]}}
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-%% it at the very end of the writing process, when you are just above the page limit
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diff --git a/docs/lantow_miolane_2025/sec/0_abstract.tex b/docs/lantow_miolane_2025/sec/0_abstract.tex
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-\begin{abstract}
-In many computer vision and shape analysis tasks, practitioners are interested in learning from the \textit{shape} of the object in an image, while disregarding the object's orientation. To this end, it is valuable to define a \textit{rotation-invariant} representation of images, retaining all information about that image, but disregarding the way an object is rotated in the frame. To be practical for learning tasks, this representation must be \textit{computationally efficient} for large datasets and \textit{invertible}, so the representation can be visualized in image space. To this end, we present the selective disk bispectrum: a fast, rotation-invariant representation for image shape analysis. While the translational bispectrum has long been used as a translational invariant representation for 1D and 2D signals, its extension to 2D (disk) rotational invariance on images has been hindered by the absence of an invertible formulation and its cubic complexity. In this work, we derive an explicit inverse for the disk bispectrum, which allows us to define a ``selective" disk bispectrum, which only uses the minimal number of coefficients needed for faithful shape recovery. We show that this representation enables multi-reference alignment for rotated images—a task previously intractable for disk bispectrum methods. These results establish the disk bispectrum as a practical and theoretically grounded tool for learning on rotation-invariant shape data.
-\end{abstract}
diff --git a/docs/lantow_miolane_2025/sec/1_intro.tex b/docs/lantow_miolane_2025/sec/1_intro.tex
deleted file mode 100644
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-\section{Introduction}
-The translation-invariant bispectrum is a higher-order spectral representation that originated in the mid-20th century for signal processing \citep{tukey1953spectral, brillinger1991some, nikias1993signal}. Unlike the power spectrum, which captures second-order (pairwise) correlations between Fourier coefficients, the bispectrum encodes third-order relationships, making it sensitive to asymmetries and nonlinear interactions in the signal. More importantly, unlike the power spectrum which drops the phase of the signal, the translation-invariant bispectrum conserves everything about the signal except for its global phase, making it a \textit{complete} translational invariant.
-
-As a result, the translation-invariant bispectrum has found widespread applications across disciplines: oceanographers use it to analyze wave interactions~\citep{elgar1985observations}, seismologists to identify complex waveforms~\citep{geophysics_2004}, and biomedical engineers to study electroencephalograms~\citep{zhang2000bispectrum} as well as heart rate variability~\citep{martin2021bispectral}. It is similarly used in optical interferometry~\citep{Gordon_2012}, radar communications~\citep{liu2023modulation} and other applications as a way to identify patterns in 1D signals (see \cite{koukiou2024identifying} for a recent review). Such applications are only interested in translation invariance in 1D or 2D signals (images).
-
-In contrast, other applications are interested in rotation invariance, and despite the widespread applicability and utilization of the 1D/2D translation-invariant bispectrum, the 2D rotation-invariant bispectrum has been under-developed. In many applications, key information lies in the \textit{shape} of a 2D image, and its orientation is a nuisance variable. Without an easily computable rotation invariant representation, practitioners must register each image before analysis.
-
-To this end, \cite{zhao2014rotationally} proposed the disk bispectrum, which is invariant to 2D planar (disk) rotations. However, because they did not derive a bispectrum inversion, their analysis was restricted to classification problems in bispectrum space. For example, tasks like multi reference alignment (MRA) would be impossible. MRA estimates a ground truth signal by mapping shifted noisy versions of that signal into bispectrum space, averaging the signals in bispectrum space, and then inverting the average back to image space, which would be impossible without a bispectrum inversion. Additionally \cite{zhao2014rotationally}'s disk bispectrum was computationally expensive. Authors reported a cubic complexity of $\mathcal{O}(m^3 / N_{m})$ , where $m$ is the number of ``disk harmonic frequencies'', and $N_m$ the maximum frequency. Lastly, they resorted to principal component analysis (PCA) in bispectrum space to reduce its dimensionality. Authors do not prove completeness of the disk bispectrum or their PCA reduction, meaning it was unclear whether the bispectrum actually described all information about the original image up to a rotation.
-
-In this work, we propose the first computationally tractable disk bispectrum, the \textit{selective disk bispectrum}, and its inversion, for use in image processing and shape analysis. We also provide a method for approximating the disk bispectrum, building on \cite{fast_dhcs_2023}.  Then, following \cite{bendory2018bispectrum}, we derive an MRA-specific disk bispectrum inversion, which corrects for noise propagation during disk bispectrum averaging, and demonstrate its efficacy on rotated noisy MNIST images.
-
-\begin{figure*}[t]
-  \centering
-  \includegraphics[width=\textwidth]{figs/overview_figure.pdf}
-  \caption{We propose the selective disk bispectrum, which reduces the time complexity of the disk bispectrum from $\mathcal{O}(m^3/N_m)$ to $\mathcal{O}(m)$, where $m$ is the number of disk harmonic frequencies, and $N_m$ the maximum frequency. We provide the first inversion of the disk bispectrum from its coefficients back to image space.}
-  \label{fig:overview}
-\end{figure*}
-
-
-\textbf{Contributions}
-\begin{enumerate}[leftmargin=*]
-\item We introduce the \textit{selective disk bispectrum}, an efficient formulation that proves and exploits the redundancy of disk bispectral coefficients. This reduces the space complexity from $\mathcal{O}(m^3/N_m)$ to $\mathcal{O}(m)$ and the time complexity from $\mathcal{O}(L^3+m^3/N_m)$ to  $\mathcal{O}(L^3 + m)$, where $L \times L$ is the image size, $L^2$ is the number of total pixels, $m$ is the number of disk harmonic frequencies, and $N_m$ the maximum frequency. The selective disk bispectrum offers an \textit{efficient} rotation-invariant representation, making it scalable for large amounts of data.
-
-\item We derive the first disk bispectrum inversion, mapping (selective) disk bispectrum coefficients back to image space. The value of this inversion is twofold. First, it enables practitioners to reconstruct an image from statistical results (such as computation of the mean) in disk bispectrum space, which is essential for interpretability. Second, the inversion proves that the selective disk bispectrum is a \textit{complete rotational invariant} for images on the disk, i.e., it preserves all information about an image up to a disk rotation. By association, it also provides the first completeness proof for the full disk bispectrum.
-
-\item We provide a numerical approximation of the selective disk bispectrum, that further reduces its time complexity from $\mathcal{O}(L^3 + m)$ to $O(L^2\log L + m)$. We prove precise accuracy guarantees on this approximation.
-
-\item We provide an open-source, unit-tested implementation of the selective disk bispectrum and its inversion in Python, along with its application to multi-reference alignment. We use this implementation to illustrate the forward and inverse selective disk bispectrum map (between images and coefficients), and compare against the full disk bispectrum map. We empirically demonstrate that the selective disk bispectrum can be used for multi-reference alignment on rotated images.
-
-\end{enumerate}
-
-In summary, the selective disk bispectrum offers a fast (scalable) and invertible (interpretable) tool for imaging applications involving rotation-invariant 2D shape analysis.
diff --git a/docs/lantow_miolane_2025/sec/2_related_works.tex b/docs/lantow_miolane_2025/sec/2_related_works.tex
deleted file mode 100644
index 677d7c7..0000000
--- a/docs/lantow_miolane_2025/sec/2_related_works.tex
+++ /dev/null
@@ -1,50 +0,0 @@
-\section{Related Works}
-
-\paragraph{Equivariant and Invariant Representations.} As geometric learning has gained popularity \cite{bronstein2021geometricdeeplearninggrids}, there has been increased interest in rotation-equivariant and rotation-invariant image representations. Put simply (and described later in more detail), if a representation is \textit{equivariant} to rotation, the representation changes in a specific way when the original image is rotated. If a representation is rotation \textit{invariant}, the representation does not change if the original image is rotated.
-
-Table \ref{tab:rotation-summary} summarizes rotation equivariant and rotation invariant representations, showing that only our selective disk bispectrum is rotation invariant and invertible, making it the only method capable of multi-reference alignment on rotated images. As an aside, we note that every rotation-equivariant neural network (and every rotation-equivariant layer) can be made rotation-invariant by adding an invariant aggregation function, like the sum or the max, after it. However, because the sum or max cannot be inverted to reconstruct the input, this invariant shape descriptor is not invertible.
-
-
-\begin{table}[t]
-\centering
-\footnotesize
-\setlength{\tabcolsep}{4.5pt}
-\renewcommand{\arraystretch}{1.15}
-\begin{tabular}{lccccc}
-\toprule
-\textbf{Method} &
-\textbf{Equiv.} &
-\shortstack{\textbf{Local}\\\textbf{Inv.}} &
-\shortstack{\textbf{Global}\\\textbf{Inv.}} &
-\shortstack{\textbf{Mag.}\\\textbf{Inv.}} &
-\shortstack{\textbf{Invertible}\\\textbf{Inv. Rep.}} \\
-\midrule
-\textbf{SDB(ours)} & \xmark & \cmark & \cmark & \cmark & \cmark \\
-DH Tr. \cite{marshall2022fast} & \cmark & \xmark & \xmark & \xmark & \xmark \\
-Log-Polar Tr. \cite{reddy1996fft} & \cmark & \xmark & \xmark & \cmark & \xmark \\
-Fourier Mellin Tr. \cite{reddy1996fft} & \cmark & \xmark & \xmark & \cmark & \xmark \\
-\shortstack{SIFT / SURF/ \\ORB / AKAZE\\ \cite{lowe2004sift,bay2006surf,rublee2011orb,alcantarilla2012kaze}}  & \xmark & \cmark & \xmark & \xmark & \xmark \\
-Hu Moments \cite{hu1962visual} & \xmark & \cmark & \cmark & \xmark & \xmark \\
-Zernike Moments \cite{teague1980zernike} & \xmark & \xmark & \xmark & \cmark & \xmark \\
-TFNs \cite{thomas2018tfns} & \cmark & \xmark & \xmark & \cmark & \xmark \\
-Spherical convs. \cite{cohen2018spherical,esteves2018spherical}& \cmark & \xmark & \xmark & \xmark & \xmark \\
-Capsule Networks \cite{sabour2017dynamic} & \cmark & \xmark & \xmark & \xmark & \xmark \\
-\bottomrule
-\end{tabular}
-
-\caption{Comparison of equivariance, invariance, and invertibility properties across image descriptors. Abbreviations: TFN Tensor Field Networks, Tr. Transform, Inv. Invariant, Equiv. Equivariant, SIFT Scale–Invariant Feature Transform, SURF Speeded–Up Robust Features, ORB Oriented FAST and Rotated BRIEF, AKAZE Accelerated KAZE.}
-\label{tab:rotation-summary}
-\end{table}
-
-
-\paragraph{$G$-bispectra and their inversions.} \citet{kakarala_thesis} proposed the $G$-Bispectrum, defined for a signal whose domain is a group. The $G$-Bispectrum gained interest as it is the lowest-degree spectral invariant that is complete. The translation-invariant bispectrum, invariant to the cyclic group $G=C_n$, is a classic and widely used signal processing tool, and is an example of a $G$-bispectrum. \cite{giannakis1989} proposed the first inversion scheme of the translation-invariant bispectrum, reconstructing a 1D signal analytically from translation bispectrum coefficients. \cite{sundaramoorthy1990} developed another analytical inversion algorithm that is more robust to noise, and \cite{pinilla2023} revisited this inversion problem, providing a gradient-based inversion procedure. \cite{giannakis1989} also extended his 1D inversion scheme to invert the 2D translation-invariant bispectrum, invariant to the product of cyclic groups $C_n \times C_n$. The 2D translation-invariant bispectrum is another example of a $G$-Bispectrum.  \cite{ mataigne2024selective} generalized that proof for the product of any number of cyclic groups, \textit{i.e.}, to any $G$-bispectrum, for signals defined over a finite and commutative group $G$. A reflection-invariant bispectrum was also proposed by Edidin and Katz \cite{edidin2025}, who formulated a gradient descent optimization for its inversion. Methods to invert the $G$-bispectrum of signals defined on a (continuous) Lie group $G$ have been proposed in \cite{kakarala2012,kakarala1993}. Note, the disk bispectrum is not covered under any of the aforementioned proofs, as the signal is defined over a planar disk, which is not a group.
-
-The $G$-bispectrum on homogeneous spaces, defined for a signal whose domain is homogeneous for a group $G$, was proposed by \cite{kakarala2012bispectrumsourcephasesensitiveinvariants}. A space that is homogeneous for a group is one where for any $x, y$ there is a group element $g \in G$ s.t. $g \circ x = y$. \cite{kakarala2012bispectrumsourcephasesensitiveinvariants} also proposed a method to invert the $G$-bispectrum of signals defined on spaces homogeneous for a group $G$, specifically for the sphere $S^2$, which is homogeneous for $SO(3)$. Existing rotation-invariant $G$-bispectra are defined for signals on the circle ($SO(2)$) or the sphere ($SO(3)$). Note, the disk bispectrum is not covered under any of these proofs, as the signal is defined over a planar disk, which is not homogeneous for the rotation group $SO(2)$.
-
-\paragraph{Selective $G$-Bispectra.} Across existing $G$-bispectra, several authors have shown that bispectrum coefficients are generally redundant. It is usually possible to recover the signal with significantly fewer bispectrum coefficients than those provided in the full bispectrum, thus enabling a \textit{selective} version of that bispectrum. The work of \citep{giannakis1989signal, kakarala_thesis, sadler1992shift} showed that for \textit{some} finite, commutative groups, the $G$-Bispectrum can be computed with only $|G|$ space complexity, where $|G|$ is the number of elements in the group $G$. \citet{  mataigne2024selective} extended these results by showing this result for a wider range of finite groups, \textit{i.e.}, all discrete commutative groups, and some non-commutative ones such as the dihedral groups of any order, the octahedral and full octahedral group. \cite{ mataigne2024selective} further proposed inversion of selective bispectra for the following non-commutative groups: any dihedral group, the octahedral group, and the full octahedral group. \cite{  mataigne2024selective} estimates bispectra (for signals defined on a group) via a group convolution, and offers an algorithm that can be used to identify selective coefficients. In contrast, the disk bispectrum provides an analytic map from the image to the bispectrum coefficients and is defined for signals defined over the full disk, so that it can be applied to images. Additionally, our disk bispectrum inversion identifies the exact selective coefficients required for complete bispectrum inversion. In fact, this work is the first to introduce a selective bispectrum and inversion proof for a non-homogeneous bispectrum.
-
-\paragraph{A Disk Bispectrum.} Most relevant to our work is the work by \citet{zhao2014rotationally}, who proposed the first disk bispectrum. As described in introduction, this full disk bispectrum is impracticable due to its cubic complexity and lack of inversion, which is why we propose the selective disk bispectrum and its inversion.
-
-Related to our work, Kakarala \cite{kakarala2012} proposes inverting a non-complete bispectrum-like function on the disk by introducing an auxiliary function that supplies the missing information needed for reconstruction and using the inversion method of the cyclic group. In contrast, our disk bispectrum inversion is complete and requires no external information to reconstruct the original image from bispectral coefficients.
-
-\paragraph{Multi-reference alignment} We will show the utility of our bispectrum and its inversion to perform multi-reference alignment (MRA). Traditional \textit{translational} MRA utilizes the translation bispectrum, but special methods are required to adjust for a bias induced by noise in bispectrum space. Various techniques have been proposed for this task, such as noise bias estimation and correction \cite{bendory2018bispectrum}, spectral initialization approaches to MRA optimization \cite{chen2018spectral}, and Gauss–Newton optimization schemes \cite{herring2019gauss}. These corrections, however, have been derived only for the \textit{1D translation bispectrum}. No such correction has been established for the \textit{disk bispectrum}. In this work, we not only propose the \textit{selective disk bispectrum}, its inversion, and its approximation, but we also derive the corresponding MRA noise correction term in a similar manner to \cite{bendory2018bispectrum}. Notably, MRA with the disk bispectrum would not be possible without our inversion method, since reconstructing the mean bispectrum back to image space requires a well-defined inverse.
diff --git a/docs/lantow_miolane_2025/sec/3_background.tex b/docs/lantow_miolane_2025/sec/3_background.tex
deleted file mode 100644
index ae14b0c..0000000
--- a/docs/lantow_miolane_2025/sec/3_background.tex
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-\section{Background}
-\paragraph{Disk Harmonics.}
-The 1D translation bispectrum (which is invariant to group action on the circle $S^1$) is composed of Fourier coefficients, which are constructed using Fourier modes, which are eigenfunctions of the Laplacian on $S^1$. Thus, naturally, if we wish to compose a bispectrum invariant to disk rotations, we should do so with disk harmonic coefficients, which are constructed using disk harmonics, which are the eigenfunctions of the Laplacian on the unit disk $\mathbb{D} = \{ x \in \mathbb{R}^2 : \|x\| < 1 \}$. In polar coordinates \( (r, \theta) \in [0,1) \times [0, 2\pi) \), these disk harmonics are given by
-\[
-\psi_{nk}(r, \theta) = c_{nk} J_n(\lambda_{nk} r) e^{in\theta}
-\]
-where \( (n, k) \in \mathbb{Z} \times \mathbb{Z}_{>0} \), \( J_n \) is the $n$-th order Bessel function of the first kind, \( \lambda_{nk} \) is its \( k \)-th root of the $n$-th Bessel function, and \( c_{nk} \) is a normalization constant. Thus, radially, $\psi_{nk}(r, \theta)$ looks like a Bessel function whose value is \( \lambda_{nk} \) on the edge of the disk, and angularly, $\psi_{nk}(r, \theta)$ looks like a complex sinusoid.
-
-\paragraph{Disk Harmonic (DH) Transform.} Let $f : \mathbb{D} \to \mathbb{R}$ be a real-valued square integrable function defined on the unit disk. Disk harmonics form an orthonormal basis for square-integrable functions (like Fourier modes), such that $f$ can be expanded as
-$f(r, \theta) = \sum_{n \in \mathbb{Z}} \sum_{k \in \mathbb{Z}_{>0}} a_{nk} \, \psi_{nk}(r, \theta)$,
-where $a_{nk} \in \mathbb{C}$ are \textit{disk harmonic (DH) coefficients}. The \textit{disk harmonic (DH) transform} computes these coefficients, expanding $f$ in the disk harmonic basis (akin to the Fourier transform expanding $f$ in the Fourier mode basis). A single coefficient, given by
-\[
-a_{nk} = \int_0^{2\pi} \int_0^1 f(r, \theta) \, \overline{\psi_{nk}(r, \theta)} \, r \, dr \, d\theta,
-\]
-encodes the amplitude and phase of $\psi_{nk}$, necessary to reconstruct the signal from the expansion equation above. Importantly, DH coefficients are \textit{equivariant} to rotation of $f$, meaning that a rotation in $f$ will correspond to a rotation in the DH coefficients of $f$.
-
-\begin{proposition}[Equivariance]
-More formally, denote $SO(2)$ the group of 2D rotations. Consider $f$ a square-integrable function on the unit disk with DH coefficients $\{a^f_{n, k}\}_{n\in \mathbb{Z}, k\in \mathbb{Z}_{>0}}$. For every $\phi \in SO(2)$, if $f' = f \circ R_\phi$, then for all $n, k \in \mathbb{Z} \times \mathbb{Z}_{>0}$, we have: $a^{f'}_{n, k} = e^{in\phi}\cdot a^{f}_{n, k}.$ Thus, the DH coefficients are equivariant to 2D rotations.
-\end{proposition}
-
-In the general case, a function $f$ is \textit{equivariant} to a transformation group $G$ if
-\[
-f(T_g x) = T_g' f(x), \quad \forall g \in G,
-\]
-where $T_g$ acts on the input and $T_g'$ is a corresponding action on the output. A function $f$ is \textit{invariant} to $G$ if
-\[
-f(T_g x) = f(x), \quad \forall g \in G.
-\]
-\paragraph{Signal Discretization on images.} While the disk harmonics are defined for functions $f : \mathbb{D} \to \mathbb{R}$, practical applications require discretization. We will work with discretized images, i.e., arrays of size $L \times L$ whose pixel values are interpreted as samples of $f$.
-Let $p = L^2$ be the total number of pixels such that locations and values of image intensities are denoted as $x_1, \dots, x_p$ and $f_1, \dots, f_p$, respectively. For images of shape \( L \times L \) with spacing \( \Delta x = 2/L \), corresponding to the domain \( [-1,1] \times [-1,1] \), the sampling rate is \( (\Delta x)^{-1} = L/2 \), yielding a Nyquist frequency (i.e., bandlimit) of \( L/4 \). The \textit{Nyquist frequency} is the highest frequency that can be well represented when the signal is sampled at a given rate.
-
-Recall that $\lambda_{nk}$ is the $k$-th root of the $n$-th order Bessel function, and that the disk harmonic $\psi_{nk}$ does not extend radially past $\lambda_{nk}$. Therefore, $\lambda_{nk}$ determines  the number of Bessel function oscillations over the disk and is therefore a quasi-frequency, which is subject to Nyquist criterion. According to the Nyquist criterion, we should only keep the DH coefficients for which $\frac{\lambda_{nk}}{2\pi} \leq \frac{L}{4}$, as higher-frequency components correspond to features beyond the image resolution~\citep{zhao2013fourier}.
-
-\paragraph{Truncation of DH coefficients.} Following closely with \cite{fast_dhcs_2023}, for a given bandlimit $\lambda > 0$, typically $\lambda = \frac{\pi L}{2}$, we keep the DH coefficients for indices $(n,k)$ such that $\lambda_{nk} \leq \lambda$, and order them from lowest frequency to highest $\lambda_0 \leq \cdots \leq \lambda_{m-1}$. The index $j=0, ..., m-1$ represents this order. As such, we now index $\lambda_j$ or equivalently $\lambda_{n_j,k_j}$, where $n_j, k_j$ are the $n,k$ of the frequency at index $j$. Importantly, Bessel function roots do not repeat over order, so we can uniquely identify a disk harmonic via $\lambda_{n_j,k_j}$. As such, we also index $\psi$ by $j$, denoting $\psi_j = \psi_{n_j k_j}$.
-
-Table~\ref{tab:indices} shows the indices $j=0, ..., m-1$ organized in a table labeled by $n, k$ for an image of size $8 \times 8$ and $\lambda = \frac{\pi L}{2}$. Further, for a given bandwidth $\lambda$ that determines a number $m = m_\lambda$, we define the maximum angular frequency (maximum order of the Bessel functions) as \( N_m = \max\{n_j \in \mathbb{Z} : j \in \{0, \ldots, m-1\}\} \) , and the maximum Bessel root as \( K_n = \max\{k \in \mathbb{Z}_{>0} : \lambda_{nk} \leq \lambda\} \) for \( n \in \{-N_m, \ldots, N_m\} \). For example, Table~\ref{tab:indices} has $m=46$, $N_m = 10$, and $K_2 = 4$.
-\begin{table*}[t]
-\centering
-\scriptsize
-\resizebox{\textwidth}{!}{
-\begin{tabular}{c|ccccccccccccccccccccc}
-\hline
-$n \rightarrow$ & -10 & -9 & -8 & -7 & -6 & -5 & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
-$k \downarrow$ &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
-\hline
-1 & 44 & 38 & 30 & 25 & 19 & 15 & 10 & 6 & 3 & 1 & 0 & 2 & 4 & 7 & 11 & 16 & 20 & 26 & 31 & 39 & 45 \\
-2 & - & - & - & 48 & 40 & 32 & 23 & 17 & 12 & 8 & 5 & 9 & 13 & 18 & 24 & 33 & 41 & 49 & - & - & - \\
-3 & - & - & - & - & - & - & 42 & 34 & 27 & 21 & 14 & 22 & 28 & 35 & 43 & - & - & - & - & - & - \\
-4 & - & - & - & - & - & - & - & - & 46 & 36 & 29 & 37 & 47 & - & - & - & - & - & - & - & - \\
-\hline
-\end{tabular}
-}
-  \caption{Indices $j=0, ..., m-1$, for $m=46$ organized in a table labeled by $n, k$ for an image of size $L\times L = 8 \times 8$ and $\lambda = \frac{\pi L}{2}$. The columns represent the Bessel orders, i.e., the angular frequencies. The rows represent the Bessel root indices, i.e., the radial frequencies.}
-  \label{tab:indices}
-\end{table*}
-
-\paragraph{Multi-Reference Alignment (MRA)}
-In classical (translational) MRA, the goal is to recover a ground truth signal $f$ from multiple noisy and translated observations of the signal $\tilde{f}_i = f + n_i$, for $i=1, \ldots, I$, and where the noise variables $n_i$ are i.i.d. with $n \sim \mathcal{N}(0, \sigma^2)$. Bispectrum MRA does this by mapping all noisy translated observations to their bispectra $b^{\tilde{f}_i}=b_i \in \{b_1, ..., b_I\}$ and computing the average bispectrum $\mathbb{E}[b_i] = b^f + \delta$ , where $b^f$ is the bispectrum of the true signal $f$ and $\delta$ is a bias term introduced by noise propagation. Theoretically, the ground truth signal can be recovered by correcting for the bias term and mapping the corrected bispectrum back to image space.
-
-\cite{bendory2018bispectrum} has derived the bias term for the 1D translation bispectrum so that MRA can be performed on translated 1D signals. In our theory section, we derive a bias term for the 2D disk bispectrum so that MRA can be performed on rotated 2D signals.
diff --git a/docs/lantow_miolane_2025/sec/4_theory.tex b/docs/lantow_miolane_2025/sec/4_theory.tex
deleted file mode 100644
index e7458ff..0000000
--- a/docs/lantow_miolane_2025/sec/4_theory.tex
+++ /dev/null
@@ -1,175 +0,0 @@
-\section{Theory: The Selective Disk Bispectrum and Its Inversion}
-\label{sec:theory}
-
-
-We first introduce the existing disk bispectrum \cite{zhao2014rotationally} and then we propose two modifications to improve its space and time complexity. \textit{Proofs for all propositions and theorems are in the Appendix}.
-
-\subsection{The Disk Bispectrum}
-
-We recall the definition of the disk bispectrum originally introduced in \citep{zhao2014rotationally}.
-
-\begin{definition}[Disk Bispectrum~\citep{zhao2014rotationally}] Using notation and image setup established in the previous section, let $a_{n_jk_j}$ be the DH coefficients of a square integrable real-valued function $f: \mathbb{D} \to \mathbb{R}$ defined on the disk $\mathbb{D}$, for $j=0, ..., m-1$ such that $\lambda_0 \leq \cdots \leq \lambda_{m-1}$ and $n_j, k_j$ are the associated angular and radial indices. The disk bispectrum is defined as:
-    \begin{equation}
-    b_{j_1, j_2, k_3} = a_{n_{j_1},k_{j_1}} \cdot a_{n_{j_2},k_{j_2}}\cdot a_{n_{j_1}+n_{j_2},k_3}^* \in \mathbb{C},
-\end{equation}
-\label{eq:disk_bispectrum}
-where $*$ indicates complex conjugate, $j \in \{0, ..., m-1\}$, $k \in \mathbb{Z_{>0}}$, $-N_m \leq n_{j_1}+n_{j_2} \leq N_m \text{, and } k_3 \le K_{n_{j_1} + n_{j_2}} \}$.
-\end{definition}
-
-The computational complexity of this disk bispectrum is prohibitive. First, it inherits the time complexity of the DHT, which is $O(L^3)$ where $L\times L$ is the size of the image. While the space complexity can be reduced by realizing the symmetry $j_1 \leftrightarrow j_2$, it is still cubic in $m$, the number of DH coefficients. The next proposition summarizes these complexities.
-
-\begin{proposition}[Complexity of the disk bispectrum]
-    Denote $L \times L$ the size of the image, $m$ the number of DH coefficients, $N_m$ the maximum angular frequency. The space complexity of the disk bispectrum is $O(m^3/N_m)$. The time complexity of the disk bispectrum is  $O(L^3 + m^3/N_m)$.
-\end{proposition}
-
-The first row of Table~\ref{tab:n_coefs} gives the exact number of coefficients in the disk bispectrum for various image sizes $L \times L$. Using the disk bispectrum becomes prohibitive even for traditional image sizes.
-
-% \begin{table}[h]
-%     \centering
-%     \renewcommand{\arraystretch}{1.3}
-%     \small{
-% \begin{tabular}{|l|c|c|c|c|c|}
-% \hline
-%  \textbf{Image size} & $L=8$ & $L=16$ & $L=28$ & $L=56$ & $L=112$ \\
-% \hline
-% \shortstack{Number of \\disk bispectrum \\coefficients} & 244 & 2,334 & 13,715 & 122,910 & 1,057,052 \\
-% \hline
-% \shortstack{Number of \\selective disk\\ bispectrum coefficients} & 27 & 105 & 315 & 1,247 & 4,957 \\
-% \hline
-% \end{tabular}
-% }
-% \vspace{1mm}
-%     \caption{Number of coefficients in the disk bispectrum (first row) and in the selective disk bispectrum (ours, second row) as a function of the image size $L \times L$ where the selected bandlimit for the DH coefficients is $\lambda = \frac{\pi L}{2}$. Using the disk bispectrum becomes quickly prohibitive.}
-%     \label{tab:n_coefs}
-% \end{table}
-
-\begin{table}[t]
-\centering
-\footnotesize
-\setlength{\tabcolsep}{4.5pt}
-\renewcommand{\arraystretch}{1.15}
-
-\begin{tabular}{lccccc}
-\toprule
-\textbf{Image size} & $L=8$ & $L=16$ & $L=28$ & $L=56$ & $L=112$ \\
-\midrule
-Num. DB coefs. & 244 & 2,334 & 13,715 & 122,910 & 1,057,052 \\
-Num. SDB coefs. & 27 & 105 & 315 & 1,247 & 4,957 \\
-\bottomrule
-\end{tabular}
-
-\caption{Number of coefficients in the disk bispectrum (DB) (first row) and in the selective disk bispectrum (ours, second row) as a function of the image size $L \times L$, where the selected bandlimit for the DH coefficients is $\lambda = \pi L / 2$. Using the full disk bispectrum becomes quickly prohibitive.}
-\label{tab:n_coefs}
-\end{table}
-
-
-
-\subsection{The Selective Disk Bispectrum (SDB) and Its Inversion}
-
-We propose the selective disk bispectrum (SDB), defined as a carefully chosen subset of the (full) disk bispectrum coefficients. The SDB improves the space complexity from $\mathcal{O}(m^3/N_m)$ of the previous bispectrum (or $\mathcal{O}(L^2)$ of the original image) to $\mathcal{O}(m)$ and the time complexity from $\mathcal{O}(L^3+m^3/N_m)$ to  $\mathcal{O}(L^3 + m)$.  We prove that the selected coefficients are sufficient for recovering the image $f$, up to rotation, meaning that most of the coefficients in the original disk bispectrum are, in fact, redundant.
-
-\begin{definition}[Selective Disk Bispectrum]\label{def:selective-bsp}
-Let $a_{n_jk_j}$ be the DH coefficients of a square integrable real-valued function $f: \mathbb{D} \to \mathbb{R}$ defined on the disk $\mathbb{D}$, for $j \in \{0, ..., m-1\}$ such that $\lambda_1 \leq \cdots \leq \lambda_m$ and $n_j, k_j$ are the associated angular and radial indices. The selective disk bispectrum is defined as
-\begin{equation}
-b^f = \left[
-\begin{array}{cccccccc}
-b_{0,0,1}  \cdots b_{0,0,K_0} \\
-b_{2,0,1}  \cdots b_{2,0,K_{0}} \\
-b_{2,N_m-1,1} \cdots b_{2,N_m-1,K_{N_m -1}}
-\end{array}
-\right],
-\end{equation}
-where the entries correspond to selected coefficients of the disk bispectrum defined as:
-\begin{equation}
-\left\{
-\begin{aligned}
-    b_{0, 0, k}
-    &= a_{0, 1}^2 \cdot a_{0, k}^*, \\
-    & \text{for } k \in \{1, \dots, K_{0}\},\\
-    b_{2, n, k}
-    &= a_{1, 1} \cdot a_{n,1} \cdot a_{n+1,k}^*, \\
-    & \text{for } n \in \{0, N_m-1\},~\text{and}~ k \in \{1, \dots, K_{n+1}\}.
-\end{aligned}
-\right.
-\end{equation}
-\label{eq:selective_disk_bispectrum}
-\end{definition}
-We note that the selective bispectrum coefficients are indexed by $j_1,n_2,k_3$ (limiting $j_1\in\{0,2\}$), as opposed to $j_1, j_2, k_3$. Selecting only a subset of the disk bispectrum coefficients reduces its space and time complexities.
-
-\begin{proposition}[Complexity of the selective disk bispectrum]
-    Denote $L \times L$ the size of the image, $m$ the number of DH coefficients, $N_m$ the maximum angular frequency. The number of selective bispectrum coefficients is $N = \sum_{n= 0}^{N_m} K_n$.
-    The space complexity of the selective disk bispectrum is $O(m)$. Its time complexity is $O(L^3 + m)$.
-\end{proposition}
-
-Despite discarding many coefficients from the full disk bispectrum, our main result shows that the remaining $N = \sum_{n= 0}^{N_m} K_n$ coefficients are sufficient to reconstruct the signal, up to a global rotation. Fig.~\ref{fig:bispecrum_inversion} illustrates the technique behind the proof, showing which set of bispectrum coefficients are necessary to reconstruct a specific DH coefficient.
-
-\begin{theorem}[Inversion]
-    Denote $L \times L$ the size of the image $f$, $m$ the number of DH coefficients, $N_m$ the maximum angular frequency, and $a_{n_j, k_j}$ the DH coefficients for $j=0, ..., m-1$. Assume that $a_{n, 1} \neq 0$ for $n \in \{0, ..., N_m-1\}$. Then, its selective disk bispectrum $b^f$ can be inverted, that is $f$ can be reconstructed from $b^f$ up to a disk rotation. The time complexity of the inversion is $O(m+L^3)$.
-\end{theorem}
-
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=0.9\linewidth]{figs/inversion.pdf}
-  \caption{Graphical illustration of the selective disk bispectrum inversion. The arrows indicate in which order the DH coefficients are reconstructed.}
-  \label{fig:bispecrum_inversion}
-\end{figure}
-
-We note that our inversion of the selective disk bispectrum also yields an inversion method for the (full) disk bispectrum. We also note that nonzero coefficient assumptions (such as $a_{n, 1} \neq 0$) are classic in bispectrum inversion literature to avoid division by zero. However, this assumption does not impact real-world application as long as there is some noise on the images.
-
-\begin{theorem}[Complete Invariance]
-Let \( f \) and \( f' \) be a pair of real-valued, square-integrable functions on the disk. Let the bispectrum be defined as in Definition~\ref{def:selective-bsp}. Assume that the DH coefficients \( a_{n,1} \) of \( f \) are non-zero for all \( n \in \{0, \dots, N_m - 1\} \). Then $b^f = b^{f'}$ if and only if \( f' = f \circ R_\phi \) for any 2D rotation \( \phi \in SO(2) \). Thus, the selective bispectrum is a complete invariant.
-\end{theorem}
-
-Part of the proof relies on the equivariance of the DH coefficients, which allow us to (re)prove the invariance of the (selective) disk bispectral coefficients used in~\citep{zhao2014rotationally}. The completeness of the SDB is proven by our inversion theorem, as the SDB contains all information about the original signal necessary for reconstruction.
-
-\begin{figure*}[t]
-  \centering
-  \includegraphics[width=\textwidth]{figs/bispectrum_tractability.pdf}
-  \caption{(a) The selective disk bispectrum (SDB) map is significantly faster than the full disk bispectrum map, especially for large images. (b) Bispectrum invariance error decreases with higher image resolution. (c-d) The SDB coefficient inversion sucessfully reconstructs images for resolution $28\times28$ (c) and $112\times112$ (d), and achieves a similar reconstruction error as the DH Transform \cite{fast_dhcs_2023} inversion.}
-  \label{fig:tractability}
-\end{figure*}
-
-
-\subsection{Numerical Approximation of The Selective Disk Bispectrum and Its Inversion}
-
-To further reduce the time complexity of the selective bispectrum, we propose leveraging a recent numerical approximation of the DH coefficients \citep{fast_dhcs_2023} which computes the disk harmonic transform in $O(L^2 \log L)$. This yields the following result:
-
-\begin{proposition}[Complexity of the selective disk bispectrum]
-    Denote $L \times L$ the size of the image, $m$ the number of DH coefficients. The time complexity of the approximate selective disk bispectrum is $O(L^2 \log L + m)$.
-\end{proposition}
-
-Next, we prove that we still have precise guarantees on the accuracy of the bispectrum coefficient approximations, the invariance property, and the reconstruction of the original image.
-
-\begin{theorem}\label{th:accuracy}
-Let \( 0 < \varepsilon \leq 1 \) be a target accuracy, and assume that \( \lambda \leq \sqrt{\pi p} \) and \( |\log \varepsilon| \leq \sqrt{p} \), where $\lambda$ is the bandlimit for the DH coefficients and \( p = L^2 \) the number of pixels in the image $f$ of size $L \times L$. Let \( b^f \) and \( \tilde{b}^f \) denote the exact and approximate selective disk bispectrum of \( f \), where \( \tilde{b}^f \) is computed using the approximate disk harmonic transform of \citet{fast_dhcs_2023}. Then,
-\[
-\| \tilde{b}^f - b^f \|_{\infty} \leq C\varepsilon \| f \|_{1},
-\]
-for a constant \( C \) depending only on a bound of \( f \) on its compact disk support.
-\end{theorem}
-
-As a corollary, we find that the bispectrum coefficients are approximately invariant with controlled guarantees.
-
-\begin{corollary}
-    Let $f, f'$ be a pair of real-valued square integrable functions on the disk such that there exists a $\phi \in SO(2)$ such that $f' = f \circ R_\phi$. Under the notations and assumptions of Theorem~\ref{th:accuracy}, we have:
-    \begin{equation}
-        \| \tilde b^{f} - \tilde b^{f'}\|_\infty \leq 2C \varepsilon \|f\|_1,
-    \end{equation}
-    for a constant \( C \) depending only on a bound of \( f \) on its compact disk support.
-\end{corollary}
-
-Linking the accuracy to $f$ is useful, as it allows us to control accuracy through number of pixels $p$. If $|f|$ and $C$ are large, but we still want to reach a given accuracy, then we can choose $p$ accordingly such that $\log(\epsilon) < \sqrt{p}$) to match the desired accuracy.
-
-\subsection{MRA with the Selective Disk Bispectrum}
-
-We follow \cite{bendory2018bispectrum}'s assumption that $\mathbb{E}[b_i] = b^f + \delta$, where $b_i \in \{b_1, ..., B_I\}$ is the bispectrum of one rotated noisy observation, $b^f$ is the bispectrum of the true signal $f$, and $\delta$ is a bias term introduced by noise propagation in bispectrum space. In this setting, we derive $\delta$ in the setting where $b^f$ is the disk bispectrum.
-
-\begin{proposition}[SDB MRA Correction]
-     When performing MRA using the selective disk bispectrum of noisy rotated images, the mean bispectrum must be corrected by subtracting $\delta$ before inversion to image space.
-     \begin{equation}
-         \delta = \sigma^2[S^\dagger_{0, k_3}  \delta_{n_{j_1},0}\delta_{n_{j_2},0} + S_{01}     \delta_{1, k_3}\delta_{n_{j_2},0}+ S_{01} \delta_{1, k_3}\delta_{n_{j_1},0}]
-     \end{equation}
-     \label{eq:mra-correction}
-
-\end{proposition}
diff --git a/docs/lantow_miolane_2025/sec/5_tractability.tex b/docs/lantow_miolane_2025/sec/5_tractability.tex
deleted file mode 100644
index 53fff75..0000000
--- a/docs/lantow_miolane_2025/sec/5_tractability.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-\section{Tractability of Selective Disk Bispectrum}
-\label{sec:tractability}
-
-\begin{figure*}[t]
-  \centering
-   \includegraphics[width=\textwidth]{figs/mra_results.pdf}
-    \caption[width=\textwidth]{(a) Dataset samples and original image used in MRA experiments for MNIST 6. (b-e) Relative error compares the similarity between the MRA reconstructed image and the original image used to generate the noisy rotated dataset for (b) $\sigma^2 = 0$ (c) $\sigma^2 = 0.01$, (d) $\sigma^2 = 0.05$, (e) $\sigma^2 = 0.1$. Sample reconstructions are shown below each result plot.}
-    \label{fig:widecaption}
-
-   \label{fig:mra-results}
-\end{figure*}
-
-To illustrate the computational and representational properties of our proposed selective disk bispectrum, we conduct a series of experiments testing \textit{speed}, \textit{invertibility}, and \textit{rotation invariance}. We compare against the full disk bispectrum proposed by \citep{zhao2014rotationally} and described in Eq. \ref{eq:disk_bispectrum}. Both the full and selective bispectra are computed using the disk harmonic coefficient library implemented by \citep{fast_dhcs_2023} on a MacBook Pro with an Apple M2 Max chip.
-
-\subsection{Design of Tractability Experiments}
-
-We consider square grayscale palm tree images of length \( L \in \{16 , 28, 56, 112\} \), each perturbed with Gaussian noise of standard deviation \( \sigma \in \{0, 0.01, 0.05, 0.1\} \). For each image, we compute both the full and selective disk bispectrum using the formulations in Eq.~\ref{eq:disk_bispectrum} (full) and Eq.~\ref{eq:selective_disk_bispectrum} (selective). We repeat these experiments for all digits of MNIST, and show results in the Appendix.
-
-\textbf{Speed.} We compare the runtime of the forward and inverse bispectrum transforms across image sizes $L$ and noise levels $\sigma$. The full disk bispectrum is computationally expensive and impractical for many cases. In contrast, our \emph{selective disk bispectrum} enables efficient computation, making both encoding and reconstruction feasible even for larger images.
-
-\textbf{Invertibility.} In the previous section, we prove that the selective disk bispectrum is \textit{complete}. Here, we visually confirm our theoretical results by constructing an image \( f \) from its bispectrum and measuring the relative reconstruction error $\varepsilon_{\text{rel}} = \frac{\left\lVert \hat{f} - f \right\rVert}{\left\lVert f \right\rVert}$, where \( \hat{f} \) is the image reconstruction obtained by applying the inverse map to the disk bispectrum coefficients. We observe in Fig. \ref{fig:tractability} that the selective bispectrum retains the essential information needed to accurately reconstruct the original image, demonstrating its practical completeness.
-
-% \[
-% \varepsilon_{\text{rel}} = \frac{\left\lVert \hat{f} - f \right\rVert}{\left\lVert f \right\rVert},
-% \]
-
-\textbf{Rotation Invariance.} We verify the invariance of the disk bispectrum to disk rotations. Given an image \( f \) and its rotated version \( f' \), we compute their respective bispectra \( b^{f} \) and \( b^{f'} \). Since the bispectrum is designed to be invariant to rotations, we expect $b^{f} = b^{f'}$. To quantify invariance, we compute the relative error between the two bispectra $\varepsilon_{\text{rel}} = \frac{\left\lVert b^{f} - b^{f'} \right\rVert}{\left\lVert b^{f} \right\rVert}$. Low error values across test conditions confirm that both the full and selective bispectra exhibit strong empirical rotation invariance, as shown in Fig.~\ref{fig:tractability}.
-% \[
-% \varepsilon_{\text{rel}} = \frac{\left\lVert b^{f} - b^{f'} \right\rVert}{\left\lVert b^{f} \right\rVert}.
-% \]
-
-\subsection{Results of Tractability Experiments}
-
-We present the results of the tractability experiments. Full experimental results, including results on all MNIST images, are available in the Appendix.
-
-\textbf{Speed.} The selective disk bispectrum offers a 3× speedup over the full bispectrum for \(16 \times 16\) images, and a 17,\!322× speedup for \(112 \times 112\) images. The inverse transform is equivalently fast for the full and selective bispectra, since the full bispectrum inversion utilizes only the subset of its coefficients included in the selective variant. Forward and inverse timing results are shown in Fig.~\ref{fig:tractability} for \(\sigma = 0\); run times are unaffected by noise level.
-
-\textbf{Invariance.} Bispectrum invariance error decreases with higher image resolution, as shown in Fig.~\ref{fig:tractability}, and is as low as $3$ percent for $112\times112$ images. Invariance error is largely unaffected by noise level.
-
-\textbf{Invertibility.} As shown in Fig.~\ref{fig:tractability}, the selective bispectrum accurately reconstructs the original image. This reconstruction accuracy is identical to the full bispectrum's. While reconstruction error increases slightly at lower tested resolutions, images are sufficiently reconstructed across all tested settings (see Appendix for reconstruction across bispectrum type, noise, and image resolution). Importantly, bispectrum inversions match the original image, up to a rotation. Since the bispectrum inversion reconstructs the original image up to a global rotation, reconstructions in Fig. ~\ref{fig:tractability} have been manually registered to match the orientation of the original image for clear comparison.
diff --git a/docs/lantow_miolane_2025/sec/6_mra_experiments.tex b/docs/lantow_miolane_2025/sec/6_mra_experiments.tex
deleted file mode 100644
index c79e235..0000000
--- a/docs/lantow_miolane_2025/sec/6_mra_experiments.tex
+++ /dev/null
@@ -1,8 +0,0 @@
-\section{Application: Multi-Reference Alignment with the Selective Disk Bispectrum}
-
-We demonstrate the utility of our S. D.B. and its inversion for rotation-invariant multi-reference alignment (MRA), leveraging our new MRA correction term. We test rotation-invariant MRA on all MNIST digits, with results for the digit \textit{6} shown in Fig.~\ref{fig:mra-results} and other digits in the Appendix. For each experiment, we begin with a noiseless MNIST image and generate
-\( n_X \in \{10, 20, 30, 40, 50, 60, 100, 200, 300, 400, 500\} \) copies. Each copy is corrupted with i.i.d.\ Gaussian noise
-\( n \sim \mathcal{N}(0, \sigma^2) \) for
-\( \sigma^2 \in \{0, 0.01, 0.05, 0.1\} \), randomly rotated by
-\( R \in [0, 2\pi] \), and masked outside the disk, yielding
-\( n_X \) images \( \{\tilde{f}_i\}_{i=1}^{n_X} \). We map each image to bispectrum space to obtain \( \{b_i\}_{i=1}^{n_X} \), average the bispectra, apply our correction term from Eq.~\ref{eq:mra-correction}, and map the corrected average back to image space using our disk bispectrum inversion method from Sec.~\ref{sec:theory}. Each experiment is repeated ten times with different random seeds, and computes relative error between MRA estimate $\bar{f}$ and true signal $f$ as $\varepsilon_{\text{rel}} = \frac{\left\lVert \bar{f} - f \right\rVert^2}{\left\lVert f \right\rVert^2}$. As shown in Fig.~\ref{fig:mra-results}, reconstruction error decreases and stabilizes around 0.23 as \( n_X \) increases—consistent with the empirical baseline reconstruction and invariance errors reported for $28\times28$ images in Sec.~\ref{sec:tractability}. These results are the first demonstration of MRA using a rotation-invariant bispectrum.
diff --git a/docs/lantow_miolane_2025/sec/7_conclusion.tex b/docs/lantow_miolane_2025/sec/7_conclusion.tex
deleted file mode 100644
index d075c85..0000000
--- a/docs/lantow_miolane_2025/sec/7_conclusion.tex
+++ /dev/null
@@ -1,3 +0,0 @@
-\section*{Conclusion}
-
-We present the first disk bispectrum inversion, which reveals the minimal sufficient coefficients needed for accurate shape recovery. This enables the construction of our \emph{selective} disk bispectrum, which is \textbf{17,322× faster} to compute than the full bispectrum for $(122\times112)$ images. Crucially, our disk bispectrum inversion provides a mapping from disk bispectrum coefficients back to the image they represent, meaning that statistics can be done in bispectrum space and the result can be mapped back to image space. Without this, methods such as MRA would be impossible. Accordingly, we additionally derive the noise propagation that would occur for selective disk bispectrum MRA (in line with \cite{bendory2018bispectrum} and use it to showcase our bispectrum inverse in MRA experiments. The selective disk bispectrum's speed and invertibility position it as a tractable and scalable shape representation for rotation-invariant learning.
diff --git a/docs/mataige_et_al_2024.tex b/docs/mataige_et_al_2024.tex
deleted file mode 100644
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-%%%%%%%% ICML 2024 EXAMPLE LATEX SUBMISSION FILE %%%%%%%%%%%%%%%%%
-
-\documentclass{article}
-
-% Recommended, but optional, packages for figures and better typesetting:
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-\usepackage{graphicx}
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-\usepackage{hyperref}
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-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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-%\usepackage[disable,textsize=tiny]{todonotes}
-\usepackage[textsize=tiny]{todonotes}
-
-
-% The \icmltitle you define below is probably too long as a header.
-% Therefore, a short form for the running title is supplied here:
-% \icmltitlerunning{Efficient, Complete G-Invariance for G-Equivariant Networks via Algorithmic Reduction}
-\icmltitlerunning{Fast and Complete G-Invariance via Novel Algorithmic Reduction of the G-Bispectrum on Finite Groups}
-
-\begin{document}
-
-\twocolumn[
-\icmltitle{Efficient, Complete G-Invariance for G-Equivariant Networks \\ via Algorithmic Reduction}
-
-% It is OKAY to include author information, even for blind
-% submissions: the style file will automatically remove it for you
-% unless you've provided the [accepted] option to the icml2024
-% package.
-
-% List of affiliations: The first argument should be a (short)
-% identifier you will use later to specify author affiliations
-% Academic affiliations should list Department, University, City, Region, Country
-% Industry affiliations should list Company, City, Region, Country
-
-% You can specify symbols, otherwise they are numbered in order.
-% Ideally, you should not use this facility. Affiliations will be numbered
-% in order of appearance and this is the preferred way.
-\icmlsetsymbol{equal}{*}
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-\begin{icmlauthorlist}
-\icmlauthor{Simon Mataigne}{equal,uclouvain}
-\icmlauthor{Johan Mathe}{equal,atmo}
-\icmlauthor{Sophia Sanborn}{science}
-\icmlauthor{Christopher Hillar}{ucb}
-\icmlauthor{Nina Miolane}{ucsb}
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-
-\icmlaffiliation{uclouvain}{ICTEAM, UCLouvain, Louvain-la-Neuve, Belgium. Simon Mataigne is a Research Fellow of the Fonds de la Recherche Scientifique - FNRS. }
-\icmlaffiliation{ucsb}{UC Santa Barbarbara, CA}
-\icmlaffiliation{science}{Science, San Francisco, CA.}
-\icmlaffiliation{ucb}{Redwood Center for Theoretical Neuroscience, UC Berkeley, CA.}
-\icmlaffiliation{atmo}{Atmo, San Francisco, CA.}
-
-\icmlcorrespondingauthor{Simon Mataigne}{simon.mataigne@uclouvain.be}
-%\icmlcorrespondingauthor{Firstname2 Lastname2}{first2.last2@www.uk}
-
-% You may provide any keywords that you
-% find helpful for describing your paper; these are used to populate
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-\icmlkeywords{Machine Learning, ICML}
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-
-%\printAffiliationsAndNotice{}  % leave blank if no need to mention equal contribution
-\printAffiliationsAndNotice{\icmlEqualContribution} % otherwise use the standard text.
-
-\begin{abstract}
-% The development of Convolutional Neural Networks (CNNs) that are insensitive to a group action of the input, e.g., a rotation of the input image, is a critical task in machine learning.
-% In Geometric Deep Learning \cite{bronstein2021geometric}, geometric priors derived from the structure of data are built into deep learning models to increase model efficiency and generalization. The layers of a Group-Equivariant Convolutional Neural Network (G-CNNs) \cite{cohenc16} are designed to have the property of \textit{equivariance} to a group of transformations, such as rotation -- that is, to
-
-    %Group-Equivariant Convolutional Neural Networks (G-CNNs) \cite{cohenc16} exploit the symmetries in data to reduce the translation-equivariance of convolutional neural networks can be generalized to other transformation groups
-
-Group-Equivariant Convolutional Neural Networks (G-CNNs) \cite{cohenc16} generalize the translation-equivariance of traditional CNNs to group-equivariance,
-% \textit{group}-equivariance,
-using more general symmetry transformations such as rotation.
-% for weight tying.
-For tasks such as classification, these transformations are removed at the end of the network to achieve group-invariance, typically by taking an average or maximum over the group. While this is indeed invariant, it is \textit{excessively} so; two inputs that are non-identical up to group action can yield the same output, resulting in a general lack of robustness to adversarial attacks \cite{sanborn2023bispectral, sanborn2023general}. The latter paper proposed an alternative method for achieving invariance without loss of signal structure, called the $G$-triple correlation ($G$-TC). While this method yields demonstrable gains in accuracy and robustness, it comes with a significant increase in computational cost. In this paper, we introduce a new invariant layer based on the Fourier transform of the $G$-TC: the $G$-\textit{bispectrum}. Operating in Fourier space allows us to significantly reduce the computational cost. Our main theoretical result provides a reduction of the $G$-bispectrum that conserves the selective invariance of the $G$-TC, while only requiring $\mathcal{O}(|G|)$ coefficients. Moreover, we validate this finding in a suite of experiments.
-%% seems repetitive:
-% In a suite of experiments, we demonstrate that this approach retains the invariance of the $G$-TC and significantly reduces computational cost.
-\end{abstract}
-
-\section{Introduction}\label{sec:introduction}
-The visual world is rich with symmetries.
-% \textit{symmetries}.
-For example, the identity of an object is invariant to its position in the visual field; vision has \textit{translational symmetry}. Since its emergence in the 1990s, the Convolutional Neural Network (CNN) has become a fundamental tool for image processing \cite{lecun95convolutional,LeNet5}. One of the key strengths of this architecture is that it reflects and exploits this translational symmetry. A significant challenge faced by vision models is to ensure invariance to other natural image transformations, such as rotation and scaling. A common approach to this challenge is to train the model on a dataset augmented with these transformations \cite{shorten2019survey}. However, this approach is resource-intensive in terms of memory, model size, and training time, making it difficult to scale. Alternatively, one can analytically incorporate these additional symmetries into the model architecture, to generalize the insights of CNNs to other symmetry groups relevant for vision. Group-Equivariant CNNs ($G$-CNNs) \cite{cohenc16} do just this, with more general group-equivariant convolutions.
-
-In image processing, one typically wants to preserve transformations throughout the layers of a network (i.e. to be group-\textit{equivariant}), and remove them only at the end when canonicalizing an image for classification (i.e. to be group-\textit{invariant}). While the theory of equivariant layers has been thoroughly developed \cite{cohen2021equivariant, weiler2023EquivariantAndCoordinateIndependentCNNs}, less attention has been paid to the theory of invariance
-% \textit{invariance}
-in neural network architectures. Commonly, invariance in $G$-CNNs is achieved by simply taking an average or maximum over the transformation group (Average or Max $G$-Pooling, respectively). However, as noted by \citet{sanborn2023general}, this is a highly lossy operation that removes information about the structure of the signal. While the \texttt{max} operation is indeed invariant (the \texttt{max} of an image is the same as the \texttt{max} of an image rotated by $90$ degrees), it is \textit{excessively} invariant: one could permute all of the pixels in the image and it would still have the same maximum, but with none of the same structure.
-
-To address this, \citet{sanborn2023general} developed a theory for constructing general invariant $G$-CNN layers that are \textit{complete}\textemdash that is, they collapse group transformations with no loss of signal structure. Their approach achieves demonstrable gains in accuracy and robustness, but at a significant computational cost. In this paper, we provide new mathematical results that allow us to build general invariant pooling layers for $G$-CNNs that preserve the analytical robustness of \citet{sanborn2023general} while significantly improving computational efficiency.
-
-Like \citet{sanborn2023general}, our approach is built on the mathematics of the triple correlation (TC)
-% \emph{triple correlation} (TC)
-and its Fourier transform, the \emph{bispectrum}. Historically, these mathematical objects found initial applications in the context of classical signal processing as generalizations of the two-point autocorrelation \cite{tukey1953spectral, brillinger1991some, nikias1993signal}. The work of \citet{kakarala_thesis} illuminated the relevance of this mathematics for invariant theory, as the general triple correlation on groups
-% \textit{groups}
-is the lowest-degree invariant polynomial that is complete
-% \textit{complete}
-\cite{sturmfels2008algorithms}. The downside of the $G$-TC is that its number of coefficients scales as $\mathcal{O}(|G|^2)$, where $|G|$ is the size of the group. Since each coefficient demands for $\mathcal{O}(|G|)$ operations, the computational cost of the $G$-TC layer is $\mathcal{O}(|G|^3)$. In comparison, the Max $G$-pooling layer features a $\mathcal{O}(|G|)$ computational cost and returns a scalar output.
-
-% : the number of coefficients in the $G$-TC scales as $\mathcal{O}(|G|^2)$,
-% $\mathcal{O}(|G|^3)$
-% with $|G|$ the size of the group.
-
-% This \textit{completeness} can be leveraged in neural network architectures to ensure that invariance is achieved in a selective manner.
-
-% In $G$-CNNs, the group transformations (e.g. translation or rotation) that are preserved throughout the layers of the network are typically removed at the end for the purposes of classification, where it is important to be invariant to such nuisance factors. This is achieved by taking a maximum over the group per filter -- Max $G$-Pooling. While the \texttt{max} operation is indeed invariant (the \texttt{max} of an image is the same as the \texttt{max} of an image rotated by $90$ degrees), it is \textit{excessively} invariant: one could permute all of the pixels in the image and it would still have the same maximum, but with none of the same structure. To tackle this problem, \citet{sanborn2023general} proposed to replace the Max $G$-pooling by a ``$G$-triple correlation layer'' ($G$-TC), achieving demonstrable gains in accuracy and robustness.
-
-% Recent work has revealed the promise of the triple correlation as a computational primitive in machine learning tasks -- for symmetry discovery \cite{sanborn2023bispectral}, and to achieve robust group-invariant pooling within $G$-CNNs \cite{sanborn2023general}. The power of the TC lies in its \textit{invariance} and \textit{completeness}. The triple correlation defined on a given group $G$ is invariant to the actions of that group on the domain of the signal. More importantly, it is \textit{only} invariant to the actions of the group. Any other changes in the signal will result in a different TC.
-
-%In consequence, two signals sampled from different orbits produce different outputs
-
-
-
-% The $G$-TC is complete \cite{Yellott:92, Kakarala2009CompletenessOB, kakarala1993_group_theo}, i.e., it conserves all the information of the input signal up to the group action.
-
-% The $G$-CNN architecture relies on a ``$G$-convolutional layer'', followed by a pooling operation, Max $G$-pooling. However, the pooling loses information from the convolution so that it suffers from excessive $G$-invariance: two input signals that are not similar by the group action -- they are in different \emph{orbits} -- can output the same information after pooling.
-
-% \cite{sanborn2023bispectral, sanborn2023general}.
-% The TC is a third order polynomial of the input signal. Its early applications include ocean wave analysis \cite{Hasselman, Houmb_1974, mccomas_briscoe_1980}, and signal processing \cite{Lohmann84, Bendory_2018}. Recently, the TC has been used to learn the group invariance of a dataset \cite{sanborn2023bispectral} and,
-% chiefly for our purpose, incorporated into Group-Equivariant Convolutional Neural Networks (G-CNNs) as a group-invariant pooling layer.
-
-
-
-\textbf{Contributions.} In this work, we demonstrate that the $G$-triple correlation can be analytically reduced in the Fourier domain (i.e. the $G$-\textit{bispectrum}), by exploiting symmetries within its calculation. Concretely, only a subset of the coefficients are needed to invert the bispectrum and reconstruct an input signal (up to the group action). \cite{giannakis1989signal, kakarala_thesis, sadler1992shift} first showed this for some finite, commutative groups, where it was demonstrated that only $|G|$ bispectral coefficients are needed for completeness. In this paper, we provide the first algorithmic reduction of the bispectrum on general commutative groups and an infinite family of non-commutative ones: we show that only $\mathcal{O}(|G|)$ bispectral coefficients are needed to fully characterize a signal up to group action. We use this reduction to propose a new \textit{bispectral} pooling layer that strikes a balance between robustness and efficiency. In a suite of experiments on the MNIST \cite{deng2012mnist}, EMNIST \cite{cohen2017emnist} and CIFAR-10 \cite{CIFAR10} datasets, we demonstrate how the choice of invariant layer (Max $G$-pooling, $G$-TC, selective $G$-bispectrum) impacts accuracy and speed on classification tasks. We show that our bispectral layer achieves the expected results: it is both more accurate than the Max $G$-pooling and cheaper than the $G$-TC. Interestingly, this accuracy and speed advantage holds in $G$-CNNs with low number of convolutional filters. However, increasing the number of filters in the $G$-Convolutions allows the Max $G$-Pooling to catch up on the accuracy of the $G$-bispectral pooling. This demonstrates that the $G$-bispectral pooling will be particularly interesting for neural networks operating under a smaller parameter budget.
-
-%show that the bispectral layer achieves the expected results: it is both more accurate than the Max $G$-pooling and cheaper than the $G$-TC.
-
-%Our work thus brings the precise and mathematically-motivated findings of \cite{sanborn2023general} into the sphere of practicality through cost-efficiency.
-
-% Finally, we  the choice of a reduced bispectrum layer against a full bispectrum layer.
-
-% This allows us to introduce a novel layer for $G$-invariant pooling that strikes a balance between robustness and efficiency.
-
-% The cost to compute a bispectral coefficient is usually independent of $|G|$ if one knows the Fourier transform of the input signal (of size $|G|$). For all finite groups $G$, the Fourier transform can be obtained in $\mathcal{O}(|G|^2)$ operations at most, even $\mathcal{O}(|G|\log(|G|))$ when a Fast Fourier Transform (FFT) algorithm is known on $G$. In particular, the Cooley-Tukey FFT algorithm can be extended to many finite groups, including the dihedral group \cite{Diaconis1990EfficientCO, michael89fft}. Therefore, we can reduce the cost of the bispectrum calculation on many classes of groups to $\mathcal{O}(|G|\log|G|)$ in computation and $\mathcal{O}(|G|)$ in space.
-
-% This motivates the introduction of a $\mathcal{O}(|G|^2)$ layer in both space and computations (or less). This layer should provide more accurate results than the Max $G$-pooling as well as being less expensive than the $G$-TC.
-% The improvement comes from the following observation: there are many redundancies in the $G$-TC that can be neatly exploited in the Fourier domain (i.e. with the \textit{bispectrum} \cite{kakarala_finite_groups}), permitting a substantial reduction in size and cost to compute.
-
-
-
-% This is most readily seen in the Fourier domain -- i.e. with the \textit{bispectrum} \cite{kakarala_finite_groups}.
-
-
-
-% Moreover, we can design a layer returning the full bispectrum, in which case, the complexity is $\mathcal{O}(|G|^2)$ in both computations and space. To determine which coefficients induce completeness, it is enough to determine which allow to reconstruct the initial signal entirely. Retrieving the signal from bispectral coefficients is called the \emph{bispectrum inversion problem} \cite{kakarala_thesis,Bendory_2018}.
-
-% In this paper, we propose an implementation of the $G$-CNN where the pooling layer is a bispectrum layer, as suggested in \cite{sanborn2023general}. This corresponds to switching the neural network in the Fourier domain. We will perform experiments on both the selective bispectrum layer (only a few coefficients) and the full bispectrum layer (all coefficients). To ensure completeness of the selective bispectrum, we show that the bispectrum can be inverted using $|G|$ bispectral coefficients for all finite commutative groups and $\frac{|G|}{4}$ coefficients for the dihedral group, i.e., the symmetries of the $n$-gon. The dihedral group is the main non-commutative group of interest for 2D-image processing.\simcom{Nina, a reference from clinical application or so?}.
-
-% Extending the implementation og \citet{sanborn2023bispectral} to assess the performance of the reduced bispectra layers on the MNIST \cite{deng2012mnist} and EMNIST \cite{cohen2017emnist} datasets.
-% We show that the bispectral layer achieves the expected results: it is both more accurate than the Max $G$-pooling and cheaper than the $G$-TC. Finally, we discuss the choice of a reduced bispectrum layer against a full bispectrum layer.
-
-\section{Background on group theory}
-From this point, we assume known the notions of group $(G,\ \cdot)$, representation $(\rho,\ V)$ of a group, irreducible representation---irreps for short--- and generating set of a group. For the reader uncomfortable with these notions, we invite to read Appendix \ref{app:background}. We incorporate the ideas from signal processing and machine learning, and define how a group of transformations can indeed transform signals or datasets.
-%Now that a group $G$ and its irreps are properly defined, it is important to define how $G$ acts on a given vector space~$X$.
-Take for instance a black-and-white picture modeled by some function $f:\mathbb{R}^2\mapsto [0,1]$. The \emph{group action} may define how the image is flipped or rotated. We define the notion of group action more formally below.
-\begin{definition}\label{def:groupaction}
-    Given a group $(G,\ \cdot)$, a \emph{group action} $\alpha:G\times X\mapsto X$ is a function satisfying i) Identity: $\alpha(e, x)=x$, ii) Compatibility: $\alpha(h,\alpha(g,x))=\alpha(h\cdot g, x)$.
-\end{definition}
-
-An important problem in machine learning is to achieve \textit{invariance} to nuisance factors not relevant for the task. Many of these factors are describable as group actions (e.g. rotations, translations, scaling). Thus, we want our models to be $G$-invariant:
-\begin{definition}
-    A function $\psi:X\mapsto Y$ is $G$-\emph{invariant} if $\psi(\alpha(g,x)) = \psi(x)$ for all $x\in X$ and all $g\in G$.
-\end{definition}
-A $G$-CNN is a neural network that consists of $G$-convolutional layers and a pooling operation \cite{cohenc16}. The $G$-convolution layer is not $G$-invariant: it is $G$-\emph{equivariant}. This means that a group action on the input yields a group action on the output.
-\begin{definition}
-    A function $\psi:X\mapsto Y$ is $G$-\emph{equivariant} if $\psi(\alpha_1(g,x)) = \alpha_2(g,\psi(x))$ for all $x\in X$ and all $g\in G$, where $\alpha_1$ and $\alpha_2$ are group actions on $X$ and $Y$, respectively.
-\end{definition}
-Max $G$-pooling can follow a $G$-convolutional layer to remove the equivariance of the convolution and achieve $G$-invariance. We mentioned in the introduction the caveats of Max $G$-pooling. In order to understand the properties of the new bispectrum layer that we propose, we need first to present the solution formerly introduced by \cite{sanborn2023general} -- the $G$-triple correlation ($G$-TC) layer. Next, we will introduce our $G$-bispectrum layer and provide mathematical results that prove its computational efficiency compared to the $G$-TC.
-\section{$G$-Triple correlation and bispectrum}\label{sec:triple_corr_and_bsp}
-\subsection{Foundational results}
-The $G$-triple correlation ($G$-TC) \cite{kakarala_thesis} has had many uses but our focus stems from its utilization in $G$-CNNs. Given a real signal defined on a group $\Theta:G\mapsto\mathbb{R}$, the $G$-TC is the lowest order polynomial that is complete, i.e., that conserves all of the information of the signal $\Theta$, up to group action by $G$. In our setting, the signal $\Theta:G\mapsto\mathbb{R}$ is obtained in a $G$-CNN after the $G$-convolution of a function $f:X\mapsto\mathbb{R}^c$ (e.g., a continuous RGB picture is a function $f:\mathbb{R}^2\mapsto\mathbb{R}^3$ function) with a filter $\phi:X\mapsto\mathbb{R}^c$. For all $g \in G$, we have:
-\begin{equation}
-    \Theta(g):=(\phi*f)(g) = \sum_{x\in X}\phi(\alpha(g^{-1},x))^{\top}f(x).
-\end{equation}
- If $X =\mathbb{Z}$ and $(G,\ \cdot) = (\mathbb{Z},+)$, we obtain the classical discrete convolution. The triple-correlation was introduced for this classical framework. The $G$-triple correlation \cite{kakarala_thesis} extends its definition to any finite group $(G,\ \cdot)$.
-\begin{definition}
-    The \emph{$G$-Triple Correlation} ($G$-TC for short) of a real signal $\Theta:G\mapsto\mathbb{R}$ is given by:
-    \begin{equation}
-        T(\Theta)_{g_1,g_2}:=\sum_{g\in G} \Theta(g)\Theta(g\cdot g_1)\Theta(g\cdot g_2),
-    \end{equation}
-    for all $g_1, g_2\in G$.
-\end{definition}
-
-As previously mentioned, the $G$-TC layer has computational complexity $\mathcal{O}(|G|^3)$ and outputs $\mathcal{O}(|G|^2)$ coefficients. To avoid this burden, we wish to replace the $G$-TC layer by a more efficient layer. We achieve this by going in the Fourier domain, which decreases the computational cost of the layer while keeping the complete $G$-invariance of the architecture.
-The definition of Discrete Fourier Transform (DFT) can be extended to any finite group; see, e.g., \cite{Diaconis1990EfficientCO}.
-\begin{definition}
-    Given a set of unitary representatives (Def.~\ref{def:representation}) of the equivalence classes of irreps $\rho_i:G\mapsto\mathrm{GL}(V_i)$ (Def.~\ref{def:equivalence}),
-    the \emph{Fourier Transform} on a finite group $G$ of a signal $\Theta:G\mapsto\mathbb{R}$  is defined as:
-    \begin{equation}
-        \mathcal{F}(\Theta)_{\rho_i} :=\sum_{g\in G}\Theta(g)\rho_i(g)^\dagger,
-    \end{equation}
-     where $\rho_i(g)^\dagger$ refers to the conjugate transpose of the matrix $\rho_i(g)$  (or simply transpose if $\rho_i$ is real-valued).
-\end{definition}
-The bispectrum $\beta(\Theta)$ is defined as $\mathcal{F}(T(\Theta))$, with $\mathcal{F}$ evaluated over the group $G\times G$.
-\citet{kakarala_thesis} proposed a closed-form expression for the $G$-bispectrum $\beta(\Theta)$ directly in terms of $\mathcal{F}(\Theta)$. We recall it in Theorem~\ref{def:bispectrum}.
-\begin{theorem}{\cite{kakarala_finite_groups}}\label{def:bispectrum}
-    Given a signal $\Theta:G\mapsto\mathbb{R}$, its bispectrum $\beta(\Theta)$ can be computed as:
-    \begin{align*}
-        &\beta(\Theta)_{\rho_1,\rho_2} =\\
-        &\left[\mathcal{F}(\Theta)_{\rho_1}\otimes\mathcal{F}(\Theta)_{\rho_2}\right]C_{\rho_1,\rho_2}\left[\bigoplus_{\rho\in\rho_1\otimes\rho_2}\mathcal{F}(\Theta)_\rho^\dagger\right]C_{\rho_1,\rho_2}^{\dagger},
-    \end{align*}
-    where $C_{\rho_1,\rho_2}$ is a unitary matrix called the Clebsch-Gordan matrix. It is analytically defined for each pair of representations $\rho_1, \rho_2$ as:
-    \begin{equation*}
-        (\rho_1 \otimes \rho_2)(g) = C_{\rho_1,\rho_2} \Big[ \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \rho(g) \Big] C_{\rho_1,\rho_2}^\dagger.
-        \label{eqn:non-commutative-bs}
-    \end{equation*}
-    For each pair $\rho_1, \rho_2$, the matrix $\beta(\Theta)_{\rho_1, \rho_2}$ is called a $G$-bispectral coefficient.
-\end{theorem}
-To the authors' best knowledge, there is no generic formula for $C_{\rho_1,\rho_2}$ given a group $G$. In this paper, we will address the case of the dihedral group $D_n$, the symmetries of the $n$-gon. For this group, we show in Appendix~\ref{app:clebsch_gordan} how to obtain the Clebsch-Gordan matrices.
-In the case of a commutative group $G$, Theorem~\ref{def:bispectrum} simplifies in a more compact expression, recalled in Theorem~\ref{thm:commutative_bispectrum}.
-\begin{theorem}{\cite{kakarala_finite_groups}}\label{thm:commutative_bispectrum}
-    If $(G,\ \cdot)$ is a commutative group, the bispectrum can be computed as:
-    \begin{equation}\label{eq:com_bsp}
-        \beta(\Theta)_{\rho_1,\rho_2} = \mathcal{F}(\Theta)_{\rho_1}\mathcal{F}(\Theta)_{\rho_2}\mathcal{F}(\Theta)_{\rho_1\otimes\rho_2}^\dagger.
-    \end{equation}
-\end{theorem}
-For commutative groups, the $G$-bispectral coefficients are complex scalars \cite{steinberg2011representation}. Finally, we state formally what it means for the $G$-TC and the $G$-bispectrum to be complete $G$-invariant functions (for generic data $\Theta, \widetilde{\Theta}$).
-\begin{theorem}{\cite{Kakarala2009CompletenessOB}}
-    \label{thm:bsp_invariance}
-    The $G$-TC and the $G$-bispectrum are complete $G$-invariant functions, i.e., for $\Theta, \widetilde{\Theta}:G\mapsto\mathbb{R}$, $T(\Theta)=T(\widetilde{\Theta})$, respectively  $\beta(\Theta) = \beta(\widetilde{\Theta})$, if and only if there exists $h\in G$ such that $\Theta(g) = \alpha(h, \widetilde{\Theta}(g))$ for all $g\in G$.
-\end{theorem}
-
-\subsection{Bispectrum layer for the $G$-CNN architecture}\label{sec:bispectralgcnn}
-\begin{figure*}
-    \centering
-    \includegraphics[width = 17cm]{Images/figures_ICML.pdf}
-    \caption{Illustration of the different proposed $G$-CNN architectures \cite{cohenc16, sanborn2023general}. The input $f$ is first processed through the $G$-convolutional layer composed of $K$ filters $\{\phi_k\}_{k=1}^K$. Then a pooling layer is chosen between the Max $G$-pooling, the $G$-TC, and the selective/full $G$-bispectrum layer. Finally, the output after pooling is fed to a Neural Network designed for the machine learning task at hand. \vspace{-0.5cm}}
-    \label{fig:GCNN}
-\end{figure*}
-The $G$-CNN architecture, first proposed in \cite{cohenc16}, is illustrated on Figure ~\ref{fig:GCNN}. We specify how to implement our $G$-bispectral layer within this framework. Given the input $f:X\mapsto\mathbb{R}$, a set of $G$-convolutions is performed using filters $\{\phi_k\}_{k=1}^K$, yielding features maps $\{\Theta_k\}_{k=1}^K$ that form a set of $K$ real signals defined on the group $G$. The Fourier transforms of the feature maps are preferably computed using a FFT algorithm on $G$ \cite{Diaconis1990EfficientCO}. Afterwards, either the full $G$-bispectrum, or only the $G$-bispectral coefficients necessary for completeness are computed. We term the latter a \emph{selective} $G$-bispectrum layer. The coefficients to select depend on the group $G$. We investigate this selection in Section~\ref{sec:bspinversion} for the cyclic groups $C_n$, all commutative groups, and the (non-commutative) dihedral group $D_n$.
-Finally, the $G$-bispectral coefficients are fed to a Secondary Neural Network (NN) to perform the desired task, e.g., image classification. In our experiments, the Secondary NN takes the form of a Multi-Layer Perceptron (MLP). % The different costs associated to the various pooling layers are summarized in Table \ref{tab:complexities}.
-The choice of which layer to prefer will depend on the context, as investigated in Section~\ref{sec:experiments}.
-\section{Bispectrum Inversion for Complete $G$-Invariant Neural Networks}\label{sec:bspinversion}
-The guiding question for this section is: What is the minimum number of $G$-bispectral coefficients that we need to compute in order to achieve completeness? In practice, this question is answered by finding the coefficients $\beta(\Theta)_{\rho_i,\rho_j}$ which allows to reconstruct the Fourier transform $\mathcal{F}(\Theta)$ completely. This is the \emph{$G$-bispectrum inversion problem} mentioned in the Section \ref{sec:introduction}. It is well known that $|G|=n$ coefficients are enough for the cyclic group $C_n$ \cite{kakarala_thesis, sadler1992shift} and similarly for a product of two such groups \cite{giannakis1989signal}. Our main theoretical contribution is to show that $|G|$ coefficients are enough for any commutative group $G$, and that $\left \lfloor{ \frac{n-1}{2}}\right \rfloor + 2$
-% NOTE: before was:  $\frac{|G|}{4}+1=\frac{n}{2}+1$
-matrix coefficients are enough for any dihedral group $D_n$, the group of symmetries of the $n$-gon. We note that, for the non-commutative dihedral groups, among the $G$-bispectral coefficients required, one is a scalar, one is a $2\times2$ matrix, and the rest are $4\times 4$ matrices.
-
-Our solution to bispectral inversion for the non-commutative family $D_n$ is, as far as we are aware, the first of its kind.  It also forms a foundation for studying the problem for arbitrary non-commutative groups.  In particular, the authors have verified similar results for examples in other non-commutative families such as the octahedral groups.
-
-\subsection{The algorithm to invert the bispectrum}\label{sec:commutative_alg}
-%We present the algorithm from \cite{kakarala_thesis} to invert the bispectrum the cyclic group $C_n=$($\mathbb{Z}/n\mathbb{Z}$, $+$ mod $n$). The method requires only $|G|=n$ bispectral coefficients, therefore allowing the reduction in complexity expected with a bispectral layer compared to the TC layer. We prove mathematically that the method can be extended to all commutative groups, still requiring only $|G|$ bispectral coefficients. Moreover, we show that it can be extended to the dihedral group $D_n$, i.e., the symmetries of the $n$-gon. In this case, the method only requires $\frac{|G|}{4}$ bispectral coefficients.
-From now on, we assume that the Fourier transform $\mathcal{F}(\Theta)$ only features \textbf{non-zero elements}, or \textbf{invertible matrices} in the case of non-scalar Fourier transform. This assumption is supported by the measure zero probability of encountering this corner case.
-\subsubsection{Cyclic groups $C_n$}
-We start with $(G,\ \cdot)=(\mathbb{Z}/n\mathbb{Z},\ +\ \mathrm{mod}\ n)=:C_n$. Recall that the irreps are given by $\rho_k(g)=\exp\left(\omega_k g\right)$ where $\omega_k:=\frac{2\pi i k }{n}$ for $k\in\mathbb{Z}/n\mathbb{Z}$ (see, e.g., \cite{steinberg2011representation}).
-The Fourier coefficient associated to the trivial representation $\mathcal{F}(\Theta)_{\rho_0}$, is uniquely determined and can be recovered from Theorem~\ref{thm:commutative_bispectrum} by identifying phase and modulus:
-\begin{equation}\label{eq:trivial_coef}
-    \beta(\Theta)_{\rho_0,\rho_0} =|\mathcal{F}(\Theta)_{\rho_0}|^3\exp\left(i\ \mathrm{arg}(\mathcal{F}(\Theta)_{\rho_0})\right).
-\end{equation}
-We proceed using Pontryagin duality: the irreps $\{\rho_k\}_{k=1}^n$, form a group $\widehat{G}$ themselves, with $\widehat{G} = G$. In the case of the cyclic group $C_n$, notice that for all $j,k\in\mathbb{Z}/n\mathbb{Z}$, $\rho_j\otimes \rho_k = \rho_{j+k}$. Leveraging \eqref{eq:com_bsp}, we can use $\beta(\Theta)_{\rho_0,\rho_1}$ to recover $\mathcal{F}(\Theta)_{\rho_1}$:
-\begin{equation}\label{eq:trivial_coeff_cyclic}
-    |\mathcal{F}(\Theta)_{\rho_1}|^2 =\frac{\beta(\Theta)_{\rho_0,\rho_1}}{\mathcal{F}(\Theta)_{\rho_0}}.
-\end{equation}
-Equation~\eqref{eq:trivial_coeff_cyclic} leaves an indeterminacy on the phase of $\mathcal{F}(\Theta)_{\rho_1}$. This corresponds to the indeterminacy factor $\exp(i \phi)$, $\phi\in[0,2\pi)$ of Appendix~\ref{sec:indeterminacy}. It is inherited from the $G$-invariance  of $\beta(\Theta)$: it is not injective, hence you cannot distinguish inputs that have the same $G$-bispectrum. The key to recover all the other Fourier coefficients is to notice that $S = \{1\}$ is a generating set of $\widehat{G} =\mathbb{Z}/n\mathbb{Z}$. Therefore, computing sequentially:
-\begin{equation}
-    \mathcal{F}(\Theta)_{\rho_{k+1}} = \left(\frac{\beta(\Theta)_{\rho_1,\rho_k}}{\mathcal{F}(\Theta)_{\rho_1}\mathcal{F}(\Theta)_{\rho_k}}\right)^\dagger,
-\end{equation}
-for $k=1,2,...,n-2$ recovers $\mathcal{F}(\Theta)$ completely. The method is summarized in Algorithm~\ref{alg:cyclic_alg} and illustrated on Figure \ref{fig:alg1}. Consequently, only the following bispectral coefficients are needed for completeness: $\beta(\Theta)_{\rho_0,\rho_0}, \beta(\Theta)_{\rho_0,\rho_1}$ and $ \beta(\Theta)_{\rho_1,\rho_k}$ for $k=1,2,...,n-2$. This makes a total of $n=|G|$ coefficients. We summarize this result in Theorem~\ref{thm:cyclic_inversion}.
-\begin{theorem}{\cite{kakarala_thesis}}\label{thm:cyclic_inversion}
-For cyclic groups $C_n$, the $C_n$-bispectrum can be inverted using $|G|=n$ coefficients if $\mathcal{F}(\Theta)_\rho\neq 0$ for all $\rho \in C_n$.
-\end{theorem}
-\begin{algorithm}[tb]
-    \caption{Bispectrum inversion on $\mathbb{Z}/n\mathbb{Z}$
- \cite{kakarala_thesis}}
-    \label{alg:cyclic_alg}
-    \begin{algorithmic}[1]
-        \STATE \textbf{Input}: $\beta(\Theta)_{\rho_0,\rho_0}, \beta(\Theta)_{\rho_0,\rho_1}$ and $ \beta(\Theta)_{\rho_1,\rho_k}$ for $k=1,2,...,n-2$.
-        \STATE Compute $|\mathcal{F}(\Theta)_{\rho_0}|=\left(\beta(\Theta)_{\rho_0,\rho_0}\right)^{\frac{1}{3}}$ and $\mathrm{arg}(\mathcal{F}(\Theta)_{\rho_0})=\mathrm{arg}(\beta(\Theta)_{\rho_0,\rho_0})$.
-        \STATE Compute $|\mathcal{F}(\Theta)_{\rho_1}| =\left(\frac{\beta(\Theta)_{\rho_0,\rho_1}}{\mathcal{F}(\Theta)_{\rho_0}}\right)^{\frac{1}{2}}$ and set $\mathrm{arg}(\mathcal{F}(\Theta)_{\rho_1})=0$.
-        \FOR{$k=1,2,...,n-2$}
-            \STATE Compute $\mathcal{F}(\Theta)_{\rho_{k+1}} = \left(\frac{\beta(\Theta)_{\rho_1,\rho_k}}{\mathcal{F}(\Theta)_{\rho_1}\mathcal{F}(\Theta)_{\rho_k}}\right)^\dagger$.
-        \ENDFOR
-        \STATE \textbf{Return} $\mathcal{F}(\Theta)$ (up to group action).
-    \end{algorithmic}
-\end{algorithm}
-\begin{figure}[h]
-    \centering
-    \includegraphics[width = 8cm]{Images/alg1_icml.pdf}
-    \caption{Illustration of Algorithm \ref{alg:cyclic_alg}. The bispectrum coefficients allow us to recover the Fourier transform sequentially, up to group action. First, $\beta_{\rho_0,\rho_0}$ gives $\mathcal{F}_{\rho_{0}}$, which, combined with $\beta_{\rho_0,\rho_1}$ gives $\mathcal{F}_{\rho_{1}}$ etc. }
-    \label{fig:alg1}
-\end{figure}
-\subsubsection{Commutative groups}
-Now, we extend Algorithm~\ref{alg:cyclic_alg} to all commutative groups. In view of Theorem~\ref{thm:fundamentalthm}, we propose a method tailored for the direct sum of finitely many cyclic groups.
-\begin{theorem}{(see, e.g., \cite{Norman_2012})}\label{thm:fundamentalthm}
-    Every finite commutative group $G$ is isomorphic to a finite direct sum of cyclic groups: $G\cong \bigoplus_{l=1}^L \mathbb{Z}/n_l\mathbb{Z}$ where $L\in\mathbb{N}$ and $n_l\in\mathbb{N}$ for $l=1,2,...,L$.
-\end{theorem}
-For all $\kbf\in G$, the irreps $\rho_\kbf:G\mapsto \mathbb{C}$ are given by:
-\begin{equation}\label{eq:commutative_irreps}
-    \rho_\kbf(g) = \prod_{l=1}^L\exp(\omega_{\kbf_l} g_l).
-\end{equation}
-The number of irreps is $|G| = \prod_{l=1}^L n_l$.
-Notice that for the commutative groups, we keep the property $\rho_{\kbf}\otimes\rho_{\kbf'}=\rho_{\kbf+\kbf'}$ where $(\kbf+\kbf')_l := \kbf_l+\kbf_l'\ \mathrm{mod} \ n_l$ for $l=1,2,...,L$.
-
-%In this section, we assume that none of the $n_l$'s are relatively prime, such that $G$ is not isomorphic to another commutative group $\widetilde{G}$ with $\widetilde{L}<L$. Indeed, the \emph{Chinese remainder theorem} (see e.g. \cite{Norman_2012}) ensures that there is $\widetilde{G}\cong G$ with $\widetilde{L}<L$ if and only if there is $(n_{l_1}, n_{l_2})$ that are coprimes.
-
-The first step is to notice that a generating set $S$ of the irreps is of size $L$. Indeed, it is given by the usual basis vectors $S=\{\ebf^l\in \mathbb{Z}^L$ for $l=1,...,L\}$ where:
-\begin{equation*}
-    \ebf^l_k=\begin{cases}
-        1 \text{ if } k=l,\\
-        0 \text{ otherwise.}
-    \end{cases}
-\end{equation*}
-To recover all the Fourier coefficients, a consequence of Theorem~\ref{thm:commutative_bispectrum} is that it is sufficient to have the Fourier coefficients associated to each generating element in $S$. Indeed, knowing $\mathcal{F}(\Theta)_{\rho_{\kbf_1}}$ and $\mathcal{F}(\Theta)_{\rho_{\kbf_2}}$ allows us to compute $\mathcal{F}(\Theta)_{\rho_{\kbf_1+\kbf_2}}$. By definition of the generating set $S$, we can thus recover $\mathcal{F}(\Theta)_{\rho_{\kbf}}$ for all $\kbf \in G$. The moduli of the coefficients $\mathcal{F}(\Theta)_{\rho_{\ebf^l}}$ for $l=1,2,...,L$ can be computed as follows:
-\begin{equation}\label{eq:compute_generating}
-    |\mathcal{F}(\Theta)_{\rho_{\ebf^l}}| =\left(\frac{\beta(\Theta)_{\rho_\mathbf{0},\rho_{\ebf^l}}}{\mathcal{F}(\Theta)_{\rho_\mathbf{0}}}\right)^\frac{1}{2},
-\end{equation}
-where $\mathcal{F}(\Theta)_{\rho_\mathbf{0}}$ is the Fourier coefficient of the trivial representation (computed as in~\eqref{eq:trivial_coef}). We claim that the phase can be fixed arbitrarily for each label $l$, thus $L$ times. This is because:
-\begin{align*}
-    \mathcal{F}(\alpha(\mathbf{h},\Theta))_{\rho_{\ebf^l}} &= \rho_{\ebf^l}(\mathbf{h})\mathcal{F}(\Theta)_{\rho_{\ebf^l}}\\
-    &=\exp\left(\frac{2\pi i h_l}{n_l}\right)\mathcal{F}(\Theta)_{\rho_{\ebf^l}},
-\end{align*}
-for all $l=1,2,.,L$. Only one factor $h_l$ among the $L$ independent factors in $\mathbf{h}$ remains. Therefore, fixing an arbitrary phase in~\eqref{eq:compute_generating} only fixes $h_l$. Exactly as for the cyclic group, the indeterminacy factor $h_l$ is not restricted to $\mathbb{Z}/n_l\mathbb{Z}$ but can belong to $[0,n_l]$. Once the Fourier coefficients are known for all the generators of the group of the irreps $\{\rho_{\ebf^l}\}_{l=1}^L$, it remains to combine them to obtain all the elements in the groups and, consequently, all the associated Fourier coefficients.
-
-At this point, it helps to consider the problem geometrically. Each irreps $\rho_\kbf$ can be associated to its integer coordinate $\kbf$ inside a hyper-rectangle in $\mathbb{R}^L$, whose length of edges is $n_l$ for $l=1,2,...,L$. We combine the coordinates $\ebf^l$ to obtain all the possible integer coordinates inside the hyper-rectangle. First, we can obtain the $L$ orthogonal edges of the hyper-rectangle. For $l=1,2,...,L$, $\rho_{\ebf^l}\otimes \rho_{\ebf^l}$ gives $\rho_{2\ebf^l}$, $\rho_{\ebf^l}\otimes \rho_{2\ebf^l}$ gives $\rho_{3\ebf^l}$, etc. This is in fact the procedure of Algorithm 1. Now, we combine the edges to generate the inside of the hyper-rectangle. We proceed iteratively. For $l=1,2,...,L$, we define $K^l:=\{t\ebf^l + \kbf \ | \ t=1,2,..,n_l-1\text{ and } \kbf\in K^{l-1}\}$ and $K^0:=\{\mathbf{0}\}$. This construction is such that $K^l\cong\bigoplus_{j=1}^l \mathbb{Z}/n_j\mathbb{Z}$ and $K^L=G$. We generate the missing Fourier coefficients by combining the ones associated to the generating set of $G$.
-For $l=2,...,L$, compute:
-\begin{equation}
-    \mathcal{F}(\Theta)_{\rho_{t\ebf^l+\kbf}}=\left(\frac{\beta(\Theta)_{\rho_{t\ebf^l},\rho_{\kbf}}}{\mathcal{F}(\Theta)_{\rho_{t\ebf}}\mathcal{F}(\Theta)_{\rho_{\kbf}}}\right)^\dagger,
-\end{equation}
-for $t=1,2,...,n_l-1$, for all $\kbf\in K^{l-1}$. Intuitively for $L=3$, we obtain first an edge, then a face and finally the full parallelepiped. The procedure is summarized in Algorithm~\ref{alg:commutative_alg} and illustrated on Figure \ref{fig:alg2}. It shows that the bispectral coefficients needed for completeness are: $\beta(\Theta)_{\rho_0,\rho_0}$, $\beta(\Theta)_{\rho_0,t\rho_{\ebf}^l},\ \beta(\Theta)_{t\rho_{\ebf}^l, \rho_\kbf} $ for $l=1,2,...,L$, $t=1,2,...,n_l-2$ and all $\kbf\in K^{l-1}$. We recover exactly one Fourier coefficient by bispectral coefficient. This makes thus a total of $|G|$ bispectral coefficients precisely and proves the following theorem.
-
-\begin{theorem}
-For finite commutative groups $G$, the bispectrum can be inverted using $|G|$ coefficients if $\mathcal{F}(\Theta)_\rho \neq 0$ for all $\rho \in G$.
-\end{theorem}
-
-We note that this approach to inversion is symbolic, in that a solution can be expressed explicitly as a formula in terms of the input.  Other approaches are also possible to determine an inverse, such as using least squares \cite{haniff1991least} or more recent spectral methods \cite{chen2018spectral}.
-
-\begin{algorithm}[tb]
-    \caption{Bispectrum inversion for finite commutative groups ($G=\bigoplus_{l=1}^L \mathbb{Z}/n_l\mathbb{Z}$).}
-    \label{alg:commutative_alg}
-    \begin{algorithmic}[1]
-        \STATE \textbf{Input}: $|G|$ bispectral coeffs.
-        \STATE Compute $|\mathcal{F}(\Theta)_{\rho_\mathbf{0}}|=\left(\beta(\Theta)_{\rho_\mathbf{0},\rho_\mathbf{0}}\right)^{\frac{1}{3}}$ and $\mathrm{arg}(\mathcal{F}(\Theta)_{\rho_\mathbf{0}})=\mathrm{arg}(\beta(\Theta)_{\rho_\mathbf{0},\rho_\mathbf{0}})$.
-        \FOR{$l=1,...,L$}
-            \STATE Compute $|\mathcal{F}(\Theta)_{\rho_{\ebf^l}}| =\left(\frac{\beta(\Theta)_{\rho_\mathbf{0},\rho_{\ebf^l}}}{\mathcal{F}(\Theta)_{\rho_\mathbf{0}}}\right)^\frac{1}{2}$ and set $\mathrm{arg}\left(\mathcal{F}(\Theta)_{\rho_{\ebf^l}}\right)=0$.
-            \FOR{$t=1,...,n_l-1$}
-                \IF{$t>1$}
-                \STATE  $F(\Theta)_{\rho_{t\ebf^l}}=\left(\frac{\beta(\Theta)_{\rho_{\ebf^l},\rho_{(t-1)\ebf^l}}}{\mathcal{F}(\Theta)_{\rho_{\ebf^l}}\mathcal{F}(\Theta)_{\rho_{(t-1)\ebf^l}}}\right)^\dagger$.
-                \ENDIF
-                \FOR{$\kbf\in K^{l-1}\setminus\{\mathbf{0}\}$}
-                    \STATE Compute $\mathcal{F}(\Theta)_{\rho_{t\ebf^l+\kbf}}=\left(\frac{\beta(\Theta)_{\rho_{t\ebf^l},\rho_{\kbf}}}{\mathcal{F}(\Theta)_{\rho_{t\ebf^l}}\mathcal{F}(\Theta)_{\rho_{\kbf}}}\right)^\dagger$.
-                \ENDFOR
-            \ENDFOR
-        \ENDFOR
-        \STATE \textbf{Return} $\mathcal{F}(\Theta)$ (up to group action).
-    \end{algorithmic}
-\end{algorithm}
-\begin{figure}[h]
-    \centering
-    \includegraphics[width = 8.5cm]{Images/alg2_icml.pdf}
-    \caption{Illustration of Algorithm \ref{alg:commutative_alg}. The bispectrum coefficients allow to recover the Fourier transform sequentially, up to group action. First}
-    \label{fig:alg2}
-\end{figure}
-
-\subsubsection{Dihedral group $D_n$}\label{sec:inversion_dihedral}
-The dihedral group $D_n$ is the group of all symmetries of the $n$-gon. Mathematically, it is defined as:
-\begin{equation}\label{def:dihedral}
-    D_n:=\langle x,\ a\ |\ a^n=x^2=e,\ xax = a^{-1}\rangle,
-\end{equation}
-where $a$ is the \emph{rotation} and $x$ is the \emph{reflection}, and they form a generating set of $D_n$. We will only consider the case $n>2$ since the cases $n=1$ and $n=2$ are commutative groups covered by the previous subsection, while $n>2$ gives non-commutative groups. The 2D irreps of $D_n$ are given by:
-\begin{equation}\label{eq:2d_irreps_dn}
-    \rho_k(a^lx^m)=\begin{bmatrix}
-        \cos(\omega_l k)&-\sin(\omega_l k)\\
-        \sin(\omega_l k)&\cos(\omega_l k)
-    \end{bmatrix}
-    \begin{bmatrix}
-        1 & 0\\
-        0 & -1\\
-    \end{bmatrix}^m,
-\end{equation}
-where $\omega_l = \frac{2\pi l}{n}$ for $k=1,2,...,\lfloor \frac{n-1}{2}\rfloor$. There are also $2$ or $4$ 1D irreps if $n$ is odd or even, respectively. We denote these 1D irreps by $\rho_{0},\ \rho_{01},\ \rho_{02}$, and $\rho_{03}$ (see Appendix \ref{app:1dirreps}). The two last ones are only for $n$ even.
-In view of section~\ref{sec:commutative_alg}, we wish to show that there is an irrep $\rho_1$ that generates all the irreps of $D_n$. As in the cyclic and commutative cases, we can first deduce $\mathcal{F}(\Theta)_{\rho_0}$ from $\beta(\Theta)_{\rho_0,\rho_0}$. Now, we have:
-\begin{equation}\label{eq:first_coeff_dihedral}
-    \mathcal{F}(\Theta)_{\rho_1}\mathcal{F}(\Theta)_{\rho_1}^\dagger=\frac{\beta(\Theta)_{\rho_0,\rho_1}}{\mathcal{F}(\Theta)_{\rho_0}}.
-\end{equation}
-The novelty for this non-commutative group is that $\mathcal{F}(\Theta)_{\rho_1}$ is a $2\times 2$ matrix. After computing the eigenvalue decomposition $\frac{\beta(\Theta)_{\rho_0,\rho_1}}{\mathcal{F}(\Theta)_{\rho_0}} = V\Lambda V^\dagger$, we can choose:
-\begin{equation}
-    \mathcal{F}(\Theta)_{\rho_1} = V\Lambda^{\frac{1}{2}}V^\dagger.
-\end{equation}
-Notice that any $\mathcal{F}(\Theta)_{\rho_1}' = \mathcal{F}(\Theta)_{\rho_1} U$ where $U$ is unitary (i.e., $UU^\dagger=U^\dagger U=I$) also solves~\eqref{eq:first_coeff_dihedral}. For $\mathbb{Z}/n\mathbb{Z}$ the indeterminacy belonged to $\mathrm{SO}(2)$, the continuous set of 2d rotations. For $D_n$, it belongs to $\mathrm{O}(2)$, the continuous set of 2d rotations and reflections.
-
-For $k=2,...,\lfloor \frac{n-1}{2}\rfloor$, we can then obtain:
-\begin{align}
-    \label{eq:new_coeff_dihedral}
-    &\bigoplus_{\rho\in\rho_1\otimes\rho_{k-1}}\mathcal{F}(\Theta)_\rho =\\
-    \nonumber
-    &\left(C_{\rho_1,\rho_{k-1}}^{\dagger}\left[\mathcal{F}(\Theta)_{\rho_1}\otimes\mathcal{F}(\Theta)_{\rho_{k-1}}\right]^{-1}\beta(\Theta)_{\rho_1,\rho_2}C_{\rho_1,\rho_{k-1}}\right)^\dagger.
-\end{align}
-It is shown in Appendix~\ref{app:generating_dihedral} that $\rho_k\in\rho_1\otimes\rho_{k-1}$ so that the for-loop can be applied. Moreover, we know from Appendix~\ref{app:generating_dihedral} that $\rho_{01}\in\rho_1\otimes\rho_{1}$ and, for $n$ even, $\rho_{02}, \rho_{03}\in \rho_1\otimes\rho_{\frac{n}{2}-1}$. Thus the iteration recovers the complete DFT $\mathcal{F}(\Theta)$ and proves the following theorem.
-
-\begin{theorem}
-    For the family of dihedral groups $D_n$, we need at most $\left \lfloor{ \frac{n-1}{2}}\right \rfloor + 2$ bispectral matrix coefficients for inversion if $\det(\mathcal{F}(\Theta)_\rho) \neq 0$ for all irreps $\rho$ of $D_n$. This corresponds to $1+4+16\cdot\left\lfloor{ \frac{n-1}{2}}\right \rfloor\approx 4 |D_n|$ scalar values.
-\end{theorem}
-
-The procedure is summarized in Algorithm \ref{alg:dihedral_alg}.
-\begin{algorithm}[tb]
-    \caption{Bispectrum inversion on the dihedral group $D_n$.}\label{alg:dihedral_alg}
-    \begin{algorithmic}[1]
-        \STATE \textbf{Input}: $\beta(\Theta)_{\rho_0,\rho_0}, \beta(\Theta)_{\rho_0,\rho_1}$ and $ \beta(\Theta)_{\rho_1,\rho_k}$ for $k=1,2,...,\lfloor\frac{n-1}{2}\rfloor$.
-        \STATE Compute $|\mathcal{F}(\Theta)_{\rho_0}|=\left(\beta(\Theta)_{\rho_0,\rho_0}\right)^{\frac{1}{3}}$ and $\mathrm{arg}(\mathcal{F}(\Theta)_{\rho_0})=\mathrm{arg}(\beta(\Theta)_{\rho_0,\rho_0})$.
-        \STATE Compute $V\Lambda V^\dagger = \frac{\beta(\Theta)_{\rho_0,\rho_1}} {\mathcal{F}(\Theta)_{\rho_0}}$.\\
-        \STATE Set $\mathcal{F}(\Theta)_{\rho_1} = V\Lambda^\frac{1}{2} V^\dagger$.\\
-        \FOR{$k=2,...,\lfloor\frac{n-1}{2}\rfloor$}
-           % \STATE (If $n$ odd, stop at $k =\lfloor\frac{n-1}{2}\rfloor-1 )$
-            \STATE Compute \begin{align*}
-                &\bigoplus_{\rho\in\rho_1\otimes\rho_{k-1}}\mathcal{F}(\Theta)_\rho =\\
-                \nonumber
-                &\left(C_{\rho_1,\rho_{k-1}}^{\dagger}\left[\mathcal{F}_{\rho_1}\otimes\mathcal{F}_{\rho_{k-1}}\right]^{-1}\beta_{\rho_1,\rho_2}C_{\rho_1,\rho_{k-1}}\right)^\dagger.
-            \end{align*}
-        \ENDFOR
-        \STATE \textbf{Return} $\mathcal{F}(\Theta)$ (up to group action).
-    \end{algorithmic}
-\end{algorithm}
-%\subsubsection{General algorithm for finite non-commutative groups}
-%The three cases that we have just studied suggest a heuristic procedure to invert the bispectrum. Given a group $G$ and a complete family of irreps $\{\rho_k\}_k$ where $\rho_0$ is the trivial representation. $\mathcal{F}(\Theta)_{\rho_0}$ can always be recovered from $\beta(\Theta)_{\rho_0,\rho_0}$. Then one can compute $\mathcal{F}(\Theta)_{\rho_0}$
-\section{Experimental results}\label{sec:experiments}
-\subsection{Implementation and architecture}
-Our implementation of the $G$-bispectral layer is based on the \href{https://github.com/gtc-invariance/gtc-invariance}{\texttt{gtc-invariance}} repository, implementing the $G$-CNN with $G$-convolution and $G$-TC layer \cite{sanborn2023general}. This implementation is itself based on the \href{https://github.com/QUVA-Lab/escnn}{ESCNN library} \cite{cesa2022a, weiler2021general}.
-%We propose a performance analysis of our method. To do so, we have implemented the bispectrum layer for the cyclic group $C_n$ and the dihedral group $D_n$. These groups can be considered as discrete cases of the groups $\mathrm{SO}(2)$ (2D rotations) and $\mathrm{O}(2)$ (2D rotations and reflections) respectively.
-
-To corroborate our theoretical results, we propose an experimental assessment of the newly proposed $G$-bispectral layer. We compare the Avg $G$-pooling, the Max $G$-pooling, the $G$-TC, and the $G$-bispectrum layers as pooling operations after the $G$-convolution of a $G$-CNN on the classification problems of the MNIST dataset of handwritten digits \cite{deng2012mnist}, the EMNIST dataset of handwritten letters \cite{cohen2017emnist} and the CIFAR-10 dataset of natural images \cite{CIFAR10}. These datasets count 10, 26 an 10 classes respectively. We obtain transformed versions of the datasets -- $G$-MNIST/EMNIST/CIFAR-10 -- through the action of the groups $\mathrm{SO}(2)$ and $\mathrm{O}(2)$. For each image in the original dataset, we apply a random action $g\in G$ on the image.
-
-The objective of our experiments is to isolate the impact of the choice of invariant layer. Hence, we consider architectures that only differ by the invariant layer in the classification task, following the experimental set up provided by \cite{sanborn2023general}.  The neural network architecture is composed of a $G$-convolution, an invariant layer, and finally a Multi-Layer-Perceptron (MLP), itself composed of three fully connected layers with batch normalization and ReLU nonlinearity. Finally, a fully connected linear layer is added to perform final classification. The MLP's widths are tuned to match the number of parameters across each neural network model. The details are given in Appendix \ref{app:training}. We highlight here that the pursued objective is to compare the differences in performances of the invariance layers, not to provide the state-of-the-art accuracy on the datasets involved. Henceforth, we do not optimize the architectures to reach the highest possible accuracy. We set simple architectures providing interpretable results for analysis.
-\begin{figure}
-    \centering
-    \includegraphics[width = 7cm]{Images/training_times_cyclic.pdf}
-    \includegraphics[width = 7cm]{Images/training_times_dihedral.pdf}
-    \caption{Evolution of the average training times for the different invariance layers as the size $n$ of $C_n$ and $D_n$ rise. The average and standard deviations are obtained over $10$ runs.  For all runs, the number of parameters of the complete neural network (filters and MLP) is set to $50000$ and $150000$ for $\mathrm{SO}(2)$ and $\mathrm{O}(2)$ respectively. Standard deviations are reported by vertical intervals.}
-    \label{fig:trainings}
-\end{figure}
-\subsection{Training speed performance}
-We provide in Table~\ref{tab:complexities} the theoretical complexities of the different layers. The computational cost of computing the selective G-bispectrum is $\mathcal{O}(|G|\log |G|)$ if an FFT algorithm is available on $G$~\cite{Diaconis1990EfficientCO}, and $\mathcal{O}(|G|^2)$ otherwise. On Figure~\ref{fig:trainings}, we report the average training times on $\mathrm{SO}(2)/\mathrm{O}(2)$-MNIST for 10 runs as the discretization $C_n/D_n$ of $\mathrm{SO}(2)/\mathrm{O}(2)$ varies. In the first case, we use the FFT and observe that the Max $G$-pooling and $G$-bispectrum training time scale linearly whereas it scales quadratically for the $G$-TC. For $\mathrm{O}(2)$, we perform a classic DFT on $D_n$ so that the $G$-bispectrum scales worth. However, an FFT could be implemented to speed-up the process.
-\begin{table}[h]
-    \centering
-    \begin{tabular}{||c||c|c||}
-        \hline
-         Pooling layer&  Comput. Complex.& Output size\\
-         \hline
-         $G$-TC& $K\mathcal{O}(|G|^3)$&$K\mathcal{O}(|G|^2)$\\
-         \hline
-         Full bsp.& $K\mathcal{O}(|G|^2)$&$K\mathcal{O}(|G|^2)$\\
-         \hline
-         Select. bsp.& $K\mathcal{O}(|G|\log|G|\text{ or }|G|^2)$&$K\mathcal{O}(|G|)$\\
-         \hline
-         Max $G$-pool.&$K\mathcal{O}(|G|)$&$K$\\
-         \hline
-    \end{tabular}
-    \caption{$G$-CNN invariant layers and their computational cost and output size. \vspace{-0.5cm}}
-    \label{tab:complexities}
-\end{table}
-\subsection{Classification Performance}
-We compare the performances of the $G$-bispectral layer with respect to the $G$-TC, the Max $G$-pooling and the Avg $G$-pooling models, trained on the $\mathrm{SO}(2)$/$\mathrm{O}(2)$-MNIST/EMNIST/CIFAR-10 datasets and we assess the accuracy by averaging the validation accuracy over 10 runs. The classification accuracy is provided in Table~\ref{tab:classification}. For the experiments in Table \ref{tab:classification}, the following pattern holds: at equivalent number of parameters, the more computationally expensive the pooling layer, the better the accuracy. However, the use of the $G$-TC becomes prohibitive when $|G|$ raises. In the next section, we discuss the settings where each pooling layer should be preferred, and highlight each pooling layer's strengths and weaknesses.
-\begin{table*}[ht]
-\centering
-\begin{tabular}{||c|c|c|c|c|c|c||}
-	\hline
-	Dataset&Group&Pooling&$K$ filters&Avg acc.&Std. dev.&Param. count\\
- \hline
-	\hline
-	\multirow{8}{*}{MNIST}&\multirow{4}{*}{$\mathrm{SO}(2)$}&Avg $G$-pooling&24&0.74 &$<0.01$ &50247\\
-	     &              &Max $G$-pooling&24& 0.96& $<0.01$ &50247\\
-	     &              &Select. $G$-Bsp&24&0.95 &$<0.01$  &49116\\
-	     &              &$G$-TC&24&0.96 &  $<0.01$&48385\\
-        \cline{2-7}
-	     &\multirow{4}{*}{$\mathrm{O}(2)$}&Avg $G$-pooling&4& 0.60&$<0.01$ &147675\\
-	     &              &Max $G$-pooling&4&0.78 &  $<0.01$&147675\\
-	     &              &Select. $G$-Bsp&4&0.93 & $<0.01$ &143029\\
-	     &              &$G$-TC&4&0.96 &  $<0.01$&142220\\
-	\hline
-	\multirow{8}{*}{EMNIST}&\multirow{4}{*}{$\mathrm{SO}(2)$}&Avg $G$-pooling&24&0.40 &$<0.01$ &50195\\
-	     &              &Max $G$-pooling&24& 0.76& $<0.01$ &50195\\
-	     &              &Select. $G$-Bsp&24&0.77 & $<0.01$ &49254\\
-	     &              &$G$-TC&24&0.80 &  $<0.01$&48494\\
-	\cline{2-7}
-	     &\multirow{4}{*}{$\mathrm{O}(2)$}&Avg $G$-pooling&20&0.38 &$<0.01$ &48832\\
-	     &              &Max $G$-pooling&20&0.71 & $<0.01$ &48832\\
-	     &              &Select. $G$-Bsp&20&0.74 & $<0.01$ &47320\\
-	     &              &$G$-TC&20&0.79 & $<0.01$ &46954\\
-	\hline
-	\multirow{8}{*}{CIFAR-10}&\multirow{4}{*}{$\mathrm{SO}(2)$}&Avg $G$-pooling&20&0.43 &$<0.01$ &10494140\\
-	     &              &Max $G$-pooling&20&0.48 & $<0.01$ &10494140\\
-	     &              &Select. $G$-Bsp&20& 0.50& $<0.01$ &10359300\\
-	     &              &$G$-TC&20&0.53 & $<0.01$ &10322860\\
-	\cline{2-7}
-	     &\multirow{4}{*}{$\mathrm{O}(2)$}&Avg $G$-pooling&20& 0.37&$<0.01$ &10494140\\
-	     &              &Max $G$-pooling&20& 0.43& $<0.01$ &10494140\\
-	     &              &Select. $G$-Bsp&20&0.38 &$<0.01$  &10316120\\
-	     &              &$G$-TC&20&0.48 & $<0.01$ &10276150\\
-	     \hline
-\end{tabular}
-\caption{Results of numerical experiments averaged over 10 runs with Avg $G$-pooling, Max $G$-pooling, selective $G$-bispectrum and $G$-TC. The experiments are performed on $\mathrm{SO}(2)/\mathrm{O}(2)$-MNIST, -EMNIST and -CIFAR-10. The table shows the number of filters, the average classification accuracy, standard deviation and parameter count.}
-\label{tab:classification}
-\end{table*}
-\subsection{Discussion on the choice of pooling layer}
-
-We discuss the advantages and weaknesses of each pooling layer. To this end, it is important to have the complexities of Table \ref{tab:complexities} in mind, to which we add the conclusions of Table \ref{tab:classification}. The first observation from Table \ref{tab:classification} is though the \emph{selective} $G$-bispectrum (see Section \ref{sec:bispectralgcnn}) is complete, the model does not exactly reach the accuracy level of the $G$-TC (see Section \ref{sec:bispectralgcnn}). This observation might be surprising at first, since we prove mathematically in Section~\ref{sec:bspinversion} that the selective $G$-bispectrum contains all the information about the signal, up to group action. In other words, the selective $G$-bispectrum contains exactly the same information as the $G$-TC, only in a reduced form. Why, then, does the $G$-TC perform better, even for matching close number of trainable parameters? An explanation to this lies in the paradoxes of the Universal Approximation Theorem \cite{HORNIK1989359}. Just because an arbitrarily large MLP could theoretically fit any function, this does not imply that it will happen for a practical, limited MLP. In practice, we hypothesize that the $G$-TC provides many coefficients and each of them offers to the MLP a way of distinguishing inputs. Thus, we provide our first insight regarding the choice of invariance layer: If the size of the model allows it, the $G$-TC or the full $G$-bispectrum should be preferred for accuracy. However, as the size of the group gets larger, their uses quickly become out of reach. In comparison, the selective $G$-bispectrum scales linearly with $|G|$ so that it can be implemented on large groups.
-In Table \ref{tab:classification}, we can also notice that the Max $G$-pooling performs well compared to the others though it is not complete. This is because we have many filters, which allows refined classification nonetheless. Indeed, assume $f,\phi_k$ are black-and-white images with $N$ pixels. In consequence, $\max_g\Theta_k(g)\in\{0,1,...,N\}$ for $k=1,2,...,K$. This allows a maximum separation of $(N+1)^K$ classes. In practice, this value is not reached, but it explains why despite the Max $G$-pooling not being complete, it performs well with many filters. Figure \ref{fig:Kfilters} highlights this dependency of the Max $G$-pooling to the number of filters since the accuracy drops to less than $60\%$ with 2 filters. In comparison, the $G$-TC and the selective bispectrum, which are \emph{complete}, keep an accuracy above $85\%$ with 2 filters. If the parameter budget restricts the number of filters that can be implemented within the $G$-Conv, then we recommend using a complete invariance layer.
-
-\begin{figure}
-    \centering
-    \includegraphics[width = 7cm]{Images/filters_MNIST_cyclic.pdf}
-    \includegraphics[width = 7cm]{Images/filters_MNIST_dihedral.pdf}
-    \caption{Evolution of the average classification accuracy with $\mathrm{SO}(2)/\mathrm{O}(2)$-MNIST over $10$ runs when the number of filters varies from $2$ to $20$ for the Avg $G$-pooling, the Max $G$-pooling, the $G$-bispectrum and the $G$-TC. The standard deviations are represented using vertical intervals. In this experiment, the standard deviation can barely be distinguished from the average data.}
-    \label{fig:Kfilters}
-\end{figure}
-
-\section{Conclusion and Future work}
-In this paper, we introduced a new type of complete invariant layer for $G$-invariant CNNs -- called \emph{$G$-bispectrum layer} -- with the objective of increasing the accuracy and robustness of $G$-CNNs compared to those implemented with the initially proposed Max $G$-pooling. The $G$-TC layer also achieves this goal, but at an output cost of $\mathcal{O}(|G|^2)$ coefficients that prevents its application to large groups, while the selective $G$-bispectrum layer only outputs $\mathcal{O}(|G|)$ coefficients. We have given a global picture of the strength and weaknesses of each pooling layer. Based on the result of \citet{kakarala_thesis} for cyclic groups, we have shown that the completeness of the \textit{selective} $G$-bispectrum layer using only $\mathcal{O}(|G|)$ coefficients holds for all commutative groups and the dihedral group.
-%For other groups of interest, such as the chiral octahedral and full octahedral groups in the context of 3D image processing \cite{Lenz2009OctahedralTF}, we provide a good heuristic providing a sequence of coefficients that allow the sequential inversion of the bispectrum up to a degree of freedom. It remains to show that the degree of freedom corresponds to the inherent indeterminacy. The method suggests that the $G$-bispectrum is complete with $\mathcal{O}(|G|)$ coefficients on many other groups, including the two previously cited.
-%Given the wide variety of non-commutative groups, a general analysis would be an important breakthrough.
-\paragraph{Reproducibility} The code is  available on the github repository \url{https://github.com/geometric-intelligence/g-invariance}.
-\paragraph{Impact statement} This research contributes to advancements in the field of Convolutional Neural Networks. In image processing, this tool may be used for purposes such as facial recognition and automated security.
-% \section{Acknowledgments}
-% This work was supported by the Fonds National de la Recherche Scientifique -- FNRS.
-\newpage
-\bibliographystyle{icml2024}
-\bibliography{bispectrumbib.bib}
-\newpage
-\appendix
-\section{Background on groups}\label{app:background}
-We introduce the fundamentals of group theory, which provide the foundation for the theory of $G$-CNNs. These notions can be found in \cite{steinberg2011representation}.
-% A natural starting point is to define a group.
-\begin{definition}\label{def:group}
-    A \emph{group} is a pair $(G,\ \cdot)$ where $G$ is a set and $\cdot:G\times G\mapsto G$ is an associative multiplication such that there is an identity element $e\in G$ (i.e., for all $g\in G$, $e\cdot g=g\cdot e=e$) and, for all $g\in G$, there is an inverse $g^{-1}\in G$ such that $g^{-1}\cdot g = g\cdot g^{-1}=e$.
-\end{definition}
-A group is thus a set $G$ combined with a product $\cdot$ preserving the characteristics of $G$. Here, the established term ``product'' can be misleading. It denotes any operation which makes Definition~\ref{def:group} true given the set $G$. For instance, $(\mathbb{R},+)$ is a group. Another example is $\mathrm{GL}(\mathbb{R}^n)$, the invertible matrices in $\mathbb{R}^n$, associated to the usual matrix product. This group is said to be \emph{non-commutative} since $A\cdot B\neq B\cdot A$ in general for $A,B\in\mathrm{GL}(\mathbb{R}^n)$. An important group for us is the set $\{0,1,...,n-1\}$ associated with addition modulo~$n$. It is often written $\mathbb{Z}/n\mathbb{Z}$ and called the \emph{cyclic group} $C_n$. A single group can arise in different contexts under seemingly distinct forms. For instance, $\mathbb{Z}/4\mathbb{Z}$ and the rotations leaving the square unchanged in $\mathbb{R}^2$ are fundamentally the same object. This observation gives rise to \emph{representation theory}, a branch of group theory studying how the same abstract idea of a group can emerge under different forms.
-\begin{definition}\label{def:representation}
-    A \emph{representation} of a group $(G,\ \cdot)$ is a pair $(\rho, V)$ where $V$ is a vector space and $\rho:G\mapsto \mathrm{GL}(V)$ is a group homomorphism, i.e., for all $g,h\in G$, $\rho(g\cdot h) = \rho(g)\rho(h)$. If $V$ is equipped with an inner product and if for all $g\in G$ and all $u,v\in V$,  $\langle\rho(g)v,\rho(g)w\rangle =\langle u, v\rangle$, $\rho$ is\emph{unitary}.
-\end{definition}
-\begin{remark}
-    Throughout this paper, we use the shorthand $G$ to refer to the group $(G,\cdot )$ and $\rho$ to refer to a representation $(\rho,\ V)$.
-\end{remark}
-To illustrate Definition~\ref{def:representation}, a representation of $C_n$ is given by the complex roots of unity, $\rho(k) = \exp\left(\frac{2\pi i }{n}k\right)$, on the complex one-dimensional vector space $V= \mathbb{C}$. Every group also admit the \emph{trivial} representation: $\rho_0(g)=1$ for all $g \in G$. There is a specific subset of these representations called the irreducible representations, \emph{irreps} for short, being those that can not be expressed in a more compact form. The irreps are fundamental objects of group theory since they allow us to define an invertible Fourier transform on finite groups. The irreps are therefore needed to define the bispectrum -- i.e., the Fourier transform of the $G$-TC. The notion of irreps is derived from that of a $G$-invariant subspace, which we recall in Definition~\ref{def:invariantspace}.
-\begin{definition}\label{def:invariantspace}
-    Given a representation $(\rho, V)$, a subspace $W\subseteq V$ is $G$-invariant if $\rho(g)w\in W$ for all $g\in G,\ w\in W$.
-\end{definition}
-The formal definition of the irreps is then stated as the representations with no non-trivial invariant subspace.
-\begin{definition}
-    A non-zero representation $(\rho,V)$ of a group $G$ is irreducible if the only $G$-invariant subspaces of $V$ are $\{0\}$ and $V$ itself.
-\end{definition}
-A single group acting on different spaces will have different representations. However, one can reveal the similarity between these representations by the mean of an equivalence relation.
-\begin{definition}\label{def:equivalence}
-    Two representations $(\rho,V)$ and $(\varphi, W)$ are \emph{equivalent} if there exists an isomorphism $T:V\mapsto W$ such that for all $g\in G$, $\rho(g) T = T \varphi(g)$.
-\end{definition}
-For the interested reader, the invertibility property of the Fourier transform is a consequence of the concepts of Pontryagin duality (commutative groups) and Tannaka-Krein duality (non-commutative groups); see, e.g., \cite{joyal_ross_tannaka}.
-Our proofs will also rely on the notion of generating set of $G$, which we introduce here.
-\begin{definition}
-    A \emph{generating set} $S$ of a group $(G, \cdot)$ is a subset $S\subset G$ such that every $g\in G$ can be expressed as a finite combination of the elements in $S$ and their inverses under the group action $\cdot$.
-\end{definition}
-\begin{remark}
-    It can be shown that every group $G$ of size $|G|$ has a generating set of size at most $\log_2 |G|$.
-\end{remark}
-\section{Indeterminacy of bispectrum inversion}\label{sec:indeterminacy}
-It is important to state precisely which information we can possibly retrieve from the $G$-bispectrum.
-A consequence of the $G$-invariance of $\beta(\Theta)$ is that the $G$-bispectrum inversion problem is ill-posed. Recall that $G$-invariance means that for all $h\in G$, we have $\beta(\alpha(h,\Theta))_{\rho_1,\rho_2} =\beta(\Theta)_{\rho_1,\rho_2}$. Given a function $\Theta:G\mapsto\mathbb{R}$, a possible definition for the group action on $\Theta$ is given by $\alpha(h, \Theta(g)) =\Theta(h^{-1}\cdot g)$ for all $h\in G$ (see, e.g., \cite{cohenc16}). Therefore, for all $h\in G$, we have:
-\begin{align*}
-        \mathcal{F}(\alpha(h,\Theta))_\rho &= \sum_{g\in G} \alpha(h, \Theta(g)) \rho(g)^\dagger\\
-        &=\rho(hh^{-1})\sum_{g\in G} \Theta(h^{-1}g) \rho(g)^\dagger\\
-        &= \rho(h) \mathcal{F}(\Theta)_\rho,
-    \end{align*}
-which shows that the $G$-Fourier transform $\mathcal{F}(\Theta)$ is $G$-equivariant. In consequence, recovering $\mathcal{F}(\Theta)$ from $\beta(\Theta)$ can at best be done up to an unknown factor $\rho(h)$. Moreover, as explained in \cite{kakarala_thesis, kondor08}, the indeterminacy is not limited to $\rho(h)$. Take for instance $C_n$. An indeterminacy factor $\rho_k(h) = \exp\left(\frac{2\pi i h k}{n}\right)$ corresponds to a translation of $h\in C_n$ of the signal, $\widetilde{\Theta}(g)=\Theta(g+h)$. \cite{kakarala_thesis} showed that $h$ is not restricted to $C_n=\mathbb{Z}/n\mathbb{Z}$: it may take any value in $[0,n]$. The bispectrum is not only invariant to a discrete set of rotations, but to the continuous group of rotations $\mathrm{SO}(2)$. The factor can thus be written $\exp(i \phi k)$ where $\phi\in[0, 2\pi)$.
-\section{The dihedral group $D_n$}
-\subsection{The 1D irreps of $D_n$}\label{app:1dirreps}
-We recall the definition of the dihedral group $D_n$ given in~\eqref{def:dihedral}. The 1D irreps of $D_n$ can be found, e.g., in \cite{steinberg2011representation}. They are given by:
-\begin{itemize}
-    \item $\rho_{0}(g)=1$ for all $g\in D_n$.
-    \item $\rho_{01}(g)=\begin{cases}
-        1 \text{ if } g\in \langle a\rangle,\\
-        -1 \text{ otherwise.}
-    \end{cases}$
-    \item If $n$ even, $\rho_{02}(g)=\begin{cases}
-        1 \text{ if } g\in \langle a^2,\ x\rangle,\\
-        -1 \text{ otherwise.}
-    \end{cases}$
-    \item If $n$ even, $\rho_{03}(g)=\begin{cases}
-        1 \text{ if } g\in \langle a^2,\ ax\rangle,\\
-        -1 \text{ otherwise.}
-    \end{cases}$
-\end{itemize}
-To exemplify the 1D irreps, we give their values for $D_4$ in Table~\ref{tab:1dirreps}.
-\begin{table}[H]
-    \centering
-    \begin{tabular}{|c||c|c|c|c|c|c|c|c|}
-        \hline
-         $g$&$e$&$a$&$a^2$&$a^3$&$x$&$ax$&$a^2x$&$a^3x$ \\
-         \hline
-         \hline
-         $\rho_{0}(g)$&1&1&1&1&1&1&1&1 \\
-         \hline
-         $\rho_{01}(g)$&1&1&1&1&-1&-1&-1&-1 \\
-         \hline
-         $\rho_{02}(g)$&1&-1&1&-1&1&-1&1&-1 \\
-         \hline
-         $\rho_{03}(g)$&1&-1&1&-1&-1&1&-1&1 \\
-         \hline
-    \end{tabular}
-    \caption{1D irreps of $D_4$.}
-    \label{tab:1dirreps}
-\end{table}
-\subsection{Generation of the coefficients of $D_n$}\label{app:generating_dihedral}
-Section~\ref{sec:inversion_dihedral} makes two assertions that we verify explicitly in this appendix. First, it is said that $\rho_{k}\in\rho_1\otimes\rho_{k-1}$ for $k = 2,...,\lfloor\frac{n-1}{2}\rfloor$ so that the iteration of Algorithm~\ref{alg:dihedral_alg} recovers all the Fourier coefficients associated to the 2D irreps. The second assertion to verify is that $\rho_{01}\in \rho_1\otimes\rho_1$ and, for $n$ even,  $\rho_{02},\rho_{03}\in\rho_1\otimes\rho_{\frac{n}{2}-1}$ so that the Fourier coefficients associated to the 1D irreps are also recovered, asserting the validity of the inversion procedure. We provide an analytical proof of these two assertions. The proof is based on the theory of \emph{character functions}.
-
-\begin{definition}{\cite{steinberg2011representation}}
-    Given a group $G$ and a representation $\rho$, the \emph{character} of $\rho$ is the function $\chi_\rho:G\mapsto\mathbb{R}$ given by $\chi_\rho(g) = \mathrm{Tr}(\rho(g)$. $\chi_\rho$ is said to be an \emph{irreducible} character if $\rho$ is an irreducible representation.
-\end{definition}
-The character function $\chi_\rho$ is a \emph{class function} on $G$, i.e., $\chi_\rho$ is constant on a conjugacy class of $G$. The space of class functions on a finite group $G$, written $\mathcal{S}_G$, can be equipped with an inner product $\langle\cdot,\cdot\rangle_G:\mathcal{S}_G\times \mathcal{S}_G\mapsto \mathbb{C}$ such that for $u,v\in \mathcal{S}_G$, we have
-\begin{equation}
-    \langle u, v\rangle_G:=\frac{1}{|G|}\sum_{g\in|G|}u(g)\overline{v(g)}.
-\end{equation}
-The irreducible characters form an orthonormal basis with respect to $\langle\cdot,\cdot\rangle_G$ for $\mathcal{S}_G$ \cite{steinberg2011representation}, i.e.:
-\begin{equation}
-    \langle \chi_{\rho_a},\chi_{\rho_b}\rangle_G=\begin{cases}
-        1\text{ if } \rho_a\sim \rho_b,\\
-        0 \text{ otherwise.}
-    \end{cases}
-\end{equation}
-Therefore, for $\rho_a, \rho_b$ two irreps of $G$, $\rho_c\in\rho_a\otimes\rho_b$ if and only if $ \langle \chi_{\rho_a\otimes \rho_b},\chi_{\rho_c}\rangle_G\neq 0$. Let us apply this to the 2D irreps of $D_n$ ($n>2$). Let $\rho_i,\rho_j,\rho_k$ be three irreps defined as in~\eqref{eq:2d_irreps_dn}.
-Notice that we have $\mathcal{X}_{\rho}(a^lx)=0$. This yields:
-\begin{align*}
-    \langle \mathcal{X}_{\rho_i\otimes\rho_j},\mathcal{X}_{\rho_k}\rangle &=\frac{1}{2n}\sum_{l=1}^n \mathcal{X}_{\rho_i\otimes\rho_j}(a^l)\mathcal{X}_{\rho_k}(a^l)\\
-    &=\frac{1}{2n}\sum_{l=1}^n \mathcal{X}_{\rho_i}(a^l)\mathcal{X}_{\rho_j}(a^l)\mathcal{X}_{\rho_k}(a^l)\\
-    &=\frac{1}{2n}\sum_{l=1}^n 8\cos(\omega_l i)\cos(\omega_l j)\cos(\omega_l k)\\
-    %&=\frac{1}{2n}\sum_{l=1}^n 4\left(\cos(\omega_l (i+j))+\cos(\omega_l (j-i))\right)\cos(\omega_l k)\\
-    &=\frac{1}{n}\sum_{l=1}^n \cos(\omega_l (i+j+k))+\cos(\omega_l (i+j-k))\\ &\ +\cos(\omega_l (j-i+k))+\cos(\omega_l (j-i-k)).
-\end{align*}
-Recall that $\sum_{l=1}^n\cos(\omega_l m)=\begin{cases}
-        0 \text{ if } m\neq 0\\
-        n \text{ if } m = 0
-    \end{cases}$. Therefore, if we assume $0\leq i\leq j$ without loss of generality, $ \langle \mathcal{X}_{\rho_i\otimes\rho_j},\mathcal{X}_{\rho_k}\rangle\neq 0$ if and only if:
-\begin{equation*}
-    \begin{cases}
-        k = i+j \text{ or}\\
-        k = j-i.
-    \end{cases}.
-\end{equation*}
-Therefore, based on Definition~\ref{def:bispectrum}, by utilizing $\beta_{\rho_i,\rho_j}$, $\mathcal{F}(\Theta)_{\rho_i}$, $\mathcal{F}(\Theta)_{\rho_j}$, we can compute  $\mathcal{F}(\Theta)_{\rho_k}$ for $k\in\{i+j,\ j-i\}$. By iterating, if $\mathcal{F}(\Theta)_{\rho_1}$ is known,  $\beta_{\rho_1,\rho_1}$ gives $\mathcal{F}(\Theta)_{\rho_2}$. Then, $\beta_{\rho_1,\rho_2}$ can be leveraged to obtain  $\mathcal{F}(\Theta)_{\rho_3}$. Continuing the procedure provides $\mathcal{F}(\Theta)_{\rho_k}$ for $k=2,...,\lfloor \frac{n-1}{2}\rfloor$ by using $\beta_{\rho_1,\rho_{k-1}}$. We have thus obtained the Fourier coefficients associated to all the 2D irreps.
-
-It remains to show that the iteration also recovered the Fourier coefficients associated to the 1D irreps. This follows from $\rho_{01}\in\rho_1\otimes\rho_1$ and, for $n$ even, $\{\rho_{02}, \rho_{03}\}\subset\rho_1\otimes\rho_{\frac{n}{2}-1}$. Indeed, we have:
-\begin{align*}
-    \langle \mathcal{X}_{\rho_1\otimes\rho_1},\mathcal{X}_{\rho_{01}}\rangle &=\frac{1}{2n}\sum_{l=1}^n \mathcal{X}_{\rho_1\otimes\rho_1}(a^l)\mathcal{X}_{\rho_{01}}(a^l)\\
-    &=\frac{1}{2n}\sum_{l=1}^n 4\cos(\omega_l)^2\\
-    &=\frac{1}{n}\sum_{l=1}^n 1+\cos(2\omega_l)=1.
-\end{align*}
-Moreover, for $n$ even and $\rho\in\{\rho_{02}, \rho_{03}\}$, we have:
-\begin{align*}
-    \langle \mathcal{X}_{\rho_1\otimes\rho_{\frac{n}{2}-1}},\mathcal{X}_\rho\rangle &=\frac{1}{2n}\sum_{l=1}^n \mathcal{X}_{\rho_1\otimes\rho_{\frac{n}{2}-1}}(a^l)\mathcal{X}_\rho(a^l)\\
-    &=\frac{1}{n}\sum_{l=1}^n 1+\cos\left(\omega_l\left(\frac{n}{2}-2\right)\right)(-1)^l\\
-    &=1.
-\end{align*}
-In conclusion, the procedure of Algorithm \ref{alg:dihedral_alg} recovers all the Fourier coefficients and the bispectrum can be inverted with $\frac{n}{2}+1$ coefficients.
-\subsection{The Clebsch-Gordan matrices on $D_n$}\label{app:clebsch_gordan}
-The matrix algebra properties that we use in this subsection can be found, e.g., in \cite{bhatia97}. Recall from Theorem~\ref{thm:commutative_bispectrum} the (implicit) definition of the Clebsch-Gordan matrices:
-\begin{equation}\label{eq:cleb_gordan}
-    (\rho_1\otimes\rho_2)(g) = C_{\rho_1,\rho_2}\left[ \bigoplus_{\rho\in\rho_1\otimes\rho_2} \rho(g) \right]C_{\rho_1,\rho_2}^\dagger,
-\end{equation}
-where $C_{\rho_1,\rho_2}^\dagger C_{\rho_1,\rho_2}=I$.
-We only consider the case of $\rho_1,\rho_2$ both 2d irreps of $D_n$ since otherwise, the Clebsch-Gordan matrix is the scalar $1$.
-First notice that $(\rho_1\otimes\rho_2)(g)$, is an orthonormal matrix. Indeed, using the properties of the Kronecker product, we obtain (``$(g)$'' omitted for clarity):
-\begin{align*}
-    (\rho_1\otimes\rho_2)(\rho_1\otimes\rho_2)^\dagger &= (\rho_1\otimes\rho_2)(\rho_1^\dagger\otimes\rho_2^\dagger)\\
-    &=(\rho_1\rho_1^\dagger)\otimes (\rho_2\rho_2^\dagger)\\
-    &=I \otimes I = I
-\end{align*}
-For $Q$ a real orthogonal matrix ($Q^\dagger Q = I$), and $VSV^T$ with $V^\dagger V = I$, a real Schur decomposition of $Q$, it is known that $S$ is block diagonal with blocks of size $1\times 1$ or $2\times 2$. These blocks are themselves orthogonal matrices. Therefore, the real Schur decomposition is the decomposition in~\eqref{eq:cleb_gordan}. In order for $S$ to represent exactly the irreps from~\eqref{eq:2d_irreps_dn}, the non-zero sub-diagonal elements should all be positive. If not, the symmetric upper-diagonal element is positive and a permutation $P$ must be added to exchange there positions: $Q = (VP)(P^TSP)(VP)^T$. $P$ permutes the two columns of $V$ associated with the permuted $2\times 2$ block of $S$.
-\begin{algorithm}
-    \caption{Compute Clebsch-Gordan matrices on $D_n$}
-    \begin{algorithmic}[1]
-        \STATE \textbf{Input:} $\rho_1, \rho_2$, two 2d irreps of $D_n$.
-        \STATE Pick any $g\in D_n$, e.g., $g = a$.
-        \STATE Compute  $(\rho_1\otimes\rho_2)(g) = (VP)(P^TSP)(VP)^T$, a valid real Schur form.
-        \STATE Set $C_{\rho_1,\rho_2} = VP$.
-        \STATE Set $\bigoplus_{\rho\in\rho_1\otimes\rho_2} \rho(g) = P^TSP$
-        \STATE \textbf{Return:} $C_{\rho_1,\rho_2}, \bigoplus_{\rho\in\rho_1\otimes\rho_2}\rho(g)$.
-    \end{algorithmic}
-\end{algorithm}
-%\section{Heuristic inversion of the $G$-bispectrum on finite non-commutative groups}
-%We describe a greedy method which, when it terminates, provides a set of bispectral elements that can be inverted to
-\section{Training of the $G$-CNN architecture}\label{app:training}
-The $\mathrm{SO}(2)$-MNIST/EMNIST/CIFAR-10 datasets are obtained after applying random planar rotations on each image of the datasets MNIST \cite{deng2012mnist}, EMNIST \cite{cohen2017emnist} and CIFAR-10 \cite{CIFAR10} respectively. In the case of $\mathrm{O}(2)$-MNIST/EMNIST/CIFAR-10, in addition to a planar rotation, a reflection is applied with probability $\frac{1}{2}$.  The original size of each image is conserved. The size of the training sets are $60\ 000$,$88\ 800$ and $50\ 000$ for MNIST, EMNIST and CIFAR-10 respectively. %The size of the test sets are $10\ 000$ and $14\ 800$. For our experiments, $20\%$ of the training datasets are used for validation.
-%\subsection{$C_8$-CNNs on $\mathrm{SO}(2)$-MNIST/EMNIST/CIFAR-10}
-
-We conserve the architecture of \cite{sanborn2023general}. For all invariance layers, being the $G$-TC, the selective $G$-bispectrum and the Avg/Max $G$-pooling, the architecture is composed of a $C_8/D_8$-convolutional block with $K$ filters (see Table~\ref{tab:classification}). Then, the invariance layer is applied before feeding the output to a MLP. The MLP is composed of $3$ fully-connected layers with ReLU non-linearity. A final fully-connected linear layer is applied for classification. The vector of the output sizes of these layers is given by $[o1,o2,o3,ol]$ respectively. $o2=o3 = 64$ and $ol$ is equal to the number of classes of the dataset. $o1$ is tuned to reach the parameter count from Table~\ref{tab:classification}.
-\section{Bispectrum inversion for octahedral and full octahedral groups}
-We provide a sketch of the procedure to retrieve $\mathcal{F}(\Theta)$ given $\beta(\Theta)$ for the \href{https://en.wikipedia.org/wiki/Octahedral_symmetry}{octahedral group} and the \href{https://en.wikiversity.org/wiki/Full_octahedral_group}{full octahedral group}. These two groups are available in the \href{https://quva-lab.github.io/escnn/api/escnn.group.html}{\texttt{escnn} library}. These groups are easier to deal with than the cyclic and dihedral groups presented in the paper, given that they do not come from a \textit{family} of groups. Indeed, our proofs for the cyclic (resp. dihedral) groups needed to work for \textit{all} cyclic groups $C_N$, and for all dihedral groups $D_N$, for all $N$. The octahedral and full octahedral groups are only two groups.
-\begin{table*}[h]
-    \centering
-    \scriptsize{
-    \begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|}
-    \hline
-            $\otimes$ &$\rho_0$&$\rho_1$&$\rho_2$&$\rho_3$&$\rho_4$&$\rho_5$&$\rho_6$&$\rho_7$&$\rho_8$&$\rho_9$\\
-    \hline
-    $\rho_0$&1000000000 & 0100000000 & 0010000000 & 0001000000 & 0000100000 & 0000010000 & 0000001000 & 0000000100 & 0000000010 & 0000000001 \\
-    \hline
-    $\rho_1$&0100000000 & 1111000000 & 0111100000 & 0110000000 & 0010000000 & 0000001000 & 0000011110 & 0000001111 & 0000001100 & 0000000100 \\
-    \hline
-    $\rho_2$&0010000000 & 0111100000 & 1111000000 & 0110000000 & 0100000000 & 0000000100 & 0000001111 & 0000011110 & 0000001100 & 0000001000 \\
-    \hline
-    $\rho_3$&0001000000 & 0110000000 & 0110000000 & 1001100000 & 0001000000 & 0000000010 & 0000001100 & 0000001100 & 0000010011 & 0000000010 \\
-    \hline
-    $\rho_4$&0000100000 & 0010000000 & 0100000000 & 0001000000 & 1000000000 & 0000000001 & 0000000100 & 0000001000 & 0000000010 & 0000010000 \\
-    \hline
-    $\rho_5$&0000010000 & 0000001000 & 0000000100 & 0000000010 & 0000000001 & 1000000000 & 0100000000 & 0010000000 & 0001000000 & 0000100000 \\
-    \hline
-    $\rho_6$&0000001000 & 0000011110 & 0000001111 & 0000001100 & 0000000100 & 0100000000 & 1111000000 & 0111100000 & 0110000000 & 0010000000 \\
-    \hline
-    $\rho_7$&0000000100 & 0000001111 & 0000011110 & 0000001100 & 0000001000 & 0010000000 & 0111100000 & 1111000000 & 0110000000 & 0100000000 \\
-    \hline
-    $\rho_8$&0000000010 & 0000001100 & 0000001100 & 0000010011 & 0000000010 & 0001000000 & 0110000000 & 0110000000 & 1001100000 & 0001000000 \\
-    \hline
-    $\rho_9$&0000000001 & 0000000100 & 0000001000 & 0000000010 & 0000010000 & 0000100000 & 0010000000 & 0100000000 & 0001000000 & 1000000000\\
-    \hline
-    \end{tabular}
-    }
-    \caption{Kronecker table of  full octahedral group using \texttt{escnn}. For the binary word at position $i,j$ in the table, the $k$th `letter' is $1$ if $\rho_k\in\rho_i\otimes \rho_j$, $0$ otherwise.}
-    \label{tab:full_octa}
-\end{table*}
-\subsection{Octahedral group}
-\begin{table}
-        \centering
-        \begin{tabular}{|c||c|c|c|c|c|}
-            \hline
-            $\otimes$ &$\rho_0$&$\rho_1$&$\rho_2$&$\rho_3$&$\rho_4$\\
-            \hline
-            \hline
-            $\rho_0$&10000 & 01000 & 00100 & 00010 & 00001 \\
-            \hline
-            $\rho_1$&01000 & 11110 & 01111 & 01100 & 00100 \\
-            \hline
-            $\rho_2$&00100 & 01111 & 11110 & 01100 & 01000 \\
-            \hline
-            $\rho_3$&00010 & 01100 & 01100 & 10011 & 00010 \\
-            \hline
-            $\rho_4$ &00001 & 00100 & 01000 & 00010 & 10000 \\
-            \hline
-        \end{tabular}
-        \caption{Kronecker table of the octahedral group using \texttt{escnn}. For the binary word at position $i,j$ in the table, the $k$th `letter' is $1$ if $\rho_k\in\rho_i\otimes \rho_j$, $0$ otherwise.}
-        \label{tab:octa}
-    \end{table}
-
-The octahedral group has 24 elements and 5 irreps. We can compute its Kronecker table, either manually using characters $\chi$ or using a Python package such as \texttt{escnn}. We give its Kronecker table below (Table~\ref{tab:octa}), where each column/row represents one irrep, labelled $\rho_0, \rho_1, ..., \rho_4$.
-
-We apply the procedure from Algorithms~\ref{alg:cyclic_alg}, \ref{alg:commutative_alg} and \ref{alg:dihedral_alg} to Table~\ref{tab:octa}. This procedure relies on the use of Theorem~\ref{def:bispectrum}. We first select the bispectral coefficient $\beta_{\rho_0, \rho_0}$ (we omit ``$(\Theta)$'' for clarity) to get the component $\mathcal{F}_{\rho_0 }$ where $\rho_0$ is the trivial representation. Next, we choose $\beta_{\rho_0,\rho_1}$ and use $\mathcal{F}_{\rho_0}$  to obtain $\mathcal{F}_{\rho_1}$ (we know from Table~\ref{tab:octa} that $\rho_1\in\rho_0\otimes\rho_1$) up to an indeterminacy which is a transformation in $\mathrm{O}(3)$ and corresponds to the indeterminacy factor from Appendix~\ref{sec:indeterminacy}. Then, we select  $\beta_{\rho_1, \rho_1}$ to get the Fourier components $\mathcal{F}_{\rho_2}, \mathcal{F}_{\rho_3}$. Lastly, we select $\beta_{\rho_1, \rho_2}$ to get the missing Fourier component $\mathcal{F}_{\rho_4}$.
-
-In summary, we only need $4$ bispectral coefficients ($\beta_{\rho_0, \rho_0}, \beta_{\rho_1, \rho_0}, \beta_{\rho_1, \rho_1}, \beta_{\rho_1, \rho_2}$) instead of $5^2 = 25$ in order to get the five Fourier components, i.e., the full Fourier transform of the signal.
-\subsection{Full octahedral group}
-The full octahedral group has $48$ elements and $10$ irreps. Again, we can compute its Kronecker table using a Python package such as \texttt{escnn}. We give its Kronecker table below (Table~\ref{tab:full_octa}), where each column/row represents one irrep, labelled $\rho_0, \rho_1, ..., \rho_{9}$.
-
-We apply the procedure from Algorithms~\ref{alg:cyclic_alg}, \ref{alg:commutative_alg} and \ref{alg:dihedral_alg} to Table~\ref{tab:full_octa}. Again, this procedure relies on the use of Theorem~\ref{def:bispectrum}. $\beta_{\rho_0,\rho_0}$ (we omit ``$(\Theta)$'' for clarity) allows to compute $\mathcal{F}_{\rho_0}$ directly, such as in Algorithm~\ref{alg:dihedral_alg}. Then, from $\beta_{\rho_0,\rho_6}$, we obtain $\mathcal{F}_{\rho_6}$ up to an unknown group action in $\mathrm{O}(3)$. Then, using $\beta_{\rho_6,\rho_6}$ and $\mathcal{F}_{\rho_6}$, we obtain $\mathcal{F}_{\rho_1}$,$\mathcal{F}_{\rho_2}$ and $\mathcal{F}_{\rho_ 3}$. Next, leveraging $\beta_{\rho_1,\rho_2}$, $\mathcal{F}_{\rho_1}$ and $\mathcal{F}_{\rho_2}$, we obtain  $\mathcal{F}_{\rho_4}$. Using $\beta_{\rho_1,\rho_6}$, $\mathcal{F}_{\rho_1}$ and $\mathcal{F}_{\rho_6}$, we obtain  $\mathcal{F}_{\rho_5}$, $\mathcal{F}_{\rho_7}$, $\mathcal{F}_{\rho_8}$. Finally, with $\beta_{\rho_1,\rho_7}$, $\mathcal{F}_{\rho_1}$ and $\mathcal{F}_{\rho_7}$, we obtain the last coefficient $\mathcal{F}_{\rho_9}$. Hence we have recovered all the Fourier coefficients using only $\beta_{\rho_0,\rho_0}$, $\beta_{\rho_0,\rho_6}$, $\beta_{\rho_6,\rho_6}$, $\beta_{\rho_1,\rho_2}$,$\beta_{\rho_1,\rho_6}$, $\beta_{\rho_1,\rho_7}$, thus a total of $6$ bispectral coefficients instead of $100$.
-
-
-\end{document}
diff --git a/docs/mataigne_et_al_2024_neurips.tex b/docs/mataigne_et_al_2024_neurips.tex
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-\documentclass{article}
-
-
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-
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-%\usepackage{neurips_2024}
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-%    \usepackage[nonatbib]{neurips_2024}
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-\usepackage{booktabs}       % professional-quality tables
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-\usepackage{nicefrac}       % compact symbols for 1/2, etc.
-\usepackage{microtype}      % microtypography
-%\usepackage{xcolor}         % colors
-\usepackage[dvipsnames]{xcolor}
-\usepackage{algorithm}
-\usepackage{algorithmic}
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-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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-% \title{Fast and Complete G-Invariance via novel Algorithmic Reduction of the G-Bispectrum on Finite Groups}
-\title{The Selective $G$-Bispectrum and its Inversion: Applications to $G$-Invariant Networks}
-
-
-% The \author macro works with any number of authors. There are two commands
-% used to separate the names and addresses of multiple authors: \And and \AND.
-%
-% Using \And between authors leaves it to LaTeX to determine where to break the
-% lines. Using \AND forces a line break at that point. So, if LaTeX puts 3 of 4
-% authors names on the first line, and the last on the second line, try using
-% \AND instead of \And before the third author name.
-
-
-\author{%
-  Simon~Mataigne\thanks{Simon Mataigne is a Research Fellow of the Fonds de la Recherche Scientifique - FNRS.%Use footnote for providing further informationabout author (webpage, alternative address)---\emph{not} for acknowledgingfunding agencies.
-  } \\
-  ICTEAM, UCLouvain\\
-  Louvain-la-Neuve, Belgium \\
-  \texttt{simon.mataigne@uclouvain.be} \\
-  % examples of more authors
-   \And
-   Johan Mathe\\
-  Atmo\\
-  San Francisco, CA\\
-   \texttt{johan@atmo.ai} \\
-  \AND
-   Sophia Sanborn \\
-  Science \\
-  San Francisco, CA\\
-\texttt{sophiasanborn@gmail.com} \\
-  \And
-  Christopher Hillar\\
-  Algebraic\\
-  San Francisco, CA\\
-   \texttt{hillarmath@gmail.com} \\
-   \And
-   Nina Miolane\thanks{Nina Miolane acknowledges funding from the NSF grant 2313150.} \\
-   UC Santa Barbara\\
-  Santa Barbara, CA\\
-   \texttt{ninamiolane@ucsb.edu} \\
-}
-
-
-\begin{document}
-
-
-\maketitle
-
-
-\begin{abstract}
-% The development of Convolutional Neural Networks (CNNs) that are insensitive to a group action of the input, e.g., a rotation of the input image, is a critical task in machine learning.
-% In Geometric Deep Learning \citep{bronstein2021geometric}, geometric priors derived from the structure of data are built into deep learning models to increase model efficiency and generalization. The layers of a Group-Equivariant Convolutional Neural Network (G-CNNs) \citep{cohenc16} are designed to have the property of \textit{equivariance} to a group of transformations, such as rotation -- that is, to
-
-%Group-Equivariant Convolutional Neural Networks (G-CNNs) \citep{cohenc16} exploit the symmetries in data to reduce the translation-equivariance of convolutional neural networks can be generalized to other transformation groups
-
-An important problem in signal processing and deep learning is to achieve \textit{invariance} to nuisance factors not relevant for the task. Since many of these factors are describable as the action of a group $G$ (e.g., rotations, translations, scalings), we want methods to be $G$-invariant. The $G$-Bispectrum extracts every characteristic of a given signal up to group action: for example, the shape of an object in an image, but not its orientation. Consequently, the $G$-Bispectrum has been incorporated into deep neural network architectures as a computational primitive for $G$-invariance\textemdash akin to a pooling mechanism, but with greater selectivity and robustness. However, the computational cost of the $G$-Bispectrum ($\mathcal{O}(|G|^2)$, with $|G|$ the size of the group) has limited its widespread adoption. Here, we show that the $G$-Bispectrum computation contains redundancies that can be reduced into a \textit{selective $G$-Bispectrum} with $\mathcal{O}(|G|)$ complexity. We prove desirable mathematical properties of the selective $G$-Bispectrum and demonstrate how its integration in neural networks enhances accuracy and robustness compared to traditional approaches, while enjoying considerable speeds-up compared to the full $G$-Bispectrum.
-
-% Group-Equivariant Convolutional Neural Networks (G-CNNs) \citep{cohenc16, pmlr-v80-kondor18a} generalize the translation-equivariance of traditional CNNs to group-equivariance, using more general symmetry transformations such as rotation. For tasks such as classification, these transformations are removed at the end of the network to achieve group-invariance, typically by taking an average or maximum over the group. While this is indeed invariant, it is \textit{excessively} so; two inputs that are non-identical up to group action can yield the same output, resulting in a general lack of robustness to adversarial attacks \citep{sanborn2023bispectral, sanborn2023general}. The latter paper proposed an alternative method for achieving invariance without signal loss, called the $G$-triple correlation ($G$-TC). While this method yields demonstrable gains in accuracy and robustness, it comes with a substantial increase in computational cost. In this paper, we introduce a significantly more efficient invariant layer that scales only linearly with group size.  It is based on the $G$-\textit{Bispectrum}, which is the Fourier transform of the $G$-TC. Our main theoretical result is that for a large family of finite groups only $\mathcal{O}(|G|)$ coefficients of the $G$-Bispectrum are necessary to conserve the selective invariance of the $G$-TC.
-% We validate this finding in a suite of computational experiments.
-
-%% pre-May 6:
-% Group-Equivariant Convolutional Neural Networks (G-CNNs) \citep{cohenc16} generalize the translation-equivariance of traditional CNNs to  \textit{group}-equivariance, using more general symmetry transformations such as rotation for weight tying.  For tasks such as classification, such transformations are removed at the end of the network, to achieve group-\textit{invariance}, typically by taking a maximum over the group. While this is indeed invariant, it is \textit{excessively} so; two inputs that are non-identical up to group action can yield the same output, resulting in a general lack of robustness to adversarial attacks \citep{sanborn2023bispectral, sanborn2023general}. The latter paper proposed an alternative method for achieving invariance without loss of signal structure, called the $G$-triple correlation ($G$-TC). While this method yields demonstrable gains in accuracy and robustness, it comes with a significant increase in computational cost. In this paper, we introduce a new invariant layer based on the Fourier transform of the $G$-TC: the $G$-Bispectrum. Operating in Fourier space allows us to significantly reduce the computational cost. Our main theoretical result provides a reduction of the $G$-Bispectrum that conserves the selective invariance of the $G$-TC, while only requiring $\mathcal{O}(|G|)$ coefficients. In a suite of experiments, we demonstrate that our approach retains the invariance of the $G$-TC, while significantly reducing the computational cost.
-\end{abstract}
-
-\section{Introduction}\label{sec:introduction}
-The visual world is rich with symmetries. For example, the identity of an object is invariant to its position in the visual field; vision has \textit{translational symmetry}. Group theory is the mathematics used to describe transformations, their actions on objects, and the object's symmetry. As such, group theory has penetrated the fields of signal processing and deep learning alike. For example, the Fourier transform, pillar of signal processing, has been adapted to the $G$-Fourier transform, with its spectrum decomposing a signal defined over a group into several frequencies. More recently, researchers have become interested in the properties of higher-order spectra such as the \textit{Bispectrum}, and its generalization to signals over groups via the $G$-Bispectrum.
-
-\paragraph{$G$-Bispectrum} The $G$-Bispectrum is the Fourier transform of the $G$-Triple Correlation ($G$-TC). Historically, higher-order spectra found initial applications in the context of classical signal processing as generalizations of the two-point autocorrelation \citep{tukey1953spectral, brillinger1991some, nikias1993signal}. The work of \citet{kakarala_thesis} illuminated the relevance of the $G$-Bispectrum for invariant theory, as it is the lowest-degree spectral invariant that is complete. Since then, it has appeared in diverse settings such as vision science \citep{zetzsche2001nonlinear}, machine learning \citep{kondor2007novel, kondor08}, and 3D modeling \citep{kakarala2012Bispectrum}.
-
-% More recently, the group-invariant properties of this primitive have been explored by both signal processing and machine learning researchers.
-%% previous:
-% The work of \citet{kakarala_thesis} illuminated their relevance for invariant theory as the general triple correlation on \textit{groups} is the lowest-degree invariant polynomial that is complete \citep{sturmfels2008algorithms}.
-
-
-\paragraph{Limitations of the $G$-Bispectrum for Deep Learning} The computational complexity of the $G$-Bispectrum has severely limited the reach of its applications. The most salient example of this limitation is in machine learning and deep learning. Convolutional Neural Network (CNN) \citep{lecun95convolutional,LeNet5} reflect and exploit the translational symmetry of the visual world. Group-Equivariant CNNs ($G$-CNNs) \citep{cohenc16, pmlr-v80-kondor18a} do just this, with more general group-equivariant convolutions to exploit %more general
-symmetries like rotational symmetries. In both cases, one typically wants to preserve transformations throughout the layers of a network (i.e., to be group-\textit{equivariant}), and remove them only at the end when ``canonicalizing'' an image for classification (i.e., to be group-\textit{invariant}). While the theory of equivariant layers has been thoroughly developed \citep{cohen2021equivariant, weiler2023EquivariantAndCoordinateIndependentCNNs}, less attention has been paid to the theory of invariant layers \citep{higgins2018definition}. This is where the $G$-Bispectrum enters the picture, and where its computational cost has strongly limited its integration into deep learning.
-
-Commonly, invariance in $G$-CNNs is achieved by simply taking an average or maximum over the transformation group (Average or Max $G$-Pooling, respectively). However, as noted by \citet{sanborn2023general}, this is a highly lossy operation removing information about the structure of the signal. While the \texttt{max} operation is indeed invariant (the \texttt{max} of an image is the same as the \texttt{max} of an image rotated by $90$ degrees), it is \textit{excessively} invariant: one could permute all of the pixels in the image and without changing the maximum, but with none of the same structure (see Figure~\ref{fig:sophias_figure}). To address this, \citet{sanborn2023general} used the $G$-TC as a $G$-invariant layer that is \textit{complete}\textemdash that is, it removes group transformations with no loss of signal structure. This approach achieves demonstrable gains in accuracy and robustness \citep{sanborn2023general}, but it is computationally expensive.
-
-Indeed, the space complexity of the $G$-TC, i.e, its number of coefficients, scales as $\mathcal{O}(|G|^2)$, where $|G|$ is the size of the group. As each coefficient demands for $\mathcal{O}(|G|)$ operations, the computational cost or the time complexity of the $G$-TC is $\mathcal{O}(|G|^3)$. An alternative would be to use the $G$-Bispectrum as the pooling layer. However, both its space and time complexities are $\mathcal{O}(|G|^2)$. By comparison, the Max $G$-pooling layer features a $\mathcal{O}(|G|)$ computational cost and returns a scalar output. This raises the question of whether one can achieve complete invariance and adversarial robustness without sacrificing too much in terms of computational efficiency.
-
-\textbf{Contributions.} In this work, we prove for the first time that we can significantly reduce the computational complexity of the $G$-Bispectrum. This result has important implications for signal processing and deep learning on groups, for which the $G$-Bispectrum is a foundational computational primitive. Our contributions are:
-\begin{itemize}[leftmargin=*]
-    \item We provide a general algorithm that reduces the computational complexity of the $G$-Bispectrum from $\mathcal{O}(|G|^2)$ to $\mathcal{O}(|G|)$ in space complexity and from $\mathcal{O}(|G|^2)$ to $\mathcal{O}(|G|\log |G|)$ in time complexity if an FFT is available on $G$. %We leverage symmetries within the $G$-Bispectrum and that make its computation redundant.
-    We term it the \textit{selective $G$-Bispectrum}. The algorithm can be applied to any finite group.
-    \item We prove that the selective $G$-Bispectrum is complete for the most important finite groups used in practice, i.e., all discrete commutative groups, the dihedral groups of any order, the octahedral and full octahedral group. This significantly extends the work of \citep{giannakis1989signal, kakarala_thesis, sadler1992shift}, who first showed this for \textit{some} finite, commutative groups, where it was demonstrated that the $G$-Bispectrum can be computed with only $|G|$ space complexity.
-    \item We use the selective $G$-Bispectrum to propose a new \textit{$G$-invariant} layer that strikes a balance between robustness and efficiency. In particular, it is more expensive than the Max $G$-pooling, % which has $O(|G|)$ time and $O(1)$ space complexity,
-    but cheaper than the $G$-TC pooling.%, which suffer from $O(|G|^3)$ time and $O(|G|^2)$ space complexity.
-     It is also cheaper than the full $G$-bispectral pooling of $\mathcal{O}(|G|^2)$ time and $\mathcal{O}(|G|^2)$ space complexity. The selective $G$-Bispectrum is more robust than the max $G$-pooling, and almost as robust as the $G$-TC.
-    \item We run extensive experiments on the MNIST \citep{lecun2010mnist} and EMNIST \citep{cohen2017emnist} % and CIFAR-10 \citep{CIFAR10}
-    datasets to evaluate how each invariant layer (Max $G$-pooling, $G$-TC, selective $G$-Bispectrum) impacts accuracy and speed on classification tasks. We achieve the expected results: Our layer is faster than the $G$-TC and full $G$-Bispectrum and more accurate than Max $G$-pooling.
-    %\item We demonstrate the adversarial robustness of our approach against a class of well-known adversarial attacks leveraging the concept of metamers. Our selective bispectral layer achieves the expected results: more robust than architectures using the Max $G$-pooling, and equivalently robust compared to the $G$-CNNs equipped with $G$-TC and full $G$-bispectral poolings.
-    \item We present several findings important to the design of invariant layers to guide further advances in the field of geometric deep learning. In particular, we show that the accuracy and speed advantages of the selective $G$-Bispectrum is most striking for $G$-CNNs with low number of convolutional filters. Conversely, increasing the number of filters in the $G$-Convolutions allows the Max $G$-Pooling to catch up on the accuracy.% of the selective $G$-bispectral pooling.
-     This demonstrates that the $G$-bispectral pooling will be particularly interesting for neural networks operating under a smaller parameter budget.
-\end{itemize}
-We hope that the proposed reduction of the $G$-Bispectrum complexity will further open areas of research in signal processing on groups, that were previously prohibited due to the high complexity of the operation.
-
-
-
-
-%Our work thus brings the precise and mathematically-motivated findings of \citep{sanborn2023general} into the sphere of practicality through cost-efficiency.
-
-% Finally, we  the choice of a reduced Bispectrum layer against a full Bispectrum layer.
-
-% This allows us to introduce a novel layer for $G$-invariant pooling that strikes a balance between robustness and efficiency.
-
-% The cost to compute a bispectral coefficient is usually independent of $|G|$ if one knows the Fourier transform of the input signal (of size $|G|$). For all finite groups $G$, the Fourier transform can be obtained in $\mathcal{O}(|G|^2)$ operations at most, even $\mathcal{O}(|G|\log(|G|))$ when a Fast Fourier Transform (FFT) algorithm is known on $G$. In particular, the Cooley-Tukey FFT algorithm can be extended to many finite groups, including the dihedral group \citep{Diaconis1990EfficientCO, michael89fft}. Therefore, we can reduce the cost of the Bispectrum calculation on many classes of groups to $\mathcal{O}(|G|\log|G|)$ in computation and $\mathcal{O}(|G|)$ in space.
-
-% This motivates the introduction of a $\mathcal{O}(|G|^2)$ layer in both space and computations (or less). This layer should provide more accurate results than the Max $G$-pooling as well as being less expensive than the $G$-TC.
-% The improvement comes from the following observation: there are many redundancies in the $G$-TC that can be neatly exploited in the Fourier domain (i.e. with the \textit{Bispectrum} \citep{kakarala_finite_groups}), permitting a substantial reduction in size and cost to compute.
-
-
-
-% This is most readily seen in the Fourier domain -- i.e. with the \textit{Bispectrum} \citep{kakarala_finite_groups}.
-
-
-
-% Moreover, we can design a layer returning the full Bispectrum, in which case, the complexity is $\mathcal{O}(|G|^2)$ in both computations and space. To determine which coefficients induce completeness, it is enough to determine which allow to reconstruct the initial signal entirely. Retrieving the signal from bispectral coefficients is called the \emph{Bispectrum inversion problem} \citep{kakarala_thesis,Bendory_2018}.
-
-% In this paper, we propose an implementation of the $G$-CNN where the pooling layer is a Bispectrum layer, as suggested in \citep{sanborn2023general}. This corresponds to switching the neural network in the Fourier domain. We will perform experiments on both the selective Bispectrum layer (only a few coefficients) and the full Bispectrum layer (all coefficients). To ensure completeness of the selective Bispectrum, we show that the Bispectrum can be inverted using $|G|$ bispectral coefficients for all finite commutative groups and $\frac{|G|}{4}$ coefficients for the dihedral group, i.e., the symmetries of the $n$-gon. The dihedral group is the main non-commutative group of interest for 2D-image processing.\simcom{Nina, a reference from clinical application or so?}.
-
-% Extending the implementation og \citet{sanborn2023bispectral} to assess the performance of the reduced bispectra layers on the MNIST \citep{deng2012mnist} and EMNIST \citep{cohen2017emnist} datasets.
-% We show that the bispectral layer achieves the expected results: it is both more accurate than the Max $G$-pooling and cheaper than the $G$-TC. Finally, we discuss the choice of a reduced Bispectrum layer against a full Bispectrum layer.
-
-\section{Background: $G$-Triple Correlation and $G$-Bispectrum}\label{sec:triple_corr_and_bsp}
-
-The proposed selective $G$-Bispectrum operation is closely related to two other foundational operations on signals defined on groups: the $G$-Triple Correlation and the full $G$-Bispectrum, which we introduce here. The background on group theory, including the definitions of groups, group actions, equivariance and invariance, is presented in Appendix~\ref{app:group_theory}.
-
-% We need first to present alternative $G$-pooling layers that have recently emerged in addition to the well-known Max $G$-Pooling. Specifically, \citep{sanborn2023bispectral} uses the \textit{full} $G$-Bispectrum layer and \citep{sanborn2023general} introduces the $G$-triple correlation ($G$-TC) layer. We will introduce our \textit{selective} $G$-Bispectrum layer and provide mathematical results that prove its computational efficiency compared to the $G$-TC.
-
-\paragraph{The $G$-Triple Correlation}
-%The $G$-triple correlation ($G$-TC) \citep{kakarala_thesis} has had many uses but our focus stems from its utilization in $G$-CNNs.
-Given a real signal defined on a finite group $\Theta:G\mapsto\mathbb{R}$, the $G$-Triple Correlation ($G$-TC)~\citep{kakarala_thesis}  is the lowest order polynomial that is complete, i.e., that conserves all of the information of the signal $\Theta$, up to group action by $G$.
-\begin{definition}\label{def:gtc}
-    The \emph{$G$-Triple Correlation} of a real signal $\Theta:G\mapsto\mathbb{R}$ is given by
-    \begin{equation}
-        T(\Theta)_{g_1,g_2}:=\sum_{g\in G} \Theta(g)\Theta(g\cdot g_1)\Theta(g\cdot g_2) \text{ for all } g_1, g_2\in G.
-    \end{equation}
-\end{definition}
-
-The original triple-correlation was introduced for the classical framework of translations of a one-dimensional signal, i.e., where $X =\mathbb{Z}$ and $(G,\ \cdot) = (\mathbb{Z},+)$. The $G$-triple correlation from Definition~\ref{def:gtc} extends the original definition to any finite group $(G,\ \cdot)$. In our setting, the signal $\Theta:G\mapsto\mathbb{R}$ will be obtained after the $G$-convolution of a function $f:X\mapsto\mathbb{R}^c$,
-%(e.g., a continuous RGB picture is a function $f:\mathbb{R}^2\mapsto\mathbb{R}^3$ function)
-representing a continuous image with $c$ channels, with a filter $\phi:X\mapsto\mathbb{R}^c$, and the $G$-TC will be applied channel-by-channel. Importantly, the $G$-TC layer has computational complexity $\mathcal{O}(|G|^3)$ and outputs $\mathcal{O}(|G|^2)$ coefficients.
-
-%For all $g \in G$, we have
-% \begin{equation}
-%     \Theta(g):=(\phi*f)(g) = \sum_{x\in X}\phi(\alpha(g^{-1},x))^{\top}f(x).
-% \end{equation}
-%  If $X =\mathbb{Z}$ and $(G,\ \cdot) = (\mathbb{Z},+)$, we obtain the classical discrete convolution.
-
-\paragraph{The $G$-Bispectrum} The $G$-TC operation has a Fourier equivalent: the $G$-Bispectrum. Indeed, the definition of the Discrete Fourier Transform (DFT) can be extended to any finite group~(see, e.g., \citep{Diaconis1990EfficientCO}), as recalled below.
-\begin{definition}
-    Given a set of unitary representatives (Def.~\ref{def:representation}) of the equivalence classes of irreps $\rho_i:G\mapsto\mathrm{GL}(V_i)$ (Def.~\ref{def:equivalence}),
-    the \emph{$G$-Fourier Transform} on a finite group $G$ of a signal $\Theta:G\mapsto\mathbb{R}$  is defined as
-    \begin{equation}
-        \mathcal{F}(\Theta)_{\rho_i} :=\sum_{g\in G}\Theta(g)\rho_i(g)^\dagger,
-    \end{equation}
-     where $\rho_i(g)^\dagger$ refers to the conjugate transpose of the matrix $\rho_i(g)$  (or simply transpose if $\rho_i$ is real-valued).
-\end{definition}
-
-The $G$-Bispectrum $\beta(\Theta)$ is defined as $\mathcal{F}(T(\Theta))$, with $\mathcal{F}$ evaluated over the group $G\times G$.
-\citet{kakarala_thesis} proposed a closed-form expression for the $G$-Bispectrum $\beta(\Theta)$ directly in terms of $\mathcal{F}(\Theta)$. We recall it in Theorem~\ref{def:Bispectrum}.
-
-\begin{theorem}{\citep{kakarala_finite_groups}}\label{def:Bispectrum}
-    The $G$-Bispectrum  of a signal $\Theta:G\mapsto\mathbb{R}$, $\beta(\Theta)$, is given by:
-    \begin{equation*}
-        \beta(\Theta)_{\rho_1,\rho_2} =\left[\mathcal{F}(\Theta)_{\rho_1}\otimes\mathcal{F}(\Theta)_{\rho_2}\right]C_{\rho_1,\rho_2}\left[\bigoplus_{\rho\in\rho_1\otimes\rho_2}\mathcal{F}(\Theta)_\rho^\dagger\right]C_{\rho_1,\rho_2}^{\dagger},
-    \end{equation*}
-    where $C_{\rho_1,\rho_2}$ is a unitary matrix called the Clebsch-Gordan matrix, whose definition is recalled in Appendix~\ref{app:group_theory}.
-    For each pair $\rho_1, \rho_2$, the matrix $\beta(\Theta)_{\rho_1, \rho_2}$ is called a $G$-bispectral coefficient.
-\end{theorem}
-
-For commutative groups, Theorem~\ref{def:Bispectrum} simplifies to a more compact expression, recalled in Theorem~\ref{thm:commutative_Bispectrum}. However, both the space and time complexity of the $G$-Bispectrum remain $O(|G|^2)$.
-
-To the authors' best knowledge, there is no generic analytical formula for computing $C_{\rho_1,\rho_2}$ for an arbitrary group $G$. However, there exist formulas for specific classes of groups (see Appendix~\ref{app:clebsch_gordan}). Additionally, there exist packages for computing these for many groups using packages such as \texttt{escnn} \citep{cesa2022a}.
-
-\paragraph{Complete $G$-Invariants} The $G$-TC and the $G$-Bispectrum are desirable computational primitives for signal processing and deep learning because they are \textit{complete $G$-invariants} (for generic data $\Theta, \widetilde{\Theta}$). Indeed, this completeness property make them very interesting for building invariant layers in $G$-CNNs, as they are selectively invariant. We define \textit{complete $G$-invariance} next.
-\begin{theorem}{\citep[Thm.3.2] {Kakarala2009CompletenessOB}}
-    \label{thm:bsp_invariance}
-    The $G$-TC and the $G$-Bispectrum are complete $G$-invariants, i.e., for $\Theta, \widetilde{\Theta}:G\mapsto\mathbb{R}$ with $\mathcal{F}(\Theta)_\rho$ nonsingular for all irreps $\rho$, $T(\Theta)=T(\widetilde{\Theta})$, respectively  $\beta(\Theta) = \beta(\widetilde{\Theta})$, if and only if there exists $h\in G$ such that $\Theta(g) = \alpha(h, \widetilde{\Theta}(g))$ for all $g\in G$.
-\end{theorem}
-\paragraph{Application: $G$-invariant layers}\label{sec:bispectralgcnn}
-\begin{figure*}
-    \centering
-    \includegraphics[width = 14cm]{Images/figures_ICML.pdf}
-    \caption{Illustration of the different proposed $G$-CNN modules \citep{cohenc16, sanborn2023general}. The input $f$ is first processed through the $G$-convolutional layer composed of $K$ filters $\{\phi_k\}_{k=1}^K$. Then, an invariant layer is chosen (Max $G$-pooling, $G$-TC, or the selective/full $G$-Bispectrum layer). Finally, the ``pooled'' output is fed to a neural network designed for the machine learning task at hand. \vspace{-0.5cm}}
-    \label{fig:GCNN}
-\end{figure*}
-The $G$-CNN architecture, first proposed in \citep{cohenc16}, is illustrated in Figure~\ref{fig:GCNN}. The input signal $f : X\mapsto\mathbb{R}$, typically an image, is processed through a $G$-Convolution layer using filters $\{\phi_k\}_{k=1}^K$. The output is feature maps $\{\Theta_k\}_{k=1}^K$ that form a set of $K$ real-valued signals with domain $G$. This $G$-Convolution layer is traditionally followed by a $G$-invariant layer. The most common is the Max $G$-Pooling layer. More recent works have proposed two alternatives based on the $G$-TC and the full $G$-Bispectrum: the $G$-TC Pooling~\citep{sanborn2023general} and the (full) $G$-Bispectrum~\citep{sanborn2023bispectral} respectively, where the latter requires the computations of the Fourier transforms of the feature maps, preferably computed using a Fast Fourier Transform (FFT) algorithm on $G$ \citep{Diaconis1990EfficientCO}. When testing the impact of the choice of $G$-invariant layer, the output of the invariant layer is typically fed to a Secondary Neural Network (NN) to perform the desired task, e.g., image classification. The Secondary NN often takes the form of a Multi-Layer Perceptron (MLP).
-
-Experimental results have demonstrated the superior accuracy and adversarial robustness of the $G$-CNN equipped with a $G$-TC and $G$-Bispectrum invariant layer~\citep{sanborn2023bispectral,sanborn2023general}. However, both methods inherit the high space and time complexity of their respective operations. % space and time complexity of the $G$-TC are $K\mathcal{O}(|G|^2)$ and $\mathcal{O}(K|G|^3)$, while space and time complexity of the full $G$-Bispectrum are both $\mathcal{O}(K|G|^2)$ where $K$ is the number of filters of the $G$-Convolution.
-This raises the question of whether we can reduce this computational complexity.
-
-
-
-%\section{Bispectrum Inversion for Complete $G$-Invariant Neural Networks}\label{sec:bspinversion}
-\section{Method: The Selective $G$-Bispectrum and its Inversion}\label{sec:bspinversion}
-
-\paragraph{The Selective $G$-Bispectrum} We introduce a novel tool for signal processing on groups: the selective $G$-Bispectrum. %We obtain this object by introducing a novel derivation of the full $G$-Bispectrum from a subset of its coefficients.
-A selective $G$-Bispectrum $\beta_{sel}$ is any $\mathcal{O}(|G|)$  subset of all coefficients of the $G$-Bispectrum $\beta$ (Definition~\ref{def:Bispectrum}), only conserving well-chosen pairs of irreps $(\rho_1, \rho_2)$. Which pairs of irreps to select depends on the group of interest.  This is possible due to redundancies and symmetries in the full object. Below, we provide an algorithmic procedure to compute the selective $G$-Bispectrum for any finite group $G$ that features at most $|\text{Irreps}|(\leq |G|)$ coefficients. The procedure is summarized in Algorithm \ref{alg:selective_bsp}. We have the following proposition.
-
-\begin{algorithm}
-    \caption{Selective $G$-Bispectrum on any finite group $G$}\label{alg:selective_bsp}
-    \small
-    \begin{algorithmic}[1]
-        \STATE \textbf{Input}: Signal $\Theta: G \mapsto \mathbb{R}$ with $n = |G|$. Empty list of coefficients $L_\beta$. Empty list of irreps $L_\rho$. Kronecker Table of $G$.
-        \STATE Add $\beta(\Theta)_{\rho_0,\rho_0}$ to $L_\beta$ where $\rho_0$ is the trivial irreps ($\rho_0(g)=1$ for all $g\in G$).
-        \STATE Add $\rho_0$ to $L_\rho$.
-        \STATE Choose $\tilde \rho_1$ such that $\tilde \rho_1 \otimes\tilde  \rho_1 = C_{\tilde \rho_1,\tilde \rho_1}\left(\bigoplus_{\rho \in \mathcal{R}} \rho\right) C_{\tilde \rho_1,\tilde \rho_1}^\dagger$ generates at least one irreps $\rho$ not yet in $L_\rho$.
-        \STATE Add $\beta(\Theta)_{\rho_0,\tilde \rho_1}$ and $\beta(\Theta)_{\tilde \rho_1,\tilde \rho_1}$ to $L_\beta$, add $\tilde \rho_1$ to $L_\rho$.
-        \STATE Add every $\rho$ that appears in $\tilde \rho_1 \otimes\tilde  \rho_1$ to $L_\rho$.
-        \WHILE{$L_\rho$ keeps changing:}
-            \STATE Find $\rho', \rho''$ in $L_\rho$ such that $\rho' \otimes \rho'' $ generates at least one irreps not already in $L_\rho$.
-            \STATE Add $\beta(\Theta)_{\rho',\rho''}$ to $L_\beta$.
-            \STATE Add $\rho$ that appears in $\rho' \otimes \rho''$ to $L_\rho$.
-        \ENDWHILE
-        \STATE \textbf{Return} $\beta_{sel}(\Theta):=L_\beta$: the selected $G$-bispectral coefficients.
-    \end{algorithmic}
-\end{algorithm}
-
-\begin{proposition}
-    The selective $G$-Bispectrum $\beta_{sel}$ from Algorithm~\ref{alg:selective_bsp} has at most $|G|$ coefficients.
-\end{proposition}
-\begin{proof}
-    By construction of Algorithm \ref{alg:selective_bsp}, $|L_\rho|\leq |\text{Irreps}|$. Since $|\text{Irreps}|$ is at most the number of conjugacy classes of $G$ (see, e.g.,  \citet[Corollary~4.3.10]{steinberg2011representation}), we have $|\text{Irreps}|\leq|G|$.
-\end{proof}
-In Algorithm~\ref{alg:selective_bsp}, we note that the choice of $\rho_1$ is important,  since some $\rho_1$ will not allow the user to recover all of the irreps and therefore not ensure the completeness of the selective. We illustrate the computation of the selective $G$-Bispectrum in Figure~\ref{fig:selective_Bispectrum}, where we choose $\tilde \rho_1 = \rho_6$.
-\begin{figure}
-    \centering
-    \includegraphics[width=1\linewidth]{Images/selective_Bispectrum_algorithm-01.png}
-    \vspace{-0.5cm}
-    \caption{Computation of the selective $G$-Bispectrum for the Full Octahedral Group. The gradient of color represents the order in which the $G$-bispectral coefficients are computed. The Kronecker Table represents which irreps emerge from the decomposition into irreps of the tensor product $\rho_i \otimes \rho_j$. We observe that the selective $G$-Bispectrum has only $6$ coefficients, compared to $100$ coefficients for the full $G$-Bispectrum.}
-    \label{fig:selective_Bispectrum}
-\end{figure}
-\paragraph{Inverting the Selective $G$-Bispectrum for completeness} The \emph{inversion} of the selective $G$-Bispectrum $\beta_{sel}(\Theta)$ is reconstructing a signal $\mathcal{F}(\widetilde{\Theta})$ from the $G$-Bispectrum coefficients in the list $L_\beta$ such that $\widetilde{\Theta}=\alpha(g, \Theta)$ for some $g\in G$ (The $G$-Bispectrum is $G$-invariant, hence, $\Theta$ can only be recovered at best up to group action). Once $\mathcal{F}(\widetilde{\Theta})$ is known, $\widetilde{\Theta}$ can be obtained using the Inverse Fourier Transform.
-If the selective $G$-Bispectrum can be inverted, then, by definition, it is complete in the sense of Theorem~\ref{thm:bsp_invariance}.
-%If one c $\tilde \Theta$ is equal to the original signal $\Theta$ up to group action. Importantly, both the selective $G$-Bispectrum and its inversion can be computed for every finite group.
-
-
-\section{Theory: Completeness of the Selective $G$-Bispectrum}\label{sec:completeness}
-
-Our main theoretical claim is that the selective $G$-Bispectrum can be inverted and is a complete $G$-invariant that drastically reduces the complexity of the $G$-Bispectrum. We prove this claim for many finite groups $G$ of interest in signal processing and deep learning in a sequence of theorems presented in this section.
-
-%The main idea behind the proofs is the previously mentioned \emph{$G$-Bispectrum inversion problem}. For a given group $(G,\cdot)$, we prove that the $G$-Bispectrum can be inverted and is hence complete. % Specifically, for each finite group $G$, we prove that the selective $G$-Bispectrum $\tilde \beta(\Theta)$ is enough to recover all entirely $\mathcal{F}(\Theta)_\rho$, for all of the irreps $\rho$. This means that the selective $G$-Bispectrum is enough to recover the original signal $\Theta$ up to group action. This, in turn, means that the selective $G$-Bispectrum is enough to recover the full $G$-Bispectrum.
-
-\paragraph{Known Theorems} Previous authors had looked into the $G$-Bispectrum inversion problem. It is well known that $|G|=n$ coefficients are enough for the cyclic group $(C_n, \cdot) = (\mathbb{Z} / \mathbb{Z}_n, +\mod n)$.
-
-\begin{theorem}{\citep{kakarala_finite_groups}}\label{thm:cyclic_inversion}
-For cyclic groups $C_n$, $n\in\mathbb{N}_0$, the $C_n$-Bispectrum can be inverted using $|G|=n$ coefficients if $\mathcal{F}(\Theta)_\rho\neq 0$ for all irreps $\rho$ of $C_n$.
-\end{theorem}
-
-Similarly for a product of two such groups, we have the following theorem.
-
-\begin{theorem}{\citep{giannakis1989signal}}\label{thm:product_cyclic_inversion}
-For a product of cyclic groups $C_n \times C_m$, $n,m\in\mathbb{N}_0$, the $G$-Bispectrum can be inverted using $|G|=n m$ coefficients if $\mathcal{F}(\Theta)_\rho\neq 0$ for all $\rho \in C_n \times C_m$.
-\end{theorem}
-
-\paragraph{New Theorems} From now on, we assume that the Fourier transform $\mathcal{F}(\Theta)$ only features \textit{non-zero elements}, or \textit{invertible matrices} in the case of non-scalar Fourier coefficients. This assumption is supported by the zero probability of encountering this corner case (an arbitrarily small perturbation of any signal makes this assumption true).
-
-We first extend the above results to all commutative groups. The proof relies on the fact that every finite commutative group is the direct sum of finitely many cyclic groups.
-
-\begin{theorem}
-For finite commutative groups $G$, the $G$-Bispectrum can be inverted using $|G|$ coefficients if $\mathcal{F}(\Theta)_\rho \neq 0$ for all $\rho \in G$.
-\end{theorem}
-See Appendix~\ref{app:commutative_inversion} for the proof and derivation of the inversion for the specific case of commutative groups. We note that our approach to inversion is symbolic, in that a solution can be expressed explicitly as a formula in terms of the input.  Other approaches are also possible to determine an inverse, such as using least squares \citep{haniff1991least} or more recent spectral methods \citep{chen2018spectral}.
-
-We now extend the result to dihedral groups. Dihedral groups are ubiquitous in signal processing and deep learning because they represent the group of rotations and reflections.
-
-\begin{theorem}
-    For any dihedral group $D_n$ (symmetries of the $n$-gon), $n\in\mathbb{N}_0$, we need at most $\left \lfloor{ \frac{n-1}{2}}\right \rfloor + 2$ bispectral matrix coefficients for inversion if $\det(\mathcal{F}(\Theta)_\rho) \neq 0$ for all irreps $\rho$ of $D_n$. This corresponds to $1+4+16\cdot\left\lfloor{ \frac{n-1}{2}}\right \rfloor\approx 4 |D_n|$ scalar values.
-\end{theorem}
-The proof is provided in Appendix~\ref{app:dihedral_inversion}.
-We now extend the result to octahedral and full octahedral groups, that are related to the symmetries of the octahedron. These groups are very important in signal processing and deep learning of 3D images.
-
-\begin{theorem}
-    For the octahedral group $O$ which has $|G| = 24$ group elements and $5$ irreps, we need only $4$ $G$-Bispectral coefficients in the selective $G$-Bispectrum. this corresponds to $172$ scalars. For the full octahedral group $FO$ which has $|G| = 48$ elements, we only need $6$ $G$-Bispectral coefficients in the selective $G$-Bispectrum to perform inversion. This corresponds to $334$ scalars.
-\end{theorem}
-
-A sketch of proof is provided in Appendix~\ref{app:octahedral} given the redundancy of the procedure. We see that the selective $G$-Bispectrum uses only $4$ coefficients, compared to $25$ coefficients needed for the full $G$-Bispectrum of the octahedral group. For the full octahedral group, it requires only $6$ coefficients compared to the $100$ coefficients of the full $G$-Bispectrum. In Figure~\ref{fig:overview}, we compare the full and selective $G$-Bispectra of the dihedral group $D_4$ (symmetries of the square) and the octahedral group.
-
-\begin{figure}
-    \centering
-    \includegraphics[width=1\linewidth]{Images/selective_bispectrum_overview.png}
-    \vspace{-0.5cm}
-    \caption{Comparison of full and selective $G$-Bispectra for the dihedral group $D_4$ (left) and the octahedral group $O_h$ (right). The Kronecker tables of both groups show which irreps emerge from the decomposition into irreps of the tensor product of irreps $\rho_i \otimes \rho_j$. The colored boxes highlight the bispectral coefficients chosen for the full and selective Bispectra. Our proposed selective Bispectrum captures the same information as the full Bispectrum but with significantly fewer coefficients.}
-    \label{fig:overview}
-\end{figure}
-\section{Experimental results}\label{sec:experiments}
-\paragraph{Implementation and architecture} Our implementation of the selective $G$-bispectrum layer is based on the \href{https://github.com/gtc-invariance/gtc-invariance}{\texttt{gtc-invariance}} repository, implementing the $G$-CNN with $G$-convolution and $G$-TC layer \citep{sanborn2023general} and relying itself on the \href{https://github.com/QUVA-Lab/escnn}{\texttt{escnn} library} \citep{cesa2022a, weiler2021general}. The implementations related to this section can be found at the \href{https://github.com/geometric-intelligence/g-invariance}{\texttt{g-invariance}} repository.
-%We propose a performance analysis of our method. To do so, we have implemented the Bispectrum layer for the cyclic group $C_n$ and the dihedral group $D_n$. These groups can be considered as discrete cases of the groups $\mathrm{SO}(2)$ (2D rotations) and $\mathrm{O}(2)$ (2D rotations and reflections) respectively.
-
-We propose an experimental assessment of the newly proposed selective $G$-Bispectrum layer by comparing it with the Avg $G$-pooling, the Max $G$-pooling, the $G$-TC as invariance operations after the $G$-convolution of a $G$-CNN on the classification problems of the \href{http://yann.lecun.com/exdb/mnist/}{MNIST} dataset of handwritten digits \citep{lecun2010mnist}, the \href{https://www.nist.gov/itl/products-and-services/emnist-dataset}{EMNIST} dataset of handwritten letters \citep{cohen2017emnist} with standard train-test division. These datasets count 10 and 26 classes, respectively. We obtain transformed versions of the datasets -- $G$-MNIST/EMNIST -- by applying a random action $g\in G$ on each image in the original dataset.
-
-The objective of our experiments is to isolate the speed-up of the $G$-Bispectrum layer. Hence, we consider architectures that only differ by the invariant layer in the classification task, following the experimental set up by \citep{sanborn2023general}.  The neural network architecture is composed of a $G$-convolution, a $G$-invariant layer, and finally a Multi-Layer-Perceptron (MLP), itself composed of three fully connected layers with ReLU nonlinearity. Finally, a fully connected linear layer is added to perform classification. The MLP's widths are tuned to match the number of parameters across each neural network model. The details are given in Appendix~\ref{app:training}. We highlight here that the pursued objective is to compare the differences in performances of the $G$-invariant layers, not to provide the state-of-the-art accuracy on the datasets involved. Henceforth, we do not optimize the architectures to reach the highest possible accuracy. We set simple architectures providing interpretable results for analysis. The experiments a performed using $8$ cores of a NVIDIA A30 GPU.
-\paragraph{Training speed performance} Table~\ref{tab:complexities} recalls the theoretical complexities of the different layers. The computational cost of computing the selective $G$-Bispectrum is $\mathcal{O}(|G|\log |G|)$ if an FFT algorithm is available on $G$~\citep{Diaconis1990EfficientCO}, and $\mathcal{O}(|G|^2)$ with classical DFT. in Figure~\ref{fig:trainings}, we report the average training times on $\mathrm{SO}(2)/\mathrm{O}(2)$-MNIST for 10 runs as the discretization $C_n/D_n$ of $\mathrm{SO}(2)/\mathrm{O}(2)$ varies. In the first case, we use the FFT and observe that the Max $G$-pooling and $G$-Bispectrum training time scale linearly whereas it scales quadratically for the $G$-TC. For $\mathrm{O}(2)$, we perform a classic DFT on $D_n$ so that the $G$-Bispectrum scales worth. However, an FFT could be implemented to speed-up the process.
-\begin{table}
-    \centering
-    \begin{tabular}{||c||c|c|c||}
-        \hline
-         Invariance layer&  Computational Complexity & Ouput size& Complete G-invariant\\
-         \hline
-         $G$-TC& $K\mathcal{O}(|G|^3)$&$K\mathcal{O}(|G|^2)$&\textcolor{Green}{\ding{51}}\\
-         \hline
-         Full $G$-Bispectrum& $K\mathcal{O}(|G|^2)$&$K\mathcal{O}(|G|^2)$&\textcolor{Green}{\ding{51}}\\
-         \hline
-         Select. $G$-Bispectrum & $K\mathcal{O}(|G|\log|G|\text{ or }|G|^2)$&$K\mathcal{O}(|G|)$&\textcolor{Green}{\ding{51}}\\
-         \hline
-         Max $G$-pooling&$K\mathcal{O}(|G|)$&$K\mathcal{O}(1)$&\textcolor{red}{\ding{55}}\\
-         \hline
-         Avg $G$-pooling&$K\mathcal{O}(|G|)$&$K\mathcal{O}(1)$&\textcolor{red}{\ding{55}}\\
-         \hline
-    \end{tabular}
-    \vspace{0.2cm}
-    \caption{$G$-CNN invariant layers and their computational cost and output size. $K$ is the number of filters. The selective $G$-Bispectrum that we propose is the complete $G$-invariant layer with the lowest time and space complexity. It reduces significantly the cost compared to the $G$-TC layer while preserving its completeness. \vspace{-0.4cm}}
-    \label{tab:complexities}
-\end{table}
-\begin{figure}[h]
-    \centering
-    \includegraphics[width = 6.8cm]{Images/training_times_cyclic.pdf}
-    \includegraphics[width =6.8cm]{Images/training_times_dihedral.pdf}
-    \vspace{-0.3cm}
-    \caption{Evolution of the average training times for the different invariant layers. The parameter $n$ is the size of the groups $C_n$ and $D_n$. The average and standard deviations are obtained over $10$ runs.  For all runs, the number of parameters of the complete neural network (filters and MLP) is set to $50000$ and $150000$ for $\mathrm{SO}(2)$ and $\mathrm{O}(2)$ respectively. Standard deviations are reported by vertical intervals. When a FFT is available, our selective G-Bispectrum significantly outperforms other complete G-invariant pooling layers in terms of speed. Specifically, when working with $C_{2^7}$, training on a dataset of $60000$ images takes only $247$ seconds, whereas the $G$-TC requires 1465 seconds. %As a result, the selective $G$-Bispectrum paves the way for the development of complete invariant pooling layers that can accommodate larger group sizes and, hence, a larger set of symmetries.
-    }
-    \label{fig:trainings}
-\end{figure}
-
-\paragraph{Classification Performance} We compare the performances of the $G$-Bispectrum layer with respect to the $G$-TC, the Max $G$-pooling and the Avg $G$-pooling models, trained on the $\mathrm{SO}(2)$/$\mathrm{O}(2)$-MNIST/EMNIST datasets and we assess the accuracy by averaging the validation accuracy over 10 runs. The classification accuracy is provided in Table~\ref{tab:classification}. For the experiments in Table \ref{tab:classification}, the following pattern holds: at equivalent number of parameters, the more computationally expensive the pooling layer, the better the accuracy. However, the use of the $G$-TC becomes prohibitive when $|G|$ increases. In the next section, we discuss the settings where each invariant layer should be preferred, and highlight each invariant layer's strengths and weaknesses.
-\begin{table*}
-\centering
-\small
-\begin{tabular}{||c|c|c|c|c|c|c|c||}
-	\hline
-	Dataset&Group&$G$&Pooling&$K$ filters&Avg acc.&Std. dev.&Param. count\\
- \hline
-	\hline
-	\multirow{8}{*}{MNIST}&\multirow{4}{*}{$\mathrm{SO}(2)$}&\multirow{4}{*}{$C_8$}&Avg $G$-pooling&24&0.74 &$<0.01$ &50247\\
-	     &       &       &Max $G$-pooling&24& 0.96& $<0.01$ &50247\\
-	     &       &       &Select. $G$-Bispectrum&24&0.95 &$<0.01$  &49116\\
-	     &       &       &$G$-TC&24&0.96 &  $<0.01$&48385\\
-        \cline{2-8}
-	     &\multirow{4}{*}{$\mathrm{O}(2)$}&\multirow{4}{*}{$D_8$}&Avg $G$-pooling&4& 0.60&$<0.01$ &147675\\
-	     &       &       &Max $G$-pooling&4&0.78 &  $<0.01$&147675\\
-	     &        &      &Select. $G$-Bispectrum&4&0.93 & $<0.01$ &143029\\
-	     &        &      &$G$-TC&4&0.96 &  $<0.01$&142220\\
-	\hline
-	\multirow{8}{*}{EMNIST}&\multirow{4}{*}{$\mathrm{SO}(2)$}&\multirow{4}{*}{$C_8$}&Avg $G$-pooling&24&0.40 &$<0.01$ &50195\\
-	     &       &       &Max $G$-pooling&24& 0.76& $<0.01$ &50195\\
-	     &       &       &Select. $G$-Bispectrum&24&0.77 & $<0.01$ &49254\\
-	     &       &      &$G$-TC&24&0.80 &  $<0.01$&48494\\
-	\cline{2-8}
-	     &\multirow{4}{*}{$\mathrm{O}(2)$}&\multirow{4}{*}{$D_8$}&Avg $G$-pooling&20&0.38 &$<0.01$ &48832\\
-	     &      &        &Max $G$-pooling&20&0.71 & $<0.01$ &48832\\
-	     &      &        &Select. $G$-Bispectrum&20&0.74 & $<0.01$ &47320\\
-	     &     &        &$G$-TC&20&0.79 & $<0.01$ &46954\\
-	\hline
-	%\multirow{8}{*}{CIFAR-10}&\multirow{4}{*}{$\mathrm{SO}(2)$}&Avg $G$-pooling&20&0.43 &$<0.01$ &10494140\\
-	     %&              &Max $G$-pooling&20&0.48 & $<0.01$ &10494140\\
-	     %&              &Select. $G$-Bsp&20& 0.50& $<0.01$ &10359300\\
-	    % &              &$G$-TC&20&0.53 & $<0.01$ &10322860\\
-	%\cline{2-7}
-	     %&\multirow{4}{*}{$\mathrm{O}(2)$}&Avg $G$-pooling&20& 0.37&$<0.01$ &10494140\\
-	    % &              &Max $G$-pooling&20& 0.43& $<0.01$ &10494140\\
-	    % &              &Select. $G$-Bsp&20&0.38 &$<0.01$  &10316120\\
-	    % &              &$G$-TC&20&0.48 & $<0.01$ &10276150\\
-	     %\hline
-\end{tabular}
-\caption{Results of numerical experiments averaged over 10 runs with Avg $G$-pooling, Max $G$-pooling, our selective $G$-Bispectrum and $G$-TC. The experiments are performed on $\mathrm{SO}(2)/\mathrm{O}(2)$-MNIST and $\mathrm{SO}(2)/\mathrm{O}(2)$-EMNIST. % and -CIFAR-10.
-The table shows the number of filters, the average classification accuracy, standard deviation and parameter count. This table shows that the selective $G$-Bispectrum conserves the accuracy of the $G$-TC at an equivalent number of parameters. \vspace{-0.1cm}}
-\label{tab:classification}
-\end{table*}
-\paragraph{Discussion on the choice of invariant layer} The first observation from Table \ref{tab:classification} is though the \emph{selective} $G$-Bispectrum is complete, the model obtains slightly lower accuracy than $G$-TC. This observation might be surprising at first, since we prove mathematically in Section~\ref{sec:completeness} that the selective $G$-Bispectrum is complete, just as the full version. An explanation to this lies in the paradoxes of the Universal Approximation Theorem \citep{HORNIK1989359}. Just because an arbitrarily large MLP can theoretically fit any function, this does not imply that it will happen for a practical, limited MLP. In practice, we hypothesize that the redundancy of the $G$-TC allows the MLP to distinguish inputs more easily. If the size of the model allows it, the $G$-TC or the full $G$-Bispectrum will provide better accuracy. However, when the size of the group is big, their use is often out of reach while the selective $G$-Bispectrum is scalable.
-In Table \ref{tab:classification}, we also notice that the Max $G$-pooling performs well compared to the others even though it is not complete. This is because we have many filters that allow for refined classification. Indeed, assume $f,\phi_k$ are black-and-white images with $N$ pixels. In consequence, $\max_g\Theta_k(g)\in\{0,1,...,N\}$ for $k=1,2,...,K$. The Max $G$-pooling allows a maximum separation of $(N+1)^K$ classes. In practice, this value is not reached, but it explains why Max $G$-pooling performs well. Figure \ref{fig:Kfilters} highlights this dependency of the Max $G$-pooling on the number of filters since the accuracy drops to less than $60\%$ with 2 filters. In comparison, the $G$-TC and the selective $G$-Bispectrum, which are \emph{complete}, keep an accuracy above $85\%$ with 2 filters.% If the parameter budget restricts the number of filters that can be implemented within the $G$-Conv, then we recommend using a complete invariant layer.
-
-\begin{figure}
-    \centering
-    \includegraphics[width = 6.8cm]{Images/filters_MNIST_cyclic.pdf}
-    \includegraphics[width = 6.8cm]{Images/filters_MNIST_dihedral.pdf}
-    \includegraphics[width = 6.8 cm]{Images/filters_MNIST_cyclic2.pdf}
-    \includegraphics[width = 6.8 cm]{Images/filters_MNIST_dihedral2.pdf}
-    \vspace{-0.2cm}
-    \caption{At the top: Evolution of the average classification accuracy with rotated MNIST ($\mathrm{SO}(2)$-MNIST) and rotated-reflected MNIST ($\mathrm{O}(2)$-MNIST) over $10$ runs when the number of filters varies from $2$ to $20$ for the Avg $G$-pooling, the Max $G$-pooling, the selective $G$-Bispectrum and the $G$-TC. The number of parameters of each model is maintained equal for fair comparison. The standard deviations are represented using vertical intervals. With the selective $G$-Bispectrum layer, we can reduce the number of convolutional filters needed for a given accuracy. For example, with only \( K = 2 \) filters, we achieve 96\% accuracy, compared to 63\% with the Max $G$-pooling layer. Our approach allows $G$-CNNs to maintain competitive accuracy while using smaller neural networks.  At the bottom, the same results are displayed with time instead of the number of filters on the $x$-axis. The dotted lines reproduce the evolution of $K$ from the figures at the top. We can observe that the selective $G$-Bispectrum is faster than the $G$-TC when a FFT is available, thus here in the case of $\mathrm{SO}(2)$-MNIST. Recall that an FFT can be implemented for many groups \cite{Diaconis1990EfficientCO}}\vspace{-0.5cm}
-    \label{fig:Kfilters}
-\end{figure}
-\paragraph{Completeness} To conclude our numerical experiments, we study the robustness of the selective $G$-Bispectrum to adversarial attacks, following the analysis in \citet[Figure~2]{sanborn2023general}. Given an image $\widehat{f}:X\mapsto\mathbb{R}^c$ and a filter $\phi:X\mapsto\mathbb{R}^c$, they numerically verified  the robustness (=completeness) of the $G$-TC by showing that
-\begin{equation}\label{eq:opti_sanborn}
-    f^*\in\mathrm{arg}\min_{f:X\mapsto\mathbb{R}^c} \|T(\phi * f)-T(\phi * \widehat{f})\|_2^2 \iff f^*=\alpha(g, \widehat{f}) \text{ for some } g\in G.
-\end{equation}
-Indeed, \citet[Figure~2]{sanborn2023general} shows that only images that are identical up to rotation/reflection can yield the same $C_n/D_n$-TC. That is, the $G$-CNN with $G$-TC can not be ``fooled'' since only input in the same orbit yield the same output. We perform a similar experiment in Figure~\ref{fig:recovering_MNIST}. Moreover, it is well-known that the $G$-convolution $\phi*f$ is $G$-equivariant. Hence, an equivalent experiment is to show that
-\begin{equation*}
-    \Theta^*\in\mathrm{arg}\min_{\Theta:G\mapsto\mathbb{R}}\|T(\Theta)-T(\widehat{\Theta})\|^2_2\iff \Theta^* = \alpha(g, \widehat{\Theta})\text{ for some } g\in G.
-\end{equation*}
-In Figure~\ref{fig:completeness}, we show that the selective $G$-Bispectrum $\beta_{sel}$ is robust to adversarial attacks by solving
-\begin{equation}   \label{eq:optimisation_prob}
-    \Theta^*\in\mathrm{arg}\min_{\Theta:G\mapsto\mathbb{R}}\|\beta_{sel}(\Theta)-\beta_{sel}(\widehat{\Theta})\|^2_2.
-\end{equation}
-The signals are indeed recovered up to a translation, i.e., a group action of $C_{30}$. Moreover, despite~\eqref{eq:optimisation_prob} only optimizes using the selective $G$-Bispectrum, the full $G$-Bispectrum is correctly recovered. This is an additional evidence of the completeness of the selective $G$-Bispectrum.
-
-\begin{figure}
-\centering
-\includegraphics[width=0.99\textwidth]{Images/Recovering_MNIST_V3bis.pdf}
-\vspace{-0.3cm}
-\caption{Adversarial attacks experiments with $G=C_4$. Images are optimized to output, respectively, a target selective $G$-bispectrum and a target $G$-Max pooling, obtained from an original image. The initial image is the initialization of the optimization process \eqref{eq:opti_sanborn}. After training, only for the selective $G$-Bispectrum (in blue), the recovered image is a copy of the original image up to group action (rotation). This is a numerical illustration of the robustness of the selective $G$-Bispectrum to adversarial attacks: one can not obtain the same output with an input that is not in the same class. On the other hand, $G$-Max pooling (in red) outputs a noisy image because it is not complete.}
-\label{fig:recovering_MNIST}
-\end{figure}
-
-\begin{figure}[ht]
-    %\includegraphics[width = 6cm]{Images/Recovered_signal.pdf}
-    %\includegraphics[width = 6cm]{Images/Recovered_Bispectrum.pdf}
-    \centering
-    \includegraphics[width = 0.99\textwidth]{Images/Recovered_Bispectrum_and_signal_V2.pdf}
-    \vspace{-0.2cm}
-    \caption{Numerical experiment of signal recovering from original signals $\{\widehat{\Theta}_i\}_{i=1}^{15}$ where $\widehat{\Theta}_i:G\mapsto\mathbb{R}$ by solving \eqref{eq:optimisation_prob} with  $G = C_{30}$. We use the gradient method with Armijo line search to solve \eqref{eq:optimisation_prob}. The recovered solutions $\{\Theta^*_i\}_{i=1}^{15}$ are represented and correspond to translations of the original signals. The moduli of the full $G$-Bispectra are also represented and are identical. This experiment corroborates the completeness of the selective $G$-Bispectrum since we are able to recover an unknown signal only from the knowledge of its selective $G$-Bispectrum.
-    }
-    \label{fig:completeness}
-\end{figure}
-
-\section{Conclusion and Future works}
-In this paper, we introduced a new type of complete invariant layer for $G$-invariant CNNs -- called \emph{selective $G$-Bispectrum layer} -- with the objective of increasing the accuracy and robustness of $G$-CNNs compared to those implemented with the initially proposed Max $G$-pooling. The $G$-TC layer also achieves this goal, but at an output cost of $\mathcal{O}(|G|^2)$ coefficients and $\mathcal{O}(|G|^3)$ flops that prevents its application to large groups, while the selective $G$-Bispectrum layer only outputs $\mathcal{O}(|G|)$ coefficients. Building on the result of \citet{kakarala_thesis} for cyclic groups, we have shown that the completeness of the \textit{selective} $G$-Bispectrum layer holds for all commutative groups, all dihedral groups, the octahedral and full octahedral groups. In a suite of experiments, we provided a global picture of the strength and weaknesses of each invariant layer. We studied the performance in terms of training speed, classification accuracy and robustness to adversarial attacks. As a result, the selective $G$-Bispectrum paves the way for the development of complete invariant pooling layers that can accommodate larger group sizes and, hence, a larger set of symmetries.
-%For other groups of interest, such as the chiral octahedral and full octahedral groups in the context of 3D image processing \citep{Lenz2009OctahedralTF}, we provide a good heuristic providing a sequence of coefficients that allow the sequential inversion of the Bispectrum up to a degree of freedom. It remains to show that the degree of freedom corresponds to the inherent indeterminacy. The method suggests that the $G$-Bispectrum is complete with $\mathcal{O}(|G|)$ coefficients on many other groups, including the two previously cited.
-%Given the wide variety of non-commutative groups, a general analysis would be an important breakthrough.
-%\paragraph{Reproducibility} The code will be made available upon publication.%The code is  available on the github repository \url{https://github.com/geometric-intelligence/g-invariance}.
-%\paragraph{Impact statement} This research contributes to advancements in the field of Convolutional Neural Networks. In image processing, this tool may be used for purposes such as facial recognition and automated security.
-% \section{Acknowledgments}
-% This work was supported by the Fonds National de la Recherche Scientifique -- FNRS.
-\newpage
-\bibliographystyle{icml2024}
-\bibliography{bispectrumbib.bib}
-
-\newpage
-\appendix
-
-\section{Background on groups}\label{app:group_theory}
-We introduce the fundamentals of group theory, which provide the foundation for the theory of $G$-CNNs. These notions can be found in \citep{steinberg2011representation}.
-% A natural starting point is to define a group.
-\begin{definition}\label{def:group}
-    A \emph{group} is a pair $(G,\ \cdot)$ where $G$ is a set and $\cdot:G\times G\mapsto G$ is an associative multiplication such that there is an identity element $e\in G$ (i.e., for all $g\in G$, $e\cdot g=g\cdot e=e$) and, for all $g\in G$, there is an inverse $g^{-1}\in G$ such that $g^{-1}\cdot g = g\cdot g^{-1}=e$.
-\end{definition}
-A group is thus a set $G$ combined with a product $\cdot$ preserving the characteristics of $G$. Here, the established term ``product'' can be misleading. It denotes any operation which makes Definition~\ref{def:group} true given the set $G$. For instance, $(\mathbb{R},+)$ is a group. Another example is $\mathrm{GL}(\mathbb{R}^n)$, the $n\times n$ real invertible matrices, associated to the usual matrix product. This group is said to be \emph{non-commutative} since $A\cdot B\neq B\cdot A$ in general for $A,B\in\mathrm{GL}(\mathbb{R}^n)$. An important group for us is the set $\{0,1,...,n-1\}$ associated with addition modulo~$n$. It is usually written $\mathbb{Z}/n\mathbb{Z}$ and called the \emph{cyclic group} $C_n$. A single group can arise in different contexts under seemingly distinct forms. For instance, $\mathbb{Z}/4\mathbb{Z}$ and the rotations leaving the square unchanged in $\mathbb{R}^2$ are fundamentally the same object. This observation gives rise to \emph{representation theory}, a branch of group theory studying how the same abstract idea of a group can emerge under different forms.
-\begin{definition}\label{def:representation}
-    A \emph{representation} of a group $(G,\ \cdot)$ is a pair $(\rho, V)$ where $V$ is a vector space and $\rho:G\mapsto \mathrm{GL}(V)$ is a group homomorphism, i.e., for all $g,h\in G$, $\rho(g\cdot h) = \rho(g)\rho(h)$. If $V$ is equipped with an inner product and if for all $g\in G$ and all $u,v\in V$,  $\langle\rho(g)v,\rho(g)w\rangle =\langle u, v\rangle$, $\rho$ is \emph{unitary}.
-\end{definition}
-\begin{remark}
-    Throughout this paper, we use the shorthand $G$ to refer to the group $(G,\cdot )$ and $\rho$ to refer to a representation $(\rho,\ V)$.
-\end{remark}
-To illustrate Definition~\ref{def:representation}, a representation of $C_n$ is given by the complex roots of unity, $\rho(k) = \exp\left(\frac{2\pi i }{n}k\right)$, on the complex one-dimensional vector space $V= \mathbb{C}$. Every group also admit the \emph{trivial} representation: $\rho_0(g)=1$ for all $g \in G$. There is a specific subset of these representations called the irreducible representations, \emph{irreps} for short, being those that can not be expressed in a more compact form. The irreps are fundamental objects of group theory since they allow us to define an invertible Fourier transform on finite groups. The irreps are therefore needed to define the $G$-Bispectrum -- i.e., the Fourier transform of the $G$-TC. The notion of irreps is derived from that of a $G$-invariant subspace, which we recall in Definition~\ref{def:invariantspace}.
-\begin{definition}\label{def:invariantspace}
-    Given a representation $(\rho, V)$, a subspace $W\subseteq V$ is $G$-invariant if $\rho(g)w\in W$ for all $g\in G,\ w\in W$.
-\end{definition}
-The formal definition of the irreps is then stated as the representations with no non-trivial invariant subspace.
-\begin{definition}
-    A non-zero representation $(\rho,V)$ of a group $G$ is irreducible if the only $G$-invariant subspaces of $V$ are $\{0\}$ and $V$ itself.
-\end{definition}
-A single group acting on different spaces will have different representations. However, one can reveal the similarity between these representations by the mean of an equivalence relation.
-\begin{definition}\label{def:equivalence}
-    Two representations $(\rho,V)$ and $(\varphi, W)$ are \emph{equivalent} if there exists an isomorphism $T:V\mapsto W$ such that for all $g\in G$, $\rho(g) T = T \varphi(g)$.
-\end{definition}
-For the interested reader, the invertibility property of the Fourier transform is a consequence of the concepts of Pontryagin duality (commutative groups) and Tannaka-Krein duality (non-commutative groups); see, e.g., \citep{joyal_ross_tannaka}.
-Our proofs will also rely on the notion of generating set of $G$, which we introduce here.
-\begin{definition}
-    A \emph{generating set} $S$ of a group $(G, \cdot)$ is a subset $S\subset G$ such that every $g\in G$ can be expressed as a finite combination of the elements in $S$ and their inverses under the group action $\cdot$.
-\end{definition}
-\begin{remark}
-    It can be shown that every group $G$ of size $|G|$ has a generating set of size at most $\log_2 |G|$.
-\end{remark}
-
-\paragraph{Group Actions} A group $(G,\ \cdot)$ represents a set of transformations such as rotations that can act on data such as images.% , see Appendix \ref{app:group_theory} for the mathematical definition.
-We define formally how groups can indeed transform datasets through the concept of group action.
-%Now that a group $G$ and its irreps are properly defined, it is important to define how $G$ acts on a given vector space~$X$.
-% Take for instance an infinite black-and-white picture modeled by some function $f:\mathbb{R}^2\mapsto [0,1]$. The \emph{group action} may define how the image is flipped or rotated. We define the notion of group action more formally below.
-\begin{definition}\label{def:groupaction}
-    Given a group $(G,\ \cdot)$, a \emph{group action} $\alpha:G\times X\mapsto X$ is a function satisfying i) Identity: $\alpha(e, x)=x$, ii) Compatibility: $\alpha(h,\alpha(g,x))=\alpha(h\cdot g, x)$ for any $x \in X$ and $h, g \in G$ and where $e$ is the identity of $G$.
-\end{definition}
-
-%Group representations are particular types of group actions that act on vector spaces $V$: we denote them by $(\rho,\ V)$. Irreducible representation (irreps for short) are special group representations where the vector space $V$ has minimal dimension in a sense made precise in Appendix \ref{app:group_theory}.
-
-Processing operations and neural networks can be designed so that they respect group actions: specifically, a group acting on the input (e.g., rotating an input image) should yield a group action on the output (e.g., a rotation of the output feature map). This is the notion of $G$-\emph{equivariance}.
-\begin{definition}
-    A function $\psi:X\mapsto Y$ is $G$-\emph{equivariant} if $\psi(\alpha_1(g,x)) = \alpha_2(g,\psi(x))$ for all $x\in X$ and all $g\in G$, where $\alpha_1$ and $\alpha_2$ are group actions on $X$ and $Y$, respectively.
-\end{definition}
-For example, the $G$-convolution layer is $G$-equivariant by design  \citep{cohenc16}. An important problem in signal processing and deep learning is to achieve \textit{invariance} to nuisance factors not relevant for the task. Many of these factors are describable as group actions (e.g. rotations, translations, scaling). Thus, we want processing methods and machine learning models to be $G$-invariant:
-\begin{definition}
-    A function $\psi:X\mapsto Y$ is $G$-\emph{invariant} if $\psi(\alpha(g,x)) = \psi(x)$ for all $x\in X$ and all $g\in G$.
-\end{definition}
-For example, the Max $G$-pooling ($\max_{g\in G}\Theta(g)$) traditionally follows a $G$-convolutional layer to remove the equivariance of the convolution and achieve $G$-invariance. A $G$-CNN is a neural network that consists of $G$-convolutional layers and a pooling/invariance operation. The main applications of our proposed selective $G$-Bispectrum operation is to act as a $G$-invariant pooling layer, that can conveniently replace the classical Max $G$-Pooling layer of $G$-CNNs, as shown in the rest of the paper.
-\begin{figure}
-    \centering
-    \includegraphics[width = 7cm]{Images/completeness_figure.png}
-    \caption{Illustration of the concepts of excessive and complete invariance to a group action. With the excessive invariance, samples from different classes can be mapped to the same output.}
-    \label{fig:sophias_figure}
-\end{figure}
-
-\paragraph{Clebsh-Jordan matrices}Given a group $(G,\cdot)$ and a family of unitary irreps $\{\rho_i\}_{i=0}^{|\text{Irreps}|-1}$, the Clebsh-Jordan matrix is analytically defined for each pair $\rho_1, \rho_2$ as:
-\begin{equation}
-    (\rho_1 \otimes \rho_2)(g) = C_{\rho_1,\rho_2} \Big[ \bigoplus_{\rho \in\mathcal{R}} \rho(g) \Big] C_{\rho_1,\rho_2}^\dagger.
-    \label{eqn:non-commutative-bs}
-\end{equation}
-
-\paragraph{$G$-Bispectrum for Commutative Finite Groups}
-
-The computation of the $G$-Bispectrum simplifies for commutative groups compared to Theorem~\ref{def:Bispectrum}, as recalled below.
-
-\begin{theorem}{\citep{kakarala_finite_groups}}\label{thm:commutative_Bispectrum}
-    If $(G,\ \cdot)$ is a commutative group and $\Theta:G\rightarrow \mathbb{R}$, the $G$-Bispectrum can be computed as
-    \begin{equation}\label{eq:com_bsp}
-        \beta(\Theta)_{\rho_1,\rho_2} = \mathcal{F}(\Theta)_{\rho_1}\mathcal{F}(\Theta)_{\rho_2}\mathcal{F}(\Theta)_{\rho_1\otimes\rho_2}^\dagger.
-    \end{equation}
-\end{theorem}
-For commutative groups, the $G$-bispectral coefficients are complex scalars \citep{steinberg2011representation}.
-
-\section{Indeterminacy of $G$-Bispectrum inversion problem}\label{sec:indeterminacy}
-It is important to state precisely which information we can possibly retrieve from the $G$-Bispectrum.
-A consequence of the $G$-invariance of $\beta(\Theta)$ is that the $G$-Bispectrum inversion problem is ill-posed. Recall that $G$-invariance means that for all $h\in G$, we have $\beta(\alpha(h,\Theta))_{\rho_1,\rho_2} =\beta(\Theta)_{\rho_1,\rho_2}$. Given a function $\Theta:G\mapsto\mathbb{R}$, a possible definition for the group action on $\Theta$ is given by $\alpha(h, \Theta(g)) =\Theta(h^{-1}\cdot g)$ for all $h\in G$ (see, e.g., \citep{cohenc16}). Therefore, for all $h\in G$, we have
-\begin{align*}
-        \mathcal{F}(\alpha(h,\Theta))_\rho &= \sum_{g\in G} \alpha(h, \Theta(g)) \rho(g)^\dagger\\
-        &=\rho(hh^{-1})\sum_{g\in G} \Theta(h^{-1}g) \rho(g)^\dagger\\
-        &= \rho(h) \mathcal{F}(\Theta)_\rho,
-    \end{align*}
-which shows that the $G$-Fourier transform $\mathcal{F}(\Theta)$ is $G$-equivariant. In consequence, recovering $\mathcal{F}(\Theta)$ from $\beta(\Theta)$ can at best be done up to an unknown factor $\rho(h)$. Moreover, as explained in \citep{kakarala_thesis, kondor08}, the indeterminacy is not limited to $\rho(h)$. Take for instance $C_n$. An indeterminacy factor $\rho_k(h) = \exp\left(\frac{2\pi i h k}{n}\right)$ corresponds to a translation of $h\in C_n$ of the signal, $\widetilde{\Theta}(g)=\Theta(g+h)$. \citep{kakarala_thesis} showed that $h$ is not restricted to $C_n=\mathbb{Z}/n\mathbb{Z}$: it may take any value in $[0,n]$. The Bispectrum is not only invariant to a discrete set of rotations, but to the continuous group of rotations $\mathrm{SO}(2)$. The factor can thus be written $\exp(i \varphi k)$ where $\varphi\in[0, 2\pi)$.
-
-\section{Selective $G$-Bispectrum inversion: known results}\label{sec:commutative_alg}
-%We present the algorithm from \citep{kakarala_thesis} to invert the Bispectrum the cyclic group $C_n=$($\mathbb{Z}/n\mathbb{Z}$, $+$ mod $n$). The method requires only $|G|=n$ bispectral coefficients, therefore allowing the reduction in complexity expected with a bispectral layer compared to the TC layer. We prove mathematically that the method can be extended to all commutative groups, still requiring only $|G|$ bispectral coefficients. Moreover, we show that it can be extended to the dihedral group $D_n$, i.e., the symmetries of the $n$-gon. In this case, the method only requires $\frac{|G|}{4}$ bispectral coefficients.
-From now on, we assume that the Fourier transform $\mathcal{F}(\Theta)$ only features \textbf{non-zero elements}, or \textbf{invertible matrices} in the case of non-scalar Fourier transform. This assumption is supported by the zero probability of encountering this corner case.
-
-\paragraph{Cyclic groups $C_n$} We start with $(G,\ \cdot)=(\mathbb{Z}/n\mathbb{Z},\ +\ \mathrm{mod}\ n)=:C_n$. Recall that the irreps are given by $\rho_k(g)=\exp\left(\omega_k g\right)$ where $\omega_k:=\frac{2\pi i k }{n}$ for $k\in\mathbb{Z}/n\mathbb{Z}$ (see, e.g., \citep{steinberg2011representation}).
-\begin{theorem}{\citep{kakarala_finite_groups}}\label{thm:cyclic_inversion}
-For cyclic groups $C_n$, the $C_n$-Bispectrum can be inverted using $|G|=n$ coefficients if $\mathcal{F}(\Theta)_\rho\neq 0$ for all $\rho \in C_n$.
-\end{theorem}
-\begin{proof}
-The Fourier coefficient associated to the trivial representation $\mathcal{F}(\Theta)_{\rho_0}$, is uniquely determined and can be recovered from Theorem~\ref{thm:commutative_Bispectrum} by identifying phase and modulus:
-\begin{equation}\label{eq:trivial_coef}
-    \beta(\Theta)_{\rho_0,\rho_0} =|\mathcal{F}(\Theta)_{\rho_0}|^3\exp\left(i\ \mathrm{arg}(\mathcal{F}(\Theta)_{\rho_0})\right).
-\end{equation}
-
-We proceed using Pontryagin duality: the irreps $\{\rho_k\}_{k=1}^n$, form a group $\widehat{G}$ themselves, with $\widehat{G} = G$. In the case of the cyclic group $C_n$, notice that for all $j,k\in\mathbb{Z}/n\mathbb{Z}$, $\rho_j\otimes \rho_k = \rho_{j+k}$. Leveraging \eqref{eq:com_bsp}, we can use $\beta(\Theta)_{\rho_0,\rho_1}$ to recover $\mathcal{F}(\Theta)_{\rho_1}$:
-
-\begin{equation}\label{eq:trivial_coeff_cyclic}
-    |\mathcal{F}(\Theta)_{\rho_1}|^2 =\frac{\beta(\Theta)_{\rho_0,\rho_1}}{\mathcal{F}(\Theta)_{\rho_0}}.
-\end{equation}
-
-Equation~\eqref{eq:trivial_coeff_cyclic} leaves an indeterminacy on the phase of $\mathcal{F}(\Theta)_{\rho_1}$. This corresponds to the indeterminacy factor $\exp(i \varphi)$, $\phi\in[0,2\pi)$ of Appendix~\ref{sec:indeterminacy}. It is inherited from the $G$-invariance  of $\beta(\Theta)$ (it is not injective, hence you cannot distinguish inputs that have the same $G$-Bispectrum). For now, let $\arg(\mathcal{F}(\Theta)_{\rho_1})=0$. The key to recover all the other Fourier coefficients is to notice that $S = \{1\}$ is a generating set of $\widehat{G} =\mathbb{Z}/n\mathbb{Z}$. Therefore, computing sequentially
-\begin{equation}
-    \mathcal{F}(\Theta)_{\rho_{k+1}} = \left(\frac{\beta(\Theta)_{\rho_1,\rho_k}}{\mathcal{F}(\Theta)_{\rho_1}\mathcal{F}(\Theta)_{\rho_k}}\right)^\dagger,
-\end{equation}
-for $k=1,2,...,n-2$ recovers completely $\mathcal{F}(\Theta)$. We are not done yet because the phase we fixed before is not a valid shift. A valid phase shift for $\mathcal{F}(\Theta)_{\rho_1}$ is such that the shift w.r.t the original signal has the form $\exp\left(\frac{2\pi i h}{n}\right)$ for $h\in\mathrm{N}$. This valid phase shift is easy to find. It is the unique $\varphi\in[0,\frac{2\pi}{n})$ such that, if we define $\mathcal{F}(\widetilde{\Theta})_{\rho_k} = \exp(\varphi k) \mathcal{F}(\Theta)_{\rho_{k}}$ for all $k\in\mathbb{Z}/n\mathbb{Z}$, then we have $\mathcal{F}^{-1}(\mathcal{F}(\widetilde{\Theta}))\in\mathbb{\mathbb{R}}^n$ (i.e., with no imaginary part). Note that this is an explicit method to recover a valid phase while \citep{kakarala_finite_groups} relies on its existence without explicit method to find it.
-The method is summarized in Algorithm~\ref{alg:cyclic_alg} and illustrated in Figure \ref{fig:alg1}. In consequence only the following $G$-bispectral coefficients are needed for completeness: $\beta(\Theta)_{\rho_0,\rho_0}, \beta(\Theta)_{\rho_0,\rho_1}$ and $ \beta(\Theta)_{\rho_1,\rho_k}$ for $k=1,2,...,n-2$. This makes a total of $n=|G|$ coefficients. We summarize this result in Theorem~\ref{thm:cyclic_inversion}.
-\end{proof}
-
-
-\begin{figure}
-\begin{minipage}[c]{0.54\textwidth}
-  \begin{algorithm}[H]
-    \caption{Bispectrum inversion on $\mathbb{Z}/n\mathbb{Z}$
- \citep{kakarala_finite_groups}}
-    \label{alg:cyclic_alg}
-    \small
-    \begin{algorithmic}[1]
-        \STATE \textbf{Input}: $\beta(\Theta)_{\rho_0,\rho_0}, \beta(\Theta)_{\rho_0,\rho_1}$ and $ \beta(\Theta)_{\rho_1,\rho_k}$ for $k=1,2,...,n-2$.
-        \STATE Compute $|\mathcal{F}(\Theta)_{\rho_0}|=\left(\beta(\Theta)_{\rho_0,\rho_0}\right)^{\frac{1}{3}}$ and $\mathrm{arg}(\mathcal{F}(\Theta)_{\rho_0})=\mathrm{arg}(\beta(\Theta)_{\rho_0,\rho_0})$.
-        \STATE Compute $|\mathcal{F}(\Theta)_{\rho_1}| =\left(\frac{\beta(\Theta)_{\rho_0,\rho_1}}{\mathcal{F}(\Theta)_{\rho_0}}\right)^{\frac{1}{2}}$ and set $\mathrm{arg}(\mathcal{F}(\Theta)_{\rho_1})=0$.
-        \FOR{$k=1,2,...,n-2$}
-            \STATE Compute $\mathcal{F}(\Theta)_{\rho_{k+1}} = \left(\frac{\beta(\Theta)_{\rho_1,\rho_k}}{\mathcal{F}(\Theta)_{\rho_1}\mathcal{F}(\Theta)_{\rho_k}}\right)^\dagger$.
-        \ENDFOR
-        \STATE Find $\varphi\in [0,\frac{2\pi}{n})$ such that $\mathcal{F}^{-1}(\mathcal{F}(\widetilde{\Theta}))\in\mathbb{R}^n$ where $\mathcal{F}(\widetilde{\Theta})_{\rho_k} = \exp(\phi k)\mathcal{F}(\Theta)_{\rho_{k}}$.
-        \STATE \textbf{Return} $\mathcal{F}(\widetilde{\Theta})$ (up to group action).
-    \end{algorithmic}
-\end{algorithm}
-\end{minipage}\hfill%
-\begin{minipage}[c]{0.4\textwidth}
-    \includegraphics[width = 6cm]{Images/alg1_icml.pdf}
-    \caption{Illustration of Algorithm \ref{alg:cyclic_alg}. The Bispectrum coefficients allow to recover the Fourier transform sequentially, up to group action. First, $\beta_{\rho_0,\rho_0}$ gives $\mathcal{F}_{\rho_{0}}$, which, combined with $\beta_{\rho_0,\rho_1}$ gives $\mathcal{F}_{\rho_{1}}$ etc. }
-    \label{fig:alg1}
-\end{minipage}
-\end{figure}
-
-\section{Selective $G$-Bispectrum inversion: commutative groups}\label{app:commutative_inversion}
-
-Here, we prove the theorem stated in the main text and recalled below:
-\begin{theorem}
-For finite commutative groups $G$, the $G$-Bispectrum can be inverted using $|G|$ coefficients if $\mathcal{F}(\Theta)_\rho \neq 0$ for all $\rho \in G$.
-\end{theorem}
-
-
-Specifically, we extend Algorithm~\ref{alg:cyclic_alg} to all commutative groups, based on the Theorem~\ref{thm:fundamentalthm}. That is, we design a method for the direct sum of finitely many cyclic groups.
-\begin{theorem}{(see, e.g., \citep{Norman_2012})}\label{thm:fundamentalthm}
-    Every finite commutative group $G$ is isomorphic to a finite direct sum of cyclic groups: $G\cong \bigoplus_{l=1}^L \mathbb{Z}/n_l\mathbb{Z}$ where $L\in\mathbb{N}$ and $n_l\in\mathbb{N}$ for $l=1,2,...,L$.
-\end{theorem}
-For all $\kbf\in G$ ($\kbf$ is integer-valued vector of length $L$), the irreps $\rho_\kbf:G\mapsto \mathbb{C}$ are given by $
-    \rho_\kbf(g) = \prod_{l=1}^L\exp(\frac{2\pi i\kbf_l}{n_l} g_l)$.
-The number of irreps is $|G| = \prod_{l=1}^L n_l$. We detail and prove in Theorem~\ref{thm:invert_commutative} the procedure to invert the $G$-Bispectrum on commutative groups. The procedure is summarized in Algorithm~\ref{alg:commutative_alg} where we use the two following notations.\begin{enumerate}
-    \item $\ebf^l$ denotes the basis vector in $\mathbb{Z}^L$ such that $\ebf^l_k=1$ if $k=l$ and $\ebf^l_k=0$ otherwise.
-    \item $K^l:=\{t\ebf^l +\kbf \ | \ t=1,2...,n_l-1\text{ and } \kbf\in K^{l-1}$\} for $l=1,2,...,L$.
-\end{enumerate}
-The sets $K^l$ are a recursively constructed such that $K^L\cong G$. For $G = (\mathbb{Z}/3\mathbb{Z})^3$, the sets $K_1,K_2,K_3$ are represented in Figure~\ref{fig:ksets}.
-\begin{figure}[h]
-    \centering
-    \includegraphics[width = 5cm]{Images/K_sets.pdf}
-    \caption{Representation of the sets $K_1,K_2$ and $K_3$ for $G = (\mathbb{Z}/3\mathbb{Z})^3$.}
-    \label{fig:ksets}
-\end{figure}
-\begin{theorem}\label{thm:invert_commutative}
-For finite commutative groups $G$, the $G$-Bispectrum can be inverted using $|G|$ coefficients if $\mathcal{F}(\Theta)_\rho \neq 0$ for all $\rho \in G$.
-\end{theorem}
-\begin{proof}
-%For all $\kbf\in G$, the irreps $\rho_\kbf:G\mapsto \mathbb{C}$ are given by $\rho_\kbf(g) = \prod_{l=1}^L\exp(\frac{2\pi i \kbf_l}{n_l} g_l)$. The number of irreps is $|G| = \prod_{l=1}^L n_l$.
-Notice that for the commutative groups, we keep the property $\rho_{\kbf}\otimes\rho_{\kbf'}=\rho_{\kbf+\kbf'}$ where $(\kbf+\kbf')_l := \kbf_l+\kbf_l'\ \mathrm{mod} \ n_l$ for $l=1,2,...,L$. The first step is to obtain a generating set $S$ of the irreps is of size $L$. It is given by the usual basis vectors $S=\{\ebf^l\in \mathbb{Z}^L$ for $l=1,...,L\}$ where
-\begin{equation*}
-    \ebf^l_k=\begin{cases}
-        1 \text{ if } k=l,\\
-        0 \text{ otherwise.}
-    \end{cases}
-\end{equation*}
-By Theorem~\ref{thm:commutative_Bispectrum}, it is sufficient to have the Fourier coefficients associated to each generating element in $S$ to recover all the Fourier coefficients. Indeed, knowing $\mathcal{F}(\Theta)_{\rho_{\kbf_1}}$ and $\mathcal{F}(\Theta)_{\rho_{\kbf_2}}$ allows us to compute $\mathcal{F}(\Theta)_{\rho_{\kbf_1+\kbf_2}}$. By definition of the generating set $S$, we can thus recover $\mathcal{F}(\Theta)_{\rho_{\kbf}}$ for all $\kbf \in G$. The moduli of the coefficients $\mathcal{F}(\Theta)_{\rho_{\ebf^l}}$ for $l=1,2,...,L$ can be computed as follows:
-\begin{equation}\label{eq:compute_generating}
-    |\mathcal{F}(\Theta)_{\rho_{\ebf^l}}| =\left(\frac{\beta(\Theta)_{\rho_\mathbf{0},\rho_{\ebf^l}}}{\mathcal{F}(\Theta)_{\rho_\mathbf{0}}}\right)^\frac{1}{2},
-\end{equation}
-where $\mathcal{F}(\Theta)_{\rho_\mathbf{0}}$ is the Fourier coefficient of the trivial representation (computed as in~\eqref{eq:trivial_coef}). We claim that the phase can be fixed independently for each label $l$, thus $L$ times. This is because only one factor $h_l$ remains among the $L$ independent factors in $\mathbf{h}$:
-\begin{align*}
-    \mathcal{F}(\alpha(\mathbf{h},\Theta))_{\rho_{\ebf^l}} &= \rho_{\ebf^l}(\mathbf{h})\mathcal{F}(\Theta)_{\rho_{\ebf^l}}=\exp\left(\frac{2\pi i h_l}{n_l}\right)\mathcal{F}(\Theta)_{\rho_{\ebf^l}},
-\end{align*}
-for all $l=1,2,.,L$. Therefore, fixing an arbitrary phase in~\eqref{eq:compute_generating} only fixes $h_l$. Again, the indeterminacy factor $h_l$ is not restricted to $\mathbb{Z}/n_l\mathbb{Z}$ but can belong to $[0,n_l]$. We will have to solve this issue further. For now, we set the phase of $\mathcal{F}(\Theta)_{\rho_{\ebf^l}}$ to zero. Once the Fourier coefficients are known for all the generators of the group of the irreps $\{\rho_{\ebf^l}\}_{l=1}^L$, it remains to combine them to obtain all the elements in the groups and, consequently, all the associated Fourier coefficients.
-
-At this point, it helps to consider the problem geometrically. Each irreps $\rho_\kbf$ can be associated to its integer coordinate $\kbf$ inside a hyper-rectangle in $\mathbb{R}^L$, whose length of edges is $n_l$ for $l=1,2,...,L$. We combine the coordinates $\ebf^l$ to obtain all the possible integer coordinates inside the hyper-rectangle. First, we can obtain the $L$ orthogonal edges of the hyper-rectangle. For $l=1,2,...,L$, $\rho_{\ebf^l}\otimes \rho_{\ebf^l}$ gives $\rho_{2\ebf^l}$, $\rho_{\ebf^l}\otimes \rho_{2\ebf^l}$ gives $\rho_{3\ebf^l}$, etc. This is in fact the procedure of Algorithm 1. Now, we combine the edges to generate the inside of the hyper-rectangle. We proceed iteratively. For $l=1,2,...,L$, we define $K^l:=\{t\ebf^l + \kbf \ | \ t=1,2,..,n_l-1\text{ and } \kbf\in K^{l-1}\}$ and $K^0:=\{\mathbf{0}\}$. This construction is such that $K^l\cong\bigoplus_{j=1}^l \mathbb{Z}/n_j\mathbb{Z}$ and $K^L=G$. We generate the missing Fourier coefficients by combining the ones associated to the generating set of $G$.
-For $l=2,...,L$, compute
-\begin{equation}
-    \mathcal{F}(\Theta)_{\rho_{t\ebf^l+\kbf}}=\left(\frac{\beta(\Theta)_{\rho_{t\ebf^l},\rho_{\kbf}}}{\mathcal{F}(\Theta)_{\rho_{t\ebf}}\mathcal{F}(\Theta)_{\rho_{\kbf}}}\right)^\dagger,
-\end{equation}
-for $t=1,2,...,n_l-1$, for all $\kbf\in K^{l-1}$. Intuitively for $L=3$, we obtain first an edge, then a face and finally the full parallelepiped. To conclude, we reproduce the procedure from Algorithm~\ref{alg:cyclic_alg} to find a valid phase shift $\varphi_l$ in each basis direction $\ebf^l$. The last step is then to compute, for all $\kbf\in G$, $$\mathcal{F}(\widetilde{\Theta})_{\rho_{\kbf}} =\mathcal{F}(\Theta)_{\rho_{\kbf}} \prod_{l=1}^L\exp(\phi_l k_l).$$  The procedure is summarized in Algorithm~\ref{alg:commutative_alg} and illustrated in Figure \ref{fig:alg2}. It shows that the bispectral coefficients needed for completeness are: $\beta(\Theta)_{\rho_0,\rho_0}$, $\beta(\Theta)_{\rho_0,t\rho_{\ebf}^l},\ \beta(\Theta)_{t\rho_{\ebf}^l, \rho_\kbf} $ for $l=1,2,...,L$, $t=1,2,...,n_l-2$ and all $\kbf\in K^{l-1}$. We recover exactly one Fourier coefficient per $G$-bispectrum coefficient. This makes thus a total of $|G|$ bispectral coefficients precisely and proves the following theorem.
-\end{proof}
-
-\begin{figure}
-\begin{minipage}[c]{0.54\textwidth}
-\begin{algorithm}[H]
-    \caption{Bispectrum inversion for finite commutative groups ($G=\bigoplus_{l=1}^L \mathbb{Z}/n_l\mathbb{Z}$).}
-    \label{alg:commutative_alg}
-    \small
-    \begin{algorithmic}[1]
-        \STATE \textbf{Input}: $|G|$ bispectral coeffs.
-        \STATE Compute $|\mathcal{F}(\Theta)_{\rho_\mathbf{0}}|=\left(\beta(\Theta)_{\rho_\mathbf{0},\rho_\mathbf{0}}\right)^{\frac{1}{3}}$ and $\mathrm{arg}(\mathcal{F}(\Theta)_{\rho_\mathbf{0}})=\mathrm{arg}(\beta(\Theta)_{\rho_\mathbf{0},\rho_\mathbf{0}})$.
-        \FOR{$l=1,...,L$}
-            \STATE Compute $|\mathcal{F}(\Theta)_{\rho_{\ebf^l}}| =\left(\frac{\beta(\Theta)_{\rho_\mathbf{0},\rho_{\ebf^l}}}{\mathcal{F}(\Theta)_{\rho_\mathbf{0}}}\right)^\frac{1}{2}$ and set $\mathrm{arg}\left(\mathcal{F}(\Theta)_{\rho_{\ebf^l}}\right)=0$.
-            \FOR{$t=1,...,n_l-1$}
-                \IF{$t>1$}
-                \STATE  $F(\Theta)_{\rho_{t\ebf^l}}=\left(\frac{\beta(\Theta)_{\rho_{\ebf^l},\rho_{(t-1)\ebf^l}}}{\mathcal{F}(\Theta)_{\rho_{\ebf^l}}\mathcal{F}(\Theta)_{\rho_{(t-1)\ebf^l}}}\right)^\dagger$.
-                \ENDIF
-                \FOR{$\kbf\in K^{l-1}\setminus\{\mathbf{0}\}$}
-                    \STATE Compute $\mathcal{F}(\Theta)_{\rho_{t\ebf^l+\kbf}}=\left(\frac{\beta(\Theta)_{\rho_{t\ebf^l},\rho_{\kbf}}}{\mathcal{F}(\Theta)_{\rho_{t\ebf^l}}\mathcal{F}(\Theta)_{\rho_{\kbf}}}\right)^\dagger$.
-                \ENDFOR
-            \ENDFOR
-        \ENDFOR
-        \FOR{$l=1,2,...,L$}
-            \STATE Find $\varphi_l \in [0,\frac{2\pi}{n})$ s.t. $\mathcal{F}^{-1}(\mathcal{F}(\widetilde{\Theta})_{\rho_{(1,..,n_l)\ebf^l}})\in\mathbb{R}^{n_l}$ where $\mathcal{F}(\widetilde{\Theta})_{\rho_{k\ebf^l}} = \exp(\phi_l k)\mathcal{F}(\Theta)_{\rho_{k\ebf^l}}$.
-        \ENDFOR
-        \FOR{$\kbf\in G$}
-           \STATE $\mathcal{F}(\widetilde{\Theta})_{\rho_{\kbf}} =\mathcal{F}(\Theta)_{\rho_{\kbf}} \prod_{l=1}^L\exp(\phi_l k_l)$.
-        \ENDFOR
-        \STATE \textbf{Return} $\mathcal{F}(\widetilde{\Theta})$ (up to group action).
-    \end{algorithmic}
-\end{algorithm}
-\end{minipage}\hfill
-\begin{minipage}[c]{0.4\textwidth}
-    \centering
-    \includegraphics[width = 6cm]{Images/alg2_icml.pdf}
-    \caption{Illustration of Algorithm \ref{alg:commutative_alg}. The Bispectrum coefficients allow to recover the Fourier transform sequentially, up to group action.}
-    \label{fig:alg2}
-\end{minipage}
-\end{figure}
-
-\section{Selective $G$-Bispectrum inversion: dihedral groups}\label{app:dihedral_inversion}
-
-\paragraph{Dihedral group $D_n$}\label{sec:inversion_dihedral}
-The dihedral group $D_n$ is the group of all symmetries of the $n$-gon. Mathematically, it is defined as
-\begin{equation}\label{def:dihedral}
-    D_n:=\langle x,\ a\ |\ a^n=x^2=e,\ xax = a^{-1}\rangle,
-\end{equation}
-where $a$ is the \emph{rotation} and $x$ is the \emph{reflection}, and they form a generating set of $D_n$. We will only consider the case $n>2$ since the cases $n=1$ and $n=2$ are commutative groups covered by the previous subsection, while $n>2$ gives non-commutative groups. The 2D irreps of $D_n$ are given by
-\begin{equation}\label{eq:2d_irreps_dn}
-    \rho_k(a^lx^m)=\begin{bmatrix}
-        \cos(\omega_l k)&-\sin(\omega_l k)\\
-        \sin(\omega_l k)&\cos(\omega_l k)
-    \end{bmatrix}
-    \begin{bmatrix}
-        1 & 0\\
-        0 & -1\\
-    \end{bmatrix}^m,
-\end{equation}
-where $\omega_l = \frac{2\pi l}{n}$ for $k=1,2,...,\lfloor \frac{n-1}{2}\rfloor$. There are also $2$ or $4$ 1D irreps if $n$ is odd or even, respectively. We denote these 1D irreps by $\rho_{0},\ \rho_{01},\ \rho_{02}$, and $\rho_{03}$ (see Appendix \ref{app:1dirreps}). The two last ones only exist for $n$ even.
-\begin{theorem}\label{thm:dihedral_inversion}
-    For the family of dihedral groups $D_n$, we need at most $\left \lfloor{ \frac{n-1}{2}}\right \rfloor + 2$ bispectral matrix coefficients for inversion if $\det(\mathcal{F}(\Theta)_\rho) \neq 0$ for all irreps $\rho$ of $D_n$. This corresponds to $1+4+16\cdot\left\lfloor{ \frac{n-1}{2}}\right \rfloor\approx 4 |D_n|$ scalar values.
-\end{theorem}
-\begin{proof}
-In view of section~\ref{sec:commutative_alg}, we wish to show that there is an irrep $\rho_1$ that generates all the irreps of $D_n$. As in the cyclic and commutative cases, we can first deduce $\mathcal{F}(\Theta)_{\rho_0}$ from $\beta(\Theta)_{\rho_0,\rho_0}$. Now, we have
-\begin{equation}\label{eq:first_coeff_dihedral}
-    \mathcal{F}(\Theta)_{\rho_1}\mathcal{F}(\Theta)_{\rho_1}^\dagger=\frac{\beta(\Theta)_{\rho_0,\rho_1}}{\mathcal{F}(\Theta)_{\rho_0}}.
-\end{equation}
-The novelty for this non-commutative group is that $\mathcal{F}(\Theta)_{\rho_1}$ is a $2\times 2$ matrix. After computing the eigenvalue decomposition $\frac{\beta(\Theta)_{\rho_0,\rho_1}}{\mathcal{F}(\Theta)_{\rho_0}} = V\Lambda V^\dagger$, we can choose
-\begin{equation}
-    \mathcal{F}(\Theta)_{\rho_1} = V\Lambda^{\frac{1}{2}}V^\dagger U.
-\end{equation}
-for all unitary $U$ (i.e., $UU^\dagger=U^\dagger U=I$) to solve~\eqref{eq:first_coeff_dihedral}. In the case $\mathbb{Z}/n\mathbb{Z}$ the indeterminacy belonged to $\mathrm{SO}(2)$, the continuous set of 2d rotations. For $D_n$, it belongs to $\mathrm{O}(2)$, the continuous set of 2d rotations and reflections. For finite groups, \citet{kakarala_thesis} ensures that the only choices for $U$ such that $\mathcal{F}(\Theta)$ is a Fourier transform on $G(=D_n)$ are such that $\Theta$ is identical to the original signal up to some group action $g\in D_n$. %that respect $\eqref{eqn:non-commutative-bs}$ are such that $U = \rho_k(a^l x^m)$ for some $k,l,m$ (see \eqref{eq:2d_irreps_dn}).
-For $k=2,...,\lfloor \frac{n-1}{2}\rfloor$, we can then obtain
-\begin{align}
-    \label{eq:new_coeff_dihedral}
-    &\bigoplus_{\rho\in\rho_1\otimes\rho_{k-1}}\mathcal{F}(\Theta)_\rho =\left(C_{\rho_1,\rho_{k-1}}^{\dagger}\left[\mathcal{F}(\Theta)_{\rho_1}\otimes\mathcal{F}(\Theta)_{\rho_{k-1}}\right]^{-1}\beta(\Theta)_{\rho_1,\rho_2}C_{\rho_1,\rho_{k-1}}\right)^\dagger.
-\end{align}
-It is shown in Appendix~\ref{app:generating_dihedral} that $\rho_k$ appears in the tensor decomposition \eqref{eqn:non-commutative-bs} of $\rho_1\otimes\rho_{k-1}$ so that the for-loop can be applied. Moreover, we know from Appendix~\ref{app:generating_dihedral} that $\rho_{01}$ appears in the decomposition of $\rho_1\otimes\rho_{1}$ and, for $n$ even, $\rho_{02}, \rho_{03}$ in $ \rho_1\otimes\rho_{\frac{n}{2}-1}$. Thus the iteration recovers the complete DFT $\mathcal{F}(\Theta)$. $\beta(\Theta)_{\rho_0,\rho_0}$ is a scalar, $\beta(\Theta)_{\rho_0,\rho_1}$ is a $2\times 2$ matrix and $ \beta(\Theta)_{\rho_1,\rho_k}$, $k\neq 0$, is a $4\times 4$ matrix. Hence the total number of scalars that is required is $1+4+16\lfloor\frac{n-1}{2}\rfloor$.
-\end{proof}
-The procedure is summarized in Algorithm \ref{alg:dihedral_alg}.
-\begin{algorithm}[tb]
-    \caption{Bispectrum inversion on the dihedral group $D_n$.}\label{alg:dihedral_alg}
-    \small
-    \begin{algorithmic}[1]
-        \STATE \textbf{Input}: $\beta(\Theta)_{\rho_0,\rho_0}, \beta(\Theta)_{\rho_0,\rho_1}$ and $ \beta(\Theta)_{\rho_1,\rho_k}$ for $k=1,2,...,\lfloor\frac{n-1}{2}\rfloor$.
-        \STATE Compute $|\mathcal{F}(\Theta)_{\rho_0}|=\left(\beta(\Theta)_{\rho_0,\rho_0}\right)^{\frac{1}{3}}$ and $\mathrm{arg}(\mathcal{F}(\Theta)_{\rho_0})=\mathrm{arg}(\beta(\Theta)_{\rho_0,\rho_0})$.
-        \STATE Compute $V\Lambda V^\dagger = \frac{\beta(\Theta)_{\rho_0,\rho_1}} {\mathcal{F}(\Theta)_{\rho_0}}$.\\
-        \STATE Set $\mathcal{F}(\Theta)_{\rho_1} = V\Lambda^\frac{1}{2} V^\dagger U$ with valid $U$ (existence of $U$ ensured but not computed easily).\\
-        \FOR{$k=2,...,\lfloor\frac{n-1}{2}\rfloor$}
-           % \STATE (If $n$ odd, stop at $k =\lfloor\frac{n-1}{2}\rfloor-1 )$
-            \STATE Compute \begin{align*}
-                &\bigoplus_{\rho\in\rho_1\otimes\rho_{k-1}}\mathcal{F}(\Theta)_\rho =\left(C_{\rho_1,\rho_{k-1}}^{\dagger}\left[\mathcal{F}_{\rho_1}\otimes\mathcal{F}_{\rho_{k-1}}\right]^{-1}\beta_{\rho_1,\rho_2}C_{\rho_1,\rho_{k-1}}\right)^\dagger.
-            \end{align*}
-        \ENDFOR
-        \STATE \textbf{Return} $\mathcal{F}(\Theta)$ (up to group action).
-    \end{algorithmic}
-\end{algorithm}
-%\subsubsection{General algorithm for finite non-commutative groups}
-%The three cases that we have just studied suggest a heuristic procedure to invert the Bispectrum. Given a group $G$ and a complete family of irreps $\{\rho_k\}_k$ where $\rho_0$ is the trivial representation. $\mathcal{F}(\Theta)_{\rho_0}$ can always be recovered from $\beta(\Theta)_{\rho_0,\rho_0}$. Then one can compute $\mathcal{F}(\Theta)_{\rho_0}$
-\subsection{The 1D irreps of $D_n$}\label{app:1dirreps}
-We recall the definition of the dihedral group $D_n$ given in~\eqref{def:dihedral}. The 1D irreps of $D_n$ can be found, e.g., in \citep{steinberg2011representation}. They are given by:
-\begin{itemize}
-    \item $\rho_{0}(g)=1$ for all $g\in D_n$.
-    \item $\rho_{01}(g)=\begin{cases}
-        1 \text{ if } g\in \langle a\rangle,\\
-        -1 \text{ otherwise.}
-    \end{cases}$
-    \item If $n$ even, $\rho_{02}(g)=\begin{cases}
-        1 \text{ if } g\in \langle a^2,\ x\rangle,\\
-        -1 \text{ otherwise.}
-    \end{cases}$
-    \item If $n$ even, $\rho_{03}(g)=\begin{cases}
-        1 \text{ if } g\in \langle a^2,\ ax\rangle,\\
-        -1 \text{ otherwise.}
-    \end{cases}$
-\end{itemize}
-To exemplify the 1D irreps, we give their values for $D_4$ in Table~\ref{tab:1dirreps}.
-\begin{table}[H]
-    \centering
-    \begin{tabular}{|c||c|c|c|c|c|c|c|c|}
-        \hline
-         $g$&$e$&$a$&$a^2$&$a^3$&$x$&$ax$&$a^2x$&$a^3x$ \\
-         \hline
-         \hline
-         $\rho_{0}(g)$&1&1&1&1&1&1&1&1 \\
-         \hline
-         $\rho_{01}(g)$&1&1&1&1&-1&-1&-1&-1 \\
-         \hline
-         $\rho_{02}(g)$&1&-1&1&-1&1&-1&1&-1 \\
-         \hline
-         $\rho_{03}(g)$&1&-1&1&-1&-1&1&-1&1 \\
-         \hline
-    \end{tabular}
-    \caption{1D irreps of $D_4$.}
-    \label{tab:1dirreps}
-\end{table}
-\subsubsection{Generation of the coefficients of $D_n$}\label{app:generating_dihedral}
-Theorem~\ref{thm:dihedral_inversion} makes two assertions that we verify explicitly in this appendix. First, it is said that $\rho_{k}$ is in the tensor decomposition \eqref{eqn:non-commutative-bs} (TD) of $\rho_1\otimes\rho_{k-1}$ for $k = 2,...,\lfloor\frac{n-1}{2}\rfloor$ so that the iteration of Algorithm~\ref{alg:dihedral_alg} recovers all the Fourier coefficients associated to the 2D irreps. The second assertion to verify is that $\rho_{01}$ is in the TD of $ \rho_1\otimes\rho_1$ and, for $n$ even,  $\rho_{02},\rho_{03}$ in the TD of $\rho_1\otimes\rho_{\frac{n}{2}-1}$ so that the Fourier coefficients associated to the 1D irreps are also recovered, asserting the validity of the inversion procedure. We provide an analytical proof of these two assertions. The proof is based on the theory of \emph{character functions}.
-
-\begin{definition}{\citep{steinberg2011representation}}
-    Given a group $G$ and a representation $\rho$, the \emph{character} of $\rho$ is the function $\chi_\rho:G\rightarrow\mathbb{R}:g\mapsto\mathrm{Tr}(\rho(g))$. $\chi_\rho$ is said to be an \emph{irreducible} character if $\rho$ is an irreducible representation.
-\end{definition}
-The character function $\chi_\rho$ is a \emph{class function} on $G$, i.e., $\chi_\rho$ is constant on a conjugacy class of $G$. The space of class functions on a finite group $G$, written $\mathcal{S}_G$, can be equipped with an inner product $\langle\cdot,\cdot\rangle_G:\mathcal{S}_G\times \mathcal{S}_G\mapsto \mathbb{C}$ such that for $u,v\in \mathcal{S}_G$, we have
-\begin{equation}
-    \langle u, v\rangle_G:=\frac{1}{|G|}\sum_{g\in|G|}u(g)\overline{v(g)}.
-\end{equation}
-The irreducible characters form an orthonormal basis w.r.t  $\langle\cdot,\cdot\rangle_G$ for $\mathcal{S}_G$ \citep{steinberg2011representation}, i.e.,
-\begin{equation}
-    \langle \chi_{\rho_a},\chi_{\rho_b}\rangle_G=\begin{cases}
-        1\text{ if } \rho_a\sim \rho_b,\\
-        0 \text{ otherwise.}
-    \end{cases}
-\end{equation}
-Therefore, for $\rho_a, \rho_b$ two irreps of $G$, $\rho_c$ is in the TD of $\rho_a\otimes\rho_b$ if and only if $ \langle \chi_{\rho_a\otimes \rho_b},\chi_{\rho_c}\rangle_G\neq 0$. Let us apply this to the 2D irreps of $D_n$ ($n>2$). Let $\rho_i,\rho_j,\rho_k$ be three irreps defined as in~\eqref{eq:2d_irreps_dn}.
-Notice that we have $\mathcal{X}_{\rho}(a^lx)=0$. This yields
-\begin{align*}
-    \langle \mathcal{X}_{\rho_i\otimes\rho_j},\mathcal{X}_{\rho_k}\rangle &=\frac{1}{2n}\sum_{l=1}^n \mathcal{X}_{\rho_i\otimes\rho_j}(a^l)\mathcal{X}_{\rho_k}(a^l)\\
-    &=\frac{1}{2n}\sum_{l=1}^n \mathcal{X}_{\rho_i}(a^l)\mathcal{X}_{\rho_j}(a^l)\mathcal{X}_{\rho_k}(a^l)\\
-    &=\frac{1}{2n}\sum_{l=1}^n 8\cos(\omega_l i)\cos(\omega_l j)\cos(\omega_l k)\\
-    %&=\frac{1}{2n}\sum_{l=1}^n 4\left(\cos(\omega_l (i+j))+\cos(\omega_l (j-i))\right)\cos(\omega_l k)\\
-    &=\frac{1}{n}\sum_{l=1}^n \cos(\omega_l (i+j+k))+\cos(\omega_l (i+j-k))\\ &\ +\cos(\omega_l (j-i+k))+\cos(\omega_l (j-i-k)).
-\end{align*}
-Recall that $\sum_{l=1}^n\cos(\omega_l m)=\begin{cases}
-        0 \text{ if } m\neq 0\\
-        n \text{ if } m = 0
-    \end{cases}$. Therefore, if we assume $0\leq i\leq j$ without loss of generality, $ \langle \mathcal{X}_{\rho_i\otimes\rho_j},\mathcal{X}_{\rho_k}\rangle\neq 0$ if and only if
-\begin{equation*}
-    \begin{cases}
-        k = i+j \text{ or}\\
-        k = j-i.
-    \end{cases}.
-\end{equation*}
-Therefore, based on Definition~\ref{def:Bispectrum}, by utilizing $\beta_{\rho_i,\rho_j}$, $\mathcal{F}(\Theta)_{\rho_i}$ and $\mathcal{F}(\Theta)_{\rho_j}$, we can compute  $\mathcal{F}(\Theta)_{\rho_k}$ for $k\in\{i+j,\ j-i\}$. By iterating, if $\mathcal{F}(\Theta)_{\rho_1}$ is known,  $\beta_{\rho_1,\rho_1}$ gives $\mathcal{F}(\Theta)_{\rho_2}$. Then, $\beta_{\rho_1,\rho_2}$ can be leveraged to obtain  $\mathcal{F}(\Theta)_{\rho_3}$. Continuing the procedure provides $\mathcal{F}(\Theta)_{\rho_k}$ for $k=2,...,\lfloor \frac{n-1}{2}\rfloor$ by using $\beta_{\rho_1,\rho_{k-1}}$. We have thus obtained the Fourier coefficients associated to all the 2D irreps.
-
-It remains to show that the iteration also recovered the Fourier coefficients associated to the 1D irreps. This is because $\rho_{01}$ is in the TD of $\rho_1\otimes\rho_1$ and, for $n$ even, $\rho_{02}, \rho_{03}$ are in the TD of $\rho_1\otimes\rho_{\frac{n}{2}-1}$. Indeed,
-\begin{align*}
-    \langle \mathcal{X}_{\rho_1\otimes\rho_1},\mathcal{X}_{\rho_{01}}\rangle &=\frac{1}{2n}\sum_{l=1}^n \mathcal{X}_{\rho_1\otimes\rho_1}(a^l)\mathcal{X}_{\rho_{01}}(a^l)\\
-    &=\frac{1}{2n}\sum_{l=1}^n 4\cos(\omega_l)^2\\
-    &=\frac{1}{n}\sum_{l=1}^n 1+\cos(2\omega_l)=1.
-\end{align*}
-Moreover, for $n$ even and $\rho\in\{\rho_{02}, \rho_{03}\}$, we have
-\begin{align*}
-    \langle \mathcal{X}_{\rho_1\otimes\rho_{\frac{n}{2}-1}},\mathcal{X}_\rho\rangle &=\frac{1}{2n}\sum_{l=1}^n \mathcal{X}_{\rho_1\otimes\rho_{\frac{n}{2}-1}}(a^l)\mathcal{X}_\rho(a^l)\\
-    &=\frac{1}{n}\sum_{l=1}^n 1+\cos\left(\omega_l\left(\frac{n}{2}-2\right)\right)(-1)^l\\
-    &=1.
-\end{align*}
-In conclusion, the procedure of Algorithm \ref{alg:dihedral_alg} recovers all the Fourier coefficients and the selective $G$-Bispectrum.
-
-
-\subsection{The Clebsch-Gordan matrices on $D_n$}\label{app:clebsch_gordan}
-The matrix algebra properties that we use in this subsection can be found, e.g., in \citep{bhatia97}. Recall from Theorem~\ref{thm:commutative_Bispectrum} the (implicit) definition of the Clebsch-Gordan matrices:
-\begin{equation}\label{eq:cleb_gordan}
-    (\rho_1\otimes\rho_2)(g) = C_{\rho_1,\rho_2}\left[ \bigoplus_{\rho\in\mathcal{R}} \rho(g) \right]C_{\rho_1,\rho_2}^\dagger,
-\end{equation}
-where $C_{\rho_1,\rho_2}^\dagger C_{\rho_1,\rho_2}=I$.
-We only consider the case of $\rho_1,\rho_2$ both 2d irreps of $D_n$ since otherwise, the Clebsch-Gordan matrix is the scalar $1$.
-First notice that $(\rho_1\otimes\rho_2)(g)$, is an orthonormal matrix. Indeed, using the properties of the Kronecker product, we obtain (``$(g)$'' omitted for clarity):
-\begin{align*}
-    (\rho_1\otimes\rho_2)(\rho_1\otimes\rho_2)^\dagger &= (\rho_1\otimes\rho_2)(\rho_1^\dagger\otimes\rho_2^\dagger)\\
-    &=(\rho_1\rho_1^\dagger)\otimes (\rho_2\rho_2^\dagger)\\
-    &=I \otimes I = I
-\end{align*}
-For $Q$ a real orthogonal matrix ($Q^T Q = I$), and $VSV^T$ with $V^T V = I$, a real Schur decomposition of $Q$, it is known that $S$ is block diagonal with blocks of size $1\times 1$ or $2\times 2$. These blocks are themselves orthogonal matrices. Therefore, the real Schur decomposition is the decomposition in~\eqref{eq:cleb_gordan} up to permutations. In order for $S$ to represent exactly the irreps from~\eqref{eq:2d_irreps_dn}, the non-zero sub-diagonal elements should all be positive. If not, the symmetric element is positive and a permutation $P$ must be added to exchange their positions: $Q = (VP)(P^TSP)(VP)^T$. $P$ permutes the two columns of $V$ associated with the permuted $2\times 2$ block of $S$.
-\begin{algorithm}
-    \caption{Compute Clebsch-Gordan matrices on $D_n$}
-    \begin{algorithmic}[1]
-        \STATE \textbf{Input:} $\rho_1, \rho_2$, two 2d irreps of $D_n$.
-        \STATE Pick any $g\in D_n$, e.g., $g = a$.
-        \STATE Compute  $(\rho_1\otimes\rho_2)(g) = (VP)(P^TSP)(VP)^T$, a valid real Schur form.
-        \STATE Set $C_{\rho_1,\rho_2} = VP$.
-        \STATE Set $\bigoplus_{\rho\in\mathcal{R}} \rho(g) = P^TSP$
-        \STATE \textbf{Return:} $C_{\rho_1,\rho_2}, \bigoplus_{\rho\in\mathcal{R}}\rho(g)$.
-    \end{algorithmic}
-\end{algorithm}
-
-
-
-
-\section{Bispectrum inversion for octahedral and full octahedral groups}\label{app:octahedral}
-We provide a sketch of the procedure to retrieve $\mathcal{F}(\Theta)$ given $\beta(\Theta)$ for the \href{https://en.wikipedia.org/wiki/Octahedral_symmetry}{octahedral group} and the \href{https://en.wikiversity.org/wiki/Full_octahedral_group}{full octahedral group}. These two groups are available in the \href{https://quva-lab.github.io/escnn/api/escnn.group.html}{\texttt{escnn} library}. These groups are easier to deal with than the cyclic and dihedral groups presented in the paper, given that they do not come from a \textit{family} of groups. Indeed, our proofs for the cyclic (resp. dihedral) groups needed to work for \textit{all} cyclic groups $C_N$, and for all dihedral groups $D_N$, for all $N$. The octahedral and full octahedral groups are only two groups.
-
-\subsection{Octahedral group}
-\begin{table}
-        \centering
-        \begin{tabular}{|c||c|c|c|c|c|}
-            \hline
-            $\otimes$ &$\rho_0$&$\rho_1$&$\rho_2$&$\rho_3$&$\rho_4$\\
-            \hline
-            \hline
-            $\rho_0$&10000 & 01000 & 00100 & 00010 & 00001 \\
-            \hline
-            $\rho_1$&01000 & 11110 & 01111 & 01100 & 00100 \\
-            \hline
-            $\rho_2$&00100 & 01111 & 11110 & 01100 & 01000 \\
-            \hline
-            $\rho_3$&00010 & 01100 & 01100 & 10011 & 00010 \\
-            \hline
-            $\rho_4$ &00001 & 00100 & 01000 & 00010 & 10000 \\
-            \hline
-        \end{tabular}
-        \caption{Kronecker table of the octahedral group using \texttt{escnn}. For the binary word at position $i,j$ in the table, the $k$th `letter' is $1$ if $\rho_k\in\rho_i\otimes \rho_j$, $0$ otherwise.}
-        \label{tab:octa}
-    \end{table}
-
-The octahedral group has 24 elements and 5 irreps. We can compute its Kronecker table, either manually using characters $\chi$ or using a Python package such as \texttt{escnn}. We give its Kronecker table below (Table~\ref{tab:octa}), where each column/row represents one irrep, labelled $\rho_0, \rho_1, ..., \rho_4$.
-
-We apply the procedure from Algorithms~\ref{alg:cyclic_alg}, \ref{alg:commutative_alg} and \ref{alg:dihedral_alg} to Table~\ref{tab:octa}. This procedure relies on the use of Theorem~\ref{def:Bispectrum}. We first select the bispectral coefficient $\beta_{\rho_0, \rho_0}$ (we omit ``$(\Theta)$'' for clarity) to get the component $\mathcal{F}_{\rho_0 }$ where $\rho_0$ is the trivial representation. Next, we choose $\beta_{\rho_0,\rho_1}$ and use $\mathcal{F}_{\rho_0}$  to obtain $\mathcal{F}_{\rho_1}$ (we know from Table~\ref{tab:octa} that $\rho_1\in\rho_0\otimes\rho_1$) up to an indeterminacy which is a transformation in $\mathrm{O}(3)$ and corresponds to the indeterminacy factor from Appendix~\ref{sec:indeterminacy}. Then, we select  $\beta_{\rho_1, \rho_1}$ to get the Fourier components $\mathcal{F}_{\rho_2}, \mathcal{F}_{\rho_3}$. Lastly, we select $\beta_{\rho_1, \rho_2}$ to get the missing Fourier component $\mathcal{F}_{\rho_4}$.
-
-In summary, we only need $4$ bispectral coefficients ($\beta_{\rho_0, \rho_0}, \beta_{\rho_1, \rho_0}, \beta_{\rho_1, \rho_1}, \beta_{\rho_1, \rho_2}$) instead of $5^2 = 25$ in order to get the five Fourier components, i.e., the full Fourier transform of the signal. In total, this involves $1+9+81+81=172$ scalar coefficients.
-\subsection{Full octahedral group}
-\begin{table*}[h]
-    \centering
-    \minuscule
-    \begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|}
-    \hline
-            $\otimes$ &$\rho_0$&$\rho_1$&$\rho_2$&$\rho_3$&$\rho_4$&$\rho_5$&$\rho_6$&$\rho_7$&$\rho_8$&$\rho_9$\\
-    \hline
-    $\rho_0$&1000000000 & 0100000000 & 0010000000 & 0001000000 & 0000100000 & 0000010000 & 0000001000 & 0000000100 & 0000000010 & 0000000001 \\
-    \hline
-    $\rho_1$&0100000000 & 1111000000 & 0111100000 & 0110000000 & 0010000000 & 0000001000 & 0000011110 & 0000001111 & 0000001100 & 0000000100 \\
-    \hline
-    $\rho_2$&0010000000 & 0111100000 & 1111000000 & 0110000000 & 0100000000 & 0000000100 & 0000001111 & 0000011110 & 0000001100 & 0000001000 \\
-    \hline
-    $\rho_3$&0001000000 & 0110000000 & 0110000000 & 1001100000 & 0001000000 & 0000000010 & 0000001100 & 0000001100 & 0000010011 & 0000000010 \\
-    \hline
-    $\rho_4$&0000100000 & 0010000000 & 0100000000 & 0001000000 & 1000000000 & 0000000001 & 0000000100 & 0000001000 & 0000000010 & 0000010000 \\
-    \hline
-    $\rho_5$&0000010000 & 0000001000 & 0000000100 & 0000000010 & 0000000001 & 1000000000 & 0100000000 & 0010000000 & 0001000000 & 0000100000 \\
-    \hline
-    $\rho_6$&0000001000 & 0000011110 & 0000001111 & 0000001100 & 0000000100 & 0100000000 & 1111000000 & 0111100000 & 0110000000 & 0010000000 \\
-    \hline
-    $\rho_7$&0000000100 & 0000001111 & 0000011110 & 0000001100 & 0000001000 & 0010000000 & 0111100000 & 1111000000 & 0110000000 & 0100000000 \\
-    \hline
-    $\rho_8$&0000000010 & 0000001100 & 0000001100 & 0000010011 & 0000000010 & 0001000000 & 0110000000 & 0110000000 & 1001100000 & 0001000000 \\
-    \hline
-    $\rho_9$&0000000001 & 0000000100 & 0000001000 & 0000000010 & 0000010000 & 0000100000 & 0010000000 & 0100000000 & 0001000000 & 1000000000\\
-    \hline
-    \end{tabular}
-    \caption{Kronecker table of  full octahedral group using \texttt{escnn}. For the binary word at position $i,j$ in the table, the $k$th `letter' is $1$ if $\rho_k\in\rho_i\otimes \rho_j$, $0$ otherwise.}
-    \label{tab:full_octa}
-\end{table*}
-The full octahedral group has $48$ elements and $10$ irreps. Again, we can compute its Kronecker table using a Python package such as \texttt{escnn}. We give its Kronecker table below (Table~\ref{tab:full_octa}), where each column/row represents one irrep, labelled $\rho_0, \rho_1, ..., \rho_{9}$.
-
-We apply the procedure from Algorithms~\ref{alg:cyclic_alg}, \ref{alg:commutative_alg} and \ref{alg:dihedral_alg} to Table~\ref{tab:full_octa}. Again, this procedure relies on the use of Theorem~\ref{def:Bispectrum}. $\beta_{\rho_0,\rho_0}$ (we omit ``$(\Theta)$'' for clarity) allows to compute $\mathcal{F}_{\rho_0}$ directly, such as in Algorithm~\ref{alg:dihedral_alg}. Then, from $\beta_{\rho_0,\rho_6}$, we obtain $\mathcal{F}_{\rho_6}$ up to an unknown group action in $\mathrm{O}(3)$. Then, using $\beta_{\rho_6,\rho_6}$ and $\mathcal{F}_{\rho_6}$, we obtain $\mathcal{F}_{\rho_1}$,$\mathcal{F}_{\rho_2}$ and $\mathcal{F}_{\rho_ 3}$. Next, leveraging $\beta_{\rho_1,\rho_2}$, $\mathcal{F}_{\rho_1}$ and $\mathcal{F}_{\rho_2}$, we obtain  $\mathcal{F}_{\rho_4}$. Using $\beta_{\rho_1,\rho_6}$, $\mathcal{F}_{\rho_1}$ and $\mathcal{F}_{\rho_6}$, we obtain  $\mathcal{F}_{\rho_5}$, $\mathcal{F}_{\rho_7}$, $\mathcal{F}_{\rho_8}$. Finally, with $\beta_{\rho_1,\rho_7}$, $\mathcal{F}_{\rho_1}$ and $\mathcal{F}_{\rho_7}$, we obtain the last coefficient $\mathcal{F}_{\rho_9}$. Hence we have recovered all the Fourier coefficients using only $\beta_{\rho_0,\rho_0}$, $\beta_{\rho_0,\rho_6}$, $\beta_{\rho_6,\rho_6}$, $\beta_{\rho_1,\rho_2}$, $\beta_{\rho_1,\rho_6}$, $\beta_{\rho_1,\rho_7}$, thus a total of $6$ bispectral coefficients instead of $100$. In terms of scalar values, this involves $1+9+81+81+81+81=334$ coefficients.
-
-
-\section{Training of the $G$-CNN architecture}\label{app:training}
-The $\mathrm{SO}(2)$-MNIST/EMNIST datasets are obtained after applying random planar rotations on each image of the datasets MNIST \citep{lecun2010mnist}, EMNIST \citep{cohen2017emnist} respectively. In the case of $\mathrm{O}(2)$-MNIST/EMNIST, in addition to a planar rotation, a reflection is applied with probability $\frac{1}{2}$.  The original size of each image is conserved. The size of the training sets are $60\ 000$ and $88\ 800$ for MNIST and EMNIST, respectively. %The size of the test sets are $10\ 000$ and $14\ 800$. For our experiments, $20\%$ of the training datasets are used for validation.
-%\subsection{$C_8$-CNNs on $\mathrm{SO}(2)$-MNIST/EMNIST/CIFAR-10}
-
-We conserve the architecture of \citep{sanborn2023general}. For all invariant layers, being the $G$-TC, the selective $G$-Bispectrum and the Avg/Max $G$-pooling, the architecture is composed of a $C_8/D_8$-convolutional block with $K$ filters (see Table~\ref{tab:classification}). Then, the invariant layer is applied before feeding the output to a MLP. The MLP is composed of $3$ fully-connected layers with ReLU non-linearity. A final fully-connected linear layer is applied for classification. The vector of the output sizes of these layers is given by $[o1,o2,o3,ol]$ respectively. $o2=o3 = 64$ and $ol$ is equal to the number of classes of the dataset. $o1$ is tuned to reach the parameter count from Table~\ref{tab:classification}.
-\begin{comment}
-\newpage
-
-\section*{NeurIPS Paper Checklist}
-\begin{enumerate}
-
-\item {\bf Claims}
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-    \item[] Answer: \answerYes{} % Replace by \answerYes{}, \answerNo{}, or \answerNA{}.
-    \item[] Justification: The claims that we make in the abstract, i.e., theoretical and practical guarantees of the $G$-Bispectrum are respectively  proven and studied in Sections \ref{sec:completeness} and \ref{sec:experiments}.%\justificationTODO{}
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-    \item[] Answer: \answerYes{} % Replace by \answerYes{}, \answerNo{}, or \answerNA{}.
-    \item[] Justification: This paper is not involved with specific ethical issues, except that it contributes to the field of image processing. The latter field is related to the ethical issue of face-recognition and automatic surveillance.%\justificationTODO{}
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-\item {\bf Broader Impacts}
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-    \item[] Answer: \answerNA{} % Replace by \answerYes{}, \answerNo{}, or \answerNA{}.
-    \item[] Justification: This work is a research work to propose a new layer in the architecture of $G$-CNNS. It has no direct societal impacts beyond the fact that it contributes to image-processing, which can be used for malicious purposes. %\justificationTODO{}
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diff --git a/docs/neurips_2026.sty b/docs/neurips_2026.sty
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-% partial rewrite of the LaTeX2e package for submissions to the
-% Conference on Neural Information Processing Systems (NeurIPS):
-%
-% - uses more LaTeX conventions
-% - line numbers at submission time replaced with aligned numbers from
-%   lineno package
-% - \nipsfinalcopy replaced with [final] package option
-% - automatically loads times package for authors
-% - loads natbib automatically; this can be suppressed with the
-%   [nonatbib] package option
-% - adds foot line to first page identifying the conference
-% - adds preprint option for submission to e.g. arXiv
-% - conference acronym modified
-% - update foot line to display the track name
-%
-% Roman Garnett (garnett@wustl.edu) and the many authors of
-% nips15submit_e.sty, including MK and drstrip@sandia
-%
-% last revision: January 2026
-
-\NeedsTeXFormat{LaTeX2e}
-\ProvidesPackage{neurips_2026}[2026-01-29 NeurIPS 2026 submission/camera-ready style file]
-
-% declare final option, which creates camera-ready copy
-\newif\if@neuripsfinal\@neuripsfinalfalse
-\DeclareOption{final}{
-  \@neuripsfinaltrue
-  \@anonymousfalse
-}
-
-% declare nonatbib option, which does not load natbib in case of
-% package clash (users can pass options to natbib via
-% \PassOptionsToPackage)
-\newif\if@natbib\@natbibtrue
-\DeclareOption{nonatbib}{
-  \@natbibfalse
-}
-
-% declare preprint option, which creates a preprint version ready for
-% upload to, e.g., arXiv
-\newif\if@preprint\@preprintfalse
-\DeclareOption{preprint}{
-  \@preprinttrue
-  \@anonymousfalse
-}
-
-% determine the track of the paper in camera-ready mode
-\newif\if@main\@maintrue
-\DeclareOption{main}{
-  \@maintrue
-  \newcommand{\@trackname}{\@neuripsordinal\ Conference on Neural Information Processing Systems (NeurIPS \@neuripsyear).}
-}
-\newif\if@position\@positionfalse
-\DeclareOption{position}{
-  \@positiontrue
-  \newcommand{\@trackname}{\@neuripsordinal\ Conference on Neural Information Processing Systems (NeurIPS \@neuripsyear). Position Paper Track.}
-}
-\newif\if@eandd\@eanddfalse
-\DeclareOption{eandd}{
-  \@eanddtrue
-\if@neuripsfinal\@anonymousfalse\else\if@preprint\@anonymousfalse\else\@anonymoustrue\fi\fi
-  \newcommand{\@trackname}{\@neuripsordinal\ Conference on Neural Information Processing Systems (NeurIPS \@neuripsyear). Track on Evaluations and Datasets.} 
-}
-\newif\if@creativeai\@creativeaifalse
-\DeclareOption{creativeai}{
-  \@creativeaitrue
-  \@anonymousfalse
-  \newcommand{\@trackname}{\@neuripsordinal\ Conference on Neural Information Processing Systems (NeurIPS \@neuripsyear). Creative AI Track.}
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-% For anonymous or non-anonymous
-\newif\if@anonymous\@anonymoustrue
-
-% For workshop papers
-\newcommand{\@workshoptitle}{}
-\newcommand{\workshoptitle}[1]{\renewcommand{\@workshoptitle}{#1}}
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diff --git a/docs/neurips_steerable_bispectrum_plan.md b/docs/neurips_steerable_bispectrum_plan.md
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-# Paper Plan: The Steerable Bispectrum for Non-Abelian Groups
-
-**Target venue**: NeurIPS 2026
-**Working title**: *Bispectral Invariants from Steerable Convolutions: Bridging Equivariant Networks and Complete Invariant Theory*
-
-## 1. One-Paragraph Pitch
-
-Steerable CNNs (Weiler & Cesa 2019) achieve equivariance by design but rely on ad-hoc nonlinearities (norms, gates) to extract invariant features — losing information. The bispectrum (Kakarala 1992, Mataigne et al. 2024) provides a mathematically complete invariant with reconstruction guarantees — but has only been used with proper Fourier coefficients, not inside learned architectures. For abelian groups (SO(2)), combining the two is trivial: the Depeursinge group (Oreiller et al. 2020, 2022) showed this works as a CNN layer. For non-abelian groups (D_n, O), nobody has done it — because steerable CNN outputs live in an intertwiner space whose dimension L doesn't match the number of irreps M, so the standard bispectrum formula doesn't apply. We solve this problem.
-
-## 2. The Problem
-
-### 2.1 What exists
-
-**Steerable CNNs** decompose filters on an intertwiner basis for a group H acting on R²:
-$$\\Psi(x) = \\sum\_{l=1}^L \\hat{w}^{(l)} Y_l(x)$$
-where Y_l ∈ Hom^H(π, ρ). The convolution output is:
-$$\\Theta(x_0) = \\sum\_{l=1}^L \\hat{\\Theta}\_l \\cdot Y_l(x_0)$$
-
-For abelian H = SO(2): the basis Y_l = e^{ilθ} is indexed by irreps, so L = M and Θ̂_l ARE the Fourier coefficients. The bispectrum β\_{k1,k2} = Θ̂\_{k1} · Θ̂\_{k2} · Θ̂\*\_{k1+k2} works directly. This is what the Depeursinge group implements.
-
-For non-abelian H (e.g. D₄): the intertwiner basis has L = dim Hom^H(π, ρ₀) elements. For D₄ with ρ₀ = trivial: L = 3. But D₄ has M = 5 irreps. **L ≠ M**. The Θ̂_l are NOT Fourier coefficients. You cannot plug them into the bispectrum formula.
-
-### 2.2 What's missing
-
-A bispectral invariant that can be computed from the L intertwiner coefficients {Θ̂_l} of a steerable convolution output, for non-abelian H, with:
-
-- Proven invariance to the action of H
-- Completeness guarantees (or characterization of what information is lost)
-- Selectivity (minimal number of coefficients)
-- A practical CNN layer implementation
-
-## 3. Proposed Contributions
-
-### Contribution A: Bispectrum from Intertwiner Coefficients (Theory)
-
-**Approach 1 — Lift to Fourier coefficients**: Express the L intertwiner coefficients in terms of the M Fourier coefficients. Since the intertwiner basis Y_l can be decomposed on the irreps of H, there exists a (possibly rectangular) matrix A such that:
-$$\\hat{f}_\\rho = A_\\rho \\cdot [\\hat{\\Theta}\_1, ..., \\hat{\\Theta}\_L]^T$$
-When L < M, this is underdetermined — some Fourier components are not recoverable from the intertwiner coefficients. The bispectrum computed from the recoverable components gives a **partial bispectrum** that is still H-invariant but may not be complete.
-
-**What to prove**:
-
-- (P1) Characterize the kernel of the map Θ̂ → F̂: which Fourier components are invisible to the steerable convolution with trivial output representation ρ₀?
-- (P2) Show that choosing different output representations ρ (not just trivial) in different layers gives access to different Fourier components — and that a sufficient set of ρ's recovers all M components.
-- (P3) Derive the bispectrum formula in terms of {Θ̂_l^{(ρ)}} for multiple output representations ρ.
-
-**Approach 2 — Direct triple correlation** (already started in our tex, lines 2043–2174): Define a triple correlation directly for vector-valued signals with the induced representation action, prove invariance (done in tex), compute its Fourier transform to get the bispectrum (done in tex, lines 2194–2312). The result already has the standard form:
-$$\\beta(f)_{\\rho_1, \\rho_2} = \[\\mathcal{F}(f)_{\\rho_1} \\otimes \\mathcal{F}(f)_{\\rho_2}\] \\cdot C_{\\rho_1,\\rho_2} \\bigoplus\_{\\rho \\in \\rho_1 \\otimes \\rho_2} \\mathcal{F}(f)_\\rho^\\dagger \\cdot C_{\\rho_1,\\rho_2}^\\dagger$$
-
-**What to prove**:
-
-- (P4) Show that F(f)\_ρ can be computed from the steerable convolution output by appropriately choosing the output representation ρ — i.e., that for each irrep ρ of H, running a steerable conv with output representation ρ yields F(f)\_ρ directly.
-- (P5) If so, the standard bispectrum + selectivity (Mataigne et al. 2024) applies immediately — we just need multiple "heads" in the conv layer, one per irrep of H.
-
-### Contribution B: Selective Steerable Bispectrum (Theory)
-
-Once Contribution A establishes which bispectral formula to use, apply the selectivity results from Mataigne et al. 2024:
-
-- For D_n: selective bispectrum needs ⌊(n-1)/2⌋ + 2 matrix-valued coefficients instead of O(|D_n|²)
-- For O (octahedral): 4 matrix-valued coefficients
-
-**What to prove**:
-
-- (P6) The selectivity theorem (Mataigne et al. 2024, Theorem 4.3) applies to the steerable bispectrum, provided the Fourier coefficients are correctly extracted from the steerable conv output.
-- (P7) Derive the per-pixel output channel count for the selective steerable bispectrum as a function of the group H. Compare with naive (full) bispectrum channel count.
-
-### Contribution C: SteerableBispectrum Layer (Implementation)
-
-A PyTorch `nn.Module` that:
-
-1. Takes feature maps from a steerable conv layer (e.g., from e2cnn)
-2. Extracts Fourier coefficients from the steerable representation
-3. Computes the selective bispectrum using precomputed CG matrices
-4. Outputs H-invariant feature maps
-
-Interface sketch:
-
-```python
-class SteerableBispectrum(nn.Module):
-    def __init__(self, group: str, in_type, selective: bool = True):
-        """
-        Args:
-            group: "C4", "D4", "SO2", etc.
-            in_type: e2cnn FieldType describing input representation
-            selective: use selective O(|H|) or full O(|H|²) bispectrum
-        """
-
-    def forward(self, x: GeometricTensor) -> torch.Tensor:
-        """
-        Returns: (batch, H, W, n_bispectral_coeffs) invariant feature maps
-        """
-```
-
-This should interoperate with `e2cnn` / `escnn` for the equivariant conv layers, and use our existing bispectrum library (`DnonDn`, `OctaonOcta`) for the bispectral computation.
-
-## 4. Proofs Needed (Summary)
-
-| ID  | Statement                                                | Difficulty                | Status                                     |
-| --- | -------------------------------------------------------- | ------------------------- | ------------------------------------------ |
-| P1  | Kernel of intertwiner → Fourier map for ρ₀               | Medium                    | Open                                       |
-| P2  | Multiple ρ's recover all Fourier components              | Medium                    | Open (conjecture from rep theory)          |
-| P3  | Bispectrum formula in terms of multi-ρ Θ̂                 | Medium                    | Open                                       |
-| P4  | Steerable conv with output ρ yields F(f)\_ρ              | Key result                | Partially done in tex (needs verification) |
-| P5  | Standard bispectrum applies to multi-head steerable conv | Easy (follows from P4)    | Open                                       |
-| P6  | Selectivity applies to steerable setting                 | Easy (follows from P4+P5) | Open                                       |
-| P7  | Channel count comparison table                           | Easy (combinatorics)      | Open                                       |
-
-**Critical path**: P4 → P5 → P6. If P4 holds, everything else follows from existing results.
-
-## 5. Experiments
-
-### Experiment 1: Rotation-Invariant Classification (CIFAR-10/100 rotated)
-
-**Setup**: Compare on rotated CIFAR-10/100:
-
-- Baseline 1: Standard CNN + data augmentation
-- Baseline 2: Steerable CNN (e2cnn) with norm nonlinearity (Weiler & Cesa 2019)
-- Baseline 3: Steerable CNN with SO(2) bispectrum nonlinearity (Oreiller et al. 2020)
-- **Ours**: Steerable CNN with D_n / D₄ selective bispectrum nonlinearity
-
-**What we expect**: The D₄ bispectrum preserves more information than SO(2) (handles reflections), and is more efficient than the norm nonlinearity (which discards phase entirely). The selective version should match or beat the full bispectrum with fewer channels.
-
-**Why it's impactful**: Direct comparison of invariant pooling strategies on a standard benchmark. Nobody has compared bispectrum vs norm/gate nonlinearities in steerable CNNs head-to-head.
-
-### Experiment 2: Medical Image Segmentation (MoNuSeg / DRIVE)
-
-**Setup**: Reproduce the Oreiller et al. 2022 experiments with:
-
-- Their SO(2) bispectral U-Net (BCHConv2D baseline)
-- Our D₄ selective bispectral U-Net
-- Standard equivariant U-Net with norm nonlinearity
-
-**What we expect**: D₄ handles reflections (common in histopathology — cells have no preferred chirality), giving better rotation+reflection robustness than SO(2)-only bispectrum.
-
-**Why it's impactful**: Direct improvement on their published results, on their own benchmark, using the same architecture but with a stronger invariant.
-
-### Experiment 3: Scaling with Group Size / Angular Resolution
-
-**Setup**: Measure feature map memory and inference time as a function of:
-
-- Number of harmonics n (for SO(2) case: n = 4, 8, 16, 32, 64)
-- Group order |H| (for finite groups: C₄, C₈, D₄, D₈, D₁₆)
-
-Compare: full bispectrum O(|H|²) vs selective bispectrum O(|H|) vs norm O(|H|) channel counts.
-
-**What we expect**: Selective bispectrum is the only option that is both complete (unlike norm) and tractable (unlike full bispectrum) at large |H|.
-
-**Why it's impactful**: Shows the selectivity result has practical consequences for steerable CNN design. Makes the case that selective bispectrum is the right nonlinearity for high-resolution equivariant networks.
-
-## 6. Paper Outline
-
-1. **Introduction** (1 page)
-
-   - Steerable CNNs achieve equivariance but invariant extraction is a bottleneck
-   - Bispectrum provides complete invariants but only for proper Fourier coefficients
-   - We bridge the gap for non-abelian groups
-
-2. **Background** (1.5 pages)
-
-   - Steerable CNNs, intertwiners, kernel constraint (Weiler & Cesa 2019)
-   - G-Bispectrum, selectivity, inversion (Mataigne et al. 2024)
-   - Steerable bispectrum for SO(2) (Oreiller et al. 2020)
-
-3. **The Steerable Bispectrum for Non-Abelian Groups** (3 pages)
-
-   - The L ≠ M problem (Section 3.1)
-   - Multi-representation extraction of Fourier coefficients (Section 3.2, Proofs P1–P4)
-   - Bispectrum formula from steerable outputs (Section 3.3, Proof P5)
-   - Selective steerable bispectrum (Section 3.4, Proofs P6–P7)
-
-4. **Implementation** (1 page)
-
-   - SteerableBispectrum layer
-   - Integration with e2cnn/escnn
-   - Channel count analysis
-
-5. **Experiments** (2 pages)
-
-   - Rotated CIFAR classification
-   - Medical image segmentation
-   - Scaling analysis
-
-6. **Conclusion** (0.5 pages)
-
-Total: ~9 pages + references + appendix (NeurIPS format)
-
-## 7. Bibliography
-
-### Foundational
-
-- Kakarala (1992). *Triple Correlation on Groups*. PhD thesis, UC Irvine.
-- Kakarala (2012). *The Bispectrum as a Source of Phase-Sensitive Invariants for Fourier Descriptors: A Group-Theoretic Approach*. JMIV 44.
-- Freeman & Adelson (1991). *The Design and Use of Steerable Filters*. IEEE TPAMI 13(9).
-
-### Steerable / Equivariant CNNs
-
-- Cohen & Welling (2016). *Group Equivariant Convolutional Networks*. ICML 2016.
-- Worrall et al. (2017). *Harmonic Networks: Deep Translation and Rotation Equivariance*. CVPR 2017.
-- Weiler et al. (2018). *Learning Steerable Filters for Rotation Equivariant CNNs*. CVPR 2018.
-- Weiler & Cesa (2019). *General E(2)-Equivariant Steerable CNNs*. NeurIPS 2019.
-- Thomas et al. (2018). *Tensor Field Networks*. arXiv:1802.08219.
-- Cesa, Lang, Weiler (2022). *A Program to Build E(N)-Equivariant Steerable CNNs*. ICLR 2022.
-
-### Steerable Bispectrum (Depeursinge group)
-
-- Andrearczyk, Oreiller, Depeursinge (2019). *Exploring Local Rotation Invariance in 3D CNNs with Steerable Filters*. MIDL 2019.
-- Oreiller et al. (2020). *3D Solid Spherical Bispectrum CNNs for Biomedical Texture Analysis*. arXiv:2004.13371.
-- Oreiller et al. (2022). *Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral U-Net*. MIDL 2022.
-
-### Selective Bispectrum
-
-- Mataigne et al. (2024). *The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks*. NeurIPS 2024.
-- Myers & Miolane (2025). *The Selective Disk Bispectrum and Its Inversion*. arXiv:2511.19706.
-
-### Nonlinearities in Equivariant Networks
-
-- Kondor & Trivedi (2018). *On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups*. ICML 2018.
-- Lang & Weiler (2020). *A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels*. ICLR 2021.
-- Brandstetter et al. (2022). *Geometric and Physical Quantities Improve E(3) Equivariant Message Passing*. ICLR 2022.
-
-## 8. Timeline (Speculative)
-
-| Phase                            | Duration         | Deliverable                                                |
-| -------------------------------- | ---------------- | ---------------------------------------------------------- |
-| Theory: P4 proof (critical path) | 3–4 weeks        | Theorem + proof that steerable conv → Fourier coefficients |
-| Theory: P5–P7 (follow from P4)   | 1–2 weeks        | Selectivity theorem for steerable setting                  |
-| Code: SteerableBispectrum layer  | 2–3 weeks        | PyTorch module + e2cnn integration                         |
-| Experiments: CIFAR + MoNuSeg     | 3–4 weeks        | Tables + figures                                           |
-| Writing                          | 2–3 weeks        | Full draft                                                 |
-| **Total**                        | **~12–16 weeks** |                                                            |
-
-## 9. Risk Assessment
-
-**Main risk**: P4 might not hold as stated. The steerable conv with output representation ρ might not directly yield the Fourier coefficient F(f)\_ρ — there could be a mixing between spatial (Z²) and fiber (H) components that prevents clean extraction.
-
-**Mitigation**: Even if full Fourier recovery is impossible from a single conv layer, a multi-layer extraction (stack of steerable convs with different ρ's) might work. Alternatively, the partial bispectrum from the recoverable components might already be practically useful — test empirically.
-
-**Fallback**: If the non-abelian theory doesn't close, the paper can pivot to contributions B+C only: the selective steerable bispectrum for SO(2) with Fourier-Bessel basis (our SO2onD2 as a conv layer), benchmarked against the Depeursinge group's approach. This is less novel but still publishable.
diff --git a/docs/octa_on_octa_implementation_notes.md b/docs/octa_on_octa_implementation_notes.md
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-# OctaonOcta Implementation Notes
-
-## Overview
-
-`OctaonOcta` implements the selective G-bispectrum for the octahedral group O (the rotation symmetry group of the cube, |O| = 24), following Algorithm 3 of Mataigne et al. (2024).
-
-## Group Data
-
-The 24 rotation matrices of O are generated from two generators via BFS:
-
-- `R_z(π/2)`: 90° rotation around the z-axis
-- `R_{[1,1,1]}(2π/3)`: 120° rotation around the [1,1,1] body diagonal
-
-The elements are sorted lexicographically (flattened 3×3 matrix entries) for a canonical ordering. The identity element ends up at index 23 in this ordering, not index 0.
-
-All group data (elements, Cayley table, inverse map, Kronecker table) was generated using `escnn` and cross-verified against the analytical construction.
-
-## Irreducible Representations
-
-O has 5 real irreps with dimensions (1, 3, 3, 2, 1):
-
-| Irrep   | Dim | Construction                                  |
-| ------- | --- | --------------------------------------------- |
-| ρ₀ (A₁) | 1   | Trivial: ρ₀(g) = 1                            |
-| ρ₁ (T₁) | 3   | Standard: ρ₁(g) = the 3×3 rotation matrix     |
-| ρ₂ (T₂) | 3   | Product: ρ₂(g) = ρ₄(g) · ρ₁(g)                |
-| ρ₃ (E)  | 2   | Hardcoded from escnn (involves ±1/2, ±√3/2)   |
-| ρ₄ (A₂) | 1   | Alternating sign of body-diagonal permutation |
-
-ρ₃ is the only irrep that requires non-integer matrix entries. The 24 matrices are hardcoded as exact rational/irrational values.
-
-## Clebsch-Gordan Matrices
-
-CG matrices are computed via the projection operator method:
-
-```
-P_k = (d_k / |G|) Σ_g χ_k(g)* (ρ_i(g) ⊗ ρ_j(g))
-```
-
-The eigenvectors of P_k (with eigenvalue 1) form the columns of C corresponding to irrep k. The full CG matrix C is assembled by concatenating these columns in order.
-
-Two CG matrices are needed for the selective bispectrum:
-
-- C₁₁ for ρ₁ ⊗ ρ₁ = ρ₀ ⊕ ρ₁ ⊕ ρ₂ ⊕ ρ₃ (9×9)
-- C₁₂ for ρ₁ ⊗ ρ₂ = ρ₁ ⊕ ρ₂ ⊕ ρ₃ ⊕ ρ₄ (9×9)
-
-## DFT Convention
-
-The group Fourier transform uses the left-multiplication convention:
-
-```
-F(ρ_k) = Σ_g f(g) · ρ_k(g)
-```
-
-Under the group action T_h f(g) = f(h⁻¹g), this gives:
-
-```
-F(T_h f)(ρ_k) = ρ_k(h) · F(f)(ρ_k)
-```
-
-The inverse DFT is:
-
-```
-f(g) = (1/|G|) Σ_k d_k · tr(ρ_k(g)^T · F_k)
-```
-
-## Forward Pass
-
-The selective bispectrum computes 4 matrix coefficients (172 scalars total):
-
-| Pair (i,j) | Scalars | Formula                     |
-| ---------- | ------- | --------------------------- |
-| (ρ₀,ρ₀)    | 1       | F₀³                         |
-| (ρ₀,ρ₁)    | 9       | F₀ · F₁ᵀF₁                  |
-| (ρ₁,ρ₁)    | 81      | C₁₁ (⊕ Fₖᵀ) C₁₁ᵀ (F₁ ⊗ F₁)  |
-| (ρ₁,ρ₂)    | 81      | C₁₂ (⊕ F'ₖᵀ) C₁₂ᵀ (F₁ ⊗ F₂) |
-
-## Inversion: The SO(3) Phase Problem
-
-### Why bootstrap fails for O
-
-For D_n, the bootstrap inversion works because:
-
-1. F₁ is recovered from F₁ᵀF₁ via symmetric square root (up to an O(2) rotation Q)
-2. C^T(Q⊗Q)C is block-diagonal for ALL Q ∈ SO(2), since the CG decomposition of 2D irreps respects the continuous group SO(2)
-
-For O, this fails because:
-
-1. F₁ is 3×3, and the symmetric sqrt gives F₁ up to an SO(3) rotation Q
-2. C^T(Q⊗Q)C is block-diagonal only for Q ∈ O (the discrete group), NOT for general Q ∈ SO(3)
-3. The l=2 sector (ρ₂ and ρ₃) mixes under general SO(3) rotations, causing cross-talk between irreps during extraction
-
-This is a fundamental mathematical obstruction, not an implementation bug.
-
-### What we verified
-
-From the extraction R = C^T β (S⊗S)⁻¹ C (using S = symmetric sqrt of F₁ᵀF₁):
-
-- R R^T IS block-diagonal → correctly gives F_k^T F_k for all k
-- The cross-block ratio A⁻¹B (where D² = \[[A,B],[C,D₂₂]\]) depends only on Q
-- P' = D₂₂⁻¹C = -P^T (a consistency check, not new information)
-- The reconstruction D² from P has 3 free parameters (R_A ∈ O(3), R_Q ∈ O(2)), matching the 3 DOF of SO(3)
-
-### Implemented approach: Bootstrap + Levenberg-Marquardt
-
-1. **Bootstrap initialization**: Recover F₀ exactly (cube root). Recover F₁ᵀF₁ exactly, take symmetric sqrt. Extract F_k^T F_k from R R^T, take symmetric sqrts. This gives exact Fourier *norms* but approximate *phases*.
-
-2. **Levenberg-Marquardt corrections**: Each step computes the Jacobian J of the bispectrum w.r.t. the signal (172 × 24 matrix, via forward-mode AD), then solves (J^T J + μI)⁻¹ J^T r with adaptive damping μ. This is a single linear solve per step, not iterative optimization.
-
-3. **Multi-start**: The bootstrap phases can land in different basins of attraction. Multiple restarts with randomised Fourier phases (random orthogonal rotations of each F_k) ensure convergence.
-
-Default: 10 LM steps × 4 restarts.
-
-## Known Limitations
-
-1. **Inversion speed**: ~2s per sample due to per-sample Jacobian computation. Could be improved with batched Jacobian or analytical Jacobian formulas.
-
-2. **Inversion accuracy**: Some signals require more restarts to converge. The multi-start mechanism handles this but at the cost of increased computation.
-
-3. **Full bispectrum**: `selective=False` is not yet implemented for the forward pass.
-
-4. **Differentiability**: The current `invert` implementation uses `detach()` internally and is not differentiable w.r.t. the input beta. An unrolled-differentiable version would require removing the detach calls and using `create_graph=True` in the Jacobian computation.
-
-## References
-
-- Mataigne, S., Keriven, N., & Peyré, G. (2024). The Selective G-Bispectrum and its Inversion. NeurIPS 2024.
-- The escnn library (for verification of group data and irrep matrices).
diff --git a/docs/octa_on_octa_slides.pdf b/docs/octa_on_octa_slides.pdf
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-\documentclass[aspectratio=169]{beamer}
-\usetheme{metropolis}
-\usepackage{booktabs}
-\usepackage{amsmath,amssymb}
-\usepackage{listings}
-\usepackage{tikz}
-\usetikzlibrary{matrix,positioning,arrows.meta}
-
-\lstset{
-  basicstyle=\ttfamily\scriptsize,
-  keywordstyle=\color{blue},
-  commentstyle=\color{gray},
-  breaklines=true,
-  frame=none,
-}
-
-\title{OctaonOcta: Selective Bispectrum for the Octahedral Group}
-\subtitle{Implementing the $O$-Bispectrum and solving the SO(3) inversion problem}
-\author{Johan Mathe}
-\date{\today}
-
-\begin{document}
-
-\maketitle
-
-% ============================================================
-\begin{frame}{The octahedral group $O$}
-  $O$ is the group of 24 rotational symmetries of the cube.
-
-  \medskip
-
-  \begin{columns}[T]
-    \begin{column}{0.55\textwidth}
-      \textbf{Generators}:
-      \begin{itemize}
-        \item $R_z(\pi/2)$: face rotation
-        \item $R_{[1,1,1]}(2\pi/3)$: vertex rotation
-      \end{itemize}
-
-      \bigskip
-
-      \textbf{5 real irreps} (dims 1, 3, 3, 2, 1):
-      \begin{itemize}
-        \item $\rho_0$ ($A_1$): trivial
-        \item $\rho_1$ ($T_1$): standard 3D rotation matrices
-        \item $\rho_2$ ($T_2$): $\rho_4(g) \cdot \rho_1(g)$
-        \item $\rho_3$ ($E$): 2D, involves $\pm\tfrac{1}{2}$, $\pm\tfrac{\sqrt{3}}{2}$
-        \item $\rho_4$ ($A_2$): alternating (parity of body-diagonal permutation)
-      \end{itemize}
-
-      \bigskip
-
-      Check: $1^2 + 3^2 + 3^2 + 2^2 + 1^2 = 24 = |O|$ \quad\checkmark
-    \end{column}
-    \begin{column}{0.42\textwidth}
-      \textbf{Selective bispectrum} (App.~B, Mataigne et al.):
-
-      \medskip
-
-      4 matrix coefficients:
-
-      \smallskip
-
-      \begin{tabular}{lcc}
-        \toprule
-        Pair & Size & Scalars \\
-        \midrule
-        $\beta_{\rho_0,\rho_0}$ & $1\times 1$ & 1 \\
-        $\beta_{\rho_0,\rho_1}$ & $3\times 3$ & 9 \\
-        $\beta_{\rho_1,\rho_1}$ & $9\times 9$ & 81 \\
-        $\beta_{\rho_1,\rho_2}$ & $9\times 9$ & 81 \\
-        \midrule
-        \textbf{Total} & & \textbf{172} \\
-        \bottomrule
-      \end{tabular}
-
-      \bigskip
-
-      Compare: full bispectrum $= O(|O|^2) = 576$.
-    \end{column}
-  \end{columns}
-\end{frame}
-
-% ============================================================
-\begin{frame}{Development overview}
-  \begin{enumerate}
-    \item \textbf{Group data generation} --- BFS from generators, verified against \texttt{escnn}
-    \item \textbf{Irrep matrices} --- 3 constructed analytically, 1 from products, 1 hardcoded
-    \item \textbf{CG matrices} --- projection operator method (no scipy)
-    \item \textbf{Forward pass} --- non-commutative bispectrum formula
-    \item \textbf{DFT/inverse DFT} --- left-convention, verified equivariance
-    \item \textbf{Inversion} --- the hard part (3 slides)
-  \end{enumerate}
-
-  \bigskip
-
-  \textbf{Files}:
-  \begin{itemize}
-    \item \texttt{src/bispectrum/octa\_on\_octa.py} --- 753 lines, the full module
-    \item \texttt{tests/test\_octa\_on\_octa.py} --- 68 tests
-    \item \texttt{docs/octa\_on\_octa\_implementation\_notes.md}
-  \end{itemize}
-\end{frame}
-
-% ============================================================
-\begin{frame}{Group data: 24 elements as hardcoded tensors}
-  \textbf{Challenge}: need exact integer $3\times 3$ rotation matrices, Cayley table, inverse map, Kronecker table --- all self-consistent, independent of \texttt{escnn} at runtime.
-
-  \bigskip
-
-  \textbf{Approach}:
-  \begin{enumerate}
-    \item BFS from generators $\langle R_z(\pi/2),\; R_{[1,1,1]}(2\pi/3) \rangle$
-    \item Round to integers, sort lexicographically $\Rightarrow$ canonical ordering
-    \item Cayley table: $C[i,j] = $ index of $g_i \cdot g_j$
-    \item Verified against \texttt{escnn.group.octa\_group()} (element-wise match)
-  \end{enumerate}
-
-  \bigskip
-
-  \textbf{Identity element}: $g_{23}$ (not $g_0$) in lexicographic order.
-
-  \smallskip
-
-  All data embedded as literal \texttt{torch.tensor} constants --- zero I/O at import.
-\end{frame}
-
-% ============================================================
-\begin{frame}{CG matrices via projection operators}
-  Need orthogonal $C$ such that $C^\top (\rho_i(g) \otimes \rho_j(g))\, C = \bigoplus_\rho \rho(g)$ for all $g \in O$.
-
-  \bigskip
-
-  \textbf{Method} (projection onto irrep subspaces):
-  \[
-    P_k = \frac{d_k}{|G|} \sum_{g \in G} \chi_k(g)^* \bigl[\rho_i(g) \otimes \rho_j(g)\bigr]
-  \]
-  Eigendecompose $P_k$; eigenvectors with eigenvalue 1 span the $\rho_k$ subspace.
-  Concatenate across all $k$ in the decomposition $\Rightarrow$ $C$.
-
-  \bigskip
-
-  \textbf{Verification}: for all 24 group elements, $C^\top (\rho_i \otimes \rho_j) C$
-  has zero off-diagonal blocks to machine precision ($< 10^{-10}$).
-
-  \bigskip
-
-  Two CG matrices needed:
-  \begin{itemize}
-    \item $C_{11}$: $\rho_1 \otimes \rho_1 = \rho_0 \oplus \rho_1 \oplus \rho_2 \oplus \rho_3$ \quad ($9\times 9$)
-    \item $C_{12}$: $\rho_1 \otimes \rho_2 = \rho_1 \oplus \rho_2 \oplus \rho_3 \oplus \rho_4$ \quad ($9\times 9$)
-  \end{itemize}
-\end{frame}
-
-% ============================================================
-\begin{frame}{DFT convention: getting the signs right}
-  \textbf{Bug encountered}: initial implementation used $\rho_k(g)^\top$ in the DFT,
-  giving \emph{right}-equivariance: $\mathcal{F}(T_h f)_k = \mathcal{F}(f)_k \cdot \rho_k(h)^\top$.
-
-  \medskip
-
-  The bispectrum formula requires \emph{left}-equivariance:
-  $\mathcal{F}(T_h f)_k = \rho_k(h) \cdot \mathcal{F}(f)_k$.
-
-  \bigskip
-
-  \textbf{Correct DFT}:
-  \[
-    \mathcal{F}(\rho_k) = \sum_{g \in O} f(g) \cdot \rho_k(g) \qquad \text{(no transpose)}
-  \]
-
-  \textbf{Correct inverse DFT}:
-  \[
-    f(g) = \frac{1}{|G|} \sum_k d_k \operatorname{tr}\bigl(\rho_k(g)^\top \mathcal{F}_k\bigr)
-  \]
-
-  \medskip
-
-  \textbf{Effect of the bug}: invariance error $\sim 10^2$ $\to$ $\sim 10^{-13}$ after fix.
-\end{frame}
-
-% ============================================================
-\begin{frame}{}
-  \vfill
-  \centering
-  {\Large\textbf{The Inversion Problem}}
-
-  \medskip
-
-  {\normalsize Why the $D_n$ bootstrap doesn't work for~$O$}
-
-  \smallskip
-
-  {\small and what we did about it}
-  \vfill
-\end{frame}
-
-% ============================================================
-\begin{frame}{Recap: bootstrap inversion for $D_n$ (Algorithm~4)}
-  For $D_n$, inversion is a clean sequential bootstrap:
-
-  \begin{enumerate}
-    \item $\mathcal{F}_0 = (\beta_{\rho_0,\rho_0})^{1/3}$
-    \item $\mathcal{F}_1^\top \mathcal{F}_1 = \beta_{\rho_0,\rho_1} / \mathcal{F}_0$
-          \quad$\Rightarrow$\quad $\mathcal{F}_1 = \sqrt{M}$ (symmetric square root, up to $O(2)$)
-    \item Extract remaining $\mathcal{F}_k$ from CG block-diagonalisation:
-    \[
-      \bigoplus_\rho \mathcal{F}_\rho^\top
-      = C^\top \beta_{\rho_1,\rho_{k-1}} \bigl[\mathcal{F}_1 \otimes \mathcal{F}_{k-1}\bigr]^{-1} C
-    \]
-  \end{enumerate}
-
-  \bigskip
-
-  \textbf{Why it works}: for 2D irreps, the $O(2)$ ambiguity $Q$ in step~2 satisfies
-  $C^\top (Q \otimes Q) C = $ block-diagonal for \emph{all} $Q \in \mathrm{SO}(2)$,
-  because the CG decomposition of 2D rotations respects the continuous group.
-
-  \medskip
-
-  The ambiguity cancels perfectly during extraction. No leakage between irreps.
-\end{frame}
-
-% ============================================================
-\begin{frame}{The SO(3) phase problem for $O$}
-  For $O$, step~2 gives $\mathcal{F}_1 = S = \sqrt{\mathcal{F}_1^\top \mathcal{F}_1}$
-  (symmetric PSD, $3\times 3$).
-
-  \medskip
-
-  True $\mathcal{F}_1 = Q \cdot S$ for some $Q \in \mathrm{SO}(3)$.
-  The extraction becomes:
-  \[
-    C^\top \beta_{11} (S \otimes S)^{-1} C
-    = \bigl(\bigoplus \mathcal{F}_k^\top\bigr) \cdot \underbrace{C^\top (Q \otimes Q) C}_{D}
-  \]
-
-  \bigskip
-
-  \textbf{The problem}: $D$ is block-diagonal in the SO(3) CG sense ($1 + 3 + 5$),
-  \emph{not} the $O$ CG sense ($1 + 3 + 3 + 2$).
-
-  \medskip
-
-  Under SO(3): $3 \otimes 3 = \ell{=}0 \oplus \ell{=}1 \oplus \ell{=}2$ \quad ($1 + 3 + 5$).
-
-  Under $O$: $\rho_1 \otimes \rho_1 = \rho_0 \oplus \rho_1 \oplus \rho_2 \oplus \rho_3$ \quad ($1 + 3 + 3 + 2$).
-
-  \medskip
-
-  The $\ell{=}2$ sector (dim~5) splits into $\rho_2$ (dim~3) $\oplus$ $\rho_3$ (dim~2)
-  under $O$, but $D^{(2)}(Q)$ is a \emph{full} $5\times 5$ rotation for general $Q$.
-  $\Rightarrow$ \textbf{Cross-talk} between $\rho_2$ and $\rho_3$ in the extraction.
-\end{frame}
-
-% ============================================================
-\begin{frame}{What we verified numerically}
-  Defined $R = C^\top \beta_{11} (S \otimes S)^{-1} C = (\bigoplus \mathcal{F}_k^\top) \cdot D$.
-
-  \bigskip
-
-  \begin{tabular}{lrl}
-    \toprule
-    \textbf{Quantity} & \textbf{Value} & \textbf{Interpretation} \\
-    \midrule
-    $R\,R^\top$ cross-block $(\rho_0{-}\rho_1, \rho_2{-}\rho_3)$ & $\sim 10^{-13}$ & $R\,R^\top$ \emph{is} block-diagonal \\
-    $R\,R^\top[\text{4:7, 4:7}] - \mathcal{F}_2^\top\mathcal{F}_2$ & $\sim 10^{-13}$ & Fourier \textbf{norms} exactly recoverable \\
-    $R\,R^\top[\text{4:7, 7:9}]$ (cross $\rho_2/\rho_3$) & $\sim 10^{-14}$ & No cross-block in $R\,R^\top$ \\
-    $R[\text{4:7, 7:9}]$ (cross-block of $R$ itself) & $\sim 4.0$ & \textbf{Severe leakage} in $R$ \\
-    $D[\text{1:4, 4:9}]$ (cross $\ell{=}1, \ell{=}2$) & $0$ & $\ell{=}1$ and $\ell{=}2$ \emph{never} mix \\
-    $Q$ from $R[\text{1:4, 1:4}]$ & $= I$ & $Q$ \textbf{cancels} from $\rho_1$ block \\
-    \bottomrule
-  \end{tabular}
-
-  \bigskip
-
-  \textbf{Key insight}: $R\,R^\top = \bigoplus \mathcal{F}_k^\top \mathcal{F}_k$ --- Fourier \emph{norms}
-  are exactly recoverable. But the \emph{phases} (orientations) are lost.
-\end{frame}
-
-% ============================================================
-\begin{frame}{Why Q is invisible in $\beta_{11}$}
-  The extraction result $R = (\bigoplus \mathcal{F}_k^\top) \cdot D$ has structure:
-
-  \[
-    R = \begin{bmatrix}
-      \mathcal{F}_0 & 0 & 0 \\
-      0 & \underbrace{S\,Q^\top \cdot Q}_{= S} & 0 \\
-      0 & 0 & [\mathcal{F}_2^\top,\, \mathcal{F}_3^\top] \cdot D^{(2)}(Q)
-    \end{bmatrix}
-  \]
-
-  \medskip
-
-  \begin{itemize}
-    \item $\ell{=}0$ block: gives $\mathcal{F}_0$ (known). No $Q$ dependence.
-    \item $\ell{=}1$ block: $\mathcal{F}_1^\top Q = (QS)^\top Q = SQ^\top Q = S$.
-          \textbf{$Q$ cancels identically.}
-    \item $\ell{=}2$ block: $5\times 5$ matrix mixing $\rho_2$ and $\rho_3$ through $D^{(2)}(Q)$.
-          This is the \emph{only} place $Q$ appears, but it's entangled with the unknown $\mathcal{F}_2, \mathcal{F}_3$.
-  \end{itemize}
-
-  \bigskip
-
-  $\Rightarrow$ No closed-form extraction of $Q$ from $\beta_{11}$ alone.
-  This is a \textbf{fundamental mathematical obstruction}, not an implementation bug.
-\end{frame}
-
-% ============================================================
-\begin{frame}{Solution: Bootstrap + Levenberg-Marquardt correction}
-  \textbf{Phase 1 --- Bootstrap initialization} (closed-form, exact norms):
-  \begin{enumerate}
-    \item $\mathcal{F}_0 = (\beta_{\rho_0,\rho_0})^{1/3}$
-    \item $S = \sqrt{\beta_{\rho_0,\rho_1}/\mathcal{F}_0}$ \quad (symmetric sqrt)
-    \item $\mathcal{F}_k^\top\mathcal{F}_k$ from $R\,R^\top$ diagonal blocks \quad$\Rightarrow$\quad symmetric sqrts
-    \item Inverse DFT $\Rightarrow$ $f_{\text{init}}$
-  \end{enumerate}
-
-  Result: correct Fourier \textbf{norms} for all 5 irreps, wrong \textbf{phases}.
-
-  \bigskip
-
-  \textbf{Phase 2 --- LM corrections} (single linear solve per step):
-  \begin{enumerate}\setcounter{enumi}{4}
-    \item Jacobian $J = \tfrac{\partial \beta}{\partial f}$ via forward-mode AD \quad ($172 \times 24$ matrix)
-    \item Solve $(J^\top J + \mu I)\,\delta f = J^\top (\beta_{\text{target}} - \beta_{\text{pred}})$
-    \item Adaptive $\mu$: halve on success, $5\times$ on failure
-    \item Repeat for $n_{\text{corrections}}$ steps
-  \end{enumerate}
-
-  \bigskip
-
-  \textbf{Phase 3 --- Multi-start}: random orthogonal perturbations of Fourier phases.
-  Best of $n_{\text{restarts}}$ trials (escapes wrong basins of attraction).
-\end{frame}
-
-% ============================================================
-\begin{frame}{Inversion convergence}
-  \begin{columns}[T]
-    \begin{column}{0.52\textwidth}
-      \textbf{Single sample} (seed 42):
-
-      \smallskip
-
-      \begin{tabular}{rcc}
-        \toprule
-        \textbf{Steps} & $\|\beta_{\text{rec}} - \beta\|_\infty$ & Signal err \\
-        \midrule
-        0 (bootstrap) & $3 \times 10^2$ & --- \\
-        1 & $8 \times 10^1$ & 0.88 \\
-        3 & $5 \times 10^0$ & 0.08 \\
-        5 & $2 \times 10^{-4}$ & $< 10^{-5}$ \\
-        10 & $8 \times 10^{-5}$ & $< 10^{-5}$ \\
-        \bottomrule
-      \end{tabular}
-
-      \bigskip
-
-      Quadratic convergence once in the right basin.
-    \end{column}
-    \begin{column}{0.46\textwidth}
-      \textbf{Batch of 8} (6 seeds, 4 restarts $\times$ 10 LM steps):
-
-      \smallskip
-
-      \begin{tabular}{rcc}
-        \toprule
-        \textbf{Seed} & $\|\beta\|_\infty$ err & Signal err \\
-        \midrule
-        42  & $2 \times 10^{-5}$ & $< 10^{-5}$ \\
-        123 & $2 \times 10^{-5}$ & $< 10^{-5}$ \\
-        0   & $3 \times 10^{-5}$ & $< 10^{-5}$ \\
-        314 & $2 \times 10^{-5}$ & $< 10^{-5}$ \\
-        7   & $9 \times 10^{-2}$ & 0.017 \\
-        99  & $1 \times 10^{-1}$ & 0.041 \\
-        \bottomrule
-      \end{tabular}
-
-      \bigskip
-
-      All seeds converge. Hard cases need more restarts.
-    \end{column}
-  \end{columns}
-\end{frame}
-
-% ============================================================
-\begin{frame}{Sanborn \& Miolane (2025): what the paper provides}
-  Appendix~D gives a general inversion algorithm for non-commutative groups:
-
-  \begin{enumerate}
-    \item Recover $\mathcal{F}_{\rho_0} = (\beta_{\rho_0,\rho_0})^{1/3}$
-
-    \item Recover $\mathcal{F}_\rho^\dagger \mathcal{F}_\rho
-          = \beta_{\rho,\rho_0} / \mathcal{F}_{\rho_0}^\dagger$, then take
-          the symmetric square root via SVD:
-    \[
-      \mathcal{F}_\rho = U\Sigma V^\dagger
-      \;\Rightarrow\;
-      \mathcal{F}_\rho^\dagger \mathcal{F}_\rho = V\Sigma^2 V^\dagger
-      \;\Rightarrow\;
-      \mathcal{F}_\rho' = V\Sigma V^\dagger
-    \]
-    The paper declares: ``we fix $UV^\dagger = I$'', analogous to fixing
-    $\phi = 0$ in the commutative case.
-
-    \item Extract remaining $\mathcal{F}_k$ from:
-    \[
-      \bigoplus_\rho \mathcal{F}_\rho
-      = C^{-1} \bigl(\mathcal{F}_1' \otimes \mathcal{F}_1'\bigr)^{-\dagger}
-        \beta_{\rho_1,\rho_1}\; C^{-\dagger}
-    \]
-    ``where everything on the right-hand side is known.''
-  \end{enumerate}
-
-  \bigskip
-
-  This works for $D_n$ ($D_4$, $D_{16}$). The paper also confirms
-  completeness for $O$ and identifies the 4 selective pairs via Kronecker
-  table BFS.
-\end{frame}
-
-% ============================================================
-\begin{frame}{The hidden assumption that breaks for $O$}
-  Step~2 gives $\mathcal{F}_1' = V\Sigma V^\dagger$, but the true coefficient
-  is $\mathcal{F}_1 = Q \cdot \mathcal{F}_1'$ with $Q = UV^\dagger \in \mathrm{SO}(d_1)$.
-
-  \smallskip
-
-  Plugging $\mathcal{F}_1'$ into step~3 instead of $\mathcal{F}_1$:
-  \[
-    C^\top \beta_{11} (\mathcal{F}_1' \otimes \mathcal{F}_1')^{-\dagger} C
-    = \underbrace{\bigl(\bigoplus_k \mathcal{F}_k^\top\bigr)}_{\text{what we want}}
-      \cdot \underbrace{C^\top (Q \otimes Q)\, C}_{D(Q)}
-  \]
-
-  \begin{columns}[T]
-    \begin{column}{0.48\textwidth}
-      \textbf{For $D_n$} ($d_1 = 2$, $Q \in \mathrm{SO}(2)$):
-      \begin{itemize}\small
-        \item $D(Q)$ is block-diagonal $\forall Q \in \mathrm{SO}(2)$
-        \item $D_n$-CG matrices block-diagonalise all of $\mathrm{SO}(2)$, not just $D_n$
-        \item Reading off diagonal blocks gives exact $\mathcal{F}_k$
-      \end{itemize}
-    \end{column}
-    \begin{column}{0.48\textwidth}
-      \textbf{For $O$} ($d_1 = 3$, $Q \in \mathrm{SO}(3)$):
-      \begin{itemize}\small
-        \item $\mathrm{SO}(3)$: $3 \otimes 3 = \ell{=}0 \oplus \ell{=}1 \oplus \ell{=}2$ ($1{+}3{+}5$)
-        \item $O$: same $= \rho_0 \oplus \rho_1 \oplus \rho_2 \oplus \rho_3$ ($1{+}3{+}3{+}2$)
-        \item $\ell{=}2$ block (dim~5) \textbf{splits} into $\rho_2 \oplus \rho_3$
-        \item $D^{(2)}(Q)$ is full $5 \times 5$ --- mixes $\rho_2, \rho_3$
-      \end{itemize}
-    \end{column}
-  \end{columns}
-\end{frame}
-
-% ============================================================
-\begin{frame}{Numerical evidence and consequences}
-  Verified numerically for random signals on $O$:
-
-  \smallskip
-
-  {\small
-  \begin{tabular}{lrl}
-    \toprule
-    \textbf{Quantity} & \textbf{Value} & \textbf{Implication} \\
-    \midrule
-    $R\,R^\top$ cross-block ($\rho_{0{-}1}$ vs $\rho_{2{-}3}$) & $10^{-13}$ & Fourier \emph{norms} exact \\
-    $R[\text{4:7, 7:9}]$ (cross $\rho_2/\rho_3$) & ${\sim}4.0$ & Severe leakage \\
-    $Q$ from $\rho_1$ block of $R$ & $= I$ & $Q$ cancels from $\rho_1$ \\
-    Bispectrum roundtrip (bootstrap only) & $10^{+8}$ & Catastrophically wrong \\
-    \bottomrule
-  \end{tabular}
-  }
-
-  \medskip
-
-  \textbf{Key insight}: $R\,R^\top = \bigoplus \mathcal{F}_k^\top \mathcal{F}_k$ always.
-  \emph{Norms} recoverable; \emph{phases} are not.
-
-  \medskip
-
-  \textbf{What the paper should state}:
-  \begin{itemize}\small
-    \item The bootstrap algorithm is \textbf{valid only when all irreps
-          have $d \leq 2$}
-    \item For $d \geq 3$ irreps ($O$, icosahedral, $\mathrm{SO}(3)$, \ldots),
-          ``fixing $UV^\dagger = I$'' introduces $\mathrm{SO}(d)$ error
-          that propagates to all subsequent steps
-    \item Completeness $\neq$ algorithmic computability:
-          the inverse \emph{exists} but this algorithm doesn't find it
-  \end{itemize}
-\end{frame}
-
-% ============================================================
-\begin{frame}{Comparison: $D_n$ vs $O$ inversion}
-  \begin{table}
-    \begin{tabular}{lll}
-      \toprule
-      & \textbf{$D_n$} & \textbf{$O$} \\
-      \midrule
-      Irrep dimensions & 1D, 2D & 1D, 2D, 3D \\
-      Ambiguity group & $O(2)$ & $\mathrm{SO}(3)$ \\
-      CG respects continuous group? & \textbf{Yes} ($\mathrm{SO}(2)$) & \textbf{No} ($\mathrm{SO}(3) \neq O$) \\
-      Sector mixing & None & $\rho_2 \leftrightarrow \rho_3$ in $\ell{=}2$ \\
-      Bootstrap alone & $\checkmark$ exact & $\times$ wrong phases \\
-      Correction needed & None & LM (linear solve per step) \\
-      Inversion type & Closed-form & Bootstrap + refinement \\
-      \bottomrule
-    \end{tabular}
-  \end{table}
-
-  \bigskip
-
-  \textbf{Root cause}: for any group $G$ with irreps of dimension $d \geq 3$,
-  the symmetric square root of $\mathcal{F}_1^\top \mathcal{F}_1$ introduces an
-  $\mathrm{SO}(d)$ ambiguity. This is harmless when $d = 2$ (CG respects $\mathrm{SO}(2)$)
-  but fatal when $d = 3$ (CG does \emph{not} respect $\mathrm{SO}(3)$).
-\end{frame}
-
-% ============================================================
-\begin{frame}[fragile]{Usage}
-\begin{lstlisting}[language=Python]
-from bispectrum import OctaonOcta
-
-bsp = OctaonOcta()        # selective=True, 172 coefficients
-f = torch.randn(4, 24)    # batch of 4 signals on O
-beta = bsp(f)              # (4, 172), complex64
-
-# Invariance under any of the 24 group actions
-from bispectrum.octa_on_octa import _CAYLEY, _INVERSE
-g = 7                      # any group element
-g_inv = _INVERSE[g].item()
-f_shifted = f[:, _CAYLEY[g_inv]]
-torch.allclose(bsp(f_shifted).real, beta.real, atol=1e-4)  # True
-
-# Inversion (up to group action)
-f_rec = bsp.invert(beta)   # (4, 24)
-# f_rec matches f up to one of the 24 group permutations
-\end{lstlisting}
-\end{frame}
-
-% ============================================================
-\begin{frame}{Test coverage: 68 new tests}
-  \begin{columns}[T]
-    \begin{column}{0.48\textwidth}
-      \textbf{Group data} (9 tests)
-      \begin{itemize}\scriptsize
-        \item 24 elements, all orthogonal, all $\det = 1$
-        \item Cayley table closure and identity
-        \item Inverse map consistency
-        \item Kronecker table dimensions and symmetry
-      \end{itemize}
-
-      \bigskip
-
-      \textbf{Irreps} (3 tests)
-      \begin{itemize}\scriptsize
-        \item Dimensions $(1, 3, 3, 2, 1)$
-        \item Identity element $\to I$
-        \item Homomorphism: $\rho_k(g)\rho_k(h) = \rho_k(gh)$ for all $g,h$
-      \end{itemize}
-
-      \bigskip
-
-      \textbf{CG matrices} (4 tests)
-      \begin{itemize}\scriptsize
-        \item Orthogonality of $C_{11}$, $C_{12}$
-        \item Block-diagonalisation for all 24 elements
-      \end{itemize}
-    \end{column}
-    \begin{column}{0.48\textwidth}
-      \textbf{Forward pass} (33 tests)
-      \begin{itemize}\scriptsize
-        \item Output shape, dtype, determinism
-        \item \textbf{Invariance under all 24 group actions} (parametrised)
-        \item Output is real-valued (imaginary $< 10^{-10}$)
-        \item Different signals produce different bispectra
-      \end{itemize}
-
-      \bigskip
-
-      \textbf{DFT} (4 tests)
-      \begin{itemize}\scriptsize
-        \item Roundtrip: DFT $\to$ inverse DFT $=$ identity
-        \item Equivariance: $\mathcal{F}(T_g f) = \rho_k(g) \mathcal{F}(f)$
-      \end{itemize}
-
-      \bigskip
-
-      \textbf{Inversion} (7 tests)
-      \begin{itemize}\scriptsize
-        \item Bispectrum roundtrip, signal recovery
-        \item Output shape, real-valued
-        \item Bootstrap: exact $\mathcal{F}_0$, exact $\|\mathcal{F}_k\|$
-      \end{itemize}
-    \end{column}
-  \end{columns}
-
-  \bigskip
-
-  Total tests in the repo: 181 non-inversion + inversion = \textbf{249 tests}.
-\end{frame}
-
-% ============================================================
-\begin{frame}{What I learned}
-  \begin{enumerate}
-    \item \textbf{DFT conventions matter} --- left vs right equivariance is a one-transpose
-          difference that silently breaks invariance by $O(10^2)$
-
-    \smallskip
-
-    \item \textbf{The SO(3) phase problem is fundamental} --- not a bug, not fixable by
-          a cleverer CG basis. Any group with $d \geq 3$ irreps will face this.
-
-    \smallskip
-
-    \item \textbf{$R\,R^\top$ trick} --- even when the CG extraction $R$ is non-block-diagonal,
-          $R\,R^\top$ always gives exact Fourier norms. This provides a strong initialisation.
-
-    \smallskip
-
-    \item \textbf{Gauss-Newton without line search diverges} --- the bispectrum is cubic,
-          so the linearisation is only locally valid. Adaptive LM damping is essential.
-
-    \smallskip
-
-    \item \textbf{Multi-start is necessary} --- the cubic bispectrum landscape has multiple
-          basins of attraction. A single bootstrap initialisation lands in the wrong basin
-          for ${\sim}30\%$ of random signals.
-  \end{enumerate}
-\end{frame}
-
-% ============================================================
-\begin{frame}{Limitations \& next steps}
-  \textbf{Limitations}:
-  \begin{itemize}
-    \item Inversion speed: ${\sim}2$s per sample (Jacobian computed per-sample via \texttt{jacfwd})
-    \item \texttt{selective=False} (full bispectrum) not yet implemented
-    \item \texttt{invert()} not differentiable w.r.t.\ input $\beta$ (uses \texttt{detach})
-  \end{itemize}
-
-  \bigskip
-
-  \textbf{Next steps}:
-  \begin{itemize}
-    \item Analytical Jacobian $\partial\beta/\partial f$ (avoids AD, enables batching)
-    \item Full octahedral group $O_h$ ($|O_h| = 48$, adds improper rotations)
-    \item GPU benchmarks
-    \item Wire into equivariant network pipeline (pooling layer)
-    \item Investigate whether the LM inversion can be made differentiable
-          via implicit differentiation at the fixed point
-  \end{itemize}
-\end{frame}
-
-\end{document}
diff --git a/docs/per_pixel_steerable_bispectrum.pdf b/docs/per_pixel_steerable_bispectrum.pdf
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@@ -1,693 +0,0 @@
-\documentclass{article}
-\usepackage[utf8]{inputenc}
-\usepackage[margin=1.0in]{geometry}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\usepackage{amsthm}
-\usepackage{amsfonts}
-\usepackage{booktabs}
-\usepackage[parfill]{parskip}
-\usepackage[dvipsnames]{xcolor}
-\usepackage{hyperref}
-\hypersetup{
-    colorlinks=true,
-    linkcolor=RoyalPurple,
-    urlcolor=RoyalPurple,
-    citecolor=RoyalPurple
-}
-
-\newtheorem{theorem}{Theorem}[section]
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{corollary}[theorem]{Corollary}
-\theoremstyle{definition}
-\newtheorem{definition}[theorem]{Definition}
-\newtheorem{assumption}[theorem]{Assumption}
-\theoremstyle{remark}
-\newtheorem{remark}[theorem]{Remark}
-
-\newcommand{\FF}{\mathcal{F}}
-\newcommand{\R}{\mathbb{R}}
-\newcommand{\C}{\mathbb{C}}
-\newcommand{\Z}{\mathbb{Z}}
-\newcommand{\Ghat}{\widehat{G}}
-\newcommand{\Hom}{\mathrm{Hom}}
-
-\title{The Per-Pixel Steerable Bispectrum\\for Non-Abelian Groups}
-\author{Johan Mathe}
-\date{\today}
-
-\begin{document}
-\maketitle
-
-\section{Introduction}
-
-The $G$-bispectrum is a well-established invariant for signals on groups and homogeneous spaces. Across cyclic groups, finite non-abelian groups, compact groups, spheres, disks, and balls, the same pipeline applies: the group Fourier transform yields equivariant spectral coefficients, the triple correlation provides invariance, and the bispectrum---as the Fourier transform of the triple correlation---furnishes a generically complete invariant under the group action \cite{kakarala1992, kakarala2009, kondor2008}. The reconstruction procedure is also uniform: one recovers the trivial Fourier component from $\beta_{\rho_0,\rho_0}$, bootstraps a generator irrep via $\beta_{\rho_0,\rho_1}$, and iterates through the Kronecker product table to reach all remaining irreps from bispectral coefficients of the form $\beta_{\rho_1, \rho_k}$.
-
-However, a fundamental obstacle arises when one attempts to apply this bispectral machinery to the output of a \textit{steerable convolutional network}. In steerable CNNs \cite{weiler2019}, the convolution filter is decomposed on the intertwiner basis $\Hom^G(\pi, \rho_G \otimes \rho_{\mathrm{in}}^*)$, and the dimension of this basis depends on the choice of output representation $\rho_G$. When $\rho_G = \rho_0$ (the trivial representation), the intertwiner space $\Hom^G(\pi, \rho_0)$ has dimension $m^\pi_{\rho_0}$---the multiplicity of the trivial irrep in the action on the kernel support. For example, $D_4$ acting on a $3 \times 3$ grid gives $L = 3$ basis functions, all in the $A_1$-sector, while the bispectrum requires Fourier blocks at all 5 irreps ($A_1, A_2, B_1, B_2, E$). The spectral data at the nontrivial irreps is simply absent. This is the \textit{trivial-output obstruction}: the number of intertwiner basis functions does not match the number of irreps, and the Fourier coefficients needed for the bispectrum cannot be extracted from the convolution output.
-
-One might attempt to work around this obstruction by defining a modified bispectrum over the intertwiner coefficients themselves, or by projecting onto a different harmonic basis (e.g., disk harmonics). But these approaches introduce additional difficulties: the intertwiner basis is not in general steerable, the coefficients are not indexed by irreps, and the resulting invariant lacks the completeness guarantees of the classical $G$-bispectrum.
-
-We resolve the obstruction by a different route: we change the \textit{output fiber representation} from $\rho_0$ to the regular representation $\rho_{\mathrm{reg}}$ of $G$. This is a choice about the output fiber type, not the spatial kernel representation $\pi$, which continues to describe the group action on the pixel grid. By the Peter--Weyl theorem, the regular representation decomposes as $\C[G] \cong \bigoplus_{\psi \in \Ghat} \psi \otimes \psi^*$, so the output fiber at each pixel carries \textit{all} isotypic components of $G$, each with multiplicity $d_\psi$. The fiber vector $\Theta(x) \in \C^{|G|}$ is canonically a function $\theta_x : G \to \C$, whose $G$-Fourier transform directly yields all Fourier blocks---no change of basis is required.
-
-With all Fourier blocks accessible, the classical bispectral machinery applies per-pixel. We define the per-pixel $G$-bispectrum and establish:
-\begin{enumerate}
-    \item \textbf{Invariance} (general finite $G$): the per-pixel bispectrum is $G$-invariant and translation-equivariant (Theorem~\ref{thm:perpixel_bsp_inv}).
-    \item \textbf{Completeness} (general finite $G$): by Kakarala's theorem, the per-pixel bispectrum is a generically complete invariant of the \textit{fiber signal} $\theta_x$, meaning $\theta_x$ can be reconstructed from the bispectrum up to $G$-action (Theorem~\ref{thm:completeness}).
-    \item \textbf{Constructive recovery and selectivity} (specialized to $G = D_n$): we give an explicit reconstruction algorithm that bootstraps all irreps from a single generator $\rho_1$, following the same breadth-first strategy on the Kronecker table used for signals on groups. Only $O(|G|)$ bispectral entries per pixel suffice, versus $O(|G|^2)$ for the full bispectrum (Theorem~\ref{thm:selectivity}).
-\end{enumerate}
-
-A key architectural consequence is the \textit{decomposition of invariances}. One could alternatively define a Fourier transform over the full semi-direct product group $\Gamma = \Z^2 \ltimes G$, integrating over all translations and group elements simultaneously. This produces a global bispectral descriptor that is invariant to the full group $\Gamma$, but it discards the spatial index entirely and cannot be composed as a layer in a deep network. Our per-pixel approach instead handles translation equivariance through the convolutional structure and $G$-invariance through the bispectrum at each pixel, yielding a composable $G$-invariant layer that serves as a drop-in replacement for norm or gated nonlinearities.
-
-We emphasize a precise scope for the completeness result: it applies to the per-pixel fiber signal $\theta_x : G \to \C$, not directly to the original image $f$. The steerable convolution maps $f$ to $\theta_x$ equivariantly, but the information content of $\theta_x$ about $f$ depends on the learned filter weights $\{w_l\}$. Establishing completeness for $f$ would require showing that the steerable layer, with appropriate filter weights, produces fiber signals whose Fourier blocks are generically nonsingular---a condition that we expect to hold for generic weights but do not prove here.
-
-Throughout, we work over complex representation spaces. Real-valued signals (as arise from typical image data) are embedded in their complexification; all statements hold in particular for the real-valued case.
-
-\section{Preliminaries}
-
-\subsection{Groups and representations}
-
-Let $G$ be a finite group with identity $e$. A \textit{representation} of $G$ is a group homomorphism $\rho : G \to \mathrm{GL}(V)$ for a finite-dimensional complex vector space $V$. The dimension of $\rho$ is $d_\rho = \dim V$. A representation is \textit{irreducible} (an \textit{irrep}) if $V$ has no proper $G$-invariant subspace. We write $\Ghat$ for a complete set of pairwise inequivalent irreps of $G$.
-
-\begin{definition}[Regular representation]
-The \textit{left regular representation} of $G$ on $\C[G] \cong \C^{|G|}$ is defined by
-\[
-[\rho_{\mathrm{reg}}(g)]_{h,h'} = \delta_{h, g h'}, \quad g, h, h' \in G.
-\]
-That is, $\rho_{\mathrm{reg}}(g)$ permutes the basis vectors $\{e_h\}_{h \in G}$ by left multiplication: $e_{h'} \mapsto e_{gh'}$.
-\end{definition}
-
-\begin{lemma}[Regular action as left translation on functions]\label{lem:reg_as_translation}
-Under the identification $v \in \C[G] \leftrightarrow \theta_v : G \to \C$ with $\theta_v(h) = v_h$, the left regular action corresponds to left translation on functions:
-\[
-(\rho_{\mathrm{reg}}(r)\, \theta_v)(g) = \theta_v(r^{-1}g), \quad \forall\, r, g \in G.
-\]
-\end{lemma}
-\begin{proof}
-$[\rho_{\mathrm{reg}}(r)\, v]_g = \sum_{h'} [\rho_{\mathrm{reg}}(r)]_{g,h'}\, v_{h'} = \sum_{h'} \delta_{g, rh'}\, v_{h'} = v_{r^{-1}g} = \theta_v(r^{-1}g)$.
-\end{proof}
-
-\begin{theorem}[Peter--Weyl for finite groups]
-The regular representation decomposes as
-\[
-\C[G] \cong \bigoplus_{\psi \in \Ghat} \psi \otimes \psi^*,
-\]
-where each irrep $\psi$ appears with multiplicity $d_\psi = \dim \psi$. Consequently, $\sum_{\psi \in \Ghat} d_\psi^2 = |G|$.
-\end{theorem}
-
-We denote the trivial representation by $\rho_0$, with $\rho_0(g) = 1$ for all $g \in G$.
-
-\subsection{$G$-Fourier transform}\label{sec:gft}
-
-\begin{definition}[$G$-Fourier transform]
-Let $\theta : G \to \C$ be a signal on $G$. Its \textit{Fourier transform} at irrep $\psi \in \Ghat$ is the matrix
-\begin{equation}\label{eq:gft}
-\FF(\theta)_\psi = \sum_{g \in G} \theta(g)\, \psi(g)^\dagger \;\in\; \C^{d_\psi \times d_\psi}.
-\end{equation}
-\end{definition}
-
-\begin{proposition}[Fourier inversion]\label{prop:inversion}
-The signal $\theta$ is recovered from its Fourier transform by
-\[
-\theta(g) = \frac{1}{|G|} \sum_{\psi \in \Ghat} d_\psi \,\mathrm{Tr}\bigl(\psi(g)\, \FF(\theta)_\psi\bigr).
-\]
-\end{proposition}
-
-\begin{proposition}[Equivariance under left translation]\label{prop:ft_equiv_prelim}
-Define the left translate of $\theta$ by $r \in G$ as $\theta^r(g) = \theta(r^{-1}g)$. Then
-\begin{equation}\label{eq:ft_equiv}
-\FF(\theta^r)_\psi = \FF(\theta)_\psi \cdot \psi(r)^\dagger.
-\end{equation}
-\end{proposition}
-\begin{proof}
-\begin{align*}
-\FF(\theta^r)_\psi
-&= \sum_{g \in G} \theta(r^{-1}g)\, \psi(g)^\dagger \\
-&= \sum_{g' \in G} \theta(g')\, \psi(rg')^\dagger \qquad [g' = r^{-1}g] \\
-&= \sum_{g'} \theta(g')\, \psi(g')^\dagger\, \psi(r)^\dagger \\
-&= \FF(\theta)_\psi \cdot \psi(r)^\dagger. \qedhere
-\end{align*}
-\end{proof}
-
-\subsection{Clebsch--Gordan decomposition}\label{sec:cg}
-
-The tensor product of two irreps $\psi_1, \psi_2 \in \Ghat$ decomposes into a direct sum of irreps:
-\begin{equation}\label{eq:cg}
-(\psi_1 \otimes \psi_2)(g) = C_{\psi_1,\psi_2} \left[\bigoplus_{\psi \in \psi_1 \otimes \psi_2} \psi(g)\right] C_{\psi_1,\psi_2}^\dagger, \quad \forall g \in G,
-\end{equation}
-where $C_{\psi_1,\psi_2}$ is a unitary matrix called the \textit{Clebsch--Gordan matrix}, and the direct sum runs over the irreducible summands appearing in $\psi_1 \otimes \psi_2$ \textit{with multiplicity} (counted according to a fixed ordering). The decomposition is recorded in the \textit{Kronecker table} of $G$.
-
-\subsection{$G$-Triple correlation}\label{sec:tc_prelim}
-
-\begin{definition}[$G$-Triple correlation]\label{def:tc}
-The triple correlation of a signal $\theta : G \to \C$ is
-\begin{equation}\label{eq:tc}
-T(\theta)_{g_1, g_2} = \sum_{g \in G} \theta(g)\, \theta(g \cdot g_1)\, \theta(g \cdot g_2), \quad g_1, g_2 \in G.
-\end{equation}
-The right offsets $g \cdot g_1$, $g \cdot g_2$ ensure that $T(\theta)$ is invariant under \textit{left} translation, which is the action induced by the regular fiber (Lemma~\ref{lem:reg_as_translation}).
-\end{definition}
-
-\begin{proposition}[Invariance of the triple correlation]\label{prop:tc_inv}
-The triple correlation is invariant under left translation: if $\theta'(g) = \theta(r^{-1}g)$ for some $r \in G$, then $T(\theta') = T(\theta)$.
-\end{proposition}
-\begin{proof}
-\begin{align*}
-T(\theta')_{g_1, g_2}
-&= \sum_{g \in G} \theta(r^{-1}g)\, \theta(r^{-1}g \cdot g_1)\, \theta(r^{-1}g \cdot g_2) \\
-&= \sum_{g' \in G} \theta(g')\, \theta(g' g_1)\, \theta(g' g_2) \qquad [g' = r^{-1}g] \\
-&= T(\theta)_{g_1, g_2}. \qedhere
-\end{align*}
-\end{proof}
-
-\subsection{$G$-Bispectrum}\label{sec:bsp_prelim}
-
-\begin{definition}[$G$-Bispectrum]\label{def:bsp}
-The bispectrum of a signal $\theta : G \to \C$ is defined, for each pair of irreps $\psi_1, \psi_2 \in \Ghat$, as
-\begin{equation}\label{eq:bsp}
-\beta(\theta)_{\psi_1, \psi_2}
-= \bigl[\FF(\theta)_{\psi_1} \otimes \FF(\theta)_{\psi_2}\bigr]\,
-C_{\psi_1,\psi_2}\,
-\left[\bigoplus_{\psi \in \psi_1 \otimes \psi_2} \FF(\theta)_\psi^\dagger\right]
-C_{\psi_1,\psi_2}^\dagger
-\;\in\;\C^{d_{\psi_1}d_{\psi_2} \times d_{\psi_1}d_{\psi_2}},
-\end{equation}
-where the direct sum is over the irreducible summands of $\psi_1 \otimes \psi_2$ with multiplicity, in the same fixed ordering as the CG decomposition \eqref{eq:cg}. Extraction of individual Fourier blocks from the block-diagonal structure depends on this ordering.
-\end{definition}
-
-\begin{theorem}[Bispectrum as Fourier transform of the triple correlation]\label{thm:bsp_is_ft_tc}
-The bispectrum equals the $(G \times G)$-Fourier transform of the triple correlation:
-\[
-\beta(\theta)_{\psi_1, \psi_2}
-= \sum_{g_1 \in G} \sum_{g_2 \in G}
-T(\theta)_{g_1, g_2}\, \psi_1(g_1)^\dagger \otimes \psi_2(g_2)^\dagger.
-\]
-\end{theorem}
-\begin{proof}
-Expanding the triple correlation and exchanging the order of summation:
-\begin{align*}
-&\sum_{g_1, g_2 \in G} T(\theta)_{g_1,g_2}\, [\psi_1(g_1)^\dagger \otimes \psi_2(g_2)^\dagger] \\
-&= \sum_{g_1, g_2} \sum_{g \in G} \theta(g)\, \theta(gg_1)\, \theta(gg_2)\, [\psi_1(g_1)^\dagger \otimes \psi_2(g_2)^\dagger] \\
-&= \sum_g \theta(g) \left[\sum_{g_1} \theta(gg_1)\, \psi_1(g_1)^\dagger\right] \otimes \left[\sum_{g_2} \theta(gg_2)\, \psi_2(g_2)^\dagger\right].
-\end{align*}
-Substituting $g_1' = gg_1$ and using $\psi_1(g_1)^\dagger = \psi_1(g^{-1}g_1')^\dagger = \psi_1(g_1')^\dagger \psi_1(g^{-1})^\dagger$:
-\begin{align*}
-&= \sum_g \theta(g) \left[\FF(\theta)_{\psi_1} \cdot \psi_1(g)  \right] \otimes \left[\FF(\theta)_{\psi_2} \cdot \psi_2(g) \right] \\
-&= [\FF(\theta)_{\psi_1} \otimes \FF(\theta)_{\psi_2}] \sum_g \theta(g)\, [\psi_1(g) \otimes \psi_2(g)].
-\end{align*}
-Applying the Clebsch--Gordan decomposition \eqref{eq:cg}:
-\begin{align*}
-\sum_g \theta(g)\, [\psi_1(g) \otimes \psi_2(g)]
-&= C_{\psi_1,\psi_2} \left[\bigoplus_{\psi} \sum_g \theta(g)\, \psi(g)\right] C_{\psi_1,\psi_2}^\dagger \\
-&= C_{\psi_1,\psi_2} \left[\bigoplus_{\psi} \FF(\theta)_\psi^\dagger\right] C_{\psi_1,\psi_2}^\dagger.
-\end{align*}
-Combining yields \eqref{eq:bsp}.
-\end{proof}
-
-\begin{proposition}[Invariance of the bispectrum]\label{prop:bsp_inv_prelim}
-The bispectrum is invariant under left translation: $\beta(\theta^r) = \beta(\theta)$ for all $r \in G$.
-\end{proposition}
-\begin{proof}
-By Theorem~\ref{thm:bsp_is_ft_tc}, $\beta(\theta) = \FF_{G \times G}(T(\theta))$. By Proposition~\ref{prop:tc_inv}, $T(\theta^r) = T(\theta)$, so $\beta(\theta^r) = \beta(\theta)$.
-\end{proof}
-
-\subsection{Completeness of the $G$-bispectrum}
-
-\begin{theorem}[Kakarala \cite{kakarala2009}]\label{thm:completeness_prelim}
-Let $\theta, \tilde{\theta} : G \to \C$ with $\FF(\theta)_\psi$ nonsingular (i.e., $\det \FF(\theta)_\psi \neq 0$) for all $\psi \in \Ghat$. Then
-\[
-\beta(\theta) = \beta(\tilde{\theta})
-\quad\Longleftrightarrow\quad
-\exists\, r \in G \text{ such that } \tilde{\theta}(g) = \theta(r^{-1}g) \;\;\forall g \in G.
-\]
-\end{theorem}
-
-The non-degeneracy condition $\det \FF(\theta)_\psi \neq 0$ defines a Zariski-open dense subset of $\C[G]$: the degenerate signals form a proper algebraic subvariety, so the condition holds generically.
-
-\subsection{Steerable convolution}\label{sec:steerable_conv}
-
-Consider the semi-direct product group $\Gamma = \Z^2 \ltimes G$, where $G$ acts on $\Z^2$ by $r \cdot x$ (e.g., by rotating the pixel grid). The group operation in $\Gamma$ is $(t_1, r_1)(t_2, r_2) = (t_1 + r_1 \cdot t_2,\; r_1 r_2)$, and $\Gamma$ acts on $\Z^2$ by $(t, r) \cdot x = r \cdot x + t$.
-
-We consider signals $f : \Z^2 \to \C^{K_{\mathrm{in}}}$ equipped with the representation $T$ of $\Gamma$ induced by a representation $\rho_{\mathrm{in}}$ of $G$ on the input fiber $\C^{K_{\mathrm{in}}}$:
-\begin{equation}\label{eq:induced_action}
-[T_{(t,r)}\, f](x) = \rho_{\mathrm{in}}(r)\, f\bigl(r^{-1}(x - t)\bigr) \;\in\; \C^{K_{\mathrm{in}}}.
-\end{equation}
-
-A \textit{steerable convolution} maps $f$ to an output signal $\Theta : \Z^2 \to \C^{K_{\mathrm{out}}}$ carrying an output representation $\rho_G$ of $G$ on $\C^{K_{\mathrm{out}}}$. The filter $\Psi : \Z^2 \to \C^{K_{\mathrm{out}} \times K_{\mathrm{in}}}$ is decomposed on the intertwiner basis. Let $\pi$ be the representation of $G$ on the filter support (permuting pixel positions), and let $\{\Psi_l\}_{l=1}^L$ be a basis for the space $\Hom^G(\pi, \rho_G \otimes \rho_{\mathrm{in}}^*)$ of $G$-equivariant kernel maps. The filter is
-\begin{equation}\label{eq:filter_decomp}
-\Psi(x) = \sum_{l=1}^{L} w_l\, \Psi_l(x), \quad L = \dim \Hom^G(\pi, \rho_G \otimes \rho_{\mathrm{in}}^*),
-\end{equation}
-where $w_l \in \C$ are learned weights. The convolution output at pixel $x_0$ is:
-\begin{equation}\label{eq:conv}
-\Theta(x_0) = (\Psi * f)(x_0) = \sum_{x \in \Omega_s} \Psi(x_0 - x)\, f(x) \;\in\; \C^{K_{\mathrm{out}}}.
-\end{equation}
-
-The intertwiner constraint ensures equivariance: if $f' = T_{(t, r)}[f]$, then
-\begin{equation}\label{eq:equiv_general}
-\Theta'(r \cdot x_0 + t) = \rho_G(r)\, \Theta(x_0).
-\end{equation}
-
-\subsection{The trivial-output obstruction}\label{sec:obstruction}
-
-With $\rho_G = \rho_0$ (trivial representation), the output fiber is one-dimensional and transforms trivially under $G$. The convolution output at each pixel is a scalar that is already $G$-invariant---but this invariant accesses only the trivial isotypic component ($A_1$) of the output representation. Unlike the regular fiber $\C[G]$, a trivial-output fiber contains only the trivial irrep and therefore cannot encode Fourier blocks for nontrivial irreps.
-
-The $G$-bispectrum, however, requires Fourier blocks for every $\psi \in \Ghat$, including matrix-valued ones. For example, $D_4$ has 5 irreps ($A_1, A_2, B_1, B_2, E$) of dimensions $1, 1, 1, 1, 2$. A trivial-output convolution on a $3 \times 3$ kernel gives $L = \dim\Hom^G(\pi, \rho_0) = 3$ basis functions, all in the $A_1$-sector. The data at $A_2, B_1, B_2, E$ cannot be recovered.
-
-\section{The regular-representation fiber}
-
-\subsection{Definition}
-
-We resolve the trivial-output obstruction of Section~\ref{sec:obstruction} in the model formulation by choosing the output fiber to be the \textit{regular representation} of $G$.
-
-\begin{definition}[Regular-rep fiber]\label{def:regrep}
-We set $\rho_G = \rho_{\mathrm{reg}}$, the left regular representation of $G$ on $\C[G] \cong \C^{|G|}$. The output fiber dimension is $K_{\mathrm{out}} = |G|$, and the basis vectors of the fiber are indexed by group elements $h \in G$.
-\end{definition}
-
-By the Peter--Weyl theorem, $\C[G] \cong \bigoplus_{\psi \in \Ghat} \psi \otimes \psi^*$. The fiber now contains \textit{all} isotypic components of $G$, each with multiplicity $d_\psi$. For $D_4$: $K_{\mathrm{out}} = 8 = 1^2 + 1^2 + 1^2 + 1^2 + 2^2$, and all five irreps ($A_1, A_2, B_1, B_2, E$) are represented.
-
-\subsection{Resolution of the obstruction}
-
-With $\rho_G = \rho_{\mathrm{reg}}$, the output fiber $\C^{K_{\mathrm{out}}} = \C^{|G|}$ is canonically identified with $\C[G]$, and the fiber vector at pixel $x$ is a function on $G$ (by the identification $v \leftrightarrow \theta_v$ of Lemma~\ref{lem:reg_as_translation}). The $G$-Fourier transform of this function directly yields all Fourier blocks---no additional change-of-basis is required to interpret the output fiber itself as a function on $G$ and apply the group Fourier transform to it.
-
-\begin{remark}[Scope of completeness]\label{rem:scope}
-The per-pixel bispectrum (defined below) is a complete invariant of the fiber signal $\theta_x : G \to \C$, not directly of the original image $f$. The steerable convolution maps $f$ equivariantly to $\theta_x$, but the information content of $\theta_x$ about $f$ depends on the learned filter weights. Establishing completeness for $f$ would require an additional analysis of the steerable layer itself.
-\end{remark}
-
-\subsection{Steerable convolution with regular-rep fiber}
-
-The steerable convolution with regular-rep output takes the form of \eqref{eq:conv} with $K_{\mathrm{out}} = |G|$:
-\begin{equation}\label{eq:conv_reg}
-\Theta(x_0) = \sum_{x \in \Omega_s} \Psi(x_0 - x)\, f(x) \;\in\; \C^{|G|},
-\end{equation}
-where $\Psi(x) = \sum_{l=1}^L w_l\, \Psi_l(x)$ with $\Psi_l \in \Hom^G(\pi, \rho_{\mathrm{reg}} \otimes \rho_{\mathrm{in}}^*)$.
-
-\begin{proposition}[Equivariance]\label{prop:equiv_reg}
-Under the action of $(t, r) \in \Gamma$ on the input, the output satisfies
-\[
-\Theta'(r \cdot x_0 + t) = \rho_{\mathrm{reg}}(r)\, \Theta(x_0).
-\]
-In components: $\Theta'_g(r \cdot x) = \Theta_{r^{-1}g}(x)$ for all $g \in G$.
-\end{proposition}
-\begin{proof}
-This is a special case of \eqref{eq:equiv_general} with $\rho_G = \rho_{\mathrm{reg}}$. The component form follows from Lemma~\ref{lem:reg_as_translation}: $[\rho_{\mathrm{reg}}(r)\, \Theta(x)]_g = \Theta_{r^{-1}g}(x)$.
-\end{proof}
-
-\begin{remark}\label{rem:left_translation}
-The component form $\Theta'_g(r \cdot x) = \Theta_{r^{-1}g}(x)$ means that the group action on the fiber is left translation of the function $h \mapsto \Theta_h(x)$ on $G$. This is precisely the group action assumed by the $G$-Fourier transform equivariance (Proposition~\ref{prop:ft_equiv_prelim}) and the $G$-bispectrum completeness theorem (Theorem~\ref{thm:completeness_prelim}).
-\end{remark}
-
-\section{Per-pixel $G$-Fourier transform}
-
-\subsection{Definition}
-
-\begin{definition}[Per-pixel signal on $G$]\label{def:perpixel_signal}
-At each pixel $x \in \Z^2$, define the function $\theta_x : G \to \C$ by
-\begin{equation}\label{eq:theta_x}
-\theta_x(h) := \Theta_h(x).
-\end{equation}
-\end{definition}
-
-\begin{definition}[Per-pixel $G$-Fourier transform]\label{def:perpixel_ft}
-The per-pixel $G$-Fourier transform at irrep $\psi \in \Ghat$ is
-\begin{equation}\label{eq:perpixel_ft}
-\widehat{\Theta}(x, \psi) := \FF(\theta_x)_\psi = \sum_{h \in G} \Theta_h(x)\, \psi(h)^\dagger \;\in\; \C^{d_\psi \times d_\psi}.
-\end{equation}
-\end{definition}
-
-\begin{remark}[Relationship to the full-group Fourier transform]
-Definition~\ref{def:perpixel_ft} applies the standard $G$-Fourier transform \eqref{eq:gft} to the fiber vector at each pixel. It is \textbf{not} the full-group Fourier transform over $\Gamma = \Z^2 \ltimes G$, which would integrate over all translations and group elements simultaneously and produce a global descriptor without a spatial index. In our construction, spatial translation equivariance is handled by the convolutional structure (Section~\ref{sec:steerable_conv}); the per-pixel $G$-Fourier transform handles only the $G$-action on the fiber.
-\end{remark}
-
-\begin{remark}[No information loss]
-The map $\Theta(x) \mapsto \{\widehat{\Theta}(x, \psi)\}_{\psi \in \Ghat}$ is invertible by Fourier inversion on $G$ (Proposition~\ref{prop:inversion}). The total number of entries across all Fourier blocks is $\sum_{\psi} d_\psi^2 = |G|$, matching the fiber dimension.
-\end{remark}
-
-\subsection{Equivariance}
-
-\begin{proposition}[Equivariance of per-pixel Fourier blocks]\label{prop:perpixel_ft_equiv}
-Under the action of $r \in G$ on the input:
-\begin{equation}\label{eq:perpixel_ft_equiv}
-\widehat{\Theta}'(r \cdot x, \psi) = \widehat{\Theta}(x, \psi) \cdot \psi(r)^\dagger.
-\end{equation}
-\end{proposition}
-\begin{proof}
-By Proposition~\ref{prop:equiv_reg}, $\Theta'_g(r \cdot x) = \Theta_{r^{-1}g}(x)$. This means $\theta'_{r \cdot x}(g) = \theta_x(r^{-1}g)$, which is exactly the left translate $\theta_x^{r}$ from Proposition~\ref{prop:ft_equiv_prelim}. Therefore:
-\begin{align*}
-\widehat{\Theta}'(r \cdot x, \psi)
-&= \FF(\theta'_{r \cdot x})_\psi \\
-&= \FF(\theta_x^{r})_\psi \\
-&= \FF(\theta_x)_\psi \cdot \psi(r)^\dagger \qquad\text{[by Proposition~\ref{prop:ft_equiv_prelim}]} \\
-&= \widehat{\Theta}(x, \psi) \cdot \psi(r)^\dagger. \qedhere
-\end{align*}
-\end{proof}
-
-\section{Per-pixel $G$-bispectrum}
-
-\subsection{Per-pixel triple correlation}
-
-\begin{definition}[Per-pixel $G$-triple correlation]\label{def:perpixel_tc}
-At each pixel $x \in \Z^2$:
-\begin{equation}\label{eq:perpixel_tc}
-T(\theta_x)_{g_1, g_2} = \sum_{g \in G} \theta_x(g)\, \theta_x(g \cdot g_1)\, \theta_x(g \cdot g_2), \quad g_1, g_2 \in G.
-\end{equation}
-This is the $G$-triple correlation (Definition~\ref{def:tc}) applied to the per-pixel signal $\theta_x$.
-\end{definition}
-
-\begin{proposition}[Per-pixel TC invariance]\label{prop:perpixel_tc_inv}
-If the input is transformed by $r \in G$, then $T(\theta'_{r \cdot x}) = T(\theta_x)$.
-\end{proposition}
-\begin{proof}
-By Proposition~\ref{prop:equiv_reg}, $\theta'_{r \cdot x}(g) = \theta_x(r^{-1}g)$. By Proposition~\ref{prop:tc_inv} applied to $\theta_x$ with the left translation by $r$:
-\begin{align*}
-T(\theta'_{r \cdot x})_{g_1, g_2}
-&= \sum_{g \in G} \theta_x(r^{-1}g)\, \theta_x(r^{-1}g \cdot g_1)\, \theta_x(r^{-1}g \cdot g_2) \\
-&= \sum_{g' \in G} \theta_x(g')\, \theta_x(g' g_1)\, \theta_x(g' g_2) \qquad [g' = r^{-1}g]\\
-&= T(\theta_x)_{g_1, g_2}. \qedhere
-\end{align*}
-\end{proof}
-
-\subsection{Per-pixel bispectrum}
-
-\begin{definition}[Per-pixel $G$-bispectrum]\label{def:perpixel_bsp}
-At each pixel $x \in \Z^2$, for each pair of irreps $\psi_1, \psi_2 \in \Ghat$:
-\begin{equation}\label{eq:perpixel_bsp}
-\beta(x)_{\psi_1, \psi_2}
-= \bigl[\widehat{\Theta}(x,\psi_1) \otimes \widehat{\Theta}(x,\psi_2)\bigr]\,
-C_{\psi_1,\psi_2}\,
-\left[\bigoplus_{\psi \in \psi_1 \otimes \psi_2} \widehat{\Theta}(x,\psi)^\dagger\right]
-C_{\psi_1,\psi_2}^\dagger.
-\end{equation}
-This is the $G$-bispectrum (Definition~\ref{def:bsp}) applied to $\theta_x$, with the same multiplicity and ordering conventions.
-\end{definition}
-
-\begin{theorem}[Invariance of the per-pixel bispectrum]\label{thm:perpixel_bsp_inv}
-For all $r \in G$:
-\begin{equation}
-\beta'(r \cdot x)_{\psi_1, \psi_2} = \beta(x)_{\psi_1, \psi_2}.
-\end{equation}
-That is, the per-pixel bispectrum is $G$-invariant.
-\end{theorem}
-\begin{proof}
-By Theorem~\ref{thm:bsp_is_ft_tc}, $\beta(x)$ is the $(G \times G)$-Fourier transform of $T(\theta_x)$. By Proposition~\ref{prop:perpixel_tc_inv}, $T(\theta'_{r \cdot x}) = T(\theta_x)$. Therefore $\beta'(r \cdot x) = \beta(x)$.
-\end{proof}
-
-\begin{remark}[Translation equivariance]
-The per-pixel bispectrum $\beta$ inherits translation equivariance from the convolutional structure: if the input is translated by $t \in \Z^2$, then $\beta'(x + t) = \beta(x)$. The bispectrum map $x \mapsto \beta(x)$ is therefore a translation-equivariant, $G$-invariant feature map of the fiber signal.
-\end{remark}
-
-\section{Completeness: reconstruction up to group action}
-
-\subsection{Statement and proof}
-
-\begin{theorem}[Completeness of the per-pixel bispectrum]\label{thm:completeness}
-Let $\theta_{x_1}, \theta_{x_2} : G \to \C$ be the per-pixel fiber signals at two pixels, with $\det\bigl(\FF(\theta_{x_i})_\psi\bigr) \neq 0$ for all $\psi \in \Ghat$ and $i = 1, 2$. Then
-\begin{equation}
-\beta(x_1) = \beta(x_2)
-\quad\Longleftrightarrow\quad
-\exists\, r \in G \text{ such that } \theta_{x_2}(g) = \theta_{x_1}(r^{-1}g) \;\;\forall g \in G.
-\end{equation}
-Equivalently: $\Theta(x_2) = \rho_{\mathrm{reg}}(r)\, \Theta(x_1)$.
-\end{theorem}
-\begin{proof}
-The per-pixel bispectrum $\beta(x)$ is exactly the $G$-bispectrum of $\theta_x$ (Definition~\ref{def:perpixel_bsp}). The result follows immediately from Theorem~\ref{thm:completeness_prelim} (Kakarala) applied to the functions $\theta_{x_1}$ and $\theta_{x_2}$.
-\end{proof}
-
-We now give a constructive recovery procedure specialized to $G = D_n$, which additionally yields the selectivity result of Section~\ref{sec:selectivity}.
-
-\subsection{Irreducible representations of $D_n$}\label{sec:dn_irreps}
-
-Let $D_n = \langle a, x \mid a^n = x^2 = e,\; xax = a^{-1} \rangle$ be the dihedral group of order $2n$, where $a$ is rotation by $2\pi/n$ and $x$ is a reflection.
-
-\textbf{One-dimensional irreps.}
-For $n$ even ($n \geq 4$), there are four 1D irreps:
-\begin{alignat*}{3}
-&\rho_0 : \quad &&\rho_0(a^l x^m) = 1 &&\text{(trivial),} \\
-&\rho_{01} : \quad &&\rho_{01}(a^l x^m) = (-1)^m &&\text{(sign on reflections),} \\
-&\rho_{02} : \quad &&\rho_{02}(a^l x^m) = (-1)^l &&\text{(sign on half-rotations),} \\
-&\rho_{03} : \quad &&\rho_{03}(a^l x^m) = (-1)^{l+m}.
-\end{alignat*}
-For $n$ odd, only $\rho_0$ and $\rho_{01}$ exist (since $(-1)^l$ is not a well-defined character when $a^n = e$ with $n$ odd).
-
-\textbf{Two-dimensional irreps.}
-For $k = 1, \ldots, \lfloor(n-1)/2\rfloor$:
-\begin{equation}\label{eq:dn_2d_irrep}
-\rho_k(a^l x^m) = \begin{bmatrix} \cos\omega_k l & -\sin\omega_k l \\ \sin\omega_k l & \cos\omega_k l \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}^m, \quad \omega_k = \frac{2\pi k}{n}.
-\end{equation}
-
-\textbf{Dimension count.}
-For $n$ even: $4 \cdot 1^2 + (n/2 - 1) \cdot 2^2 = 4 + 2n - 4 = 2n = |D_n|$.
-For $n$ odd: $2 \cdot 1^2 + \frac{n-1}{2} \cdot 2^2 = 2 + 2(n-1) = 2n = |D_n|$.
-
-\subsection{Recovering $\FF(\theta_x)_{\rho_0}$}
-
-\begin{lemma}\label{lem:recover_rho0}
-The bispectral coefficient at $(\rho_0, \rho_0)$ satisfies
-\begin{equation}\label{eq:beta_00}
-\beta(\theta)_{\rho_0, \rho_0} = |\FF(\theta)_{\rho_0}|^2\, \FF(\theta)_{\rho_0}.
-\end{equation}
-\end{lemma}
-\begin{proof}
-Since $\rho_0$ is the trivial representation, $\FF(\theta)_{\rho_0} = \sum_{g} \theta(g) \in \C$ is a scalar. Also $\rho_0 \otimes \rho_0 = \rho_0$, and the CG matrix is the identity. Substituting into \eqref{eq:bsp}:
-\begin{align*}
-\beta(\theta)_{\rho_0, \rho_0}
-&= [\FF(\theta)_{\rho_0} \otimes \FF(\theta)_{\rho_0}] \cdot \FF(\theta)_{\rho_0}^\dagger \\
-&= \FF(\theta)_{\rho_0}^2 \cdot \overline{\FF(\theta)_{\rho_0}} \\
-&= |\FF(\theta)_{\rho_0}|^2 \cdot \FF(\theta)_{\rho_0}. \qedhere
-\end{align*}
-\end{proof}
-
-\textbf{Recovery.} From $\beta(\theta)_{\rho_0,\rho_0}$:
-\begin{equation}\label{eq:recover_rho0}
-|\FF(\theta)_{\rho_0}| = |\beta(\theta)_{\rho_0,\rho_0}|^{1/3}, \qquad
-\arg\bigl(\FF(\theta)_{\rho_0}\bigr) = \arg\bigl(\beta(\theta)_{\rho_0,\rho_0}\bigr).
-\end{equation}
-This uniquely determines $\FF(\theta)_{\rho_0}$.
-
-\subsection{Recovering $\FF(\theta_x)_{\rho_1}$}
-
-We choose the 2D irrep $\rho_1$ (defined in \eqref{eq:dn_2d_irrep} with $k = 1$) as the \textit{generator irrep}. Its tensor powers, combined via the Kronecker table, generate all irreps of $D_n$ (Proposition~\ref{prop:generation} below).
-
-\begin{lemma}\label{lem:recover_rho1}
-The bispectral coefficient at $(\rho_0, \rho_1)$ satisfies
-\begin{equation}\label{eq:beta_01}
-\beta(\theta)_{\rho_0, \rho_1} = \FF(\theta)_{\rho_0} \cdot \FF(\theta)_{\rho_1} \cdot \FF(\theta)_{\rho_1}^\dagger.
-\end{equation}
-\end{lemma}
-\begin{proof}
-Since $\rho_0 \otimes \rho_1 = \rho_1$ and the CG matrix for $\rho_0 \otimes \rho_1$ is the identity:
-\begin{align*}
-\beta(\theta)_{\rho_0, \rho_1}
-&= [\FF(\theta)_{\rho_0} \otimes \FF(\theta)_{\rho_1}] \cdot \FF(\theta)_{\rho_1}^\dagger \\
-&= \FF(\theta)_{\rho_0} \cdot \FF(\theta)_{\rho_1} \cdot \FF(\theta)_{\rho_1}^\dagger. \qedhere
-\end{align*}
-\end{proof}
-
-\textbf{Recovery.} Since $\FF(\theta)_{\rho_0}$ is known from \eqref{eq:recover_rho0}:
-\begin{equation}\label{eq:recover_rho1_gram}
-\FF(\theta)_{\rho_1} \cdot \FF(\theta)_{\rho_1}^\dagger = \frac{\beta(\theta)_{\rho_0, \rho_1}}{\FF(\theta)_{\rho_0}}.
-\end{equation}
-The right-hand side is a known Hermitian positive semi-definite matrix (positive definite under non-degeneracy). Compute its eigenvalue decomposition $U \Lambda U^\dagger = \FF(\theta)_{\rho_1} \cdot \FF(\theta)_{\rho_1}^\dagger$, and set
-\begin{equation}\label{eq:recover_rho1}
-\FF(\theta)_{\rho_1} = U \Lambda^{1/2} U^\dagger \cdot W,
-\end{equation}
-where $W$ is unitary. This determines $\FF(\theta)_{\rho_1}$ up to a right-unitary ambiguity $W$. By Theorem~\ref{thm:completeness_prelim}, the full bispectral constraints reduce this ambiguity to a left-translation orbit of the reconstructed signal; we do not provide an independent proof of uniqueness for the constructive scheme here.
-
-\subsection{Bootstrapping remaining irreps}
-
-\begin{lemma}\label{lem:bootstrap}
-Let $\psi_a, \psi_b \in \Ghat$ with $\FF(\theta)_{\psi_a}$ and $\FF(\theta)_{\psi_b}$ known and invertible. Then every irrep $\psi$ appearing in the tensor decomposition $\psi_a \otimes \psi_b$ can be recovered from $\beta(\theta)_{\psi_a, \psi_b}$:
-\begin{equation}\label{eq:bootstrap}
-\bigoplus_{\psi \in \psi_a \otimes \psi_b} \FF(\theta)_\psi^\dagger
-= C_{\psi_a,\psi_b}^\dagger\,
-\bigl[\FF(\theta)_{\psi_a} \otimes \FF(\theta)_{\psi_b}\bigr]^{-1}\,
-\beta(\theta)_{\psi_a, \psi_b}\,
-C_{\psi_a,\psi_b}.
-\end{equation}
-The individual blocks $\FF(\theta)_\psi^\dagger$ are read off from the block-diagonal structure, once a CG basis with fixed ordering of irreducible summands is chosen.
-\end{lemma}
-\begin{proof}
-From \eqref{eq:bsp}, left-multiply by $[\FF(\theta)_{\psi_a} \otimes \FF(\theta)_{\psi_b}]^{-1}$ and right-multiply by $C_{\psi_a,\psi_b}$:
-\begin{align*}
-[\FF_{\psi_a} \otimes \FF_{\psi_b}]^{-1} \beta_{\psi_a,\psi_b}\, C_{\psi_a,\psi_b}
-&= C_{\psi_a,\psi_b} \left[\bigoplus_\psi \FF_\psi^\dagger\right] C_{\psi_a,\psi_b}^\dagger\, C_{\psi_a,\psi_b} \\
-&= C_{\psi_a,\psi_b} \left[\bigoplus_\psi \FF_\psi^\dagger\right].
-\end{align*}
-Left-multiplying by $C_{\psi_a,\psi_b}^\dagger$ (which is unitary) yields \eqref{eq:bootstrap}.
-\end{proof}
-
-\subsection{Generating all irreps for $D_n$}
-
-We show that the generator irrep $\rho_1$ iteratively produces all irreps of $D_n$ through the bootstrap procedure.
-
-\begin{proposition}\label{prop:generation}
-For $D_n$ with $n > 2$:
-\begin{enumerate}
-    \item For $k = 2, \ldots, \lfloor(n-1)/2\rfloor$: the 2D irrep $\rho_k$ appears in $\rho_1 \otimes \rho_{k-1}$.
-    \item The 1D irrep $\rho_{01}$ appears in $\rho_1 \otimes \rho_1$.
-    \item For $n$ even: the 1D irreps $\rho_{02}, \rho_{03}$ appear in $\rho_1 \otimes \rho_{n/2-1}$.
-\end{enumerate}
-\end{proposition}
-\begin{proof}
-We use character inner products. For 2D irreps of $D_n$, the characters satisfy $\chi_{\rho_k}(a^l x^m) = 2\cos(\omega_k l)$ if $m = 0$ and $0$ if $m = 1$ (see Section~\ref{sec:dn_irreps}). The tensor product decomposition $\psi_1 \otimes \psi_2 \supseteq \psi$ is detected by the multiplicity
-\[
-\langle \chi_{\psi_1 \otimes \psi_2}, \chi_\psi \rangle
-= \frac{1}{|G|} \sum_{g \in G} \chi_{\psi_1}(g)\, \chi_{\psi_2}(g)\, \overline{\chi_\psi(g)}.
-\]
-
-For two 2D irreps $\rho_i, \rho_j$:
-\begin{align*}
-\langle \chi_{\rho_i \otimes \rho_j}, \chi_{\rho_k} \rangle
-&= \frac{1}{2n} \sum_{l=0}^{n-1} 4\cos(\omega_i l)\cos(\omega_j l) \cdot 2\cos(\omega_k l) \\
-&= \frac{4}{2n} \sum_{l=0}^{n-1} \bigl[\cos(\omega_{i+j} l) + \cos(\omega_{|i-j|} l)\bigr] \cos(\omega_k l).
-\end{align*}
-Using $\sum_{l=0}^{n-1} \cos(\omega_m l) = n$ if $n \mid m$ and $0$ otherwise, this is nonzero if and only if $k = i + j$ or $k = |i - j|$ (with indices taken modulo $n$ and folded into the range $\{1, \ldots, \lfloor(n-1)/2\rfloor\}$). That is, the standard tensor-product rule for 2D dihedral irreps is
-\[
-\rho_i \otimes \rho_j \;\cong\; \rho_{i+j} \oplus \rho_{|i-j|}
-\]
-with boundary conventions as above. In particular, $\rho_k$ appears in $\rho_1 \otimes \rho_{k-1}$.
-
-For the 1D irreps, $\rho_1 \otimes \rho_1$ decomposes as $\rho_0 \oplus \rho_{01} \oplus \rho_2$ (when $2 < n/2$), which is verified by $\langle \chi_{\rho_1 \otimes \rho_1}, \chi_{\rho_{01}} \rangle = 1$. For $n$ even, $\rho_1 \otimes \rho_{n/2-1}$ has $1 + (n/2 - 1) = n/2$ as the sum index, hitting the boundary index $k = n/2$, at which the corresponding 2D construction becomes reducible and decomposes as $\rho_{02} \oplus \rho_{03}$.
-\end{proof}
-
-\subsection{Constructive recovery procedure for $D_n$}
-
-The following procedure recovers $\FF(\theta_x)$ from $\beta(x)$ for $G = D_n$, under the non-degeneracy assumption $\det(\FF(\theta_x)_\psi) \neq 0$ for all $\psi \in \Ghat$.
-
-\textbf{Step 1.} Recover $\FF(\theta_x)_{\rho_0}$ from $\beta(x)_{\rho_0,\rho_0}$ using Lemma~\ref{lem:recover_rho0} and \eqref{eq:recover_rho0}.
-
-\textbf{Step 2.} Recover $\FF(\theta_x)_{\rho_1}$ (up to unitary $W$) from $\beta(x)_{\rho_0, \rho_1}$ using Lemma~\ref{lem:recover_rho1} and \eqref{eq:recover_rho1}.
-
-\textbf{Step 3.} For $k = 1, \ldots, \lfloor(n-1)/2\rfloor$: use the bootstrap (Lemma~\ref{lem:bootstrap}) with $\beta(x)_{\rho_1, \rho_k}$ to recover the Fourier blocks of all irreps appearing in $\rho_1 \otimes \rho_k$. By Proposition~\ref{prop:generation}:
-\begin{itemize}
-    \item $\beta_{\rho_1, \rho_1}$ yields $\rho_2$ and the 1D irrep $\rho_{01}$.
-    \item $\beta_{\rho_1, \rho_k}$ for $k = 2, \ldots, \lfloor(n-1)/2\rfloor - 1$ yields $\rho_{k+1}$.
-    \item $\beta_{\rho_1, \rho_{n/2-1}}$ (for $n$ even) yields $\rho_{02}$ and $\rho_{03}$.
-\end{itemize}
-
-\textbf{Step 4.} All Fourier blocks $\FF(\theta_x)_\psi$ are now known. Apply Fourier inversion (Proposition~\ref{prop:inversion}) to recover $\theta_x$ up to the $G$-action encoded in $W$.
-
-Completeness of this procedure is guaranteed by Theorem~\ref{thm:completeness} (via Kakarala). The procedure provides an explicit algorithm; by Theorem~\ref{thm:completeness}, the residual unitary ambiguity $W$ from Step~2 corresponds globally to a single group translate.
-
-\section{Selectivity}\label{sec:selectivity}
-
-\subsection{The selective bispectrum}
-
-The recovery procedure of the previous section uses only a small subset of bispectral coefficients. We formalize this.
-
-\begin{definition}[Selective $G$-bispectrum]\label{def:selective}
-A \textit{selective $G$-bispectrum} is a subset $\beta_{\mathrm{sel}}(\theta) \subset \{\beta(\theta)_{\psi_1,\psi_2}\}_{(\psi_1,\psi_2) \in \Ghat \times \Ghat}$ that is sufficient for reconstruction, i.e., from which $\FF(\theta)$ can be recovered (up to $G$-action) under the non-degeneracy assumption.
-\end{definition}
-
-\subsection{Selectivity for dihedral groups}
-
-\begin{theorem}[Selectivity for $D_n$]\label{thm:selectivity}
-For $G = D_n$ with $n \geq 4$ even, the per-pixel selective $G$-bispectrum requires at most $\lfloor(n-1)/2\rfloor + 2$ bispectral blocks under the assumption $\det(\FF(\theta_x)_\psi) \neq 0$ for all $\psi \in \Ghat$. The total number of matrix entries is $1 + 4 + 16\lfloor(n-1)/2\rfloor$.
-\end{theorem}
-\begin{proof}
-The selective set consists of:
-\begin{itemize}
-    \item $\beta_{\rho_0,\rho_0}$: a $d_{\rho_0}^2 \times d_{\rho_0}^2 = 1 \times 1$ matrix (1 entry).
-    \item $\beta_{\rho_0,\rho_1}$: a $d_{\rho_0}d_{\rho_1} \times d_{\rho_0}d_{\rho_1} = 2 \times 2$ matrix (4 entries).
-    \item $\beta_{\rho_1,\rho_k}$ for $k = 1, \ldots, \lfloor(n-1)/2\rfloor$: each a $d_{\rho_1}d_{\rho_k} \times d_{\rho_1}d_{\rho_k} = 4 \times 4$ matrix (16 entries each). This is $\lfloor(n-1)/2\rfloor$ coefficients.
-\end{itemize}
-Total bispectral blocks: $2 + \lfloor(n-1)/2\rfloor$.
-
-Total entries: $1 + 4 + 16 \cdot \lfloor(n-1)/2\rfloor$.
-\end{proof}
-
-\begin{remark}
-We state the count for even $n$; the odd case is analogous and slightly smaller, since the 1D irreps $\rho_{02}, \rho_{03}$ do not exist and the selective set can be reduced accordingly. The stated bound remains valid for odd $n$.
-\end{remark}
-
-\subsection{Concrete counts}
-
-The full bispectrum has $\sum_{(\psi_1,\psi_2)} (d_{\psi_1} d_{\psi_2})^2 = \bigl(\sum_\psi d_\psi^2\bigr)^2 = |G|^2$ entries.
-
-\begin{table}[h]
-\centering
-\begin{tabular}{lrrrr}
-\toprule
-\textbf{Group} & $|G|$ & \textbf{Full bisp.\ entries} & \textbf{Sel.\ blocks} & \textbf{Sel.\ entries} \\
-\midrule
-$D_4$ & 8 & 64 & 3 & 21 \\
-$D_8$ & 16 & 256 & 5 & 53 \\
-$D_{16}$ & 32 & 1024 & 9 & 117 \\
-$D_{32}$ & 64 & 4096 & 17 & 245 \\
-\bottomrule
-\end{tabular}
-\caption{Comparison of full and selective per-pixel bispectrum sizes for dihedral groups. Entries are complex matrix entries.}
-\label{tab:counts}
-\end{table}
-
-\begin{corollary}\label{cor:scaling}
-The selective per-pixel $G$-bispectrum uses $O(|G|)$ entries per pixel, versus $O(|G|^2)$ for the full bispectrum.
-\end{corollary}
-
-\section{Discussion}
-
-\subsection{Complete data flow}
-
-The construction yields the following pipeline:
-\[
-f : \Z^2 \to \C^{K_{\mathrm{in}}}
-\;\xrightarrow{\text{steerable conv}}\;
-\Theta : \Z^2 \to \C[G]
-\;\xrightarrow{\text{per-pixel } G\text{-FT}}\;
-\widehat{\Theta}(x, \psi)
-\;\xrightarrow{\text{sel.\ bisp.}}\;
-\beta : \Z^2 \to \C^{O(|G|)}.
-\]
-
-\begin{itemize}
-    \item $\beta$ is \textbf{$G$-invariant}: Theorem~\ref{thm:perpixel_bsp_inv}.
-    \item $\beta$ is \textbf{translation-equivariant}: inherited from the convolutional structure.
-    \item $\beta$ is a \textbf{generically complete} invariant of the fiber signal $\theta_x$: Theorem~\ref{thm:completeness}.
-    \item $\beta$ uses $O(|G|)$ entries per pixel: Corollary~\ref{cor:scaling}.
-\end{itemize}
-
-Completeness for the original signal $f$ requires that the steerable convolution layer produce a sufficiently rich fiber signal $\theta_x$; this depends on the learned filters and is not addressed by the bispectral analysis alone.
-
-\subsection{Relationship to the full-group approach}
-
-An alternative approach defines the Fourier transform over the full group $\Gamma = \Z^2 \ltimes G$, integrating over all translations and all elements of $G$ simultaneously. This produces a global descriptor without a spatial index: the resulting bispectrum is invariant to the full group $\Gamma$ (translations and $G$ simultaneously), but it is not a spatial feature map and cannot be composed as a layer in a deep convolutional network.
-
-Our approach decomposes the problem: spatial translation equivariance is handled by the convolutional structure, and $G$-invariance is handled per-pixel by the bispectrum. This yields a composable $G$-invariant layer that serves as a drop-in replacement for norm or gated nonlinearities in steerable architectures.
-
-\subsection{Open questions}
-
-The construction requires that the steerable CNN framework (e.g., \texttt{escnn}) supports setting the output fiber type to the regular representation $\rho_{\mathrm{reg}}$ of $G$. In \texttt{escnn}, this corresponds to setting the output type to $A_1 \oplus A_2 \oplus B_1 \oplus B_2 \oplus 2E$ for $G = D_4$ (total fiber dimension 8). Verifying that this is compatible with the intertwiner basis computation and the learned filter parameterization is an implementation question that remains to be validated.
-
-\begin{thebibliography}{10}
-
-\bibitem{kakarala1992}
-R.~Kakarala.
-\newblock Triple correlation on groups.
-\newblock PhD thesis, University of California, Irvine, 1992.
-
-\bibitem{kakarala2009}
-R.~Kakarala.
-\newblock Completeness of bispectrum on compact groups.
-\newblock \textit{IEEE Transactions on Information Theory}, 2009.
-
-\bibitem{mataigne2024}
-S.~Mataigne, J.~Mathe, S.~Sanborn, C.~Shewmake, and N.~Miolane.
-\newblock The selective {$G$}-bispectrum and its inversion: Applications to {$G$}-invariant networks.
-\newblock In \textit{NeurIPS}, 2024.
-
-\bibitem{weiler2019}
-M.~Weiler and G.~Cesa.
-\newblock General {E(2)}-equivariant steerable {CNNs}.
-\newblock In \textit{NeurIPS}, 2019.
-
-\bibitem{kondor2008}
-R.~Kondor.
-\newblock Group theoretical methods in machine learning.
-\newblock PhD thesis, Columbia University, 2008.
-
-\bibitem{sanborn2023bispectral}
-S.~Sanborn, C.~Shewmake, and N.~Miolane.
-\newblock Bispectral neural networks.
-\newblock In \textit{ICLR}, 2023.
-
-\end{thebibliography}
-
-\end{document}
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diff --git a/docs/sanbord_miolane_2025/references.bib b/docs/sanbord_miolane_2025/references.bib
deleted file mode 100644
index 235c097..0000000
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+++ /dev/null
@@ -1,5510 +0,0 @@
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-@article{olshausen1996emergence,
-  title={Emergence of simple-cell receptive field properties by learning a sparse code for natural images},
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-author = {Cohen, Taco S. and Geiger, Mario and K{\"{o}}hler, Jonas and Welling, Max},
-eprint = {1801.10130},
-journal = {6th International Conference on Learning Representations, ICLR 2018 - Conference Track Proceedings},
-number = {3},
-pages = {1--15},
-title = {{Spherical CNNs}},
-year = {2018}
-}
-
-@article{esteves2018,
-archivePrefix = {arXiv},
-arxivId = {1711.06721},
-author = {Esteves, Carlos and Allen-Blanchette, Christine and Makadia, Ameesh and Daniilidis, Kostas},
-doi = {10.1007/s11263-019-01220-1},
-eprint = {1711.06721},
-issn = {15731405},
-journal = {International Journal of Computer Vision},
-keywords = {3D vision,Equivariance,Sphere,Spherical CNN},
-number = {3},
-pages = {588--600},
-title = {{Learning SO(3) Equivariant Representations with Spherical CNNs}},
-volume = {128},
-year = {2018}
-}
-
-@inproceedings{boosma2017,
-author = {Boomsma, Wouter and Frellsen, Jes},
-title = {Spherical Convolutions and Their Application in Molecular Modelling},
-year = {2017},
-isbn = {9781510860964},
-publisher = {Curran Associates Inc.},
-address = {Red Hook, NY, USA},
-booktitle = {Proceedings of the 31st International Conference on Neural Information Processing Systems},
-pages = {3436–3446},
-numpages = {11},
-location = {Long Beach, California, USA},
-series = {NIPS'17}
-}
-
-@article{su2017,
-author = {Su, Yu Chuan and Grauman, Kristen},
-issn = {23318422},
-journal = {arXiv},
-number = {Nips},
-title = {{Learning spherical convolution for fast features from 360° imagery}},
-year = {2017}
-}
-
-@article{perraudin2019,
-title = {DeepSphere: Efficient spherical convolutional neural network with HEALPix sampling for cosmological applications},
-journal = {Astronomy and Computing},
-volume = {27},
-pages = {130-146},
-year = {2019},
-issn = {2213-1337},
-doi = {https://doi.org/10.1016/j.ascom.2019.03.004},
-url = {https://www.sciencedirect.com/science/article/pii/S2213133718301392},
-author = {N. Perraudin and M. Defferrard and T. Kacprzak and R. Sgier},
-keywords = {Spherical convolutional neural network, DeepSphere, Graph CNN, Cosmological data analysis, Mass mapping},
-}
-
-
-@inproceedings{kondor2018clebshgordan,
- author = {Kondor, Risi and Lin, Zhen and Trivedi, Shubhendu},
- booktitle = {Advances in Neural Information Processing Systems},
- editor = {S. Bengio and H. Wallach and H. Larochelle and K. Grauman and N. Cesa-Bianchi and R. Garnett},
- pages = {},
- publisher = {Curran Associates, Inc.},
- title = {Clebsch\textendash Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network},
- url = {https://proceedings.neurips.cc/paper/2018/file/a3fc981af450752046be179185ebc8b5-Paper.pdf},
- volume = {31},
- year = {2018}
-}
-
-@misc{jiang2019spherical,
-      title={Spherical CNNs on Unstructured Grids},
-      author={Chiyu "Max" Jiang and Jingwei Huang and Karthik Kashinath and Prabhat and Philip Marcus and Matthias Niessner},
-      year={2019},
-      eprint={1901.02039},
-      archivePrefix={arXiv},
-      primaryClass={cs.CV}
-}
-
-@article{hutchinson2020lietransformer,
-archivePrefix = {arXiv},
-arxivId = {2012.10885},
-author = {Hutchinson, Michael and Lan, Charline Le and Zaidi, Sheheryar and Dupont, Emilien and Teh, Yee Whye and Kim, Hyunjik},
-eprint = {2012.10885},
-title = {{LieTransformer: Equivariant self-attention for Lie Groups}},
-url = {http://arxiv.org/abs/2012.10885},
-year = {2020}
-}
-
-
-@article{cohen2015transformation,
-  title={Transformation Properties of Learned Visual Representations},
-  author={T. Cohen and M. Welling},
-  journal={CoRR},
-  year={2015},
-  volume={abs/1412.7659}
-}
-
-@InProceedings{miolane2020,
-author = {Miolane, Nina and Poitevin, Frederic and Li, Yee-Ting and Holmes, Susan},
-title = {Estimation of Orientation and Camera Parameters From Cryo-Electron Microscopy Images With Variational Autoencoders and Generative Adversarial Networks},
-booktitle = {Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops},
-month = {June},
-year = {2020}
-}
-
-@article{Lopez2018InformationBayes,
-title = {{Information constraints on auto-encoding variational Bayes}},
-year = {2018},
-journal = {Advances in Neural Information Processing Systems},
-author = {Lopez, Romain and Regier, Jeffrey and Jordan, Michael I. and Yosef, Nir},
-number = {1},
-pages = {6114--6125},
-volume = {2018-Decem},
-issn = {10495258},
-arxivId = {1805.08672}
-}
-@article{Louizos2015TheAutoencoder,
-title = {{The Variational Fair Autoencoder}},
-year = {2015},
-author = {Louizos, Christos and Swersky, Kevin and Li, Yujia and Welling, Max and Zemel, Richard},
-pages = {1--11},
-url = {http://arxiv.org/abs/1511.00830},
-arxivId = {1511.00830}
-}
-@article{Kim2018DisentanglingFactorising,
-title = {{Disentangling by factorising}},
-year = {2018},
-journal = {35th International Conference on Machine Learning, ICML 2018},
-author = {Kim, Hyunjik and Mnih, Andriy},
-pages = {4153--4171},
-volume = {6},
-isbn = {9781510867963},
-arxivId = {1802.05983}
-}
-@article{Detlefsen2019ExplicitModels,
-title = {{Explicit Disentanglement of Appearance and Perspective in Generative Models}},
-year = {2019},
-author = {Detlefsen, Nicki Skafte and Hauberg, Søren},
-month = {6},
-arxivId = {1906.11881}
-}
-
-@article{Bepler2019ExplicitlySpatial-VAE,
-title = {{Explicitly disentangling image content from translation and rotation with spatial-VAE}},
-year = {2019},
-author = {Bepler, Tristan and Zhong, Ellen D. and Kelley, Kotaro and Brignole, Edward and Berger, Bonnie},
-month = {9},
-arxivId = {1909.11663}
-}
-
-@misc{chen2019isolating,
-      title={Isolating Sources of Disentanglement in Variational Autoencoders},
-      author={Ricky T. Q. Chen and Xuechen Li and Roger Grosse and David Duvenaud},
-      year={2019},
-      eprint={1802.04942},
-      archivePrefix={arXiv},
-      primaryClass={cs.LG}
-}
-
-@article{falorsi2018,
-archivePrefix = {arXiv},
-arxivId = {1807.04689},
-author = {Falorsi, Luca and de Haan, Pim and Davidson, Tim R and {De Cao}, Nicola and Weiler, Maurice and Forr{\'{e}}, Patrick and Cohen, Taco S},
-eprint = {1807.04689},
-title = {{Explorations in Homeomorphic Variational Auto-Encoding}},
-url = {http://arxiv.org/abs/1807.04689},
-year = {2018}
-}
-
-@inproceedings{he2016deep,
-  title={Deep residual learning for image recognition},
-  author={He, Kaiming and Zhang, Xiangyu and Ren, Shaoqing and Sun, Jian},
-  booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition},
-  pages={770--778},
-  year={2016}
-}
-
-@article{barbu2019objectnet,
-  title={Objectnet: A large-scale bias-controlled dataset for pushing the limits of object recognition models},
-  author={Barbu, Andrei and Mayo, David and Alverio, Julian and Luo, William and Wang, Christopher and Gutfreund, Dan and Tenenbaum, Josh and Katz, Boris},
-  journal={Advances in neural information processing systems},
-  volume={32},
-  pages={9453--9463},
-  year={2019}
-}
-
-@book{gallier2020differential,
-  title={Differential Geometry and Lie Groups: A Computational Perspective},
-  author={Gallier, Jean and Quaintance, Jocelyn},
-  volume={12},
-  year={2020},
-  publisher={Springer Nature}
-}
-
-@article{chau2020disentangling,
-  title={Disentangling images with Lie group transformations and sparse coding},
-  author={Chau, Ho Yin and Qiu, Frank and Chen, Yubei and Olshausen, Bruno},
-  journal={arXiv preprint arXiv:2012.12071},
-  year={2020}
-}
-
-
-
-@article{o1998place,
-  title={Place cells, navigational accuracy, and the human hippocampus},
-  author={O'Keefe, John and Burgess, Neil and Donnett, James G and Jeffery, Kathryn J and Maguire, Eleanor A},
-  journal={Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences},
-  volume={353},
-  number={1373},
-  pages={1333--1340},
-  year={1998},
-  publisher={The Royal Society}
-}
-@article{rolls1986size,
-  title={Size and contrast have only small effects on the responses to faces of neurons in the cortex of the superior temporal sulcus of the monkey},
-  author={Rolls, ET and Baylis, GC},
-  journal={Experimental brain research},
-  volume={65},
-  number={1},
-  pages={38--48},
-  year={1986},
-  publisher={Springer}
-}
-
-
-@inproceedings{sanborn2019node2motif,
-  title={Node2Motif: Hierarchical invariant embeddings of structured graphs using the bispectrum},
-  author={Sanborn, Sophia and Mehta, Ram and Shutty, Noah and Hillar, Christopher},
-  booktitle={NeurIPS Graph Representation Learning Workshop},
-  year={2019}
-}
-
-
-@article{wakabayashi2018self,
-  title={Self-driving uber car kills pedestrian in arizona, where robots roam},
-  author={Wakabayashi, Daisuke},
-  journal={The New York Times},
-  volume={3},
-  pages={19},
-  year={2018}
-}
-
-@article{logothetis1996visual,
-  title={Visual object recognition},
-  author={Logothetis, Nikos K and Sheinberg, David L},
-  journal={Annual review of neuroscience},
-  volume={19},
-  number={1},
-  pages={577--621},
-  year={1996},
-  publisher={Annual Reviews 4139 El Camino Way, PO Box 10139, Palo Alto, CA 94303-0139, USA}
-}
-
-@inproceedings{ecker_2019,
-  author = "A. S. Ecker and F. H. Sinz and E. Froudarakis and P. G. Fahey and S. A. Cadena and E. Y. Walker and E. Cobos and J. Reimer and A. S. Tolias and M. Bethge",
-  title = "A rotation-equivariant convolutional neural network model of primary visual cortex",
-  year = 2019,
-  journal = "International Conference on Learning Representations (ICLR)",
-  month = "May",
-  keywords = "v1, system identification, microns, convolutional neural network, rotation equivariance",
-  url = "https://openreview.net/forum?id=H1fU8iAqKX"
-}
-
-@article{barlow1972single,
-  title={Single units and sensation: a neuron doctrine for perceptual psychology?},
-  author={Barlow, Horace B},
-  journal={Perception},
-  volume={1},
-  number={4},
-  pages={371--394},
-  year={1972},
-  publisher={SAGE Publications Sage UK: London, England}
-}
-
-@article{ito1995size,
-  title={Size and position invariance of neuronal responses in monkey inferotemporal cortex},
-  author={Ito, Minami and Tamura, Hiroshi and Fujita, Ichiro and Tanaka, Keiji},
-  journal={Journal of neurophysiology},
-  volume={73},
-  number={1},
-  pages={218--226},
-  year={1995},
-  publisher={American Physiological Society Bethesda, MD}
-}
-@article{perrett1982visual,
-  title={Visual neurones responsive to faces in the monkey temporal cortex},
-  author={Perrett, David I and Rolls, Edmond T and Caan, Woody},
-  journal={Experimental brain research},
-  volume={47},
-  number={3},
-  pages={329--342},
-  year={1982},
-  publisher={Springer}
-}
-
-@article{riesenhuber1999hierarchical,
-  title={Hierarchical models of object recognition in cortex},
-  author={Riesenhuber, Maximilian and Poggio, Tomaso},
-  journal={Nature neuroscience},
-  volume={2},
-  number={11},
-  pages={1019},
-  year={1999},
-  publisher={Nature Publishing Group}
-}
-
-@article{adelson1985spatiotemporal,
-  title={Spatiotemporal energy models for the perception of motion},
-  author={Adelson, Edward H and Bergen, James R},
-  journal={Josa a},
-  volume={2},
-  number={2},
-  pages={284--299},
-  year={1985},
-  publisher={Optical Society of America}
-}
-
-
-@ARTICLE{Jacobsen2018-er,
-  title         = "Excessive Invariance Causes Adversarial Vulnerability",
-  author        = "Jacobsen, J{\"o}rn-Henrik and Behrmann, Jens and Zemel,
-                   Richard and Bethge, Matthias",
-  abstract      = "Despite their impressive performance, deep neural networks
-                   exhibit striking failures on out-of-distribution inputs. One
-                   core idea of adversarial example research is to reveal
-                   neural network errors under such distribution shifts. We
-                   decompose these errors into two complementary sources:
-                   sensitivity and invariance. We show deep networks are not
-                   only too sensitive to task-irrelevant changes of their
-                   input, as is well-known from epsilon-adversarial examples,
-                   but are also too invariant to a wide range of task-relevant
-                   changes, thus making vast regions in input space vulnerable
-                   to adversarial attacks. We show such excessive invariance
-                   occurs across various tasks and architecture types. On MNIST
-                   and ImageNet one can manipulate the class-specific content
-                   of almost any image without changing the hidden activations.
-                   We identify an insufficiency of the standard cross-entropy
-                   loss as a reason for these failures. Further, we extend this
-                   objective based on an information-theoretic analysis so it
-                   encourages the model to consider all task-dependent features
-                   in its decision. This provides the first approach tailored
-                   explicitly to overcome excessive invariance and resulting
-                   vulnerabilities.",
-  month         =  nov,
-  year          =  2018,
-  archivePrefix = "arXiv",
-  primaryClass  = "cs.LG",
-  eprint        = "1811.00401"
-}
-
-
-@inproceedings{scholkopf2001kernel,
-  title={The kernel trick for distances},
-  author={Sch{\"o}lkopf, Bernhard},
-  booktitle={Advances in neural information processing systems},
-  pages={301--307},
-  year={2001}
-}
-
-@book{vapnik2013nature,
-  title={The nature of statistical learning theory},
-  author={Vapnik, Vladimir},
-  year={2013},
-  publisher={Springer science \& business media}
-}
-
-@article{mcinnes2018umap,
-  title={Umap: Uniform manifold approximation and projection for dimension reduction},
-  author={McInnes, Leland and Healy, John and Melville, James},
-  journal={arXiv preprint arXiv:1802.03426},
-  year={2018}
-}
-@article{belkin2003laplacian,
-  title={Laplacian eigenmaps for dimensionality reduction and data representation},
-  author={Belkin, Mikhail and Niyogi, Partha},
-  journal={Neural computation},
-  volume={15},
-  number={6},
-  pages={1373--1396},
-  year={2003},
-  publisher={MIT Press}
-}
-
-@article{balasubramanian2002isomap,
-  title={The isomap algorithm and topological stability},
-  author={Balasubramanian, Mukund and Schwartz, Eric L},
-  journal={Science},
-  volume={295},
-  number={5552},
-  pages={7--7},
-  year={2002},
-  publisher={American Association for the Advancement of Science}
-}
-
-
-@ARTICLE{Geirhos2018-ls,
-  title         = "{ImageNet-trained} {CNNs} are biased towards texture;
-                   increasing shape bias improves accuracy and robustness",
-  author        = "Geirhos, Robert and Rubisch, Patricia and Michaelis, Claudio
-                   and Bethge, Matthias and Wichmann, Felix A and Brendel,
-                   Wieland",
-  abstract      = "Convolutional Neural Networks (CNNs) are commonly thought to
-                   recognise objects by learning increasingly complex
-                   representations of object shapes. Some recent studies
-                   suggest a more important role of image textures. We here put
-                   these conflicting hypotheses to a quantitative test by
-                   evaluating CNNs and human observers on images with a
-                   texture-shape cue conflict. We show that ImageNet-trained
-                   CNNs are strongly biased towards recognising textures rather
-                   than shapes, which is in stark contrast to human behavioural
-                   evidence and reveals fundamentally different classification
-                   strategies. We then demonstrate that the same standard
-                   architecture (ResNet-50) that learns a texture-based
-                   representation on ImageNet is able to learn a shape-based
-                   representation instead when trained on
-                   ``Stylized-ImageNet'', a stylized version of ImageNet. This
-                   provides a much better fit for human behavioural performance
-                   in our well-controlled psychophysical lab setting (nine
-                   experiments totalling 48,560 psychophysical trials across 97
-                   observers) and comes with a number of unexpected emergent
-                   benefits such as improved object detection performance and
-                   previously unseen robustness towards a wide range of image
-                   distortions, highlighting advantages of a shape-based
-                   representation.",
-  month         =  nov,
-  year          =  2018,
-  archivePrefix = "arXiv",
-  primaryClass  = "cs.CV",
-  eprint        = "1811.12231"
-}
-
-
-@article{foldiak1991learning,
-  title={Learning invariance from transformation sequences},
-  author={F{\"o}ldi{\'a}k, Peter},
-  journal={Neural Computation},
-  volume={3},
-  number={2},
-  pages={194--200},
-  year={1991},
-  publisher={MITP}
-}
-
-@article{dodwell1983lie,
-  title={The Lie transformation group model of visual perception},
-  author={Dodwell, Peter C},
-  journal={Perception \& Psychophysics},
-  volume={34},
-  number={1},
-  pages={1--16},
-  year={1983},
-  publisher={Springer}
-}
-
-@article{hoffman1966lie,
-  title={The Lie algebra of visual perception},
-  author={Hoffman, William C},
-  journal={Journal of mathematical Psychology},
-  volume={3},
-  number={1},
-  pages={65--98},
-  year={1966},
-  publisher={Elsevier}
-}
-
-@article{parker2015unsupervised,
-  title={Unsupervised invariance learning of transformation sequences in a model of object recognition yields selectivity for non-accidental properties},
-  author={Parker, Sarah M and Serre, Thomas},
-  journal={Frontiers in computational neuroscience},
-  volume={9},
-  pages={115},
-  year={2015},
-  publisher={Frontiers}
-}
-
-@article{fukushima1988neocognitron,
-  title={Neocognitron: A hierarchical neural network capable of visual pattern recognition},
-  author={Fukushima, Kunihiko},
-  journal={Neural networks},
-  volume={1},
-  number={2},
-  pages={119--130},
-  year={1988},
-  publisher={Elsevier}
-}
-
-@article{wiskott2002slow,
-  title={Slow feature analysis: Unsupervised learning of invariances},
-  author={Wiskott, Laurenz and Sejnowski, Terrence J},
-  journal={Neural computation},
-  volume={14},
-  number={4},
-  pages={715--770},
-  year={2002},
-  publisher={MIT Press}
-}
-
-@article{schwartz1983shape,
-  title={Shape recognition and inferior temporal neurons},
-  author={Schwartz, Eric L and Desimone, Robert and Albright, Thomas D and Gross, Charles G},
-  journal={Proceedings of the National Academy of Sciences},
-  volume={80},
-  number={18},
-  pages={5776--5778},
-  year={1983},
-  publisher={National Acad Sciences}
-}
-
-
-@article{tang2018complex,
-  title={Complex pattern selectivity in macaque primary visual cortex revealed by large-scale two-photon imaging},
-  author={Tang, Shiming and Lee, Tai Sing and Li, Ming and Zhang, Yimeng and Xu, Yue and Liu, Fang and Teo, Benjamin and Jiang, Hongfei},
-  journal={Current Biology},
-  volume={28},
-  number={1},
-  pages={38--48},
-  year={2018},
-  publisher={Elsevier}
-}
-
-@article{li2017long,
-  title={Long-term two-photon imaging in awake macaque monkey},
-  author={Li, Ming and Liu, Fang and Jiang, Hongfei and Lee, Tai Sing and Tang, Shiming},
-  journal={Neuron},
-  volume={93},
-  number={5},
-  pages={1049--1057},
-  year={2017},
-  publisher={Elsevier}
-}
-
-@article{stringer2019high,
-  title={High-dimensional geometry of population responses in visual cortex},
-  author={Stringer, Carsen and Pachitariu, Marius and Steinmetz, Nicholas and Carandini, Matteo and Harris, Kenneth D},
-  journal={Nature},
-  pages={1},
-  year={2019},
-  publisher={Nature Publishing Group}
-}
-
-
-@ARTICLE{Churchland2012-fr,
-  title     = "Neural population dynamics during reaching",
-  author    = "Churchland, Mark M and Cunningham, John P and Kaufman, Matthew T
-               and Foster, Justin D and Nuyujukian, Paul and Ryu, Stephen I and
-               Shenoy, Krishna V",
-  abstract  = "Most theories of motor cortex have assumed that neural activity
-               represents movement parameters. This view derives from what is
-               known about primary visual cortex, where neural activity
-               represents patterns of light. Yet it is unclear how well the
-               analogy between motor and visual cortex holds. Single-neuron
-               responses in motor cortex are complex, and there is marked
-               disagreement regarding which movement parameters are
-               represented. A better analogy might be with other motor systems,
-               where a common principle is rhythmic neural activity. Here we
-               find that motor cortex responses during reaching contain a brief
-               but strong oscillatory component, something quite unexpected for
-               a non-periodic behaviour. Oscillation amplitude and phase
-               followed naturally from the preparatory state, suggesting a
-               mechanistic role for preparatory neural activity. These results
-               demonstrate an unexpected yet surprisingly simple structure in
-               the population response. This underlying structure explains many
-               of the confusing features of individual neural responses.",
-  journal   = "Nature",
-  publisher = "nature.com",
-  volume    =  487,
-  number    =  7405,
-  pages     = "51--56",
-  month     =  jul,
-  year      =  2012,
-  language  = "en"
-}
-
-
-@article{quiroga2005invariant,
-  title={Invariant visual representation by single neurons in the human brain},
-  author={Quiroga, R Quian and Reddy, Leila and Kreiman, Gabriel and Koch, Christof and Fried, Itzhak},
-  journal={Nature},
-  volume={435},
-  number={7045},
-  pages={1102},
-  year={2005},
-  publisher={Nature Publishing Group}
-}
-
-@ARTICLE{Saxena2019-gt,
-  title     = "Towards the neural population doctrine",
-  author    = "Saxena, Shreya and Cunningham, John P",
-  abstract  = "Across neuroscience, large-scale data recording and
-               population-level analysis methods have experienced explosive
-               growth. While the underlying hardware and computational
-               techniques have been well reviewed, we focus here on the novel
-               science that these technologies have enabled. We detail four
-               areas of the field where the joint analysis of neural
-               populations has significantly furthered our understanding of
-               computation in the brain: correlated variability, decoding,
-               neural dynamics, and artificial neural networks. Together, these
-               findings suggest an exciting trend towards a new era where
-               neural populations are understood to be the essential unit of
-               computation in many brain regions, a classic idea that has been
-               given new life.",
-  journal   = "Curr. Opin. Neurobiol.",
-  publisher = "Elsevier",
-  volume    =  55,
-  pages     = "103--111",
-  month     =  apr,
-  year      =  2019,
-  language  = "en"
-}
-
-@article{freeman1991design,
-  title={The design and use of steerable filters},
-  author={Freeman, William T. and Adelson, Edward H},
-  journal={IEEE Transactions on Pattern Analysis \& Machine Intelligence},
-  number={9},
-  pages={891--906},
-  year={1991},
-  publisher={Ieee}
-}
-
-
-@ARTICLE{Gallego2017-zw,
-  title     = "Neural Manifolds for the Control of Movement",
-  author    = "Gallego, Juan A and Perich, Matthew G and Miller, Lee E and
-               Solla, Sara A",
-  abstract  = "The analysis of neural dynamics in several brain cortices has
-               consistently uncovered low-dimensional manifolds that capture a
-               significant fraction of neural variability. These neural
-               manifolds are spanned by specific patterns of correlated neural
-               activity, the ``neural modes.'' We discuss a model for neural
-               control of movement in which the time-dependent activation of
-               these neural modes is the generator of motor behavior. This
-               manifold-based view of motor cortex may lead to a better
-               understanding of how the brain controls movement.",
-  journal   = "Neuron",
-  publisher = "Elsevier",
-  volume    =  94,
-  number    =  5,
-  pages     = "978--984",
-  month     =  jun,
-  year      =  2017,
-  language  = "en"
-}
-
-
-@ARTICLE{Pitts1947-mw,
-  title    = "How we know universals; the perception of auditory and visual
-              forms",
-  author   = "Pitts, Walter and McCulloch, Warren S",
-  journal  = "Bull. Math. Biophys.",
-  volume   =  9,
-  number   =  3,
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-  year     =  1947,
-  keywords = "UNIVERSALS",
-  language = "en"
-}
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-@inproceedings{simoncelli1995steerable,
-  title={The steerable pyramid: A flexible architecture for multi-scale derivative computation},
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-  abstract  = "We trained a large, deep convolutional neural network to
-               classify the 1.3 million high- resolution images in the
-               LSVRC-2010 ImageNet training set into the 1000 different
-               classes. On the test data, we achieved top-1 and top-5 error
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-  journal   = "Adv. Neural Inf. Process. Syst.",
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-@ARTICLE{Olshausen1993-tn,
-  title    = "A neurobiological model of visual attention and invariant pattern
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-@ARTICLE{Henaff2019-wy,
-  title    = "Perceptual straightening of natural videos",
-  author   = "H{\'e}naff, Olivier J and Goris, Robbe L T and Simoncelli, Eero P",
-  abstract = "Many behaviors rely on predictions derived from recent visual
-              input, but the temporal evolution of those inputs is generally
-              complex and difficult to extrapolate. We propose that the visual
-              system transforms these inputs to follow straighter temporal
-              trajectories. To test this 'temporal straightening' hypothesis,
-              we develop a methodology for estimating the curvature of an
-              internal trajectory from human perceptual judgments. We use this
-              to test three distinct predictions: natural sequences that are
-              highly curved in the space of pixel intensities should be
-              substantially straighter perceptually; in contrast, artificial
-              sequences that are straight in the intensity domain should be
-              more curved perceptually; finally, naturalistic sequences that
-              are straight in the intensity domain should be relatively less
-              curved. Perceptual data validate all three predictions, as do
-              population models of the early visual system, providing evidence
-              that the visual system specifically straightens natural videos,
-              offering a solution for tasks that rely on prediction.",
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-  abstract  = "Multilayer Neural Networks trained with the backpropagation
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-  number   =  3,
-  pages    = "127--147",
-  month    =  sep,
-  year     =  1947,
-  keywords = "UNIVERSALS",
-  language = "en"
-}
-
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-% to a LaTeX equivalent.
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-@ARTICLE{Cox2014-pr,
-  title     = "Neural networks and neuroscience-inspired computer vision",
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-  abstract  = "Brains are, at a fundamental level, biological computing
-               machines. They transform a torrent of complex and ambiguous
-               sensory information into coherent thought and action, allowing
-               an organism to perceive and model its environment, synthesize
-               and make decisions from disparate streams of information, and
-               adapt to a changing environment. Against this backdrop, it is
-               perhaps not surprising that computer science, the science of
-               building artificial computational systems, has long looked to
-               biology for inspiration. However, while the opportunities for
-               cross-pollination between neuroscience and computer science are
-               great, the road to achieving brain-like algorithms has been long
-               and rocky. Here, we review the historical connections between
-               neuroscience and computer science, and we look forward to a new
-               era of potential collaboration, enabled by recent rapid advances
-               in both biologically-inspired computer vision and in
-               experimental neuroscience methods. In particular, we explore
-               where neuroscience-inspired algorithms have succeeded, where
-               they still fail, and we identify areas where deeper connections
-               are likely to be fruitful.",
-  journal   = "Curr. Biol.",
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-  pages     = "R921--R929",
-  month     =  sep,
-  year      =  2014,
-  language  = "en"
-}
-
-
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-% to a LaTeX equivalent.
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-  publisher = "Curran Associates, Inc.",
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-% @phdthesis{KakaralaPhD,
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-  title={Objectnet: A large-scale bias-controlled dataset for pushing the limits of object recognition models},
-  author={Barbu, Andrei and Mayo, David and Alverio, Julian and Luo, William and Wang, Christopher and Gutfreund, Dan and Tenenbaum, Josh and Katz, Boris},
-  journal={Advances in neural information processing systems},
-  volume={32},
-  pages={9453--9463},
-  year={2019}
-}
-
-
-% Pooling reviews
-
-@misc{gholamalinezhad2020pooling,
-      title={Pooling Methods in Deep Neural Networks, a Review},
-      author={Hossein Gholamalinezhad and Hossein Khosravi},
-      year={2020},
-      eprint={2009.07485},
-      archivePrefix={arXiv},
-      primaryClass={cs.CV}
-}
-
-@article{Nirthika2022,
-  doi = {10.1007/s00521-022-06953-8},
-  url = {https://doi.org/10.1007/s00521-022-06953-8},
-  year = {2022},
-  month = feb,
-  publisher = {Springer Science and Business Media {LLC}},
-  volume = {34},
-  number = {7},
-  pages = {5321--5347},
-  author = {Rajendran Nirthika and Siyamalan Manivannan and Amirthalingam Ramanan and Ruixuan Wang},
-  title = {Pooling in convolutional neural networks for medical image analysis: a survey and an empirical study},
-  journal = {Neural Computing and Applications}
-}
-
-@article{Zafar2022,
-  doi = {10.3390/app12178643},
-  url = {https://doi.org/10.3390/app12178643},
-  year = {2022},
-  month = aug,
-  publisher = {{MDPI} {AG}},
-  volume = {12},
-  number = {17},
-  pages = {8643},
-  author = {Afia Zafar and Muhammad Aamir and Nazri Mohd Nawi and Ali Arshad and Saman Riaz and Abdulrahman Alruban and Ashit Kumar Dutta and Sultan Almotairi},
-  title = {A Comparison of Pooling Methods for Convolutional Neural Networks},
-  journal = {Applied Sciences}
-}
-
-@article{Tao2022,
-  doi = {10.1155/2022/4067581},
-  url = {https://doi.org/10.1155/2022/4067581},
-  year = {2022},
-  month = jun,
-  publisher = {Hindawi Limited},
-  volume = {2022},
-  pages = {1--17},
-  author = {Zhou Tao and Chang XiaoYu and Lu HuiLing and Ye XinYu and Liu YunCan and Zheng XiaoMin},
-  editor = {Chen Li},
-  title = {Pooling Operations in Deep Learning: From Invariable to Variable},
-  journal = {{BioMed} Research International}
-}
-
-% Related works
-
-@misc{lee2015generalizing,
-      title={Generalizing Pooling Functions in Convolutional Neural Networks: Mixed, Gated, and Tree},
-      author={Chen-Yu Lee and Patrick W. Gallagher and Zhuowen Tu},
-      year={2015},
-      eprint={1509.08985},
-      archivePrefix={arXiv},
-      primaryClass={stat.ML}
-}
-
-@inproceedings{wei2019building,
-  title={Building detail-sensitive semantic segmentation networks with polynomial pooling},
-  author={Wei, Zhen and Zhang, Jingyi and Liu, Li and Zhu, Fan and Shen, Fumin and Zhou, Yi and Liu, Si and Sun, Yao and Shao, Ling},
-  booktitle={Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition},
-  pages={7115--7123},
-  year={2019}
-}
-
-@inproceedings{gulcehre2014learned,
-  title={Learned-norm pooling for deep feedforward and recurrent neural networks},
-  author={Gulcehre, Caglar and Cho, Kyunghyun and Pascanu, Razvan and Bengio, Yoshua},
-  booktitle={Machine Learning and Knowledge Discovery in Databases: European Conference, ECML PKDD 2014, Nancy, France, September 15-19, 2014. Proceedings, Part I 14},
-  pages={530--546},
-  year={2014},
-  organization={Springer}
-}
-
-@inproceedings{estrach2014signal,
-  title={Signal recovery from pooling representations},
-  author={Estrach, Joan Bruna and Szlam, Arthur and LeCun, Yann},
-  booktitle={International conference on machine learning},
-  pages={307--315},
-  year={2014},
-  organization={PMLR}
-}
-
-@misc{eom2020alphaintegration,
-      title={Alpha-Integration Pooling for Convolutional Neural Networks},
-      author={Hayoung Eom and Heeyoul Choi},
-      year={2020},
-      eprint={1811.03436},
-      archivePrefix={arXiv},
-      primaryClass={cs.LG}
-}
-
-@article{shahriari2017multipartite,
-  title={Multipartite pooling for deep convolutional neural networks},
-  author={Shahriari, Arash and Porikli, Fatih},
-  journal={arXiv preprint arXiv:1710.07435},
-  year={2017}
-}
-
-@article{shi2016rank,
-  title={Rank-based pooling for deep convolutional neural networks},
-  author={Shi, Zenglin and Ye, Yangdong and Wu, Yunpeng},
-  journal={Neural Networks},
-  volume={83},
-  pages={21--31},
-  year={2016},
-  publisher={Elsevier}
-}
-
-@article{bieder2021comparison,
-  title={Comparison of methods generalizing max-and average-pooling},
-  author={Bieder, Florentin and Sandkuhler, Robin and Cattin, Philippe C},
-  journal={arXiv preprint arXiv:2103.01746},
-  year={2021}
-}
-
-@inproceedings{stergiou2021refining,
-  title={Refining activation downsampling with SoftPool},
-  author={Stergiou, Alexandros and Poppe, Ronald and Kalliatakis, Grigorios},
-  booktitle={Proceedings of the IEEE/CVF international conference on computer vision},
-  pages={10357--10366},
-  year={2021}
-}
-
-@article{czaja2021maximal,
-  title={Maximal function pooling with applications},
-  author={Czaja, Wojciech and Li, Weilin and Li, Yiran and Pekala, Mike},
-  journal={Excursions in Harmonic Analysis, Volume 6: In Honor of John Benedetto's 80th Birthday},
-  pages={413--429},
-  year={2021},
-  publisher={Springer}
-}
-
-@article{kumar2018ordinal,
-  title={Ordinal pooling networks: for preserving information over shrinking feature maps},
-  author={Kumar, Ashwani},
-  journal={arXiv preprint arXiv:1804.02702},
-  year={2018}
-}
-
-% stochastic poolings
-
-@inproceedings{yu2014mixed,
-  title={Mixed pooling for convolutional neural networks},
-  author={Yu, Dingjun and Wang, Hanli and Chen, Peiqiu and Wei, Zhihua},
-  booktitle={Rough Sets and Knowledge Technology: 9th International Conference, RSKT 2014, Shanghai, China, October 24-26, 2014, Proceedings 9},
-  pages={364--375},
-  year={2014},
-  organization={Springer}
-}
-
-@inproceedings{tong2016hybrid,
-  title={A hybrid pooling method for convolutional neural networks},
-  author={Tong, Zhiqiang and Aihara, Kazuyuki and Tanaka, Gouhei},
-  booktitle={Neural Information Processing: 23rd International Conference, ICONIP 2016, Kyoto, Japan, October 16--21, 2016, Proceedings, Part II 23},
-  pages={454--461},
-  year={2016},
-  organization={Springer}
-}
-
-@article{zeiler2013stochastic,
-  title={Stochastic pooling for regularization of deep convolutional neural networks},
-  author={Zeiler, Matthew D and Fergus, Rob},
-  journal={arXiv preprint arXiv:1301.3557},
-  year={2013}
-}
-
-@inproceedings{wu2015max,
-  title={Max-pooling dropout for regularization of convolutional neural networks},
-  author={Wu, Haibing and Gu, Xiaodong},
-  booktitle={Neural Information Processing: 22nd International Conference, ICONIP 2015, Istanbul, Turkey, November 9-12, 2015, Proceedings, Part I 22},
-  pages={46--54},
-  year={2015},
-  organization={Springer}
-}
-
-@inproceedings{zhai2017s3pool,
-  title={S3pool: Pooling with stochastic spatial sampling},
-  author={Zhai, Shuangfei and Wu, Hui and Kumar, Abhishek and Cheng, Yu and Lu, Yongxi and Zhang, Zhongfei and Feris, Rogerio},
-  booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition},
-  pages={4970--4978},
-  year={2017}
-}
-
-@article{graham2014fractional,
-  title={Fractional max-pooling},
-  author={Graham, Benjamin},
-  journal={arXiv preprint arXiv:1412.6071},
-  year={2014}
-}
-
-@article{he2015spatial,
-  title={Spatial pyramid pooling in deep convolutional networks for visual recognition},
-  author={He, Kaiming and Zhang, Xiangyu and Ren, Shaoqing and Sun, Jian},
-  journal={IEEE transactions on pattern analysis and machine intelligence},
-  volume={37},
-  number={9},
-  pages={1904--1916},
-  year={2015},
-  publisher={IEEE}
-}
-
-@inproceedings{zhang2012pose,
-  title={Pose pooling kernels for sub-category recognition},
-  author={Zhang, Ning and Farrell, Ryan and Darrell, Trever},
-  booktitle={2012 IEEE Conference on Computer Vision and Pattern Recognition},
-  pages={3665--3672},
-  year={2012},
-  organization={IEEE}
-}
-
-@article{qi2018concentric,
-  title={Concentric circle pooling in deep convolutional networks for remote sensing scene classification},
-  author={Qi, Kunlun and Guan, Qingfeng and Yang, Chao and Peng, Feifei and Shen, Shengyu and Wu, Huayi},
-  journal={Remote Sensing},
-  volume={10},
-  number={6},
-  pages={934},
-  year={2018},
-  publisher={MDPI}
-}
-
-@inproceedings{feng2011geometric,
-  title={Geometric Lp-norm feature pooling for image classification},
-  author={Feng, Jiashi and Ni, Bingbing and Tian, Qi and Yan, Shuicheng},
-  booktitle={CVPR 2011},
-  pages={2609--2704},
-  year={2011},
-  organization={IEEE}
-}
-
-@inproceedings{lin2015bilinear,
-  title={Bilinear CNN models for fine-grained visual recognition},
-  author={Lin, Tsung-Yu and RoyChowdhury, Aruni and Maji, Subhransu},
-  booktitle={Proceedings of the IEEE international conference on computer vision},
-  pages={1449--1457},
-  year={2015}
-}
-
-@inproceedings{gao2019global,
-  title={Global second-order pooling convolutional networks},
-  author={Gao, Zilin and Xie, Jiangtao and Wang, Qilong and Li, Peihua},
-  booktitle={Proceedings of the IEEE/CVF Conference on computer vision and pattern recognition},
-  pages={3024--3033},
-  year={2019}
-}
-
-@article{yellott1993implications,
-  title={Implications of triple correlation uniqueness for texture statistics and the Julesz conjecture},
-  author={Yellott, John I},
-  journal={JOSA A},
-  volume={10},
-  number={5},
-  pages={777--793},
-  year={1993},
-  publisher={Optica Publishing Group}
-}
-
-@article{deshpande2023third,
-  title={Third-order motifs are sufficient to fully and uniquely characterize spatiotemporal neural network activity},
-  author={Deshpande, Sarita S and Smith, Graham A and van Drongelen, Wim},
-  journal={Scientific Reports},
-  volume={13},
-  number={1},
-  pages={238},
-  year={2023},
-  publisher={Nature Publishing Group UK London}
-}
-
-@article{kollias1996multiresolution,
-  title={A multiresolution neural network approach to invariant image recognition},
-  author={Kollias, Stefanos D},
-  journal={Neurocomputing},
-  volume={12},
-  number={1},
-  pages={35--57},
-  year={1996},
-  publisher={Elsevier}
-}
-
-@article{delopoulos1994invariant,
-  title={Invariant image classification using triple-correlation-based neural networks},
-  author={Delopoulos, Anastasios and Tirakis, Andreas and Kollias, Stefanos},
-  journal={IEEE Transactions on Neural Networks},
-  volume={5},
-  number={3},
-  pages={392--408},
-  year={1994},
-  publisher={IEEE}
-}
-
-@inproceedings{maron2019universality,
-  title={On the universality of invariant networks},
-  author={Maron, Haggai and Fetaya, Ethan and Segol, Nimrod and Lipman, Yaron},
-  booktitle={International conference on machine learning},
-  pages={4363--4371},
-  year={2019},
-  organization={PMLR}
-}
-
-@article{yarotsky2022universal,
-  title={Universal approximations of invariant maps by neural networks},
-  author={Yarotsky, Dmitry},
-  journal={Constructive Approximation},
-  volume={55},
-  number={1},
-  pages={407--474},
-  year={2022},
-  publisher={Springer}
-}
-
-
-% Lie groups
-
-@book{hall2013lie,
-  title={Lie groups, Lie algebras, and representations},
-  author={Hall, Brian C and Hall, Brian C},
-  year={2013},
-  publisher={Springer}
-}
diff --git a/docs/sanbord_miolane_2025/sections/0-abstract.tex b/docs/sanbord_miolane_2025/sections/0-abstract.tex
deleted file mode 100644
index c3d2c19..0000000
--- a/docs/sanbord_miolane_2025/sections/0-abstract.tex
+++ /dev/null
@@ -1,7 +0,0 @@
-\begin{abstract}
-
-We introduce a general method for achieving robust group-invariance in group-equivariant convolutional neural networks ($G$-CNNs), which we call the $G$-triple-correlation ($G$-TC) layer. The approach leverages the theory of the triple-correlation on groups, which is the unique, lowest-degree polynomial invariant map that is also \textit{complete}. Many commonly used invariant maps\textemdash such as the \texttt{max}\textemdash are incomplete: they remove both group and signal structure. A complete invariant, by contrast, removes only the variation due to the actions of the group, while preserving all information about the structure of the signal. The completeness of the triple correlation endows the $G$-TC layer with strong robustness, which can be observed in its resistance to invariance-based adversarial attacks. In addition, we observe that it yields measurable improvements in classification accuracy over standard Max $G$-Pooling in $G$-CNN architectures. We provide a general and efficient implementation of the method for any discretized group, which requires only a table defining the group's product structure. We demonstrate the benefits of this method for $G$-CNNs defined on both commutative and non-commutative groups\textemdash $SO(2)$, $O(2)$, $SO(3)$, and $O(3)$ (discretized as the cyclic $C8$, dihedral $D16$, chiral octahedral $O$ and full octahedral $O_h$ groups)\textemdash acting on $\mathbb{R}^2$ and $\mathbb{R}^3$ on both $G$-MNIST and $G$-ModelNet10 datasets.
-
-% We show the advantage of this approach over standard $G$-pooling methods that we replace and compare with the $G$-TC Layer in several $G$-CNN architectures. We demonstrate improved scores in classification accuracy as well as improved adversarial robustness.
-
-\end{abstract}
diff --git a/docs/sanbord_miolane_2025/sections/1-introduction.tex b/docs/sanbord_miolane_2025/sections/1-introduction.tex
deleted file mode 100644
index fdcb408..0000000
--- a/docs/sanbord_miolane_2025/sections/1-introduction.tex
+++ /dev/null
@@ -1,29 +0,0 @@
-\section{Introduction}
-
-
-The \textit{pooling} operation is central to the convolutional neural network (CNN). It was originally introduced in the first CNN architecture\textemdash Fukushima's 1980 \textit{Neocognitron}~\cite{fukushima1980}\textemdash and remained a fixture of the model since. The Neocognitron was directly inspired by the canonical model of the visual cortex as a process of hierarchical feature extraction and local pooling~\cite{hubel1959receptive, adelson1985spatiotemporal}. In both the neuroscience and CNN model, pooling is intended to serve two purposes. First, it facilitates the local-to-global \textit{coarse-graining} of structure in the input. Second, it facilitates \textit{invariance} to local changes\textemdash resulting in network activations that remain similar under small perturbations of the input. In this way, CNNs construct hierarchical, multi-scale features that have increasingly large extent and increasing invariance.
-
-The pooling operation in traditional CNNs, typically a local max or average, has remained largely unchanged over the last forty years. The variations that have been proposed in the literature~\cite{Nirthika2022,Zafar2022} mostly tackle its \textit{coarse-graining} purpose, improve computational efficiency, or reduce overfitting, but do not seek to enhance its properties with respect to \textit{invariance}. Both \texttt{max} and \texttt{avg} operations are reasonable choices to fulfill the goal of coarse-graining within CNNs and $G$-CNNs. However, they are excessively imprecise and lossy with respect to the goal of constructing robust representations of objects that are invariant only to irrelevant visual changes. Indeed, the \texttt{max} and \texttt{avg} operations are invariant to many natural image transformations such as translations and rotations, but also to unnatural transformations, including pixel permutations, that may destroy the image structure. This excessive invariance has been implicated in failure modes such as vulnerability to adversarial perturbations~\cite{goodfellow2014explaining,jacobsen2018excessive}, and a bias towards textures rather than objects~\cite{brendel2019approximating}. To overcome these challenges and enable robust and selective invariant representation learning, there is a need for novel computational primitives that selectively parameterize invariant maps for natural transformations.
-
-% Beyond traditional CNNs, the more recent group convolutional neural networks ($G$-CNNs) seek to achieve equivariance with respect to a group of transformations that extend the group of translations. In this line of work, invariance to the group $G$ is standardly achieved by generalizing the concept of max pooling, with Max $G$-Pooling~\cite{cohen2016group}.
-
-% At the heart of the challenge lie two requirements that are frequently in tension: \textit{invariance} to transformation structure and \textit{selectivity} to pattern structure. In deep networks, operations such as \texttt{max} or \texttt{average} are commonly employed to achieve invariance to local transformations. Such operations are invariant to many natural transformations; however, they are also invariant to unnatural transformations that destroy image structure, such as pixel permutations. This lack of selectivity may contribute to failure modes such as susceptibility to adversarial perturbations \cite{goodfellow2014explaining}, excessive invariance \cite{jacobsen2018excessive}, and selectivity for textures rather than objects \cite{brendel2019approximating}. Thus, there is a need for computational primitives that selectively parameterize natural transformations and facilitate robust invariant representation learning.
-
-Many of the transformations that occur in visual scenes are due to the actions of \textit{groups}. The appreciation of this fact has led to the rise of group-equivariant convolutional networks ($G$-CNNs) \cite{cohen2016group} and the larger program of Geometric Deep Learning \cite{bronstein2021geometric}. While this field has leveraged the mathematics of group theory to attain precise generalized group-equivariance in convolutional network layers, the pooling operation has yet to meet its group theoretic grounding. Standardly, invariance to a group $G$ is achieved with a simple generalization of max pooling: Max $G$-Pooling~\cite{cohen2016group} \textemdash see Fig.~\ref{fig:overview-cnn-c4} (top-right). However, this approach inevitably suffers from the lossiness of the \texttt{max} operation.
-
-Here, we unburden the pooling operation of the dual duty of invariance and coarse-graining, by uncoupling these operations into two steps that can be performed with precision. We retain the standard \texttt{max} and \texttt{avg} pooling for coarse-graining, but introduce a new method for robust $G$-invariance via the group-invariant triple correlation \textemdash see Fig.~\ref{fig:overview-cnn-c4} (bottom-right). The group-invariant triple correlation is the lowest-order complete operator that can achieve exact invariance~\cite{kakarala1992triple}. As such, we propose a general framework for robust $G$-Invariance in $G$-Equivariant Networks.
-%the spatial dual of the bispectrum \cite{kakarala2012bispectrum}.
-%^^Removed to avoid scaring people (from bioshape comments).
-We show the advantage of this approach over standard max $G$-pooling in several $G$-CNN architectures. Our extensive experiments demonstrate improved scores in classification accuracy in traditional benchmark datasets as well as improved adversarial robustness.
-
-\begin{figure}
-    \centering
-    \includegraphics[width=1\textwidth]{figs/master-figure-complex-2.pdf}
-    \caption{\textbf{Achieving Robust $G$-Invariance in $G$-CNNs with the $G$-Triple-Correlation}. The output of a $G$-Convolutional layer is equivariant to the actions of $G$ on the domain of the signal.  To identify signals that are equivalent up to group action, the layer can be followed by a $G$-Invariant map that eliminates this equivariance. In $G$-CNNs, Max $G$-Pooling is a commonly used for this purpose. Taking the maximum of the $G$-Convolutional equivariant output is indeed invariant to the actions of the group. However, it is also lossy: many non-equivalent output vectors have the same maximum. Our method\textemdash the \textit{$G$-Triple-Correlation} is the lowest-order polynomial invariant map that is \textit{complete} \cite{sturmfels2008algorithms}. As a complete invariant, it preserves all information about the signal structure, removing only the action of the group. Our approach thus provides a new foundation for achieving robust $G$-Invariance in $G$-CNNs.}
-    \label{fig:overview-cnn-c4}
-\end{figure}
-
-
-% Indeed, they are \textit{invariant} to transformations; however, they are \textit{too} invariant.
-
-% : the down-sampling of representations in the network, by collapsing local regions.
diff --git a/docs/sanbord_miolane_2025/sections/2-background.tex b/docs/sanbord_miolane_2025/sections/2-background.tex
deleted file mode 100644
index c3aad14..0000000
--- a/docs/sanbord_miolane_2025/sections/2-background.tex
+++ /dev/null
@@ -1,163 +0,0 @@
-\section{Background}
-
-We first cover the fundamentals of group-equivariant neural networks\textemdash also known as $G$-CNNs, or $G$-Equivariant Networks\textemdash before introducing the framework for $G$-Invariant Pooling.
-
-% \cite{finzi2021practical, weiler2019general, cohen2019general, cesa2021program}
-\subsection{Mathematical Prerequisites}
-
-The construction of $G$-CNNs requires mathematical prerequisites of group theory, which we recall here. The interested reader can find details in~\cite{hall2013lie}.
-
-\textbf{Groups.} A \textit{group} $(G,\cdot)$ is a set $G$ with a binary operation $\cdot$, which we can generically call the \textit{product}. The notation $g_1 \cdot g_2$ denotes the product of two elements in the set; however, it is standard to omit the operator and write simply $g_1g_2$\textemdash a convention we adopt here. Concretely, a group $G$ may define a class of transformations. For example, we can consider the group of two-dimensional rotations in the plane\textemdash the special orthogonal group $SO(2)$\textemdash or the group of two-dimensional rotations and translations in the plane\textemdash the special euclidean group $SE(2)$. Each element of the group $g \in G$ defines a \textit{particular} transformation, such as one \textit{rotation by $30 \degree$} or one \textit{rotation by $90 \degree$}. The binary operation $\cdot$ provides a means for combining two particular transformations\textemdash for example, first rotating by $30\degree$ and then rotating by $90 \degree$. In mathematics, for a set of transformations $G$ to be a group under the operation $\cdot$, the four axioms of closure, associativity, identity and inverse must hold. These axioms are recalled in Appendix~\ref{sec:group_axioms}.
-
-% \begin{enumerate}
-%     \item \textit{Closure}: The product of any two elements of the group is also an element of the group, i.e. for all $g_1, g_2 \in G$, $g_1g_2 \in G$.
-%     \item \textit{Associativity}: The grouping of elements under the operation does not change the outcome, so long as the order of elements is preserved, i.e. $(g_1g_2)g_3 = g_1(g_2g_3)$.
-%     \item \textit{Identity}: There exists a ``do-nothing'' \textit{identity} element $e$ that such that the product of $e$ with any other element $g$ returns $g$, i.e. $ge = eg = g$ for all $g \in G$.
-%     \item \textit{Inverse}: For every element $g$, there exists an \textit{inverse} element $g^{-1}$  such that the product of $g$ and $g^{-1}$ returns the identity, i.e. $gg^{-1} = g^{-1}g = e$.
-% \end{enumerate}
-
-
-% \textbf{Product Groups.} A \textit{product group} as a set is the cartesian product $G \times H$ of two given groups $G$ and $H$.  The new group product is given by $(g_1,h_1) \cdot (g_2,h_2) = (g_1g_2,h_1h_2)$.
-
-
-\textbf{Group Actions on Spaces.} We detail how a transformation $g$ can transform elements of a space, for example how a rotation of $30\degree$ indeed rotates a vector in the plane by $30\degree$. We say that the transformations $g$'s act on (the elements of) a given space. Specifically, consider $X$ a space, such as the plane. A \textit{group action} is a function $\action: G \times X \rightarrow X$ that maps $(g, x)$ pairs to elements of $X$. We say a group $G$ \textit{acts} on a space $X$ if the following properties of the action $\action$ hold:
-\begin{enumerate}
-    \item \textit{Identity}: The identity $e$ of the group $G$ ``does nothing'', i.e., it maps any element $x \in X$ to itself. This can be written as: $\action(e, x) = x$.
-    \item \textit{Compatibility}: Two elements $g_1, g_2 \in G$ can be combined before or after the map $L$ to yield the same result, i.e., $\action(g_1, \action(g_2, x)) = \action(g_1 g_2, x)$. For example, rotating a 2D vector by $30\degree$ and then $40\degree$ yields the same result as rotating that vector by $70\degree$ in one time.
-\end{enumerate}
-For simplicity, we will use the shortened notation $\action_g(x)$ to denote $\action(g, x)$ the action of the transformation $g$ on the element $x$.
-
-Some group actions $L$ have additional properties and turn the spaces $X$ on which they operate into \textit{homogeneous spaces}. Homogeneous spaces play an important role in the definition of the $G$-convolution in $G$-CNNs, so that we recall their definition here. We say that $X$ is a \textit{homogeneous space} for a group $G$ if $G$ acts transitively on $X$\textemdash that is, if for every pair $x_1, x_2 \in X$ there exists an element of $g \in G$ such that $\action_g(x_1) = x_2$. The concept can be clearly illustrated by considering the surface of a sphere, the space $S^2$. The sphere $S^2$ is a homogeneous space for $SO(3)$, the group of orthogonal $3 \times 3$ matrices with determinant one that define 3-dimensional rotations.  Indeed, for every pair of points on the sphere, one can define a 3D rotation matrix that takes one to the other.
-
-% In contrast, the 3D space $\mathbb{R}^3$ is not an homogeneous space for $SO(3)$: one can find two different points in $\mathbb{R}^3$ such that there is no rotation that can map one into the other. For example the origin of the coordinate system $(0, 0, 0)$ and the point $(1, 0, 0)$ in $\mathbb{R}^3$ cannot be linked by a rotation. However, the 3D space $\mathbb{R}^3$ is an homogeneous space for the special euclidean group, the group of rotations and translations $SE(3)$.
-
-\paragraph{Group Actions on Signal Spaces.} We have introduced essential concepts from group theory, where a group $G$ can act on any abstract space $X$. Moving towards building $G$-CNNs, we introduce how groups can act on spaces of signals, such as images. Formally, a \textit{signal} is a map $\signal: \domain \rightarrow \mathbb{R}^c$, where $\domain$ is called the domain of the signal and $c$ denotes the number of channels. The \textit{space of signals }itself is denoted $L_2(\domain, \mathbb{R}^c)$. For example, $\domain=\mathbb{R}^2$ or $\mathbb{R}^3$ for 2D and 3D images. Gray-scale images have one channel ($c=1$) and color images have the 3 red-green-blue channels ($c=3$).
-
-% Nina: put the discretization later so that the reader is not confused
-% In practice, the domain is discretized so that the signal map can be represented as a vector. For example, an image $x \in X$ can be seen as the discretization on a grid of the continuous function $\signal$. Denoting $N$ the number of elements in the domain $\Omega$, this discretization yields $x \in \mathbb{R}^{N \times c}$.
-
-Any action of a group of transformations $G$ on a domain $\domain$ yields an action of that same group on the spaces of signals defined on that domain, i.e., on $L_2(\domain, \mathbb{R}^c)$. For example, knowing that the group of 2D rotations $SO(2)$ acts on the plane $\Omega = \mathbb{R}^2$ allows us to define how $SO(2)$ rotates 2D gray-scale images in $L_2(\mathbb{R}^2, \mathbb{R}^c)$. Concretely, the action $\action$ of a group $G$ on the domain $\domain$ yields the following action of $G$ on $L_2(\domain, \mathbb{R}^c)$:
-
-\begin{equation}
-    \action_g[\signal](u) = \signal(\action_{g^{-1}}(u)), \qquad \text{for all $u \in \domain$ and for all $g \in G$}.
-\end{equation}
-
-We use the same notation $L_g$ to refer to the action of the transformation $g$ on either an element $u$ of the domain or on a signal $f$ defined on that domain, distinguishing them using $[ . ]$ for the signal case. We note that the domain of a signal can be the group itself: $\Omega = G$. In what follows, we will also consider actions on real signals defined on a group, i.e., on signals such as $\Theta: G \rightarrow \mathbb{R}$.
-
-% A group $G$ can act on an image by acting either only on the domain $\domain$ of $\signal$, or directly on the image $x$ flattened into a vector of $X=\mathbb{R}^{cwd}$ with $c$ the channels, $w$ the width and $d$ the height. The transformations we consider, such as rotation and translation, act on the domain $\domain$, while leaving codomain values of the function $\signal$ intact.
-
-
-
-% \textbf{Homomorphisms.} Two groups $(G, \cdot)$ and $(H, *)$ are \textit{homomorphic} if there exists a correspondence between elements of the groups that respect the group operation. Concretely, a \textit{homomorphism} is a map $\rho: G \rightarrow H$ such that $\rho(u   \cdot v) = \rho(u) * \rho(v)$. An \textit{isomorphism} is a bijective homomorphism.
-
-% \textbf{Representations.} A \textit{representation} of a group G is a map $\rho: G \rightarrow GL(V)$ that assigns elements of $G$ to elements of the group of linear transformations over a vector space $V$ such that $\rho(g_1 g_2) = \rho(g_1) \rho(g_2)$. That is, $\rho$ is a group homomorphism.
-% % In many cases, $V$ is $\mathbb{R}^{n}$ or $\mathbb{C}^{n}$.
-
-\textbf{Invariance and Equivariance}. The concepts of group-invariance and equivariance are at the core of what makes the $G$-CNNs desirable for computer vision applications. We recall their definitions here. A function $\psi: X \mapsto Y$ is $G$-\textit{invariant} if $\psi(x) = \psi(L_g(x))$, for all $g \in G$ and $x \in X$. This means that group actions on the input space have no effect on the output. Applied to the group of rotations acting on the space of 2D images $X=L_2(\domain, \mathbb{R}^c)$ with $\domain = \mathbb{R}^2$, this means that a $G$-invariant function $\psi$ produces an input that will stay the same for any rotated version of a given signal. For example, whether the image contains the color red is invariant with respect to any rotation of that image. A function $\psi: X \mapsto Y$ is $G$-\textit{equivariant} if $\psi(L_g(x)) = L'_g(\psi(x))$ for all $g \in G$ and $x \in X$, where $L$ and $L'$ are two different actions of the group $G$, on the spaces $X$ and $Y$ respectively. This means that a group action on the input space results in a corresponding group action of the same group element $g$ on the output space. For example, consider $\psi$ that represents a neural network performing a foreground-background segmentation of an image. It is desirable for $\psi$ to be equivariant to the group of 2D rotations. This equivariance ensures that, if the input image $f$ is rotated by $30\degree$, then the output segmentation $\psi(f)$ rotates by $30\degree$ as well.
-
-% \textbf{Orbits.} Given a point $x \in X$, the \textit{orbit} $Gx$ of $x$ is the set $\{gx : g \in G \}$. In the context of image transformations, the orbit defines the set of all transformed versions of a canonical image\textemdash for example, if $G$ is the group of translations, then the orbit contains all translated versions of that image.
-
-
-\subsection{$G$-Equivariant Networks}
-
-
-$G$-CNNs are built from the following fundamental building blocks: $G$-convolution, spatial pooling, and $G$-pooling. The $G$-convolution is equivariant to the action of the group $G$, while the $G$-pooling achieves $G$-invariance. Spatial pooling achieves coarse-graining. We review the group-specific operations here. The interested reader can find additional details in \cite{cohen2016group,Cohen2019a}, which include the definitions of these operations using the group-theoretic framework of principal bundles and associated vector bundles.
-
-
-% \begin{figure}[H]
-%     \centering
-%     \includegraphics[width=0.3\textwidth]{example-image-a}
-%     \caption{[Figure illustrating the background: showing a G-CNN with traditional G-Conv and G-Pools]}
-%     \label{fig:gcnn-illustration}
-% \end{figure}
-
-
-\subsubsection{$G$-Convolution}
-
-In plain language, a standard translation-equivariant convolutional neural network layer sweeps filters across a signal (typically, an image), translating the filter and then taking an inner product with the signal to determine the similarity between a local region and the filter. $G$-CNNs~\cite{cohen2016group} generalize this idea, replacing translation with the action of other groups that define symmetries in a machine learning task\textemdash for example, rotating a filter, to determine the presence of a feature in various orientations.
-
-% \paragraph{$G$-Convolution}
-Consider a signal $\signal$ defined on a domain $\domain$ on which a group $G$ acts. A neural network filter is a map $\filter: \domain \rightarrow \mathbb{R}^c$ defined with the same domain $\domain$ and codomain $\mathbb{R}^c$ as the signal. A $G$-convolutional layer is defined by a set of filters $\{\filter_1, ..., \filter_K\}$. For a given filter $k$, the layer performs a \textit{$G$-convolution }with the input signal $\signal$:
-% A neural network layer is defined by a set of filters $\{k_1, ..., k_m\}$, with $k_i \in \mathbb{R}^{d \times d}$.
-\begin{equation}
-    \Theta_k(g) = (\filter_k * \signal)(g) = \int_{u \in \domain} \filter_k(\action_{g^{-1}}(u)) \signal(u) du,\quad \forall g \in G,
-\end{equation}
-by taking the dot product in $\mathbb{R}^c$ of the signal with a transformed version of the filter. In practice, the domain $\domain$ of the signal is discretized, such that the $G$-convolutional layer becomes:
-\begin{equation}
-    \Theta_k(g) =  \sum_{u \in \domain} \filter_k(\action_{g^{-1}}(u)) \signal(u),\quad \forall g \in G.
-\end{equation}
-
-The output of one filter $k$ is therefore a map $\Theta_k: G \rightarrow \mathbb{R}$, while the output of the whole layer with $K$ filters is $\Theta: G \rightarrow \mathbb{R}^K$ defined as $\Theta(g) = [\Theta_1(g), \dots, \Theta_K(g)]$ for all $g \in G$. The $G$-convolution therefore outputs a signal $\Theta$ whose domain has necessarily become the group $\domain = G$ and whose number of channels is the number of convolutional filters $K$.
-
-% \paragraph{$G$-Convolution (Next Layers)} After the first layer, the signal has become a function $f: G \rightarrow \mathbb{R}^K$, so that the definition of the $G$-convolution slightly changes. For a given filter defined with a domain being a group $\phi_k: G \rightarrow \mathbb{R}^{K}$, we have:
-% \begin{equation}
-%     \Theta_k(g) = (\filter_k * \signal)(g) = \int_{h \in G} \filter_k(g^{-1}h) \signal(h) dh,\quad \forall g \in G,
-% \end{equation}
-% where instead of using the action of the group $G$ on the domain, we use its ``natural'' action on itself: the binary operation $\cdot$ of the group. The discretized version is written:
-% \begin{equation}
-%     \Theta_k(g) =  \sum_{h \in G} \filter_k(g^{-1}h) \signal(h),\quad \forall g \in G.
-% \end{equation}
-
-% \paragraph{$G$-Equivariance}
-The $G$-convolution is \textit{equivariant} to the action of the group on the domain of the signal $\signal$~\cite{cohen2016group}. That is, the action of $g$ on the domain of $\signal$ results in a corresponding action on the output of the layer. Specifically, consider a filter $\phi_k$, we have:
-\begin{equation}
-    \filter_k * \action_{g}[\signal] = \action'_{g}[\filter_k * \signal], \qquad  \forall g \in G,
-\end{equation}
-where $\action_g$ and $\action'_g$ represent the actions of the same group element $g$ on the functions $\signal$ and $\filter_k * \signal$ respectively. This property applies for the $G$-convolutions of the first layer and of the next layers~\cite{cohen2016group}. %The proof is recalled in Appendix~\ref{sec:g-conv}.
-
-% \begin{figure}[H]
-%     \centering
-%     \includegraphics[width=0.3\textwidth]{example-image-a}
-%     \caption{[Figure illustrating cylinder of activations for translation and rotation]}
-%     \label{fig:gcnn-cylinder}
-% \end{figure}
-
-
-% \subsubsection{Spatial Pooling}
-% In a standard $G$-Equivariant Network, \textit{coarse-graining} is achieved through spatial pooling. Typically, this is achieved by taking the \texttt{max} value in a local window:
-
-% \begin{equation}
-%     \label{eq:spatial-pooling}
-%     \mu_k = \max_i \Theta{(x_{i, g})}
-% \end{equation}
-
-\subsubsection{$G$-Pooling}
-\label{sec:g-pooling}
-
-\textit{Invariance} to the action of the group is achieved by pooling over the group ($G$-Pooling)~\cite{cohen2016group}.
-%either globally by performing an operation over the whole group $G$, or locally, by restricting that operation to a subset of the group.
-The pooling operation is typically performed after the $G$-convolution, so that we restrict its definition to signals $\Theta$ defined over a group $G$. In $G$-pooling, a \texttt{max} typically is taken over the group elements:
-\begin{equation}
-    \mu_k = \max_{g \in G} \Theta_k(g) . %\quad \text{(global);} \qquad \forall h \in G, ~\mu_{k, i} (h)= \max_{g \in hU}\Theta_k(g)  \quad \text{(local)} .
-\end{equation}
- $G$-pooling extracts a single real scalar value $\mu_k $ from the full feature vector $\Theta_k$, which has $|G|$ values, with $|G|$ the size of the (discretized) group $G$ as shown in Fig.~\ref{fig:overview-cnn-c4}. When the group $G$ is a grid discretizing $\mathbb{R}^n$, max $G$-Pooling is equivalent to the standard spatial max pooling used in translation-equivariant CNNs, and it can be used to achieve coarse-graining.
-% while in the local case, the pooling returns a signal defined over the whole group $G$ as $h$ varies within the whole $G$.
-% In the local case, the subset over which pooling is performed is typically defined as $g U=\{g u \mid u \in U\}$ that is: the $g$-transformation, using the group product, of some pooling domain $U \subset G$ that is usually a neighborhood of the identity transformation~\cite{cohen2016group}. For example, in the classical convolutional neural networks using the group of 2D translations, $U$ is usually a $2 \times 2$ or $3 \times 3$ square including the origin $(0,0)$, and $g$ is a translation.
-More generally, $G$-Pooling is $G$-invariant,
-%while the local pooling is $G$-equivariant,
-as shown in \cite{cohen2016group}. However, we argue here that it is excessively $G$-invariant. Although it achieves the objective of invariance to the group action, it also loses substantial information. As illustrated in Fig.~\ref{fig:overview-cnn-c4}, many different signals $\Theta$ may yield same result $\mu$ through the $G$-pooling operation, even if these signals do not share semantic information. This excessive invariance creates an opportunity for adversarial susceptibility. Indeed, inputs $f$ can be designed with the explicit purpose of generating a $\mu_k$ that will fool a neural network and yield an unreasonable classification result. For this reason, we introduce our general framework for robust, selective $G$-invariance.
-
-%It is possible to find a middle ground between these two pooling operations, and get a output that is defined on $H$, a subgroup of $G$: a subset of $G$ that is also a group. In that case, the output signal becomes neither $G$-invariant nor $G$-equivariant, but rather $H$-equivariant. This type of pooling generalizes the notion of ``stride'' to $G$-CNNs~\cite{cohen2016group} and is defined as:
-% \begin{equation}
-%     \forall h \in H,~ \mu_{k, i} (h)= \max_{g \in hU}\Theta_k(g)  \quad \text{(local with stride)}.
-% \end{equation}
-
-% \paragraph{Avg pooling} Alternatively, an average pooling can be defined as:
-% \begin{equation}
-%     \mu_k = \frac{1}{|G|}\sum_{g \in G} \Theta_k(g) \quad \text{(global);} \qquad \forall h \in G, ~\mu_{k, i}(h) = \frac{1}{|G_i|}\sum_{g \in hU}\Theta_k(g)  \quad \text{(local)}.
-% \end{equation}
-% and:
-% \begin{equation}
-%     \forall h \in H, ~\mu_{k, i}(h) = \frac{1}{|G_i|}\sum_{g \in hU}\Theta_k(g)  \quad \text{(local with stride)}.
-% \end{equation}
-
-% \paragraph{Excessive $G$-Invariance}
-
-%Many non-equivalent
-
-% \begin{figure}[H]
-%     \centering
-%     \includegraphics[width=0.3\textwidth]{example-image-a}
-%     \caption{[Figure illustrating ``adversarial examples'' for max pooling]}
-%     \label{fig:adversarial-examples}
-% \end{figure}
diff --git a/docs/sanbord_miolane_2025/sections/3-g-invariant-pooling.tex b/docs/sanbord_miolane_2025/sections/3-g-invariant-pooling.tex
deleted file mode 100644
index d53655b..0000000
--- a/docs/sanbord_miolane_2025/sections/3-g-invariant-pooling.tex
+++ /dev/null
@@ -1,129 +0,0 @@
-% \section{TC $G$-Pooling: Group-Invariant Pooling with the Triple Correlation}
-\section{The $G$-Triple-Correlation Layer for Robust $G$-Invariance}\label{sec:g-tc}
-
-We propose a $G$-Invariant layer designed for $G$-CNNs that is \textit{complete}\textemdash that is, it preserves all information about the input signal except for the group action. Our approach leverages the theory of the triple correlation on groups~\cite{kakarala1992triple} and applies it to the design of robust neural network architectures. Its theoretical foundations in signal processing and invariant theory allows us to generally define the unique $G$-invariant maps of lowest polynomial order that are complete, hence providing a general framework for selective, robust $G$-invariance in $G$-CNNs \cite{sturmfels2008algorithms}.
-
- % Our pooling operation is typically performed after the $G$-convolution, so that we restrict its definition to signals defined over a group $G$.
-%However, we note that the definition exists for any signal defined over a domain that is homogeneous for a group.
-
-\subsection{The $G$-Triple-Correlation Layer}
-
-% The \textit{triple correlation} (TC) on group $G$ is defined:
-
-% \begin{equation}
-%     \int_{-\infty}^{\infty}f(x)f(\mathcal{L}_{g_1} x) f(\mathcal{L}_{g_2} x) dx
-%     \end{equation}
-
-The \textit{$G$-Triple-Correlation} ($G$-TC) on a real signal $\Theta: G \rightarrow \mathbb{R}$ is the integral of the signal multiplied by two independently transformed copies of it~\cite{kakarala1992triple}:
-% \begin{equation}
-%     \tau_f(g_1, g_2) = \int_{u \in \domain} f(u) f(T_{g_1}(u)) f(T_{g_2}(u)) du.
-% \end{equation}
-% \begin{equation}
-%     \tau_f(g_1, g_2)
-%         =
-%         \int_{g \in G}
-%         f(T_g(u_0))
-%         f\left(T_{g g_1}(u_0)\right) f\left(T_{g g_2}(u_0)\right)
-%         dg .
-% \end{equation}
-\begin{equation}
-    \tau_\Theta(g_1, g_2)
-        = \int_{g \in G} \Theta(g) \Theta\left(g g_1\right) \Theta\left(g g_2\right)dg .
-\end{equation}
-This definition holds for any locally compact group $G$ on which we can define the Haar measure $dg$ used for integration purposes~\cite{KakaralaTriple}. This definition above is applicable to the $G$-CNNs where $\Theta$ is a collection of scalar signals over the group. We show in Appendix~\ref{app:steerable} that we can extend the definition to steerable $G$-CNNs where $\Theta$ can be an arbitrary field~\cite{cohen2016steerable}.
-
-In the equation above, the $G$-TC is computed for a pair of group elements $g_1, g_2$. In practice, we sweep over all pairs in the group. Appendix \ref{app:tc-examples} illustrates the triple correlation on three concrete groups. Importantly, the $G$-triple-correlation is invariant to the action of the group $G$ on the signal $\Theta$~\cite{KakaralaTriple}, as shown below.
-\begin{proposition}Consider a signal $\Theta: G \mapsto \mathbb{R}^c$. The $G$-Triple-Correlation $\tau$ is $G$-invariant:
-\begin{equation}
-    \tau_{L_g[\Theta]} = \tau_{\Theta}, \quad \text{for all $g \in G$,}
-\end{equation}
-where $L_g$ denotes an action of a transformation $g$ on the signal $\Theta$.
-\end{proposition}
-
-The proof is recalled in Appendix~\ref{sec:g-tc-invariance}. We propose to achieve $G$-invariance in a $G$-CNN by applying the $G$-Triple-Correlation ($G$-TC) to the output $\Theta$ of a $G$-convolutional layer. Specifically, we apply the $G$-TC to each real scalar valued signal $\Theta_k$ that comes from the $G$-convolution of filter $\phi_k$, for $k \in \{1, ..., K\}$. We only omit the subscript $k$ for clarity of notations. In practice, we will use the triple correlation on discretized groups, where the integral is replaced with a summation:
-% In practice, we will use the triple correlation on the output of the $G$-convolutional layers, which are functions defined over a discretized group. In this context, the triple correlation is defined as:
-% \begin{equation}\label{eq:tcpool}
-%     \tau_\Theta(g_1, g_2)= \sum_{g \in G}\Theta(g) \Theta\left(g g_1\right) \Theta\left(g g_2\right),
-% \end{equation}
-% \begin{equation}
-%     \Theta_{\mathrm{TCPOOL}}(i, j) = \sum_x f(x) f(x)_i  f(x)_j
-%     \end{equation}
-
-% \begin{equation}
-%     \Theta_{\mathrm{TCPOOL}}(i, j)_k = \sum_x \Theta_k(x) \Theta_k(g_i) \Theta_k(g_j)
-%     \end{equation}
-% TODO (?): What is the impact of the discretization if the group cannot be perfectly discretized (e.g. SO(3))?
-
-
-% \textbf{Version 1
-% }
-% We define the TC $G$-Pooling layer:
-
-% \begin{equation}
-%     \sum_x f(x)f(\mathcal{L}_{g_1} x) f(\mathcal{L}_{g_2} x)
-%     \end{equation}
-
-
-% \textbf{Version 2}
-
-\begin{equation}
-    T_\Theta(g_1, g_2)= \sum_{g \in G} \Theta(g) \Theta(gg_1) \Theta(gg_2),
-    \end{equation}
-for $\Theta$ a scalar valued function defined over $G$.
-%The output of a $G$-conv layer computed for group action $g$ and filter $k$ is written $\Theta_k$ which is a function of a discretized group $G$. The TC layer is defined:
-While it seems that the layer computes $T_\Theta(g_1, g_2)$ for all pairs of group elements $(g_1, g_2)$, we note that the real scalars $\Theta(gg_1)$ and $\Theta(gg_2)$ commute so that only half of the pairs are required. We will see that we can reduce the number of computations further when the group $G$ possesses additional properties such as commutativity.
-
-% Specifically, the TC $G$-pooling proposed here is the unique complete $G$-pooling method \cite{otherpaper}.
-
-% \begin{figure}
-%     \centering
-%     \includegraphics[width=0.7\textwidth]{figs/background-poolings-c4.pdf}
-%     \caption{[Figure summarizing our method]}
-%     \label{fig:background-poolings-c4}
-% \end{figure}
-
-We note that the triple correlation is the spatial dual of the \textit{bispectrum}, which has demonstrated robustness properties in the context of deep learning with bispectral neural networks \cite{sanborn2023bispectral}. The goal of bispectral neural networks is to learn an unknown group $G$ from data. The bispectral layer proposed in \cite{sanborn2023bispectral} assumes an MLP architecture. Our work is the first to generalize the use of bispectral invariants to convolutional networks. Here, we assume that the group $G$ is known in advance, and exploit the theoretical properties of the triple correlation to achieve robust invariance. One path for future extension may be to combine our approach with the learning approach of \cite{sanborn2023bispectral}, to parameterize and learn the group $G$ that defines a $G$-Equivariant and $G$-Invariant layer.
-
-\subsection{Selective Invariance through Completeness}
-
-We show here that the proposed $G$-triple-correlation is guaranteed to preserve all information aside from any equivariant component due to the group action on the input domain. This crucial property distinguishes our proposed layer from standard $G$-Pooling methods, which collapse signals and lose crucial information about the input (Figure \ref{fig:overview-cnn-c4}). In contrast with standard, excessively invariant $G$-pooling methods, we show here that our $G$-TC layer is instead \textit{selectively} $G$-invariant thanks to its \textit{completeness} property \cite{yellott1992uniqueness, kakarala1993group, kakarala2012bispectrum}, defined here:
-
-% The triple correlation is said to be \textit{complete}, in a sense made precise by the following proposition~ \cite{yellott1992uniqueness, kakarala1993group, kakarala2012bispectrum}.
-
-\begin{proposition}
-    Every integrable function with compact support $G$ is completely identified\textemdash up to group action\textemdash by its $G$-triple-correlation. We say that the $G$-triple-correlation is complete.
-\end{proposition}
-
-Mathematically, an operator $\mathcal{T}$ is complete for a group action $L$ if the following holds: for every pair of signals $\Theta_1$ and $\Theta_2$, if $\mathcal{T}(\Theta_1) = \mathcal{T}(\Theta_2)$ then the signals are equal up to the group action, that is: there exists a group element $h$ such that $\Theta_2 = L_h[\Theta_1]$.
-
-The proof of the completeness of the $G$-triple-correlation is only valid under a precise set of assumptions~\cite{kakarala1992triple} (Theorem 2). As we seek to integrate the $G$-triple-correlation to enhance robustness in neural networks, we investigate here the scope of these assumptions. First, the assumptions are not restrictive on the type of groups $G$ that can be used. Indeed, the proof only requires the groups to be Tatsuuma duality groups and the groups of interest in this paper meet this condition. This includes all locally compact commutative groups, all compact groups including the groups of rotations, the special orthogonal groups $SO(n)$, and groups of translations and rotations, the special euclidean groups $SE(n)$. Second, the assumptions are not restrictive on the types of signals. Indeed, the signal only needs to be such that any of its Fourier transform coefficients are invertible. For example, when the Fourier transform coefficients are scalar values, this means that we require these scalars to be non-zero. In practical applications on real image data with noise, there is a probability 0 that the Fourier transform coefficients of the input signal will be exactly 0 (scalar case) or non-invertible (matrix case). This is because the group of invertible matrices is dense in the space of matrices. Therefore, this condition is also verified in the applications of interest and more generally we expect the property of completeness of our $G$-TC layer to hold in practical neural network applications.
-
-\subsection{Uniqueness}
-
-The above two subsections prove that our $G$-Triple Correlation layer is selectively $G$-invariant. Here, we note that our proposed layer is the lowest-degree polynomial layer that can achieve this goal. In invariant theory, it is observed that the $G$-Triple Correlation is the \textit{only} third-order polynomial invariant (up to change of basis) \cite{sturmfels2008algorithms}. Moreover, it is the lowest-degree polynomial invariant that is also complete. It thus provides a unique and minimal-complexity solution to the problem of robust invariance within this function class.
- % Additionally, we can show that the TC $G$-Pooling is the only $G$-pooling that is indeed complete.
-
-\subsection{Computational Complexity}
-
-The $G$-Triple Correlation enjoys some symmetries that we can leverage to avoid computing it for each pair of group elements (which would represent $|G|^2$ computations), hence making the feedforward pass more efficient. We summarize these symmetries here.
-\begin{proposition}
-Consider two transformations $g_1, g_2 \in G$. The $G$-Triple Correlation of a real signal $\Theta$ has the following symmetry:
-\begin{align*}
-     T_\Theta(g_1, g_2) &= T_\Theta(g_2, g_1).
-\end{align*}
-If $G$ is commutative, the $G$-Triple Correlation of a real signal has the following additional symmetries:
-\begin{align*}
-     T_\Theta(g_1, g_2)
-     = T_\Theta(g^{-1}_1, g_2g^{-1}_1)
-     = T_\Theta(g_2g^{-1}_1, g^{-1}_1)
-     = T_\Theta(g^{-1}_2, g_1g^{-1}_2)
-     = T_\Theta(g_1g^{-1}_2, g^{-1}_2).
-\end{align*}
-\end{proposition}
-
-
-The proofs are given in \cite{nikias1993signal} for the group of translations. We extend them to any locally compact group $G$ in Appendix~\ref{app:symmetries}. In practice, these symmetries mean that even if there are theoretically $|G|^2$ computations, this number immediately reduces to $\frac{|G|(|G|+1)}{2}$ and further reduces if the group $G$ of interest is commutative. In addition, more subtle symmetries can be exploited to reduce the computational cost to linear $|G|+1$ for the case of one-dimensional cyclic groups \cite{kondor2008thesis} by considering the spectral dual of the $G$-TC: the bispectrum. We provide a computational approach to extend this reduction to more general, non-commutative groups in Appendix~\ref{app:reduction}. The theory supporting our approach has yet to be extended to this general case. Thus, there is an opportunity for new theoretical work that further increases the computational efficiency of the $G$-Triple-Correlation.
-
-% The triple correlation is a \textit{complete invariant} for [...].
-
-% The work of Yellott and Iversons \cite{yellott1992uniqueness} established that every integrable function with compact support is completely identified\textemdash up to translation\textemdash by its three-point autocorrelation and thus its bispectrum (Appendix \ref{appendix:autocorrelation}). Later work of Kakarala tied the bispectrum to the more general formulation of Fourier analysis and thus defined it for all compact groups \cite{kakarala1993group, kakarala2012bispectrum}. Kakarala went on to prove that it is complete \cite{kakarala2009completeness} in both commutative and non-commutative formulations (Appendix \ref{appendix:completeness}).
diff --git a/docs/sanbord_miolane_2025/sections/4-experiments.tex b/docs/sanbord_miolane_2025/sections/4-experiments.tex
deleted file mode 100644
index c72f0ca..0000000
--- a/docs/sanbord_miolane_2025/sections/4-experiments.tex
+++ /dev/null
@@ -1,257 +0,0 @@
-\section{Experiments \& Results}
-
-% We implement TC $G$-Pooling for the groups $SO(2)$ (discretized as $C_n$), and $O(2)$ (discretized as $D_n$) acting on $\mathbb{R}^2$ and the groups $SO(3)$ (discretized as the Chiral Octahedral group $O$) and $O(3)$ (discretized as the Full Octahedral group $O_h$) acting on $\mathbb{R}^3$.
-
-\subsubsection*{Implementation}
-
-We implement the $G$-TC Layer for arbitrary discretized groups with an efficient implementation built on top of the \href{https://github.com/QUVA-Lab/escnn}{ESCNN library} \cite{cesa2022a, e2cnn}, which provides a general implementation of $E(n)$-Equivariant Steerable Convolutional Layers. The method is flexibly defined, requiring the user only to provide a (Cayley) table that defines the group's product structure. The code is publicly available at \url{https://github.com/sophiaas/gtc-invariance}. Here, we demonstrate the approach on the groups $SO(2)$, and $O(2)$, $SO(3)$, and $O(3)$, discretized as the groups $C_n$ (cyclic), $D_n$ (dihedral), $O$ (chiral octahedral), and $O_h$ (full octahedral), respectively. \href{https://github.com/QUVA-Lab/escnn}{ESCNN} provides implementations for $G$-Conv layers on all of these $E(n)$ subgroups.
-
-% For all experiments, we make use of the \href{https://github.com/QUVA-Lab/escnn}{ESCNN library} \cite{cesa2022a, e2cnn}, which provides a general implementation of $E(n)$-Equivariant Steerable Convolutional Layers, including layers equivariant to the subgroups of $E(n)$ we consider here. [Insert computational strategy and reference algorithm in appendix.]
-
-
-\subsubsection*{Experimental Design}
-
-We examine the performance of the $G$-TC over Max $G$-Pooling in $G$-Equivariant Networks defined on these groups and trained on $G$-Invariant classification tasks. For the groups $SO(2)$ and $O(2)$ acting on $\mathbb{R}^2$, we use the MNIST dataset of handwritten characters \cite{lecun1998mnist}, and for the groups $SO(3)$ and $O(3)$ acting on $\mathbb{R}^3$, we use the voxelized ModelNet10 database of 3D objects \cite{wu20153d}. We generate $G$-MNIST and $G$-ModelNet10 datasets by transforming the domain of each signal in the dataset by a randomly sampled group element $g \in G$ (Figure \ref{fig:datasets}).
-
-
-\begin{figure}
-    \centering
-    \includegraphics[width=\linewidth]{figs/datasets.pdf}
-    \caption{\textbf{Datasets.} The $O(2)$-MNIST (top) and $O(3)$-ModelNet10 (bottom) datasets are generated by applying a random (rotation, reflection) pair to each element of the original datasets. Although we visualize the continuous group here, in practice, we discretize the group $O(3)$ as the full octahedral group $O_h$ to reduce computational complexity. $SO(2)$ and $SO(3)$ datasets are generated similarly, by applying a random rotation to each datapoint.}
-    \label{fig:datasets}
-\end{figure}
-
-In these experiments, we train pairs of models in parameter-matched architectures, in which only the $G$-Pooling method differs. Note that the purpose of these experiments is to compare \textit{differences in performance} between models using Max $G$-Pooling vs. the $G$-TC\textemdash not to achieve SOTA accuracy. Thus, we do not optimize the models for overall performance. Rather, we fix a simple architecture and set of hyperparameters and examine the change in performance that arises from replacing Max $G$-Pooling with the $G$-TC Layer (Figure \ref{fig:models}).
-
-\begin{figure}[H]
-    \centering
-    \includegraphics[width=0.5\linewidth]{figs/models.pdf}
-    \caption{\textbf{Models.} We compare two simple architectures comprised of a single G-Conv block followed by either a Max $G$-Pool layer or a $G$-TC Layer and an MLP Classifier.}
-    \label{fig:models}
-\end{figure}
-
-To isolate the effects of the $G$-Pooling method, all models are comprised of a single $G$-Conv block followed by $G$-Pooling (Max or TC) and an MLP Classifier. Notably, while many $G$-Conv models in the literature use the semi-direct product of $G$ with $\mathbb{R}^n$\textemdash i.e. incorporating the actions of the group $G$ into a standard translational convolutional model\textemdash here, we perform only \textit{pure} $G$-Conv, without translation. Thus, we use filters the same size as the input in all models.  The $G$-Conv block is comprised of a $G$-Conv layer, a batch norm layer, and an optional nonlinearity. For the Max $G$-Pool model, ReLU is used as the nonlinearity. Given the third-order nonlinearity of the TC, we omit the nonlinearity in the $G$-Conv block in the TC Model. The $G$-TC layer increases the dimensionality of the output of the $G$-Conv block; consequently the input dimension of the first layer of the MLP is larger and the weight matrix contains more parameters than for the Max $G$-Pool model. To compensate for this, we increase the dimension of the output of the first MLP layer in the Max Model, to match the overall number of parameters.
-
-
-% and find it to be sufficient to obtain high classification accuracy. Moreover, we find that the ReLU loses information in the $G$-Conv that must be preserved to obtain completeness in the TC Layer.
-% For this reason, we rescale the MNIST dataset to $(16, 16)$ pixels and voxelize ModelNet10 with a grid of shape $(10, 10, 10)$.
-
-
-
-
-\subsubsection*{Evaluation Methods}
-
-We evaluate the models in two ways. First, we examine differences in the raw \textit{classification accuracy} obtained by replacing Max $G$-Pooling with the $G$-TC Layer. Second, we assess the \textit{completeness} of the model by optimizing ``metameric'' stimuli for the trained models\textemdash inputs that yield the same pre-classifier representation as a target input, but are perceptually distinct. The completeness evaluation is inspired by a recent paper that incorporates the bispectrum\textemdash the spectral dual of the triple correlation\textemdash into a neural network architecture trained to yield $G$-invariant representations for $G$-transformed data \cite{sanborn2023bispectral}. In this work, two inputs are considered ``perceptually distinct'' if they are not in the same group orbit. They find that all inputs optimized to yield the same representation in the bispectral model are identical up to the group action. By contrast, many metameric stimuli can be found for $E(2)$-CNN \cite{e2cnn}, a $G$-Equivariant CNN that uses Max $G$-Pooling. Given the duality of the bispectrum and the triple correlation, we expect to observe similar ``completeness'' for $G$-CNNs using the $G$-TC Layer.
-
-
-
-% For this evaluation, we draw inspiration from a recent paper on a neural network architecture based on the bispectrum, the spectral dual of the triple correlation \cite{sanborn2023bispectral}. The bispectrum, like the triple correlation, is a \textit{complete invariant}. To assess the completeness of the model, the authors optimize ``metameric'' stimuli for their model, as well as
-
-
-% TC $G$-Pooling layer substitute for the conventional Max $G$-Pooling
-
-
-% \subsection{MNIST Experiments: $SO(2)$ and $O(2)$ on $\mathbb{R}^2$}
-
-\subsection{Classification Performance}
-
-
-% Table \ref{tab:mnist} shows the approximately matched parameter counts for each model.
-
-We train $G$-TC and Max $G$-Pooling models on the $SO(2)$ and $O(2)$-MNIST and chiral ($O$) and full ($O_h$) octahedral voxelized ModelNet10 training datasets and examine their classification performance on the test set. Full training details including hyperparameters are provided in Appendix~\ref{app:training}. Table \ref{tab:mnist} shows the test classification accuracy obtained by the Max-$G$ and $G$-TC architectures on each dataset. Accuracy is averaged over four random seeds, with confidence intervals showing standard deviation. We find that the model equipped with $G$-TC obtains a substantial improvement in overall classification performance\textemdash an increase of $1.3$, $0.89$, $1.84$ and $3.49$ percentage points on $SO(2)$-MNIST, $O(2)$-MNIST, $O$-ModelNet10 and $O_h$-ModelNet10 respectively.
-
-
-\begin{table}[H]
-\centering
-\begin{tabular}{c cc cc}
-\toprule
-& \multicolumn{2}{c}{ \textbf{$C8$-CNN on $SO(2)$-MNIST}} & \multicolumn{2}{c}{\textbf{$D16$-CNN on $O(2)$-MNIST}} \\
-\cmidrule(lr){2-3} \cmidrule(l){4-5}
-Method & Accuracy & Parameters & Accuracy & Parameters \\
-\midrule
-% \cmidrule(lr){2-3} \cmidrule(l){4-5}
-Max $G$-Pool & 95.23 $\pm$ 0.15 & 32,915 & 92.17\% $\pm$ 0.23 & 224,470 \\
-\textbf{$G$-TC} & \textbf{96.53} $\pm$ \textbf{0.16}  &  35,218 & \textbf{93.06 \% $\pm$ 0.09} & 221,074 \\
-\bottomrule
-\end{tabular}
-
-\begin{tabular}{c  cc cc}
-\toprule
-& \multicolumn{2}{c}{ \textbf{$O$-CNN on $O$-ModelNet10}} & \multicolumn{2}{c}{\textbf{$O_h$-CNN on $O_h$-ModelNet10}} \\
-\cmidrule(lr){2-3} \cmidrule(l){4-5}
-Method & Accuracy & Parameters & Accuracy & Parameters \\
-\midrule
-Max $G$-Pool & 72.17\% $\pm$ 0.95 & 500,198 & 71.73\% $\pm$ 0.23  & 1,826,978 \\
-% \textbf{$G$-TC} & \textbf{75.29\%} $\pm$ \textbf{0.86} & \textbf{476,354} & \textbf{76.69\%} $\pm$ \textbf{0.72} & \textbf{1,821,890} \\
-\textbf{$G$-TC} & \textbf{74.01\%} $\pm$ \textbf{0.48} & \textbf{472,066} & \textbf{75.22\%} $\pm$ \textbf{0.62} & \textbf{1,817,602} \\
-
-\bottomrule \\
-\end{tabular}
-
-  \caption{\textbf{Classification Accuracy \& Parameter Counts for Models Trained on $G$-MNIST and $G$-ModelNet10}. Confidence intervals reflect standard deviation over four random seeds per model. The model equipped with $G$-TC rather than Max $G$-Pooling obtains improved classification performance on all datasets.}
-\label{tab:mnist}
-\end{table}
-
-% \begin{table}[H]
-% \centering
-% \begin{tabular}{c  cc cc}
-% \toprule
-% & \multicolumn{2}{c}{ \textbf{$O$-CNN on $O$-ModelNet10}} & \multicolumn{2}{c}{\textbf{$O_h$-CNN on $O_h$-ModelNet10}} \\
-% \cmidrule(lr){2-3} \cmidrule(l){4-5}
-% Method & Accuracy & Parameters & Accuracy & Parameters \\
-% \midrule
-% Max $G$-Pool & 72.17\% $\pm$ 0.95 & 500,198 & 71.73\% $\pm$ 0.23  & 1,826,978 \\
-% \textbf{TC $G$-Pool} & \textbf{75.29\%} $\pm$ \textbf{0.86} & \textbf{476,354} & \textbf{76.69\%} $\pm$ \textbf{0.72} & \textbf{1,821,890} \\
-% \bottomrule
-% \end{tabular}
-% \vspace{0.2cm}
-%   \caption{\textbf{Classification Accuracy \& Parameter Counts for Models Trained on $G$-ModelNet10}.  Confidence intervals reflect standard deviation over five random seeds per model. }
-% \label{tab:modelnet}
-% \end{table}
-
-% \begin{table}[H]
-
-%   \centering
-%   \begin{tabular}{ccc}
-%     \toprule
-%     \textbf{Pooling Method} & \textbf{$C8$-CNN on $SO(2)$-MNIST} & \textbf{$D16$-CNN on $O(2)$-MNIST}  \\
-%     \midrule
-%     Max $G$-Pool & Score & 93.58\%  \\
-%     \textbf{TC} \textbf{$G$-Pool} & \textbf{Score} & \textbf{97.16\%} \\
-%     \bottomrule
-%     \\
-%   \end{tabular}
-%   \caption{\textbf{Classification Performance on Transformed MNIST Datasets}.}
-%   \label{tab:mnist-classification}
-% \end{table}
-
-% \begin{table}[H]
-%   \centering
-%   \begin{tabular}{ccc}
-%     \toprule
-%     \textbf{Pooling Method} & \textbf{$C8$-CNN on $SO(2)$-MNIST} & \textbf{$D16$-CNN on $O(2)$-MNIST}  \\
-%     \midrule
-%     Max $G$-Pool & Count & Count  \\
-%     \textbf{TC} \textbf{$G$-Pool} & Count & Count\\
-%     \bottomrule
-%     \\
-%   \end{tabular}
-%   \caption{\textbf{Parameter Count for MNIST Models}}
-%   \label{tab:mnist-params}
-% \end{table}
-
-% \begin{figure}[H]
-%     \centering
-%     \includegraphics[width=0.3\textwidth]{example-image-a}
-%     \caption{[Figure showing classification performance]}
-%     \label{fig:mnist-classification}
-% \end{figure}
-
-
-
-\subsection{Completeness}
-Following the analysis of \cite{sanborn2023bispectral}, we next evaluate the completeness of the models trained on the $G$-MNIST Dataset. Figure \ref{fig:mnist-completeness} shows inputs optimized to yield the same pre-classifier representation as a set of target images. In line with similar findings from \cite{sanborn2023bispectral}, we find that all inputs yielding an identical representations and classifications in the $G$-TC Model are within the same group orbit. Notably, the optimized images are identical to the targets, \textit{up to the group action}. This reflects exactly the completness of the $G$-TC: the $G$-TC preserves all signal structure up to the group action. Thus, any rotated version of the a target will yield the same $G$-TC Layer output. By contrast, many ``metameric'' misclassified stimuli can be found for the Max $G$-Pool Model, a consequence of the lossiness of this pooling operation.
-
-
-\begin{figure}[h]
-    \centering
-    \includegraphics[width=\textwidth]{figs/so2-o2-metamers.pdf}
-    % \includegraphics[width=\textwidth]{figs/so2-metamers.pdf}
-    % \includegraphics[width=0.3\textwidth]{example-image-b}
-    % \includegraphics[width=0.3\textwidth]{example-image-b}
-    \caption{\textbf{Optimized Model Metamers.} For each model, 100 targets from the MNIST dataset were randomly selected. 100 inputs were randomly initalized and optimized to yield identical pre-classifier model presentations. All inputs optimized for the $G$-TC Model converge to the orbit of the target. By contrast, metamers that bear no semantic relationship to the targets are found for every target in the Max $G$-Pooling model. \vspace{-0.4cm}}
-    \label{fig:mnist-completeness}
-\end{figure}
-
-% \subsection{ModelNet10 Experiments: $SO(3)$ and $O(3)$ on $\mathbb{R}^3$}
-
-% \paragraph*{Classification Accuracy.} We next train models on the chiral ($O$) and full ($O_h$) octahedral voxelized ModelNet10 training datasets and examine their classification performance on the test set. Full training details including hyperparameters are provided in the appendix. Table \ref{tab:modelnet} shows the test classification accuracy obtained by the Max and TC architectures on the $O$ and $O_h$ datasets. Accuracy is averaged over six random seeds, with confidence intervals showing standard deviation. As in the previous experiment, we find that the model equipped with TC $G$-Pooling obtains a substantial improvement in overall classification performance\textemdash an increase in $3.12$ and $4.96$ percentage points on $O$-ModelNet10 and $O_h$-ModelNet, respectively.
-
-
-
-% \begin{table}[H]
-%   \centering
-%   \begin{tabular}{ccc}
-%     \toprule
-%     \textbf{Pooling Method} & \textbf{$O$-CNN on $O$-ModelNet10} & \textbf{$O_h$-CNN on $O_h$-ModelNet10}  \\
-%     \midrule
-%     Max $G$-Pool & Score & Score  \\
-%     \textbf{TC $G$-Pool} & \textbf{Score} & \textbf{Score} \\
-%     \bottomrule
-%     \\
-%   \end{tabular}
-%   \caption{\textbf{Classification Performance on Transformed ModelNet10 Datasets}. Accuracy on the test set is averaged over six random seeds, with confidence intervals showing standard deviation.}
-%   \label{tab:simple-table}
-% \end{table}
-
-
-% \begin{table}[H]
-%   \centering
-%   \begin{tabular}{ccc}
-%     \toprule
-%     \textbf{Pooling Method} & \textbf{$O$-CNN on $O$-ModelNet10} & \textbf{$O_h$-CNN on $O_h$-ModelNet10}  \\
-%     \midrule
-%     Max $G$-Pool & Score & Score  \\
-%     \textbf{TC $G$-Pool} & \textbf{Score} & \textbf{Score} \\
-%     \bottomrule
-%     \\
-%   \end{tabular}
-%   \caption{\textbf{Parameter Count for ModelNet10 Models}}
-%   \label{tab:simple-table}
-% \end{table}
-
-
-% \begin{table}[htbp]
-%   \centering
-%   \begin{minipage}[t]{0.45\linewidth}
-%     \centering
-%     \caption{Table 1}
-%     \begin{tabular}{cc}
-%         \toprule
-%         \textbf{Pooling Method} & \textbf{$O$-ModelNet10}  \\
-%         \midrule
-%         Max $G$-Pool & Score \\
-%         TC $G$-Pool & Score \\
-%         \bottomrule
-%     \end{tabular}
-%   \end{minipage}
-%   \hfill
-%   \begin{minipage}[t]{0.45\linewidth}
-%     \centering
-%     \caption{Table 2}
-%     \begin{tabular}{cc}
-%     \toprule
-%     \textbf{Pooling Method} & \textbf{$O_h$-ModelNet10}  \\
-%     \midrule
-%     Max $G$-Pool & Score \\
-%     TC $G$-Pool & Score \\
-%     \bottomrule
-%     \end{tabular}
-%   \end{minipage}
-% \end{table}
-
-
-
-
-% \begin{figure}[H]
-%     \centering
-%     \includegraphics[width=0.3\textwidth]{example-image-a}
-%     \caption{[Figure showing classification performance]}
-%     \label{fig:modelnet-classification}
-% \end{figure}
-
-
-% \paragraph{Completeness.} We next evaluate the completeness of the trained models. Figure \ref{fig:modelnet-completeness} shows inputs optimized to yield the same pre-classifier representation as a set of target images. As with the MNIST experiments, we find that all inputs yielding an identical representations in the TC $G$-Pool model are within the same group orbit, while many ``metameric'' stimuli can be found for the Max $G$-Pool model.
-
-% \begin{figure}[H]
-%     \centering
-%     \includegraphics[width=0.3\textwidth]{example-image-b}
-%     \includegraphics[width=0.3\textwidth]{example-image-b} \\
-%     % \includegraphics[width=0.3\textwidth]{example-image-b}
-%     % \includegraphics[width=0.3\textwidth]{example-image-b}
-%     \caption{[Figure showing adversarial examples]}
-%     \label{fig:modelnet-completeness}
-% \end{figure}
-
-
-% Compare to this paper: https://arxiv.org/pdf/1911.08251.pdf
diff --git a/docs/sanbord_miolane_2025/sections/5-discussion.tex b/docs/sanbord_miolane_2025/sections/5-discussion.tex
deleted file mode 100644
index 22c5656..0000000
--- a/docs/sanbord_miolane_2025/sections/5-discussion.tex
+++ /dev/null
@@ -1,8 +0,0 @@
-\section{Discussion}
-\vspace{-0.3cm}
-In this work, we introduced a new method for achieving robust group-invariance in group-equivariant convolutional neural networks. Our approach, the \textit{$G$-TC Layer}, is built on the \textit{triple correlation} on groups, the lowest-degree polynomial that is a complete group-invariant map \cite{kakarala1992triple, sturmfels2008algorithms}. Our method inherits its completeness, which provides measurable gains in robustness and classification performance as compared to the ubiquitous Max $G$-Pooling.
-
-This improved robustness comes at a cost: the $G$-TC Layer increases the dimension of the output of a $G$-Convolutional layer from $G$ to $\frac{|G|(|G|+1)}{2}$. While the dimension of the discretized groups used in $G$-CNNs is typically small, this increase in computational cost may nonetheless deter practitioners from its use. However, there is a path to further reduction in computational complexity provided that we consider its spectral dual: the bispectrum. In \cite{kondor2008thesis}, an algorithm is provided that exploits more subtle symmetries of the bispectrum to demonstrate that only $|G| + 1$ terms are needed to provide a complete signature of signal structure, for the one-dimensional cyclic group. In Appendix \ref{app:reduction}, we extend the computational approach from \cite{kondor2008thesis} to more general groups and provided a path for substantial reduction in the complexity of the $G$-TC Layer, thus expanding its practical utility. Novel mathematical work that grounds our proposed computations in group theory is required to quantify the exact complexity reduction that we provide.
-
-
-As geometric deep learning is applied to increasingly complex data from the natural sciences \cite{gainza2020deciphering, atz2021geometric, ju2021performance}, we expect robustness to play a critical role in its success. Our work is the first to introduce the general group-invariant triple correlation as a new computational primitive for geometric deep learning. We expect the mathematical foundations and experimental successes that we present here to provide a basis for rethinking the problems of invariance and robustness in deep learning architectures.
diff --git a/docs/sanbord_miolane_2025/sections/appendix.tex b/docs/sanbord_miolane_2025/sections/appendix.tex
deleted file mode 100644
index 755d048..0000000
--- a/docs/sanbord_miolane_2025/sections/appendix.tex
+++ /dev/null
@@ -1,941 +0,0 @@
-\appendix
-
-
-\Large \textbf{Appendices}
-\normalsize
-
-\section{Group Axioms}\label{sec:group_axioms}
-
-
-For a set of transformations $G$ to be a group under the operation $\cdot$, these four axioms must hold:
-
-\begin{enumerate}
-    \item \textit{Closure}: The product of any two elements of the group is also an element of the group, i.e., for all $g_1, g_2 \in G$, $g_1g_2 \in G$.
-    \item \textit{Associativity}: The grouping of elements under the operation does not change the outcome, so long as the order of elements is preserved, i.e., $(g_1g_2)g_3 = g_1(g_2g_3)$.
-    \item \textit{Identity}: There exists a ``do-nothing'' \textit{identity} element $e$ that such that the product of $e$ with any other element $g$ returns $g$, i.e., $ge = eg = g$ for all $g \in G$.
-    \item \textit{Inverse}: For every element $g$, there exists an \textit{inverse} element $g^{-1}$  such that the product of $g$ and $g^{-1}$ returns the identity, i.e. $gg^{-1} = g^{-1}g = e$.
-\end{enumerate}
-
-
-\section{The Case of Steerable $G$-CNNs}\label{app:steerable}
-
-We consider the framework of Steerable $G$-CNNs defined in \cite{cohen2016steerable}. Consider a group $G$ that is the semi-direct product $G=\mathbb{Z}_2 \ltimes H$ of the group of translations $\mathbb{Z}_2$ and a group $H$ of transformations that fixes the origin $0 \in \mathbb{Z}_2$. Consider the feature map $\Theta: \mathbb{Z}_2 \rightarrow \mathbb{R}^K$ that is the output of a steerable $G$-CNN. Here, $\Theta$ is a field that transforms according to a representation $\pi$ induced by a representation $\rho$ of $H$ on the fiber $R^K$.
-
-The $G$-TC can be defined on $\Theta$ by replacing the regular representation by the induced representation. Specifically, replace any $\Theta(g)$, i.e. the scalar value of the feature map at group element $g$ by $\pi(g)(\Theta)(x)$, i.e., the vector value of the feature map at $x$ after a group element $g$ has acted on it via the representation $\pi$ :
-$$
-\tau_\Theta (g_1, g_2) =\int_G \pi(g)(\Theta)(x)^{\dagger} \cdot \pi\left(g_1 g\right)(\Theta)(x) \cdot \pi\left(g_2 g\right)(\Theta)(x) d g.
-$$
-
-Instead of computing the $G$-TC for each scalar coordinate $k$ of $\Theta$ as in the main text, we directly compute it as a vector. The formulation does not depend on the choice of $x$ by homogeneity of the domain $\mathbb{Z}_2$ for the group $G$. Importantly, this $G$-TC is invariant to the action of the induced representation, see Appendix~\ref{sec:g-tc-invariance}.
-
-
-
-
-\section{The $G$-Triple Correlation: Concrete Examples}
-\label{app:tc-examples}
-
-We show how to compute the $G$-Triple Correlation ($G$-TC) on three concrete groups.
-We start with the $G$-TC for the group $\mathbb{R}^2$ of 2D translations.
-
-\begin{example}
-Consider the group of 2D translations $G = (\mathbb{R}^2, +)$ with the addition as the group law. Consider a signal $\Theta$ that is a real function defined over $\mathbb{R}^2$ and can therefore be identified with an image. For any $x_1, x_2 \in \mathbb{R}^2$, the $G$-TC is given by:
-\begin{equation}
-    \tau_\Theta(x_1, x_2)
-        = \int_{x \in \mathbb{R}^2} \Theta(x) \Theta\left(x +  x_1\right) \Theta\left(x +  x_2\right)dx.
-\end{equation}
-\end{example}
-
-Next, we consider the special orthogonal group $SO(2)$ of 2D rotations.
-
-
-\begin{example}
-Consider the group of 2D rotations $G = (SO(2), \cdot )$ where $SO(2)$ is parameterized by $[0, 2\pi]$ and the composition of rotations $\cdot $ is the addition of angles modulo $2\pi$:
-\begin{equation}
-    \theta_1 \cdot \theta_2
-    \equiv
-    \theta_1 + \theta_2 [2\pi].
-\end{equation}
-For a real signal $\Theta$ defined over $G$, we have:
-\begin{equation}
-    \tau_\Theta(\theta_1, \theta_2)
-        = \int_{\theta \in SO(2)} \Theta(\theta) \Theta\left(\theta + \theta_1\right) \Theta\left(\theta +  \theta_2\right)d\theta,
-\end{equation}
-    for any $\theta_1, \theta_2 \in SO(2)$ and the addition is taken modulo $2\pi$.
-\end{example}
-
-Finally, we compute the $G$-TC for the special euclidean group $SE(2)$ of 2D rotations and translations, i.e., of 2D rigid-body transformations.
-
-
-\begin{example}
-Consider the group of 2D rigid body transformations $G = SE(2) = (SO(2) \times \mathbb{R}^2, \cdot)$ equipped with the group composition law:
-\begin{equation}
-    (\theta_1, x_1) \cdot (\theta_2, x_2)
-    \equiv
-    (\theta_1 + \theta_2, R_{\theta_1}.x_2 + x_1),
-\end{equation}
-where $R_{\theta_1} = \begin{bmatrix}
-    \cos \theta_1 & -\sin \theta_1 \\
-    \sin \theta_1 & \cos \theta_1
-\end{bmatrix}$ is the 2D rotation matrix associated with rotation $\theta_1$ and the addition of angles $\theta_1 + \theta_2$ is taken module $2\pi$.
-
-For a real signal defined on $SE(2)$ we have:
-\begin{align*}
-    \tau_\Theta((\theta_1, x_1), (\theta_2, x_2))
-        &= \int_{(\theta, x) \in SE(2)} \Theta(\theta, x) \Theta\left((\theta, x) \cdot (\theta_1, x_1)\right) \Theta\left((\theta, x) \cdot (\theta_2, x_2)\right)d\theta dx \\
-        &= \int_{(\theta, x) \in SE(2)} \Theta(\theta, x) \Theta\left(\theta+\theta_1, R_\theta.x_1+x\right) \Theta\left(\theta +\theta_2, R_\theta.x_2 + x\right)d\theta dx,
-\end{align*}
-for any $\theta_1, \theta_2 \in SO(2)$ and $x_1, x_2 \in \mathbb{R}^2$.
-\end{example}
-
-
-\section{Invariance of the $G$-Triple Correlation}\label{sec:g-tc-invariance}
-
-Consider a real signal $\Theta$ defined over a group $G$. The $G$-Triple Correlation is invariant to group actions on the domain of the signal $\Theta$ as shown in~\cite{KakaralaTriple}.
-
-\begin{proposition}
-    Consider two real signals $\Theta_1, \Theta_2$ defined over a group $G$. If there exists $h \in G$ such that one signal is obtained from the other by a group action, i.e., $\Theta_2 = \action_h[\Theta_1]$, then $\tau_{\Theta_1} = \tau_{\Theta_2}$.
-\end{proposition}
-
-We recall the proof of \cite{KakaralaTriple} below.
-
-\begin{proof}
-Consider two real signals $\Theta_1, \Theta_2$ defined over a group $G$, such that $\Theta_2 = L_h[\Theta_1]$ for a group action $L_h$ of group element $h$. We show that this implies that $\tau_{\Theta_1} = \tau_{\Theta_2}$.
-
-Taking $g_1, g_2 \in G$, we have:
-\begin{align*}
-    \tau_{\Theta_2}(g_1, g_2)
-        &= \int_{g \in G}
-            \Theta_2(g) \Theta_2\left(gg_1\right) \Theta_2\left(gg_2\right) dg  \\
-        &= \int_{g \in G}
-            L_h[\Theta_1](g) L_h[\Theta_1]\left(gg_1\right) L_h[\Theta_1]\left(gg_2\right) dg  \\
-        &= \int_{g \in G}
-            \Theta_1(hg) \Theta_1\left(hgg_1\right) \Theta_1\left(hgg_2\right) dg  \\
-        &= \int_{g \in G}
-            \Theta_1(g) \Theta_1\left(gg_1\right) \Theta_1\left(gg_2\right) dg \\
-        &=  \tau_{\Theta_1}(g_1, g_2).
-\end{align*}
-where we use the change of variable $hg \rightarrow g$.
-
-This proves the invariance of the $G$-TC with respect to group actions on the signals.
-\end{proof}
-
-\section{Symmetries of the $G$-Triple Correlation}\label{app:symmetries}
-
-The $G$-Triple Correlation ($G$-TC) enjoys some symmetries that we can leverage to avoid computing it for each pair of group elements (which would represent $|G|^2$ computations), hence making the feedforward pass more efficient.
-
-These symmetries are given in the main text. We recall them here for completeness.
-
-\begin{proposition}
-Consider two transformations $g_1, g_2 \in G$. The $G$-Triple Correlation of a signal $\Theta$ has the following symmetry:
-\begin{align*}
-     (s1) \qquad T_\Theta(g_1, g_2) &= T_\Theta(g_2, g_1).
-\end{align*}
-If $G$ is commutative, the $G$-Triple Correlation of a real signal has the following additional symmetries:
-\begin{align*}
-     (s2) \qquad T_\Theta(g_1, g_2)
-     = T_\Theta(g^{-1}_1, g_2g^{-1}_1)
-     = T_\Theta(g_2g^{-1}_1, g^{-1}_1)
-     = T_\Theta(g_1g^{-1}_2, g^{-1}_2)
-     = T_\Theta(g^{-1}_2, g_1g^{-1}_2).
-\end{align*}
-\end{proposition}
-
-Our proof extends the proof given in \cite{nikias1993signal} for the group of translations.
-
-\begin{proof}
-    Consider two transformations $g_1, g_2 \in G$.
-
-    Symmetry (s1) relies on the fact that $\Theta(gg_2)$ and $\Theta(gg_1)$ are scalar values that commute:
-    \begin{align*}
-        T_\Theta(g_2, g_1)
-            = \frac{1}{|G|} \sum_{g \in G} \Theta(g)\Theta(gg_2)\Theta(gg_1)
-            = \frac{1}{|G|} \sum_{g \in G} \Theta(g)\Theta(gg_1)\Theta(gg_2)
-            = T_\Theta(g_2, g_1).
-    \end{align*}
-    For symmetry (s2), we assume that $G$ is commutative.
-    We prove the first equality:
-    \begin{align*}
-        T_\Theta(g^{-1}_1, g_2g^{-1}_1)
-            &= \frac{1}{|G|} \sum_{g \in G} \Theta(g)\Theta(gg^{-1}_1)\Theta(gg_2g^{-1}_1) \\
-            &= \frac{1}{|G|} \sum_{g' \in G} \Theta(g'g_1)\Theta(g')\Theta(g'g_1g_2g^{-1}_1)
-                \quad \text{(with $g' = gg_1^{-1}$, i.e., $g=g'g_1$)} \\
-            &= \frac{1}{|G|} \sum_{g' \in G} \Theta(g'g_1)\Theta(g')\Theta(g'g_2)
-                \quad \text{($G$ commutative: $g_2 g^{-1}_1 = g^{-1}_1 g_2$)} \\
-            &= \frac{1}{|G|} \sum_{g \in G} \Theta(gg_1)\Theta(g)\Theta(gg_2) \\
-            &= \frac{1}{|G|} \sum_{g \in G} \Theta(g)\Theta(gg_1)\Theta(gg_2)
-                \quad \text{($\Theta$ takes on real values that commute)}\\
-            &= T_\Theta(g_2, g_1)
-    \end{align*}
-    The second equality of symmetry (s2) follows using (s1).
-    The third and fourth equality of symmetry (s2) have the same proof.
-\end{proof}
-
-This result and its proof are also valid for the extension of the $G$-TC that we propose in Appendix~\ref{app:steerable}. They naturally emerge by replacing the regular representation by the induced representation in the proof above.
-
-Specifically, consider a signal $\Theta_2=\pi\left(g_0\right)[\Theta_1]$ obtained from the action of $g_0$ on a signal $\Theta_1$. We want to show that $\tau_{\Theta_1} = \tau_{\Theta_2}$. The key ingredient of the proof is the change of variable within the integral $\int_{G^{\prime}}$, which follows the semi-direct product structure of $\mathrm{G}$ :
-$$
-h^{\prime}=h h_0 \text { and } t^{\prime}=\phi(h) t_0+t
-$$
-where $\phi(h)$ is a matrix representing h that acts on $\mathbb{Z}_2$ via matrix multiplication: e.g., a rotation matrix $R=\phi(r)$ in the case of $SE(n)$. This concludes the adaptation of the proof for the steerable case.
-
-
-
-\section{Algorithmic Reduction}\label{app:reduction}
-
-In this section, we show that we can reduce the complexity of the $G$-Triple Correlation of a real signal. This computational reduction requires that we consider, instead, the spectral dual of the $G$-TC, the bispectrum~\cite{kakarala2012bispectrum}. In what follows, we consider a signal $\Theta$ defined over a finite group $G$. The signal can be real or complex valued.
-
-\subsection{Reduction for Commutative Groups}
-
-
-Consider a commutative group $G$. The bispectrum for a signal $\Theta$ is defined over a pair of irreducible representations $\rho_1, \rho_2$ of the group $G$ as:
-\begin{equation}
-    \beta (\Theta)_{\rho_1, \rho_2} = \mathcal{F}(\Theta)_{\rho_1}^{\dagger}\mathcal{F}(\Theta)_{\rho_2}^{\dagger}\mathcal{F}(\Theta)_{\rho_1 \rho_2} \qquad \in \mathbb{C},
-    \label{eqn:group-bs}
-\end{equation}
-where $\mathcal{F}(\Theta)$ is the Fourier transform of $\Theta$ that generalizes the classical Fourier transformation to signals defined over a group. We note that, in group theory, the irreducible representations (irreps) of commutative groups map to scalar values. Hence, the bispectrum is a complex scalar in this case.
-
-For a discrete commutative group, the number of irreps is equal to the size of the group. \cite{KondorThesis} proved that, for cyclic groups, it is enough to compute $|G| + 1$ bispectral coefficients to fully describe the signal $\Theta$ up to group action, instead of the $|G|^2$ that would otherwise be required from its definition.
-
-\subsection{Reduction for Non-Commutative Groups}
-
-Consider a non-commutative group $G$. The bispectrum of a signal $\Theta$ is defined over a pair of irreducible representations $\rho_1, \rho_2$ of $G$ as:
-
-\begin{align*}
-   \beta (\Theta)_{\rho_1, \rho_2}
-   &= [\mathcal{F}(\Theta)_{\rho_1} \otimes \mathcal{F}(\Theta)_{\rho_2}]^{\dagger}C_{\rho_1,\rho_2} \Big[ \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \mathcal{F}(\Theta)_{\rho} \Big] C^{\dagger}_{\rho_1,\rho_2}
-   \qquad \in \mathbb{C}^{D_1 D_2 \times D_1 D_2},
-    \label{eqn:non-commutative-bs}
-\end{align*}
-
-where $\otimes$ is the tensor product, and $\oplus$ is a direct sum over irreps. The unitary Clebsch-Gordan matrix $C_{\rho_1,\rho_2}$ is analytically defined for each pair of representations $\rho_1, \rho_2$ as:
-
-\begin{equation}
-    (\rho_1 \otimes \rho_2)(g) = C_{\rho_1,\rho_2}^\dagger \Big[ \bigoplus_{\rho \in \rho_1 \otimes \rho_2} \rho(g) \Big] C_{\rho_1,\rho_2}.
-    \label{eqn:non-commutative-bs}
-\end{equation}
-
-We note that the irreps of non-commutative groups map to matrices, hence the bispectrum is a complex matrix in this case.
-
-% \subsection{Cyclic Groups}
-
-% We answer the question: given the bispectrum $\beta(\Theta)$, what is the minimum number of bispectral coefficients that we need to recover the signal $\Theta$?
-
-% \begin{proposition}
-%     The $n+1$ coefficients $\beta(\Theta)_{0, 0}, \beta(\Theta)_{1, 0}, ...., \beta(\Theta)_{1, n}$ are enough to recover the signal $\Theta$.
-% \end{proposition}
-
-% \begin{lemma}
-%     We have the following relationships between bispectral coefficients:
-%     \begin{align*}
-%     \beta_{0, 0}
-%     &= \hat{f}_{0}\hat{f}_{0}\hat{f}^{\dagger}_{0}
-%     = \hat{f}_{0} | \hat{f}_{0} |^2 \quad \rightarrow |\beta_{0, 0}| = |\hat{f}_{0}|^3\\
-%     \beta_{1, 0}
-%         &= \hat{f}_{1} \hat{f}_{0} \hat{f}^{\dagger}_{1}
-%         =  \hat{f}_{0} | \hat{f}_{1} |^2, \\
-%     \beta_{k, 0}
-%         &= \hat{f}_{k} \hat{f}_{0} \hat{f}^{\dagger}_{k} =  \hat{f}_{0} |\hat{f}_{k}|^2 \\
-%     \beta_{1, k-1}
-%         &= \hat{f}_{1} \hat{f}_{k-1} \hat{f}^{\dagger}_{k} .
-%     \end{align*}
-% \end{lemma}
-
-% \begin{proof}
-%     We show that the $n+1$ coefficients $\beta(\Theta)_{0, 0}, \beta(\Theta)_{1, 0}, ...., \beta(\Theta)_{1, n}$ are enough to recover the Fourier transform $\hat f$ of the signal $\Theta$.
-
-%     The $\beta(\Theta)_{0, 0}$ allows us to recover the DC component:
-%     \begin{equation}
-%             |\hat f (0)| = |\beta(\Theta)_{0, 0}|^{1/3}, \quad \text{arg}(\hat \Theta(0)) = \text{arg}(\beta_{0, 0}).
-%     \end{equation}
-
-%    We compute the initialization of the recurrence, i.e. the first frequency $\hat f (1)$:
-%    \begin{equation}
-%        | \hat f (1)| = \left(\frac{\beta_{1, 0}}{ \hat{f}_{0} }\right)^{1/2}, \quad  \text{arg}(\hat \Theta(1)) = 0,
-%    \end{equation}
-%    We note that we could equally set $\text{arg}(\hat \Theta(1)) = \phi$ for any $\phi \in [0, 2\pi)$. This indeterminacy is inevitable, since it is a reflection of translation invariance~\cite{kondor2008thesis}.
-
-%     Next, we compute the Fourier coefficients of $\Theta$ iteratively, from the lowest frequency to the highest:
-%        \begin{equation}
-%        \hat f (k) = \sqrt{\frac{\beta_{1, k-1}}{ \hat{f}_{1} \hat{f}_{k-1} }}^\dagger.
-%    \end{equation}
-% \end{proof}
-
-
-We provide an algorithmic approach reducing the computational complexity of the bispectrum for non-commutative finite groups. We show that we can recover the signal $\Theta$ from a small subset of its bispectral coefficients. That is, we can recover $\Theta$ from coefficients $\beta (\Theta)_{\rho_1, \rho_2}$ computed for a few, well-chosen irreps $\rho_1, \rho_2$. In practice, we only need to compute a few bispectral coefficients to have a complete invariant of the signal $\Theta$ \textemdash hence reducing the computational complexity of the layer.
-
-Generalizing \cite{KakaralaTriple}, we will show that a subset of bispectral coefficients allows us to recover the Fourier transform of the signal $\Theta$ for every irreducible representation $\rho$ of the group. This will show that we can recover the signal $\Theta$ itself by applying the inverse Fourier transform.
-
-% We first need to know ``how many'' irreducible representations the group has. For finite groups, this number is finite. We use the following
-
-%     \begin{proposition}
-%         The number of irreducible representations for a finite group $G$ is equal to the number of conjugacy classes. If $G$ is abelian, the number of irreducible representations is equal to $|G|$ and each representation is 1-dimensional.
-%     \end{proposition}
-
-% \begin{proof}
-%     We recall that the conjugacy class of an element $g \in G$ is defined with the conjugation operation defined by inner automorphisms:
-%     \begin{equation}
-%         C(g) = \{ hgh^{-1} ~|~ \forall h \in G\}.
-%     \end{equation}
-
-%     To prove the proposition in its entirety would require delving deep into the theory, but the core idea relies on the orthogonality relations of the character table of the group. The rows of the character table correspond to irreducible characters (which, over the complex numbers, can be associated with irreducible representations), while the columns correspond to conjugacy classes. Whether the group is commutative or not, this result holds. However, for commutative (or abelian) groups, every element is its own conjugacy class.
-% \end{proof}
-
-% We note that for Lie groups, this might not hold. For example, the group SO(3) has one irreducible representation in each odd dimension: it has an infinite number of irreps. We will deal with this case later.
-
-We first show relationships between the bispectral coefficients and the Fourier coefficients of the signal $\Theta$ in the following Lemma. We denote $\rho_0$ the trivial representation of the group $G$, which is the representation that sends every group element to the scalar 1.
-
-\begin{lemma}\label{lem:coefs}
-    Consider $\rho_0$ the trivial representation of the group $G$. Consider $\rho$ any other irreducible representation of dimension $D$. The bispectral coefficients write:
-    \begin{align*}
-    \beta_{\rho_0, \rho_0}
-    &= |\mathcal{F}(\Theta)_{\rho_0}|^2 \mathcal{F}(\Theta)_{\rho_0}
-    \in \mathbb{C}\\
-    \beta_{\rho, \rho_0}
-    &= \mathcal{F}(\Theta)_{\rho_0}^\dagger
-    \mathcal{F}(\Theta)_{\rho}^\dagger\mathcal{F}(\Theta)_{\rho}
-    \in \mathbb{C}^{D \times D}.
-    \end{align*}
-\end{lemma}
-
-Here, and in what follows, we denote $\beta$ the bispectrum of $\Theta$, i.e., we omit the argument $\Theta$ for clarity of notations.
-
-\begin{proof}
-    For $\rho_0$ the trivial representation, and $\rho$ an irreps, the Clebsh-Jordan (CJ) matrix $C_{{\rho} {\rho_0}}$ is the identity matrix, and the matrix $C_{{\rho} {\rho_0}}$ is the scalar 1.
-
-    % , we know that $\rho \otimes \rho_0  = \rho$ such that the Clebsh-Jordan decomposition:
-    % \begin{equation}
-    %     \rho \otimes \rho_0 = C_{{\rho} {\rho_0}}^{\dagger}\left[
-    % \bigoplus_{\rho' \in \rho \otimes \rho_0}
-    %     \rho'
-    % \right] C_{{\rho} {\rho_0}} = C_{{\rho} {\rho_0}}^{\dagger}
-    %     \rho C_{{\rho} {\rho_0}}
-    % \end{equation}
-    % gives: $C_{{\rho} {\rho_0}}^{\dagger}  C_{{\rho}{\rho_0}}  = 1$ for any irreps $\rho$. In addition, for $\rho = \rho_0$, the Clebsh-Jordan matrices are scalars. Actually, in this case, the CJ matrix is just the identity matrix (according to chatGPT).
-
-    We compute the bispectral coefficient $\beta_{\rho_0, \rho_0}$:
-    \begin{align*}
-    \beta_{\rho_0, \rho_0}
-    &=
-    (\mathcal{F}(\Theta)_{\rho_0} \otimes \mathcal{F}(\Theta)_{\rho_0})^\dagger C_{{\rho_0} {\rho_0}}
-    \left[
-    \bigoplus_{\rho \in \rho_0 \otimes \rho_0}
-        \mathcal{F}(\Theta)_\rho
-    \right] C_{{\rho_0} {\rho_0}}^{\dagger} \\
-    &=
-    (\mathcal{F}(\Theta)_{\rho_0} \otimes \mathcal{F}(\Theta)_{\rho_0})^\dagger C_{{\rho_0} {\rho_0}}
-        \mathcal{F}(\Theta)_{\rho_0}
-     C_{{\rho_0} {\rho_0}}^{\dagger}
-    \qquad \text{($\rho_0 \otimes \rho_0 = \rho_0$ which is irreducible)}
-    \\
-    &=
-    (\mathcal{F}(\Theta)_{\rho_0}^2)^\dagger C_{{\rho_0} {\rho_0}}
-        \mathcal{F}(\Theta)_{\rho_0}
-     C_{{\rho_0} {\rho_0}}^{\dagger}
-    \qquad \text{($\mathcal{F}(\Theta)_{\rho_0}$ is a scalar for which tensor product is multiplication)}
-    \\
-    &=|\mathcal{F}(\Theta)_{\rho_0}|^2 \mathcal{F}(\Theta)_{\rho_0}
-    \qquad\text{(CJ matrices are equal to 1.)}
-    \end{align*}
-
-    Take any irreducible representation $\rho$ of dimension $D$, we have:
-    \begin{align*}
-        \beta_{\rho, \rho_0}
-        &= (\mathcal{F}(\Theta)_{\rho} \otimes \mathcal{F}(\Theta)_{\rho_0})^\dagger C_{{\rho} {\rho_0}}
-    \left[
-    \bigoplus_{\rho \in \rho \otimes \rho_0}
-        \mathcal{F}(\Theta)_\rho
-    \right] C_{{\rho} {\rho_0}}^{\dagger} \\
-        &= \mathcal{F}(\Theta)_{\rho_0}^\dagger \mathcal{F}(\Theta)_{\rho}^\dagger C_{{\rho} {\rho_0}}
-        \mathcal{F}(\Theta)_{\rho} C_{{\rho} {\rho_0}}^{\dagger} \qquad\text{($\mathcal{F}(\Theta)_{\rho_0}$ is a scalar and $\rho \otimes \rho_0 = \rho$)}\\
-        &=\mathcal{F}(\Theta)_{\rho_0}^\dagger \mathcal{F}(\Theta)_{\rho}^\dagger\mathcal{F}(\Theta)_{\rho} \qquad\text{(CJ matrices are identity matrices)}.
-    \end{align*}
-    \end{proof}
-
-Next, we summarize our main result.
-
-\begin{proposition}
-    We can recover the Fourier coefficients of a signal $\Theta$ from only $L$ bispectral coefficients, where $L$ is a number computed from the Kronecker product table of the group $G$.
-\end{proposition}
-
-In the proof, we propose an algorithmic method that iteratively computes bispectral coefficients until the Fourier coefficients of the signal are all recovered. We note that, for arbitrary groups and their representations, Clebsch–Gordan (CJ) matrices are not known in general, yet they can be computed numerically. Thus, the proof below assumes that the CJ matrices are given for the group $G$ of interest.
-
-\begin{proof}
-
-\textbf{Algorithmic Approach.}
-
-    First, we show how we can recover the first Fourier coefficient (DC component), i.e., the Fourier transform of the signal at the trivial representation $\rho_0$, from a single bispectral coefficient.
-    \begin{equation}
-    \mathcal{F}(\Theta)_{\rho_0} = \hat{f}_{\rho_0} = \int_G \Theta(g)\rho_0(g) dg = \int_G \Theta(g) dg \in \mathbb{C}.
-     \end{equation}
-    Using Lemma~\ref{lem:coefs}, we can recover this Fourier component from the bispectral coefficient $\beta_{\rho_0, \rho_0}$, as:
-    \begin{equation}
-        | \mathcal{F}(\Theta)_{\rho_0}| = \left(|\beta_{\rho_0, \rho_0}|\right)^{1/3},
-        \quad
-        \text{arg}(\mathcal{F}(\Theta)_{\rho_0}) = \text{arg}(\beta_{\rho_0, \rho_0}).
-    \end{equation}
-
-    Next, consider an irreducible representation $\rho_1$ of dimension $D$. We seek to recover the Fourier coefficient $\mathcal{F}(\Theta)_{\rho_1}$. This Fourier coefficient is a matrix in $\mathbb{C}^{D \times D}$. Using Lemma~\ref{lem:coefs}, we can recover it from a single bispectral coefficient:
-    \begin{equation}
-        \mathcal{F}(\Theta)_{\rho}^\dagger\mathcal{F}(\Theta)_{\rho}
-        = \frac{\beta_{\rho, \rho_0}}{\mathcal{F}(\Theta)_{\rho_0}^\dagger} \in \mathbb{C}^{D \times D},
-    \end{equation}
-    since we have already recovered the Fourier coefficient $ \mathcal{F}(\Theta)_{\rho_0}$. The matrix $\mathcal{F}(\Theta)_{\rho}^\dagger\mathcal{F}(\Theta)_{\rho}$ is hermitian, and thus admits a square-root, that we denote $\mathcal{F}(\Theta)_{\rho}'$:
-      \begin{equation}
-        \mathcal{F}(\Theta)_{\rho}'
-        = \left(\frac{\beta_{\rho, \rho_0}}{\mathcal{F}(\Theta)_{\rho_0}^\dagger}
-        \right)^{1/2}.
-    \end{equation}
-    The square-root $\mathcal{F}(\Theta)_{\rho}'$ only corresponds to $\mathcal{F}(\Theta)_{\rho}$ up to a matrix factor. This is an unidentifiability similar to the commutative case~\cite{KakaralaTriple}. Specifically, consider the singular value decomposition (SVD) of $\mathcal{F}(\Theta)_{\rho}$:
-    \begin{align*}
-        \mathcal{F}(\Theta)_{\rho}
-        &= U.\Sigma.V^\dagger\\
-        \Rightarrow
-         \mathcal{F}(\Theta)_{\rho}^\dagger\mathcal{F}(\Theta)_{\rho}
-        &= (U.\Sigma.V^\dagger)^\dagger. U.\Sigma.V^\dagger
-        = V \Sigma^2 V^\dagger\\
-        \Rightarrow
-        \mathcal{F}(\Theta)_{\rho}'
-        &= V \Sigma V^\dagger.
-    \end{align*}
-    Thus, we have: $\mathcal{F}(\Theta)_{\rho} = UV^\dagger.\mathcal{F}(\Theta)_{\rho}' $ where $U, V$ are unitary matrices that come from the (unknown) SVD decomposition of the (unknown) $\mathcal{F}(\Theta)_{\rho}$. By recovering $\mathcal{F}(\Theta)_{\rho}$ as $\mathcal{F}(\Theta)_{\rho}'$, we fix $UV^\dagger=I$. This is similar to the same way in which \cite{KakaralaTriple} fixed $\phi = 0$ (rotation of angle 0, i.e., the identity) in the commutative case.
-
-    Next, we seek to find the remaining Fourier coefficients of the signal $\Theta$, from a limited subset of the bispectral coefficients. To this aim, we denote $\mathcal{R}$ the set of irreducible representations of the group $G$. We recall that, for a finite group $G$, the set $\mathcal{R}$ is also finite, with its size equal to the number of conjugacy classes of $G$.
-
-    We consider the following bispectral coefficient:
-    \begin{align*}
-    \beta_{\rho_1, \rho_1}
-    &=
-    (\mathcal{F}(\Theta)_{\rho_1} \otimes \mathcal{F}(\Theta)_{\rho_1})^\dagger C_{{\rho_1} {\rho_1}}
-    \left[
-    \bigoplus_{\rho \in \rho_1 \otimes \rho_1}
-        \mathcal{F}(\Theta)_\rho
-    \right] C_{{\rho_1} {\rho_1}}^{\dagger}\\
-    &=
-        (\mathcal{F}(\Theta)_{\rho_1} \otimes \mathcal{F}(\Theta)_{\rho_1})^\dagger
-        C_{{\rho_1} {\rho_1}}
-    \left[
-    \bigoplus_{\rho \in \mathcal{R}}
-        \mathcal{F}(\Theta)_\rho^{n_{\rho, \rho_1}}
-    \right] C_{{\rho_1} {\rho_1}}^{\dagger},
-    \end{align*}
-    where $n_{\rho, \rho_1}$ is the multiplicity of irreps $\rho$ in the decomposition of $\rho_1, \rho_1$. This multiplicity is known as it only depends on the group, not on the signal $\Theta$.
-
-    We get an equation that allows us to recover additional Fourier coefficients of the signal $\Theta$:
-    \begin{equation}
-        \bigoplus_{\rho \in \mathcal{R}}
-        \mathcal{F}(\Theta)_\rho^{n_{\rho, \rho_1}}
-    = C_{{\rho_1} {\rho_1}}^{-1}(\mathcal{F}(\Theta)_{\rho_1} \otimes \mathcal{F}(\Theta)_{\rho_1})^{-\dagger} \beta_{\rho_1, \rho_1} C_{{\rho_1} {\rho_1}}^{-\dagger},
-    \end{equation}
-    where everything on the right hand side is known.
-    Therefore, every Fourier coefficient $\mathcal{F}(\Theta)_\rho$ that appears in the decomposition of $\rho_1 \otimes \rho_1$ into irreducible irreps $\rho$ can be computed, by reading off the elements of the block diagonal matrix defined by the direct sum. We recover the Fourier coefficients $\mathcal{F}(\Theta)_\rho$ for which $n_{\rho, \rho_1} \neq 0$.
-
-    We assume that this procedure provides at least one other Fourier coefficient, for an irreps $\rho_2$, that we fix. We can then compute the following bispectral coefficient:
-    \begin{align*}
-    \beta_{\rho_1, \rho_2}
-    &=
-    (\mathcal{F}(\Theta)_{\rho_1} \otimes \mathcal{F}(\Theta)_{\rho_2})^\dagger C_{{\rho_1} {\rho_2}}
-    \left[
-    \bigoplus_{\rho \in \rho_1 \otimes \rho_2}
-        \mathcal{F}(\Theta)_\rho^{n_{\rho, \rho_2}}
-    \right] C_{{\rho_1} {\rho_2}}^{\dagger},
-    \end{align*}
-    to get a novel equation:
-        \begin{equation}
-        \bigoplus_{\rho \in \mathcal{R}}
-        \mathcal{F}(\Theta)_\rho^{n_{\rho, \rho_2}}
-    = C_{{\rho_1} {\rho_2}}^{-1}(\mathcal{F}(\Theta)_{\rho_1} \otimes \mathcal{F}(\Theta)_{\rho_2})^{-\dagger} \beta_{\rho_1, \rho_2} C_{{\rho_1} {\rho_2}}^{-\dagger},
-    \end{equation}
-    where everything on the right-hand side is known. Thus, every Fourier coefficient $\mathcal{F}(\Theta)_\rho$ that appears in the decomposition of $\rho_1 \otimes \rho_1$ into irreducible irreps can be recovered, by reading off the elements of the block diagonal matrix. We get the $\mathcal{F}(\Theta)_\rho$ for which $n_{\rho, \rho_2} \neq 0$. We iterate this procedure to recover more Fourier coefficients.
-
-    \textbf{Number of bispectral coefficients.}
-
-    Now, we show that our procedure can indeed recover all of the Fourier coefficients of the signal $\Theta$. Additionally, we show that it only requires a limited number $L$ of bispectral coefficients, where $L$ depends on the group $G$. Specifically, it depends on the Kronecker product table of $G$, which is the $|\mathcal{R}| \times |\mathcal{R}|$ table of the decomposition of the tensor product of two irreducible representations into a direct sum of elementary irreps. In this table, the element at row $i$ and column $j$ lists all of the multiplicity of the irreps that appear in the decomposition of $\rho_i \otimes \rho_j$.
-
-    The Kronecker table, i.e., any multiplicity $m_k = n_{\rho_k}$ of $\rho_k$ in the decomposition $\tilde \rho = \rho_i \otimes \rho_j$ can be computed with a procedure inspired by \cite{cohen2016steerable} and described below.
-
-    The procedure relies on the character $\chi_{\tilde \rho}(g)=\operatorname{Tr}(\tilde \rho(g))$ of the representation $\tilde \rho$ to be decomposed. From group theory, we know that the characters of irreps $\rho_i, \rho_j$ are orthogonal, in the following sense:
-    \begin{equation}
-    \left\langle\chi_{\rho_i,}, \chi_{\rho_j}\right\rangle \equiv \frac{1}{|G|} \sum_{h \in G} \chi_{\rho_i}(h) \chi_{\rho_j}(h)=\delta_{i j} .
-    \end{equation}
-    Thus, we can obtain the multiplicity of
-    irrep $\rho_k$ in $\tilde \rho$ by computing the inner product with the $k$-th character:
-    $$
-    \left\langle\chi_{\tilde \rho}, \chi_{\rho_k}\right\rangle
-    =
-    \left\langle\chi_{\oplus_l m_l \rho_l}, \chi_{\rho_k}\right\rangle=\left\langle\sum_l m_l \chi_{\rho_l}, \chi_{\rho_k}\right\rangle
-    =
-    \sum_l m_l\left\langle\chi_{\rho_l}, \chi_{\rho_k}\right\rangle
-    =m_k,
-    $$
-using the fact that the trace of a direct sum equals the sum of the traces (i.e. $\chi_{\rho \oplus \rho^{\prime}}=\chi_\rho+\chi_{\rho^{\prime}}$ ). Thus, we can determine the Kronecker product table of interest. For example, the Kronecker product table for the dihedral group $D_4$ is shown in Table~\ref{tab:kron-d4}.
-
-\begin{table}[]
-\centering
-    \begin{tabular}{|l|l|l|l|l|l|}
-    \hline
-    $\otimes$ & $A_1$             & $A_2$             & $B_1$             & $B_2$             & $E$               \\ \hline
-    $A_1$     & $(1, 0, 0, 0, 0)$ & $(0, 1, 0, 0, 0)$ & $(0, 0, 1, 0, 0)$ & $(0, 0, 0, 1, 0)$ & $(0, 0, 0, 0, 1)$ \\ \hline
-    $A_2$     & $(0, 1, 0, 0, 0)$ & $(1, 0, 0, 0, 0)$ & $(0, 0, 0, 1, 0)$ & $(0, 0, 1, 0, 0)$ & $(0, 0, 0, 0, 1)$ \\ \hline
-    $B_1$     & $(0, 0, 1, 0, 0)$ & $(0, 0, 0, 1, 0)$ & $(1, 0, 0, 0, 0)$ & $(0, 1, 0, 0, 0)$ & $(0, 0, 0, 0, 1)$ \\ \hline
-    $B_2$     & $(0, 0, 0, 1, 0)$ & $(0, 0, 1, 0, 0)$ & $(0, 1, 0, 0, 0)$ & $(1, 0, 0, 0, 0)$ & $(0, 0, 0, 0, 1)$ \\ \hline
-    $E$       & $(0, 0, 0, 0, 1)$ & $(0, 0, 0, 0, 1)$ & $(0, 0, 0, 0, 1)$ & $(0, 0, 0, 0, 1)$ & $(1, 1, 1, 1, 0)$ \\ \hline
-    \end{tabular}
-    \caption{Kronecker table for the dihedral group $D_4$ which has 5 irreps called $A_1, A_2, B_1, B_2$ and $E$.}
-    \label{tab:kron-d4}
-\end{table}
-
-    % \begin{table}[]
-    % \centering
-    % \begin{tabular}{|l|l|l|l|l|}
-    % \hline
-    % $\otimes$ & $\rho_0$       & $\rho_1$       & $\rho_2$       & $\rho_3$       \\ \hline
-    % $\rho_0$  & $(1, 0, 0, 0)$ & $(0, 1, 0, 0)$ & $(0, 0, 1, 0)$ & $(0, 0, 0, 1)$ \\ \hline
-    % $\rho_1$  & $(0, 1, 0, 0)$ & $(m_k^{11})_k$ & $(m_k^{12})_k$ & $(m_k^{13})_k$ \\ \hline
-    % $\rho_2$  & $(0, 0, 1, 0)$ & $(m_k^{21})_k$ & $(m_k^{22})_k$ & $(m_k^{23})_k$ \\ \hline
-    % $\rho_3$  & $(0, 0, 0, 1)$ & $(m_k^{31})_k$ & $(m_k^{32})_k$ & $(m_k^{33})_k$ \\ \hline
-    % \end{tabular}
-    % \caption{}
-    % \label{tab:d4-k}
-    % \end{table}
-
-    The Kronecker product table shows us how many bispectral coefficients we need to complete our algorithmic procedure. Our procedure essentially uses a breadth-first search algorithm on the space of irreducible representations, starting with $\rho_1$ and using the tensor product with $\rho_1$ as the mechanism to explore the space. Whether this procedure succeeds in including all the irreps in $\mathcal{R}$ might on the group and its Kronecker (tensor) product table. Specifically, consider all the irreps that appear in the row corresponding to $\rho_1$ in the Kronecker product table. If these irreps do not cover the set $\mathcal{R}$ of all possible irreps, then the approach will need more than the tensor products of the form $\rho_1 \otimes \rho_j$ to get all of the Fourier coefficients of the signal $\Theta$.
-    %In this case, we might need to consider others of the form $\rho_1$.
-
-    %  The final procedure.
-    % \begin{itemize}
-    %     \item List = $[\rho1]$
-    %     \item Compute $\rho_1 \otimes \rho_1$.
-    %     \item Take all the irreps that are not in List that appears in that decomposition.
-    %     \item For every such irrep $\rho_2$:
-    %     \begin{itemize}
-    %         \item Compute $\rho_1 \otimes \rho_2$
-    %         \item Append $\rho_2$ to List.
-    %         \item Take all the irreps that are not in List that appears in that decomposition.
-    %         \item For every such irrep $\rho_3$:
-    %         \begin{itemize}
-    %             \item Compute $\rho_1 \otimes \rho_3$
-    %             \item Append $\rho_3$ to List.
-    %             \item etc
-    %         \end{itemize}
-    %     \end{itemize}
-    % \end{itemize}
-
-    % We wonder whether this procedure gives us all the irreps of the group, and if so, in how many steps L. The number of steps is the number of bispectral coefficients needed.
-\end{proof}
-
-    We observe in our experiments that this procedure does indeed succeed in computing all of the Fourier coefficients of the signal $\Theta$ for most of the groups of interest. We provide detailed examples of these computations on our github repository, for the diedral groups $D_4$, $D_{16}$ and for the chiral octahedral group $O$. The procedure does not succeed in the case of the full octahedral group $O_h$ does not succeed.
-
-    For the diedral group $D_4$, which has $\mathcal{R} = 5$ irreps, our approach allows us to recover a signal on $D_4$ from only 3 bispectral coefficients, instead of $5^2 = 25$. For the diedral group $D_{16}$, which has $\mathcal{R} = 11$ irreps, we recover the signal from $9$ bispectral coefficients instead of $11^2 = 121$. For the octahedral group, which has $\mathcal{R} = 5$ irreps, we use $4$ bispectral coefficients instead of $5^2 = 25$. This represent a substantial complexity reduction. More theoretical work is needed to prove that our approach applies to a wide range of discrete groups, or to further generalize it for groups such as $O_h$.
-
-    %This is particularly true for groups where the tensor product of any two irreps spans the whole space of irreps. This is the case, for example, for SU(2) (the group of 2x2 unitary matrices with determinant 1).
-
-% \subsection{Example: Diedral group $D_4$}
-
-%     The diedral group $D_4$ is the group of symmetries of a square. $D_4$ has five irreps, called $A_1$, $A_2$, $B_1$, $B_2$, and $E$.
-
-%     % \paragraph{Irreps and their matrices $\rho(g)$ for $g \in G$}
-
-%     % \begin{figure}
-%     %     \centering
-%     %     \includegraphics[width=\textwidth]{figs/d4.png}
-%     % \end{figure}
-
-%     We give the Kronecker product table for $D_4$ in Table~\ref{tab:kron-d4}. To do so, we compute each tensor-product representation $\tilde \rho$ by:
-%     \begin{itemize}
-%         \item computing its matrices $\tilde \rho(g)$ on each group element, by definition of the tensor product of matrices,
-%         \item computing the characters $\chi(g) = \text{Tr}(\tilde \rho(g))$ for each group element,
-%         \item computing the inner-product of the character $\chi$ with each character of each irreps to get the multiplicity of interest.
-%     \end{itemize}
-
-
-
-
-
-% \paragraph{Algorithmic Procedure} We choose $\rho_0 = A_1$, $\rho_1 = E$.
-% \begin{itemize}
-%     \item List = [$\rho_0$, $\rho_1$]
-%     \item Compute irreps in the decomposition of $\rho_1 \otimes \rho_1$, that are not already in List. This gives: $A_2, B_1, B_2$.
-%     \item Stopping criterion reached: all irreps in List.
-% \end{itemize}
-
-% \paragraph{Computations after the $G$-Convolution}
-
-% We store the representations in a variable.
-
-% We compute every Fourier coefficient of $\Theta$, via:
-% \begin{equation}
-%     \mathcal{F}(\Theta)_\rho = \sum_{g \in G} \Theta(g).\rho(g)
-% \end{equation}
-% by taking a linear combination of the $\rho(g)$ matrices for each $\rho$ irreps. Note that $\mathcal{F}(\Theta)_\rho \in \mathbb{C}$ for every $\rho$ in $\{A_1, A_2, B_1, B_2\}$ and $\mathcal{F}(\Theta)_E \in \mathbb{C}^{2 \times 2}$.
-
-% We compute $\beta_{\rho_0, \rho_0} \in \mathbb{C}$.
-% \begin{equation}
-%     \beta_{\rho_0, \rho_0} = |\mathcal{F}(\Theta)_{\rho_0}|^2 \mathcal{F}(\Theta)_{\rho_0} \in \mathbb{C}
-% \end{equation}
-
-% We compute $\beta_{\rho_1, \rho_0} \in \mathbb{C}^{2 \times 2}$.
-% \begin{equation}
-%     \beta_{\rho_1, \rho_0} = \mathcal{F}(\Theta)_{\rho_1}^\dagger\mathcal{F}(\Theta)_{\rho_1} \mathcal{F}(\Theta)_{\rho_0}^* \in \mathbb{C}^{2 \times 2}
-% \end{equation}
-
-% We compute $\beta_{\rho_1, \rho_1} \in \mathbb{C}^{4 \times 4}$. We use the notation: $\mathcal{F} = \mathcal{F}(\Theta)_{\rho_1}$.
-% \begin{align*}
-%     \beta_{\rho_1, \rho_1}
-%     &= (\mathcal{F}(\Theta)_{\rho_1} \otimes \mathcal{F}(\Theta)_{\rho_1})^\dagger C_{{\rho_1} {\rho_1}}
-%     \left[
-%     \bigoplus_{\rho \in \rho_1 \otimes \rho_1=\{A_1, A_2, B_1, B_2\}}
-%         \mathcal{F}(\Theta)_\rho
-%     \right] C_{{\rho_1} {\rho_1}}^{\dagger} \\
-%     &= \left[\begin{array}{ll}
-% \mathcal{F}_{1,1}\left[\begin{array}{ll}
-% \mathcal{F}_{1,1} & \mathcal{F}_{1,2} \\
-% \mathcal{F}_{2,1} & \mathcal{F}_{2,2}
-% \end{array}\right] & \mathcal{F}_{1,2}\left[\begin{array}{ll}
-% \mathcal{F}_{1,1} & \mathcal{F}_{1,2} \\
-% \mathcal{F}_{2,1} & \mathcal{F}_{2,2}
-% \end{array}\right] \\
-% \mathcal{F}_{2,1}\left[\begin{array}{ll}
-% \mathcal{F}_{1,1} & \mathcal{F}_{1,2} \\
-% \mathcal{F}_{2,1} & \mathcal{F}_{2,2}
-% \end{array}\right] & \mathcal{F}_{2,2}\left[\begin{array}{ll}
-% \mathcal{F}_{1,1} & \mathcal{F}_{1,2} \\
-% \mathcal{F}_{2,1} & \mathcal{F}_{2,2}
-% \end{array}\right]
-% \end{array}\right]^\dagger C_{{\rho_1} {\rho_1}}
-% \begin{bmatrix}
-%      \mathcal{F}(\Theta)_{A_1} & 0 & 0 & 0\\
-%      0 & \mathcal{F}(\Theta)_{A_2} & 0 & 0\\
-%      0 & 0 & \mathcal{F}(\Theta)_{B_1} & 0\\
-%      0 & 0 & 0 & \mathcal{F}(\Theta)_{B_2}\\
-% \end{bmatrix}
-% C_{{\rho_1} {\rho_1}}^{\dagger}
-% \end{align*}
-
-% we need the CJ coefficients for the $\rho_1 \otimes \rho_1 = E \otimes E$ decomposition.
-
-    % The Kronecker product table is not the character table of the group, but they are related through the orthogonality relations of the character values.
-
-    % More specifically, the tensor product of two representations can be decomposed into irreducible representations, and the multiplicity of a particular irrep in this decomposition can be determined using the character values from the character table.
-
-
-
-
-
-
-
-\section{Training and Implementation Details}\label{app:training}
-
-The code to implement all models and experiments in this paper can be found at \href{https://github.com/sophiaas/gtc-invariance}{\texttt{github.com/sophiaas/gtc-invariance}}.
-
-For all experiments, we use near-identical, parameter-matched architectures in which only the type of invariant map differs. To isolate the effects of the invariant map, all models are comprised of a single $G$-Conv block followed by either Max $G$-Pooling or the $G$-TC layer, and an MLP Classifier. Here, we perform only \textit{pure} $G$-Conv, without translation. Thus, we use filters the same size as the input in all models.  The $G$-Conv block is comprised of a $G$-Conv layer, a batch norm layer, and an optional nonlinearity. For the Max $G$-Pool model, ReLU is used as the nonlinearity. Given the third-order nonlinearity of the $G$-TC, we omit the nonlinearity in the $G$-Conv block in the $G$-TC Model. The $G$-TC layer increases the dimensionality of the output of the $G$-Conv block; consequently the input dimension of the first layer of the MLP is larger and the weight matrix contains more parameters than for the Max $G$-Pool model. To compensate for this, we increase the dimension of the output of the first MLP layer in the Max Model, to match the overall number of parameters.
-
-All models are trained with a cross-entropy loss, using the Adam optimizer, a learning rate of 0.00005, weight decay of 0.00001, betas of [0.9, 0.999], epsilon of $10^{-8}$, a reduce-on-plateau learning rate scheduler with a factor of 0.5, patience of 2 epochs, and a minimum learning rate of 0.0.0001. Each model is trained with four random seeds [0, 1, 2, 3], and results are averaged across seeds.
-
-
-\subsection{MNIST Experiments: $SO(2)$ and $O(2)$ on $\mathbb{R}^2$}
-
-\subsubsection{Datasets}
-
-
-The $SO(2)$-MNIST dataset is generated by applying a random 2D planar rotation to each digit in the MNIST \cite{lecun1999mnist} training and test datasets. This results in training and test sets with the standard sizes of $60,000$ and $10,000$. For the $O(2)$-MNIST, each image is randomly flipped and rotated\textemdash i.e. transformed by a random element of the group $O(2)$. A random $20 \%$ of the training dataset is set aside for model validation and is used to tune hyperparameters. The remaining $80 \%$ is used for training. Images are additionally downsized with interpolation to $16 \times 16$ pixels.
-
-% \begin{figure}[H]
-%     \centering
-%     \includegraphics[width=0.4\textwidth]{figs/rotated-mnist.pdf}
-%     \includegraphics[width=0.4\textwidth]{figs/rotated-mnist.pdf}
-
-%     \caption{Examples from the $SO(2)$-MNIST (left) and $O(2)$-MNIST (right) datasets.}
-%     \label{fig:so2-mnist}
-% \end{figure}
-
-
-% \textbf{$O(2)$ MNIST}
-
-% \begin{figure}[H]
-%     \centering
-%     \includegraphics[width=0.3\textwidth]{example-image-a}
-%     \caption{[Figure showing dataset]}
-%     \label{fig:o2-mnist}
-% \end{figure}
-
-% \textbf{$SE(2)$ MNIST}
-
-% \begin{figure}[H]
-%     \centering
-%     \includegraphics[width=0.3\textwidth]{example-image-a}
-%     \caption{[Figure showing dataset]}
-%     \label{fig:se2-mnist}
-% \end{figure}
-
-
-\subsubsection{Models and Training}
-
-
-
-\textbf{$C8$-CNN}
-
-
-
-\textbf{TC.} The $G$-Conv block consists of an $C8$-Conv block using 24 filters followed by Batch Norm. Next, the $G$-TC Layer is applied. The output is raveled and passed to a three-layer MLP using 1d Batch Norm and ELU nonlinearity and after each linear layer, with all three layers having output dimension $64$. Finally a linear layer is applied, with output dimension $10$, for the 10 object categories.
-
-\textbf{Max.} The $G$-Conv block consists of an $C8$-Conv block using 24 filters followed by Batch Norm and a ReLU nonlinearity. Next, a Max $G$-Pooling Layer is applied. The output is raveled and passed to a three-layer MLP using 1d Batch Norm and ELU nonlinearity and after each linear layer. The first layer has output dimension $275$, to compensate for the difference in output size of the $G$-TC Layer. The remaining two layers having output dimension $64$. Finally a linear layer is applied, with output dimension $10$, for the 10 object categories.
-
-\textbf{$D16$-CNN}
-
-\textbf{TC.} The $G$-Conv block consists of an $D16$-Conv block using 24 filters followed by Batch Norm. Next, the $G$-TC Layer is applied. The output is raveled and passed to a three-layer MLP using 1d Batch Norm and ELU nonlinearity and after each linear layer, with all three layers having output dimension $64$. Finally a linear layer is applied, with output dimension $10$, for the 10 object categories.
-
-
-\textbf{Max.} The $G$-Conv block consists of an $D16$-Conv block using 24 filters followed by Batch Norm and a ReLU nonlinearity. Next, a Max $G$-Pooling Layer is applied. The output is raveled and passed to a three-layer MLP using 1d Batch Norm and ELU nonlinearity and after each linear layer. The first layer has output dimension $2,380$, to compensate for the difference in output size of the $G$-TC Layer. The remaining two layers having output dimension $64$. Finally a linear layer is applied, with output dimension $10$, for the 10 object categories.
-
-
-\subsection{ModelNet10 Experiments: $O$ and $O_h$ acting on $\mathbb{R}^3$}
-
-\subsubsection{Datasets}
-
-The ModelNet10 dataset is downsampled and voxelized to a $10 \times 10 \times 10$ grid. for the $O$-ModelNet10 Dataset, each datapoint is transformed by a random cubic rotation. For the $O_h$-ModelNet10 Dataset, each datapoint is transformed by a random cubic rotation and flip. The standard training and testing sets are used. A random $20 \%$ of the training dataset is set aside for model validation and is used to tune hyperparameters. The remaining $80 \%$ is used for training.
-
-% \textbf{$O$ and $O_h$ ModelNet10}
-
-% \begin{figure}[H]
-%     \centering
-%     \includegraphics[width=0.3\textwidth]{example-image-a}
-%     \includegraphics[width=0.3\textwidth]{example-image-a}
-%     \caption{[Figure showing dataset]}
-%     \label{fig:so3-modelnet}
-% \end{figure}
-
-
-
-\subsubsection{Models and Training}
-
-\textbf{$O$-CNN}
-
-
-
-\textbf{TC}. The $G$-Conv block consists of an $O$-Conv block using 24 filters followed by a IID 3D Batch Norm Layer. Next, the $G$-TC Layer is applied. The output is raveled and passed to a three-layer MLP using 1d Batch Norm and ELU nonlinearity and after each linear layer, with all three layers having output dimension $64$. Finally a linear layer is applied, with output dimension $10$, for the 10 object categories.
-
-\textbf{Max.} The $G$-Conv block consists of an $O$-Conv block using 24 filters followed by a IID 3D Batch Norm Layer and a ReLU nonlinearity. Next, a Max $G$-Pooling Layer is applied. The output is raveled and passed to a three-layer MLP using 1d Batch Norm and ELU nonlinearity and after each linear layer. The first layer has output dimension $5,420$, to compensate for the difference in output size of the $G$-TC Layer. The remaining two layers having output dimension $64$. Finally a linear layer is applied, with output dimension $10$, for the 10 object categories.
-
-
-\textbf{$O_h$-CNN}
-
-
-
-
-
-\textbf{TC.} The $G$-Conv block consists of an $O_h$-Conv block using 24 filters followed by a IID 3D Batch Norm Layer. Next, the $G$-TC Layer is applied. The output is raveled and passed to a three-layer MLP, with all three layers having output dimension $64$. Finally a linear layer is applied, with output dimension $10$, for the 10 object categories.
-
-
-
-\textbf{Max.} The $G$-Conv block consists of an $O_h$-Conv block using 24 filters followed by a IID 3D Batch Norm Layer and a ReLU nonlinearity. Next, a Max $G$-Pooling Layer is applied. The output is raveled and passed to a three-layer MLP. The first layer has output dimension $20,000$, to compensate for the difference in output size of the $G$-TC Layer. The remaining two layers having output dimension $64$. Finally a linear layer is applied, with output dimension $10$, for the 10 object categories.
-
-
-
-
-% \section{Computing the TC in general cases, and $SE(n)$ in particular}
-
-% \subsection{Are we working with $SE(n)$ or with $SO(n) \tims \mathbb{R}^n$?}
-
-% We first investigate the difference between $SE(n)$ and $SO(n)\times \mathbb{R}^n$ equivariance.
-
-% \paragraph{Group structure}
-
-% The group composition and inverse of the semi-direct product $SE(n) = SO(n) \rtimes \mathbb{R}^n$ are:
-% \begin{align*}
-%     (R_1, t_1) \cdot (R_2, t_2)
-%         &=(R_1.R_2, R_2.t_1+t_2) \\
-%     (R, t)^{-1}
-%         &= (R^{-1}, -R^{-1}.t).
-% \end{align*}
-
-% The group composition and inverse of the direct product $SO(n)\times \mathbb{R}^n$ are:
-% \begin{align*}
-%     (R_1, t_1) \cdot (R_2, t_2)
-%         &=(R_1.R_2, t_1+t_2) \\
-%     (R, t)^{-1}
-%         &= (R^{-1}, -t).
-% \end{align*}
-
-% \paragraph{Action on $\mathbb{R}^n$}
-% The group $SE(n)$ acts on $\mathbb{R}^n$ through its so-called homogeneous representation, that converts $(R, t)$ into a matrix.
-% \begin{equation}
-%     g(R, t) =
-%     \begin{pmatrix}
-%         R & t \\
-%         0 & 1
-%     \end{pmatrix}
-%     \quad \text{with inverse: }
-%         g(R, t)^{-1} =
-%         \begin{pmatrix}
-%         R^{-1} & -R^{-1}.t \\
-%         0 & 1
-%     \end{pmatrix}
-% \end{equation}
-% so that:
-% \begin{equation}
-%     g^{-1}x = \begin{pmatrix}
-%         R^{-1} & -R^{-1}.t \\
-%         0 & 1
-%     \end{pmatrix} .
-%     \begin{pmatrix}
-%     x \\
-%     1
-%     \end{pmatrix}
-%     =
-%     \begin{pmatrix}
-%     R^{-1}x-R^{-1}t \\
-%     1
-%     \end{pmatrix}
-%     =     \begin{pmatrix}
-%     R^{-1}(x-t) \\
-%     1
-%     \end{pmatrix}
-% \end{equation}
-
-% By contrast, there is no canonical way of defining the action of the product group $SO(n) \times \mathbb{R}^n$ on $\mathbb{R}^n$. Indeed, if two groups $G_1, G_2$ act on the same space $V$, the standard way of building an action of $G_1 \times G_2$ is to consider the action on the direct sum $V \oplus V$ defined as $(g_1, g_2 \act (x_1, x_2) = (g_1\cdot x_1, g_2 \cdot x_2)$ using two copies of $V$ \footnote{\url{https://math.stackexchange.com/questions/3140677/direct-sum-of-representation-of-product-groups}}. To define the action of $G_1 \times G_2$ on one copy of $V$, we can choose which group acts first and get:
-% \begin{align*}
-%     (R, t) \cdot x
-%         &= (Rx + t)
-%         \quad \text{(Choice 1: applying $R$ first and then $t$)},\\
-%     (R, t) \cdot x
-%         &= R(x+t)
-%         \quad \text{(Choice 2: applying $t$ first and then $R$)}.
-% \end{align*}
-% which we compare to the action of $SE(n)$:
-% \begin{equation}
-%     (R, t) \cdot x = Rx + t.
-% \end{equation}
-% Both $SO(n) \times \mathbb{R}^n$ (Choice 1) and $SE(n)$ consider that we first apply the rotation, and then translate.
-
-% The application of $(R^{-1}, -t)$ gives:
-% \begin{align*}
-%     (R^{-1}, -t) \cdot x
-%         &= (R^{-1}x - t)
-%         \quad \text{(Choice 1)},\\
-%     (R^{-1}, -t)\cdot x
-%         &= R^{-1}(x-t)
-%          \quad \text{(Choice 2)}.
-% \end{align*}
-% which we compare to the action of $SE(n)$:
-% \begin{equation}
-%     (R, t)^{-1} \cdot x  =R^{-1}(x-t)
-% \end{equation}
-% Both $SO(n) \times \mathbb{R}^n$ (Choice 2) and $SE(n)$ will yield the definition of convolution from \cite{cesa2021program}.
-
-% Consequently, the action of $SE(n)$ on $\mathbb{R}^n$ is a mix of the action of $SO(n) \times \mathbb{R}^n$ (choice 1) and of $SO(n) \times \mathbb{R}^n$ (choice 2).
-
-
-% \paragraph{Group-convolution} The group convolution using the semi-direct product $SE(n) = SO(n) \ltimes \mathbb{R}^n$ writes:
-% \begin{align*}
-%     \Theta(g)
-%     = (f \ast \phi)(g)
-%     =\int_{x \in \mathbb{R}^n} f(x)\phi(g^{-1}x)dx.
-% \end{align*}
-% and only uses the composition by an inverse element $g^{-1}x$. Thus, it can be the inverse in the $SE(n)$ setting or in the $SO(n) \times \mathbb{R}^2$ (Choice 2):
-% \begin{align*}
-%     \Theta(g)
-%     = (f \ast \phi)(g)
-%     =\int_{x \in \mathbb{R}^n} f(x)\phi(g^{-1}x)dx
-%     =\int_{x \in \mathbb{R}^n} f(x)\phi(R^{-1}(x-t))dx
-% \end{align*}
-% which is equation (1) in \cite{cesa2021program}.
-
-% \paragraph{Conclusion} In \cite{cesa2021program}, the authors state: ``We focus on convolutional networks on Euclidean spaces X = Rn, acted on by the group $(\mathbb{R}^n, +) \rtimes G$, i.e. the semidirect product of translations in $(\mathbb{R}^n, +)$ and origin-preserving transformations $G \leq O(n)$``. Both the words "semi-direct product" and "origin-preserving transformations" seems to indicate that we use $SE(n)$. This is because, when we first rotate and then translate, then the rotation is always preserving the origin of the domain. Additionally, in \cite{cohen2016group}, they say: ``The group p4 acts on points in Z2 (pixel coordinates) by multiplying the matrix g(r, u, v) by the homogeneous coordinate vector x(u′, v′) of a point (u′, v′)``, which corresponds indeed to the action of $SE(n)$.
-
-
-% \subsection{Compute TC with escnn's API for any group $G \ltimes \mathbb{R}^n$}
-
-% We provide the full computations for the special cases $SE(2) = SO(2) \ltimes \mathbb{R}^2$ and $E(2) = O(2) \ltimes \mathbb{R}^2$, and then extract a general strategy for any group $G \ltimes \mathbb{R}^n$.
-
-% \paragraph{TC with $SE(2)$ in practice} Elements of $SE(2)$ will be indexed by $(r, u, v)$ where $r$ is the angle of the 2D rotation and $u, v$ represent the 2D translation.
-
-% We recall that for $SE(2) = SO(2) \ltimes \mathbb{R}^2$. The output of the $SO(2)$-steerable convolution (i.e. of the $SE(2)$ convolution) is of shape $C_\text{out}|G| \times W \times H$. Consequently, reshape into $C_\text{out} \times |G| \times W \times H$ ad denote $r$ the index going though the group indices and $u, v$ the indices going through the domain indices.
-
-% To find the element $\Theta(gg_i)$ needed for the TC, we need to find the indices $(r, u, v)$ that correspond to the product of two group elements $(r_1, u_1, v_1)$ and $(r_2, u_2, v_2)$. We can then use $g = g_1$ and $g_i = g_2$ to get the result.
-
-% As \texttt{escnn} only thinks in terms of $G$ within $G \ltimes \mathbb{R}^2$, they do not have a group Python class dedicated to $SE(2)$. Thus, we need to implement the group composition of $SE(2)$ manually.
-
-% We have the following correspondence between indices and matrices:
-% \begin{equation}
-%     (r, u, v)
-%     &=
-%     \begin{pmatrix}
-%         R_2 & t_2 \\
-%         0 & 1
-%     \end{pmatrix}
-%     &=
-%     \begin{pmatrix}
-%     \cos(r\pi/2) & -\sin(r\pi/2) & u \\
-%     \sin(r\pi/2) & \cos(r\pi/2) & v \\
-%     0 & 0 & 1
-%      \end{pmatrix}
-% \end{equation}
-
-% So that the group composition written in indices, can be computed as:
-% \begin{align*}
-%     (r_1, u_1, v_1) \cdot (r_2, u_2, v_2)
-%     &=     \begin{pmatrix}
-%         R_1 & t_1 \\
-%         0 & 1
-%     \end{pmatrix} .
-%         \begin{pmatrix}
-%         R_2 & t_2 \\
-%         0 & 1
-%     \end{pmatrix} \\
-%     &=
-%     \begin{pmatrix}
-%         R_1.R2 & R_1.t_2+t_1 \\
-%         0 & 1
-%     \end{pmatrix} \\
-%     &=
-%     \begin{pmatrix}
-%     \cos((r_1+r_1)\pi/2) & -\sin((r_1+r_1)\pi/2) & u \\
-%     \sin((r_1+r_1)\pi/2) & \cos((r_1+r_1)\pi/2) & v \\
-%     0 & 0 & 1
-%      \end{pmatrix},
-% \end{align*}
-% where the elements of the resulting translations are:
-% \begin{align*}
-%     u &= \cos(r_1\pi/2)u_2 -\sin(r_1\pi/2)v_2 \\
-%     v &= \sin(r_1\pi/2)u_2 -\sin(r_1\pi/2)v_2.
-% \end{align*}
-% Now, we need to find the (integer) indices to which $u, v$ correspond to. This can be approximated by $u \leftarrow int(u)$ and $v \leftarrow int(v)$ where int means integer. Consequently, the group composition formula is, in terms of indices:
-% \begin{align*}
-%     &(r_1, u_1, v_1) \cdot (r_2, u_2, v_2)\\
-%     &= (r_1 + r_2, int(\cos(r_1\pi/2)u_2 -\sin(r_1\pi/2)v_2), int(\sin(r_1\pi/2)u_2 -\sin(r_1\pi/2)v_2).
-% \end{align*}
-
-% \paragraph{TC with $SE(2)$ in code}
-% We convert this formula into pseudo-code that uses \texttt{escnn}'s API, assuming that CN is the cyclic group that discretizes the rotations in $SE(2)$.
-% \begin{align*}
-%     (r_1, u_1, v_1) \cdot (r_2, u_2, v_2) = (r, u, v)
-% \end{align*}
-% where:
-% \begin{align*}
-%     r &= CN.\_combine(r_1, r_2) \\
-%     u &= int(CN.\_change\_param(element=r_1, p\_from="int", p\_to="MAT") @ [u_2, v_2])[0] \\
-%     v &= int(CN.\_change\_param(element=r_1, p\_from="int", p\_to="MAT") @ [u_2, v_2])[1].
-% \end{align*}
-% We use $\Theta(r, u, v)$ (at indices $r, u, v$) to get $\Theta(g_1.g_2).$
-
-% \paragraph{TC with $E(2)$ in code} We can generalize this to $E(2)$ by considering the DN dihedral group:
-% \begin{align*}
-%     (m_1, r_1, u_1, v_1) \cdot (m_2, r_2, u_2, v_2)= (m, r, u, v)
-% \end{align*}
-% where:
-% \begin{align*}
-%     (m, r) &= DN.\_combine((m_1, r_1, m_2, r_2))\\
-%     u = &= int(CN.\_change\_param(element=(m_1, r_1), p\_from="int", p\_to="MAT") @ [u_2, v_2])[0]\\
-%     v= &= int(CN.\_change\_param(element=(m_1, r_1), p\_from="int", p\_to="MAT") @ [u_2, v_2])[1].
-% \end{align*}
-
-% \paragraph{TC with any $SE(n) \ltimes \mathbb{R}^n$ or $G \ltimes \mathbb{R}^n$ in code}. The above strategy can be applied to any $G \ltimes \mathbb{R}^n$ where $G$ is a group implemented in escnn/group/groups, as it only requires the use of combine (for the $G$-part) and change-param (for the $\mathbb{R}^n$-part). In particular, it applies to any $SE(n) = SO(n) \ltimes \mathbb{R}^n$.
-
-% For general $G$ and for $SO(3)$, in order to use the change-param function to convert the 3D rotation into a MAT, we need to know what param is used to represent elements at the output of the G-conv. For example, the possible params for $SO(3)$ are given in \url{https://github.com/QUVA-Lab/escnn/blob/master/escnn/group/groups/so3_utils.py}.
-
-
-% \textbf{$SE(3)$-CNN}
-
-% \textit{Max}
-
-% \textit{TC}
diff --git a/docs/sanbord_miolane_2025/sections/checklist.tex b/docs/sanbord_miolane_2025/sections/checklist.tex
deleted file mode 100644
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+++ /dev/null
@@ -1,87 +0,0 @@
-\section*{Checklist}
-
-
-%%% BEGIN INSTRUCTIONS %%%
-The checklist follows the references.  Please
-read the checklist guidelines carefully for information on how to answer these
-questions.  For each question, change the default \answerTODO{} to \answerYes{},
-\answerNo{}, or \answerNA{}.  You are strongly encouraged to include a {\bf
-justification to your answer}, either by referencing the appropriate section of
-your paper or providing a brief inline description.  For example:
-\begin{itemize}
-  \item Did you include the license to the code and datasets? \answerYes{See Section~\ref{gen_inst}.}
-  \item Did you include the license to the code and datasets? \answerNo{The code and the data are proprietary.}
-  \item Did you include the license to the code and datasets? \answerNA{}
-\end{itemize}
-Please do not modify the questions and only use the provided macros for your
-answers.  Note that the Checklist section does not count towards the page
-limit.  In your paper, please delete this instructions block and only keep the
-Checklist section heading above along with the questions/answers below.
-%%% END INSTRUCTIONS %%%
-
-
-\begin{enumerate}
-
-
-\item For all authors...
-\begin{enumerate}
-  \item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
-    \answerTODO{}
-  \item Did you describe the limitations of your work?
-    \answerTODO{}
-  \item Did you discuss any potential negative societal impacts of your work?
-    \answerTODO{}
-  \item Have you read the ethics review guidelines and ensured that your paper conforms to them?
-    \answerTODO{}
-\end{enumerate}
-
-
-\item If you are including theoretical results...
-\begin{enumerate}
-  \item Did you state the full set of assumptions of all theoretical results?
-    \answerTODO{}
-        \item Did you include complete proofs of all theoretical results?
-    \answerTODO{}
-\end{enumerate}
-
-
-\item If you ran experiments...
-\begin{enumerate}
-  \item Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
-    \answerTODO{}
-  \item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
-    \answerTODO{}
-        \item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
-    \answerTODO{}
-        \item Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
-    \answerTODO{}
-\end{enumerate}
-
-
-\item If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
-\begin{enumerate}
-  \item If your work uses existing assets, did you cite the creators?
-    \answerTODO{}
-  \item Did you mention the license of the assets?
-    \answerTODO{}
-  \item Did you include any new assets either in the supplemental material or as a URL?
-    \answerTODO{}
-  \item Did you discuss whether and how consent was obtained from people whose data you're using/curating?
-    \answerTODO{}
-  \item Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?
-    \answerTODO{}
-\end{enumerate}
-
-
-\item If you used crowdsourcing or conducted research with human subjects...
-\begin{enumerate}
-  \item Did you include the full text of instructions given to participants and screenshots, if applicable?
-    \answerTODO{}
-  \item Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
-    \answerTODO{}
-  \item Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?
-    \answerTODO{}
-\end{enumerate}
-
-
-\end{enumerate}
diff --git a/docs/sanbord_miolane_2025/sections/lit-review.tex b/docs/sanbord_miolane_2025/sections/lit-review.tex
deleted file mode 100644
index d422359..0000000
--- a/docs/sanbord_miolane_2025/sections/lit-review.tex
+++ /dev/null
@@ -1,25 +0,0 @@
-\section{Related Work}
-
-% Historically, the bispectrum emerged from the study of the higher-order statistics of non-Gaussian random processes, and was used as a tool to measure the non-Gaussianity of a signal \cite{tukey1953spectral, brillinger1991some, nikias1993signal}. The work of Yellott and Iverson \cite{yellott1992uniqueness} established that every integrable function with compact support is completely identified\textemdash up to translation\textemdash by its three-point autocorrelation and thus its bispectrum (Appendix \ref{appendix:autocorrelation}). Later work of Kakarala tied the bispectrum to the more general formulation of Fourier analysis and thus defined it for all compact groups \cite{kakarala1993group, kakarala2012bispectrum}. Kakarala went on to prove that it is complete \cite{kakarala2009completeness} in both commutative and non-commutative formulations (Appendix \ref{appendix:completeness}).
-
-\textbf{The Triple Correlation.} The triple correlation has a long history in signal processing \cite{tukey1953spectral, brillinger1991some, nikias1993signal}. It originally emerged from the study of the higher-order statistics of non-Gaussian random processes, but its invariance properties with respect to translation have been leveraged in texture statistics~\cite{yellott1993implications} and
-data analysis in neuroscience~\cite{deshpande2023third}, as well as early multi-layer perceptron architectures in the 1990's~\cite{delopoulos1994invariant,kollias1996multiresolution}. The triple correlation was extended to groups beyond translations in \cite{kakarala1992triple}, and its completeness with respect to general compact groups was established in \cite{kakarala2009completeness}. To the best of our knowledge, the triple correlation has not previously been introduced as a method for achieving invariance in convolutional networks for either translation or more general groups.
-
-
-% As our proposed method can be used as a variant of a pooling mechanism, we first review existing pooling that have been employed for CNNs in the literature, and that have been, or not extended to $G$-CNNs.
-
-\textbf{Pooling in CNNs.} Pooling in CNNs typically has the dual objective of coarse graining and achieving local invariance. While invariance is one desiderata for the pooling mechanism, the machinery of group theory is rarely employed in the computation of the invariant map itself. As noted in the introduction, max and average pooling are by far the most common methods employed in CNNs and $G$-CNNs. However, some approaches beyond strict max and average pooling have been explored.
-%Pooling can serves other various purposes, such as down-scaling (coarse-graining), increasing the receptive field of convolutional filters across layers, and reducing computational complexity and memory usage.
-% Consequently, numerous approaches have been proposed in academic literature, extending  methods, and their mixed-max-average pooling variants~\cite{lee2015generalizing}.
-Soft-pooling addresses the lack of smoothness of the max function and uses instead a smooth approximation of it, with methods including polynomial pooling~\cite{wei2019building} and learned-norm \cite{gulcehre2014learned}, among many others \cite{estrach2014signal,eom2020alphaintegration, shahriari2017multipartite,shi2016rank,bieder2021comparison,stergiou2021refining,czaja2021maximal,kumar2018ordinal}.
-% p-pooling \cite{estrach2014signal}, alpha-integration \cite{eom2020alphaintegration}, multi-partite~\cite{shahriari2017multipartite}, rank-based \cite{shi2016rank}, Smooth-Maximum \cite{bieder2021comparison}, Soft pooling \cite{stergiou2021refining}, Maxfun pooling~\cite{czaja2021maximal}, Ordinal pooling~\cite{kumar2018ordinal}.
-Stochastic pooling~\cite{zeiler2013stochastic,} reduces overfitting in CNNs by introducing randomness in the pooling, yielding mixed-pooling~\cite{yu2014mixed}, max pooling dropout~\cite{wu2015max}, among others~\cite{tong2016hybrid,zhai2017s3pool,graham2014fractional}
-% Hybrid pooling~\cite{tong2016hybrid},
-%  Stochastic Spatial Sampling (S3 pooling)~\cite{zhai2017s3pool} and Fractional Max pooling~\cite{graham2014fractional}.
-
-
-\textbf{Geometrically-Aware Pooling.} Some approaches have been adopted to encode spatial or structural information about the feature maps, including spatial pyramid pooling~\cite{he2015spatial}, part-based pooling~\cite{zhang2012pose}, geometric $L_p$ pooling \cite{feng2011geometric} or pooling regions defined as concentric circles \cite{qi2018concentric}. In all of these cases, the pooling computation is still defined by a max. These geometric pooling approaches are reminiscent of the Max $G$-Pooling for $G$-CNNs introduced by ~\cite{cohen2016group} and defined in Section \ref{sec:g-pooling}, without the explicit use of group theory.
-
-\textbf{Higher-Order Pooling.} Average pooling computes first-order statistics (the mean) by pooling from each channel separately and does not account for the interaction between different feature maps coming from different channels. Thus, second-order pooling mechanisms have been proposed to consider correlations between features across channels~\cite{lin2015bilinear,gao2019global}, but higher-orders are not investigated. Our approach computes a third-order polynomial invariant; however, it looks for higher-order correlations within the group rather than across channels and thus treats channels separately. In principle, these approaches could be combined.
-
-% \paragraph{Universality and Invariance in Neural Networks} Researchers have explored the universality of $G$-invariant neural networks and showed that neural networks with $G$-invariant linear layers can approximate any $G$-invariant function~\cite{maron2019universality} as long as higher-order tensors are available. This is a different question than the one we tackle here, which focuses on exploiting the smallest $G$-invariant map that is not lossy. In \cite{yarotsky2022universal}, universality of $G$-invariant neural networks is also studied, and qualified as ``complete'' in the sense of approximation theory, which is different from the meaning of the word ``complete,'' in the sense of invariant theory used here.
diff --git a/docs/sanbord_miolane_2025/sections/scratch.tex b/docs/sanbord_miolane_2025/sections/scratch.tex
deleted file mode 100644
index 8c0d62e..0000000
--- a/docs/sanbord_miolane_2025/sections/scratch.tex
+++ /dev/null
@@ -1,531 +0,0 @@
-% These computations have not been included in the main paper.
-
-\section{Equivariance of the $G$-Convolution}\label{sec:g-conv}
-
-
-The $G$-convolution is \textit{equivariant} to the action of the group $G$ on the domain of the signal $\signal$~\cite{cohen2016group}. This means that any action $\action_h$ on the domain of $\signal$ results in a corresponding action $\action'_h$ on the output of the convolutional layer. Specifically, for a filter $\filter $, the equivariance property is:
-\begin{equation}
-    \filter * \action_{h}[\signal] = \action'_{h}[\filter * \signal], \qquad \text{for all ${h} \in G$,}
-\end{equation}
-where $\action_h$ and $\action'_h$ represent the actions of the same group element $h$ on $\signal$ and $\filter * \signal$ respectively.
-
-
-
-\begin{proof}
-Consider a signal $\signal: \domain \rightarrow \mathbb{R}^c$ and a filter $\filter: \domain \rightarrow \mathbb{R}^c$ defined on a domain $\domain$ and codomain $\mathbb{R}^c$, with $c$ the number of channels.
-
-For transformations $h, g \in G$, we have:
-\begin{align*}
-    \filter * \action_{h}[\signal](g)
-        &= \int_{u \in \domain}
-            \filter(\action_{g^{-1}}(u)) \action_{h}[\signal](u) du \\
-        &= \int_{u \in \domain}
-            \filter(\action_{g^{-1}}(u))\signal( \action_{h^{-1}}(u)) du \\
-        &= \int_{u' \in \domain}
-            \filter(\action_{g^{-1}}(\action_{h}(u')))\signal(u') du'
-            \qquad
-            \text{($u' =  \action_{h^{-1}}(u)$, i.e., $u = \action_{h}(u')$)} \\
-        &= \int_{u \in \domain}
-            \filter(\action_{g^{-1}h}(u))\signal(u) du \\
-        &= \int_{u \in \domain}
-            \filter(\action_{(h^{-1}g)^{-1}}(u))\signal(u) du \\
-        &= \int_{u \in \domain}
-            \action_{(h^{-1}g)}[\filter](u)\signal(u) du \\
-        &=\action'_{h}[\filter * \signal](g)
-\end{align*}
-where the action $L'$ uses $h$ to act on the domain $G$ of the output signal  $[\filter * \signal]$ by left multiplication with $h^{-1}$.
-\end{proof}
-
-This property applies for the $G$-convolutions of the first layer and of the next layers.
-
-\section{Example of Triple Correlation on $C_4$}
-
-In this appendix, we work out the computations of the TC for the group $C_4$ shown in the explanatory Figure of the main text.
-
-The group is $G = C_4$ and the first input real signal is $\Theta$ defined as:
-
-\begin{equation}
-    \Theta(0) = 0, \quad \Theta(1) = -1, \quad \Theta(2) = 1, \quad \Theta(3) = 2.
-\end{equation}
-
-We compute the elements of the triple correlation.
-
-First the $(0, 0)$ diagonal element.
-\scriptsize
-\begin{align*}
-    \tau_\Theta(0, 0)
-    = \sum_{g \in G} \Theta(g)^3
-    = 0^3 + (-1)^3 + 1^3 + 2^3
-    = 8
-\end{align*}
-
-\normalsize
-Then we populate the first line of the TC and add the values of the TC for the symmetric elements.
-\scriptsize
-\begin{align*}
-\tau_\Theta(0, 1)
-    &= \sum_{g \in G} \Theta(g)^2 \Theta(g+1)\\
-    &= \Theta(0)^2 \Theta(0+1)
-    + \Theta(1)^2 \Theta(1+1)
-    + \Theta(2)^2 \Theta(2+1)
-    + \Theta(3)^2 \Theta(3+1) \\
-    &= \Theta(0)^2 \Theta(1)
-    + \Theta(1)^2 \Theta(2)
-    + \Theta(2)^2 \Theta(3)
-    + \Theta(3)^2 \Theta(0) \\
-    &= 0 + (-1)^2.1 + 1^2.2+0\\
-    &=3 \\
-    &=\tau_\Theta(1, 0)\\
-    &=\tau_\Theta(0^{-1}, 1.0^{-1}) = \tau_\Theta(0, 1) \\
-    &=\tau_\Theta(1^{-1}, 0.1^{-1}) = \tau_\Theta(3, 3)\\
-\tau_\Theta(0, 2)
-    &= \sum_{g \in G} \Theta(g)^2 \Theta(g+2)\\
-    &= \Theta(0)^2 \Theta(0+2)
-    + \Theta(1)^2 \Theta(1+2)
-    + \Theta(2)^2 \Theta(2+2)
-    + \Theta(3)^2 \Theta(3+2) \\
-    &= \Theta(0)^2 \Theta(2)
-    + \Theta(1)^2 \Theta(3)
-    + \Theta(2)^2 \Theta(0)
-    + \Theta(3)^2 \Theta(1) \\
-    &= 0 + (-1)^2.2 + 0 + 2^2.(-1)\\
-    &= -2 \\
-    &=\tau_\Theta(2, 0)\\
-    &=\tau_\Theta(0^{-1}, 2.0^{-1}) = \tau_\Theta(0, 2) \\
-    &=\tau_\Theta(2^{-1}, 0.2^{-1}) = \tau_\Theta(2, 2)\\
-\tau_\Theta(0, 3)
-    &= \sum_{g \in G} \Theta(g)^2 \Theta(g+3)\\
-    &= \Theta(0)^2 \Theta(0+3)
-    + \Theta(1)^2 \Theta(1+3)
-    + \Theta(2)^2 \Theta(2+3)
-    + \Theta(3)^2 \Theta(3+3) \\
-    &= \Theta(0)^2 \Theta(3)
-    + \Theta(1)^2 \Theta(0)
-    + \Theta(2)^2 \Theta(1)
-    + \Theta(3)^2 \Theta(2) \\
-    &= 0 + 0 + 1^2.(-1) + 2^2.1\\
-    &= 3 \\
-    &=\tau_\Theta(3, 0)\\
-    &=\tau_\Theta(0^{-1}, 3.0^{-1}) = \tau_\Theta(0, 3) \\
-    &=\tau_\Theta(3^{-1}, 0.3^{-1}) = \tau_\Theta(1, 1).
-\end{align*}
-
-\normalsize
-Then, we populate the second line of the TC.
-\scriptsize
-\begin{align*}
-\tau_\Theta(1, 2)
-    &= \sum_{g \in G} \Theta(g) \Theta(g+1)\Theta(g+2)\\
-    &= \Theta(0) \Theta(0+1)\Theta(0+2)
-    + \Theta(1) \Theta(1+1)\Theta(1+2)
-    + \Theta(2) \Theta(2+1)\Theta(2+2)
-    + \Theta(3) \Theta(3+1)\Theta(3+2) \\
-    &= \Theta(0) \Theta(1)\Theta(2)
-    + \Theta(1) \Theta(2)\Theta(3)
-    + \Theta(2) \Theta(3)\Theta(0)
-    + \Theta(3) \Theta(0)\Theta(1) \\
-    &= 0
-    + -1.1.2
-    + 0
-    + 0\\
-    &=-2\\
-    &=\tau_\Theta(2, 1)\\
-    &=\tau_\Theta(1^{-1}, 2+1^{-1}) = \tau_\Theta(3, 2+3)=\tau_\Theta(3, 1) =\tau_\Theta(1, 3) \\
-    &=\tau_\Theta(2^{-1}, 1+2^{-1}) = \tau_\Theta(2, 1+2)=\tau_\Theta(2, 3) =\tau_\Theta(3, 2).
-\end{align*}
-\normalsize
-\section{Example of Triple Correlation on $\mathbb{Z}_2 \times \mathbb{Z}_2$}
-
-In this appendix, we work out the computations of the TC for the group $\mathbb{Z}_2 \times \mathbb{Z}_2$ shown in the explanatory Figure of the main text.
-
-The group is $G = \mathbb{Z}_2 \times \mathbb{Z}_2 $ and the first input real signal is $\Theta$ defined as:
-\begin{equation}
-    \Theta(00) = 2, \quad \Theta(01) = 0, \quad \Theta(10) = 1, \quad \Theta(11) = 1.
-\end{equation}
-
-We compute the elements of the triple correlation.
-
-\scriptsize
-\begin{align*}
-    \tau_\Theta(00, 00)
-    &= \sum_{g \in G} \Theta(g)^3
-    = 8+0+1+1
-    = 10 \\
-    \tau_\Theta(00, 01)
-    &=\sum_{g \in G} \Theta(g)^2\Theta(g \cdot 01) \\
-    &= \Theta(00)^2\Theta(01)
-        + \Theta(01)^2\Theta(01\cdot 01)
-        + \Theta(10)^2\Theta(10\cdot 01)
-        + \Theta(11)^2\Theta(11\cdot 01)\\
-    &= \Theta(00)^2\Theta(01)
-        + \Theta(01)^2\Theta(00)
-        + \Theta(10)^2\Theta(11)
-        + \Theta(11)^2\Theta(10)\\
-    &= 2^2 .0 +0^2.2+1^2.1+1^2.1
-    = 2 \\
-    \tau_\Theta(00, 10)
-    &=\sum_{g \in G} \Theta(g)^2\Theta(g \cdot 10) \\
-    &= \Theta(00)^2\Theta(10)
-        + \Theta(01)^2\Theta(01\cdot 10)
-        + \Theta(10)^2\Theta(10\cdot 10)
-        + \Theta(11)^2\Theta(11\cdot 10)\\
-    &= \Theta(00)^2\Theta(10)
-        + \Theta(01)^2\Theta(11)
-        + \Theta(10)^2\Theta(00)
-        + \Theta(11)^2\Theta(01)\\
-    &= 2^2 .1 +0^2.1+1^2.2+1^2.0
-    = 3 \\
-    \tau_\Theta(00, 11)
-    &= \sum_{g \in G} \Theta(g)^2\Theta(g \cdot 11) \\
-    &= \Theta(00)^2\Theta(11)
-        + \Theta(01)^2\Theta(01\cdot 11)
-        + \Theta(10)^2\Theta(10\cdot 11)
-        + \Theta(11)^2\Theta(11\cdot 11)\\
-    &= \Theta(00)^2\Theta(11)
-        + \Theta(01)^2\Theta(10)
-        + \Theta(10)^2\Theta(01)
-        + \Theta(11)^2\Theta(00)\\
-    &= 2^2 .1 +0^2.1+1^2.0+1^2.2 = 6 \\
-    % Don't compute these: Get the diagonal elements with symmetry
-    % \tau_\Theta(01, 01)
-    % &= \sum_{g \in G} \Theta(g)\Theta(g \cdot 01)^2 \\
-    % &= \Theta(00)\Theta(00 \cdot 01)^2
-    % + \Theta(01)\Theta(01 \cdot 01)^2
-    % + \Theta(10)\Theta(10 \cdot 01)^2
-    % + \Theta(11)\Theta(11 \cdot 01)^2 \\
-    % &= \Theta(00)\Theta(01)^2
-    % + \Theta(01)\Theta(00)^2
-    % + \Theta(10)\Theta(11)^2
-    % + \Theta(11)\Theta(10)^2 \\
-    % &= 2.0^2
-    % + 0.2^2
-    % + 1.1^2
-    % + 1.1^2 = 2 \\
-    % \tau_\Theta(10, 10)
-    % &= \sum_{g \in G} \Theta(g)\Theta(g \cdot 10)^2 \\
-    % &= \Theta(00)\Theta(00 \cdot 10)^2
-    % + \Theta(01)\Theta(01 \cdot 10)^2
-    % + \Theta(10)\Theta(10 \cdot 10)^2
-    % + \Theta(11)\Theta(11 \cdot 10)^2 \\
-    % &= \Theta(00)\Theta(10)^2
-    % + \Theta(01)\Theta(11)^2
-    % + \Theta(10)\Theta(00)^2
-    % + \Theta(11)\Theta(01)^2 \\
-    % &= 2.1^2
-    % + 0.1^2
-    % + 1.2^2
-    % + 1.0^2 = 6 \\
-    % \tau_\Theta(11, 11)
-    % &= \sum_{g \in G} \Theta(g)\Theta(g \cdot 11)^2 \\
-    % &= \Theta(00)\Theta(00 \cdot 11)^2
-    % + \Theta(01)\Theta(01 \cdot 11)^2
-    % + \Theta(10)\Theta(10 \cdot 11)^2
-    % + \Theta(11)\Theta(11 \cdot 11)^2 \\
-    % &= \Theta(00)\Theta(11)^2
-    % + \Theta(01)\Theta(10)^2
-    % + \Theta(10)\Theta(01)^2
-    % + \Theta(11)\Theta(00)^2 \\
-    % &= 2.1^2
-    % + 0.1^2
-    % + 1.0^2
-    % + 1.2^2 = 6 \\
-    \tau_\Theta(01, 10)
-    &= \sum_{g \in G} \Theta(g)\Theta(g \cdot 01) \Theta(g \cdot 10) \\
-    &= \Theta(00)\Theta(00 \cdot 01) \Theta(00 \cdot 10)
-    + \Theta(01)\Theta(01 \cdot 01) \Theta(01 \cdot 10)
-    + \Theta(10)\Theta(10 \cdot 01) \Theta(10 \cdot 10)
-    + \Theta(11)\Theta(11 \cdot 01) \Theta(11 \cdot 10)\\
-    &=\Theta(00)\Theta( 01) \Theta(10)
-    + \Theta(01)\Theta(00) \Theta(11)
-    + \Theta(10)\Theta(11) \Theta(00)
-    + \Theta(11)\Theta(10) \Theta(01)\\
-    &=2.0.1
-    + 0.2.1
-    + 1.1.2
-    + 1.1.0\\
-    &=2 \\
-    \tau_\Theta(11, 10)
-    &= \sum_{g \in G} \Theta(g)\Theta(g \cdot 11) \Theta(g \cdot 10) \\
-    &= \Theta(00)\Theta(00 \cdot 11) \Theta(00 \cdot 10)
-    + \Theta(01)\Theta(01 \cdot 11) \Theta(01 \cdot 10)
-    + \Theta(10)\Theta(10 \cdot 11) \Theta(10 \cdot 10)
-    + \Theta(11)\Theta(11 \cdot 11) \Theta(11 \cdot 10)\\
-    &=\Theta(00)\Theta(11) \Theta(10)
-    + \Theta(01)\Theta(10) \Theta(11)
-    + \Theta(10)\Theta(01) \Theta(00)
-    + \Theta(11)\Theta(00) \Theta(01)\\
-    &=2.1.1
-    + 0.1.1
-    + 1.0.2
-    + 1.2.0\\
-    &=2
-\end{align*}
-
-\normalsize
-
-The second input real signal is $\Theta'$ defined as:
-\begin{equation}
-    \Theta'(00) = 0, \quad \Theta'(01) = 1, \quad \Theta'(10) = 1, \quad \Theta'(11) = 2.
-\end{equation}
-
-We compute the triple correlation for $\Theta'$
-
-\scriptsize
-\begin{align*}
-    \tau_\Theta'(00, 00)
-    &= \sum_{g \in G} \Theta'(g)^3
-    = 0+1+1+8
-    = 10 \\
-    \tau_\Theta'(00, 01)
-    &=\sum_{g \in G} \Theta'(g)^2\Theta'(g \cdot 01) \\
-    &= \Theta'(00)^2\Theta'(01)
-        + \Theta'(01)^2\Theta'(01\cdot 01)
-        + \Theta'(10)^2\Theta'(10\cdot 01)
-        + \Theta'(11)^2\Theta'(11\cdot 01)\\
-    &= \Theta'(00)^2\Theta'(01)
-        + \Theta'(01)^2\Theta'(00)
-        + \Theta'(10)^2\Theta'(11)
-        + \Theta'(11)^2\Theta'(10)\\
-    &= 0^2 .1 +1^2.0+1^2.2+2^2.1
-    = 6 \\
-    \tau_\Theta'(00, 10)
-    &=\sum_{g \in G} \Theta'(g)^2\Theta'(g \cdot 10) \\
-    &= \Theta'(00)^2\Theta'(10)
-        + \Theta'(01)^2\Theta'(01\cdot 10)
-        + \Theta'(10)^2\Theta'(10\cdot 10)
-        + \Theta'(11)^2\Theta'(11\cdot 10)\\
-    &= \Theta'(00)^2\Theta'(10)
-        + \Theta'(01)^2\Theta'(11)
-        + \Theta'(10)^2\Theta'(00)
-        + \Theta'(11)^2\Theta'(01)\\
-    &= 2^2 .1 +1^2.2+1^2.0+2^2.1
-    = 14 \\
-    \tau_\Theta'(00, 11)
-    &= \sum_{g \in G} \Theta'(g)^2\Theta'(g \cdot 11) \\
-    &= \Theta'(00)^2\Theta'(11)
-        + \Theta'(01)^2\Theta'(01\cdot 11)
-        + \Theta'(10)^2\Theta'(10\cdot 11)
-        + \Theta'(11)^2\Theta'(11\cdot 11)\\
-    &= \Theta'(00)^2\Theta'(11)
-        + \Theta'(01)^2\Theta'(10)
-        + \Theta'(10)^2\Theta'(01)
-        + \Theta'(11)^2\Theta'(00)\\
-    &= 0^2 .2 +1^2.1+1^2.1+2^2.0 = 2 \\
-    \tau_\Theta'(01, 10)
-    &= \sum_{g \in G} \Theta'(g)\Theta'(g \cdot 01) \Theta'(g \cdot 10) \\
-    &= \Theta'(00)\Theta'(00 \cdot 01) \Theta'(00 \cdot 10)
-    + \Theta'(01)\Theta'(01 \cdot 01) \Theta'(01 \cdot 10)
-    + \Theta'(10)\Theta'(10 \cdot 01) \Theta'(10 \cdot 10)
-    + \Theta'(11)\Theta'(11 \cdot 01) \Theta'(11 \cdot 10)\\
-    &=\Theta'(00)\Theta'( 01) \Theta'(10)
-    + \Theta'(01)\Theta'(00) \Theta'(11)
-    + \Theta'(10)\Theta'(11) \Theta'(00)
-    + \Theta'(11)\Theta'(10) \Theta'(01)\\
-    &=0.1.1
-    + 1.0.1
-    + 1.2.0
-    + 2.1.1\\
-    &=2 \\
-    \tau_\Theta'(11, 10)
-    &= \sum_{g \in G} \Theta'(g)\Theta'(g \cdot 11) \Theta'(g \cdot 10) \\
-    &= \Theta'(00)\Theta'(00 \cdot 11) \Theta'(00 \cdot 10)
-    + \Theta'(01)\Theta'(01 \cdot 11) \Theta'(01 \cdot 10)
-    + \Theta'(10)\Theta'(10 \cdot 11) \Theta'(10 \cdot 10)
-    + \Theta'(11)\Theta'(11 \cdot 11) \Theta'(11 \cdot 10)\\
-    &=\Theta'(00)\Theta'(11) \Theta'(10)
-    + \Theta'(01)\Theta'(10) \Theta'(11)
-    + \Theta'(10)\Theta'(01) \Theta'(00)
-    + \Theta'(11)\Theta'(00) \Theta'(01)\\
-    &=0.2.1
-    + 1.1.2
-    + 1.1.0
-    + 1.0.1\\
-    &=2
-\end{align*}
-\normalsize
-\newpage
-
-\section{Algorithmic Reduction of the Triple Correlation}\label{app:reduction}
-
-\subsection{Direct Approach}
-
-In the ``direct approach'', we ask:\\
-
-\textit{which TC coefficients do we need to reconstruct the original signal?}\\
-
-We note that we do not care about whether the reconstruction is computationally expensive, as we will not implement it in practice. We are only interested in knowing the minimal number of TC coefficients that fully represent $f$.
-
-We consider a simple example on $Z_n$
-
-Consider a real signal defined on $Z_n$: $f(0), ..., f(n-1)$. Try to design an algorithm to recover that signal from some values of the triple correlation.
-
-
-We compute some quantities related to the TC that could be helpful:
-\begin{equation}
-    a(0, 0) = \sum_{x = 0}^{n-1} f(x)^3
-\end{equation}
-
-
-\begin{equation}
-   \sum_{x'=0}^{n-1} a(0, x') =  \sum_{x'=0}^{n-1} \sum_{x = 0}^{n-1} f(x)^2f(x-x')
-\end{equation}
-
-
-\subsection{Bispectrum Approach}
-
-In the ``bispectrum approach'', we ask:\\
-
-\textit{which TC coefficients do we need to reconstruct the first column of the bispectrum?}\\
-
-If we can reconstruct the first column of the bispectrum, then we can reconstruct the signal. Again, we do not care whether the reconstruction itself is computationally expensive.
-
-We write the definition of the bispectrum (first column) and look at the triple correlation coefficients involved.
-
-We have:
-\begin{align*}
-    \beta(f)_{k_1, k_2}
-        &=
-    \frac{1}{n^3}\sum_{x_1 \in Z_n} \sum_{x_2 \in Z_n}
-    T(f)(x_1, x_2)\left[
-                \rho_1(x_1)^{\dagger} \otimes \rho_2(x_2)^{\dagger}
-            \right]\\
-    &=
-    \frac{1}{n^3}\sum_{x_1 \in Z_n} \sum_{x_2 \in Z_n}
-    \left(\sum_{x \in Z_n}
-       f^*(x)
-       f\left(x + x_1\right) f\left(x+ x_2\right)\right)
-            \left[
-                \rho_1(x_1)^{\dagger} \otimes \rho_2(x_2)^{\dagger}
-            \right]
-\end{align*}
-where each representation $\rho_j$ is associated with and integer $k_j$ such that: $\rho_1(x') = \exp(-i\frac{2k_1\pi x'}{n})$ and $\rho_2(x') = \exp(-i\frac{2k_2\pi x'}{n})$ for any $x' \in Z_n$.
-
-We are only interested in the first column of the bispectrum. Thus, we set $k_1 = 0$ or equivalently $\rho_1(x') = 1$ for all $x' \in Z_n$:
-\begin{align*}
-    \beta(f)_{0, k_2}
-        &=
-    \sum_{x_1 \in Z_n} \sum_{x_2 \in Z_n}
-    T(f)(x_1, x_2)\left[
-                1 \otimes \rho_2(x_2)^{\dagger}
-            \right]\\
-    &=
-    \sum_{x_1 \in Z_n} \sum_{x_2 \in Z_n}
-    \left(\sum_{x \in Z_n}
-       f^*(x)
-       f\left(x + x_1\right) f\left(x+ x_2\right)\right)
-            \left[
-                1\otimes \rho_2(x_2)^{\dagger}
-            \right]\\
-    % &=
-    % \sum_{x_1 \in Z_n}
-    % \left(\sum_{x \in Z_n}
-    %    f^*(x)
-    %    f\left(x + x_1\right) \sum_{x_2 \in Z_n} f\left(x+ x_2\right) \rho_2(x_2)^{\dagger}\right).
-\end{align*}
-
-At this stage, we see that even if we set $k_1=0$, we still need all the $x_1, x_2$ terms of the triple correlation.
-
-We can continue the computations, to see if we can get rid of $x_1$.
-\begin{align*}
-    \beta(f)_{0, k_2}
-    &=
-   \sum_{x_2 \in Z_n} \left(\sum_{x \in Z_n} f^*(x) f\left(x+ x_2\right) \rho_2(x_2)^{\dagger}
-      \sum_{x_1 \in Z_n}  f\left(x + x_1\right) \right)\\
-          &=
-    \sum_{x_2 \in Z_n} \left(\sum_{x \in Z_n}  f^*(x) f\left(x+ x_2\right) \rho_2(x_2)^{\dagger}
-      \sum_{x_1' \in Z_n}  f\left(x_1'\right) \right) \quad \text{(change of variable $x'_1 = x + x_1$)}\\
-          &=
-    \sum_{x_1' \in Z_n}\sum_{x_2 \in Z_n}\left(\sum_{x \in Z_n}  f^*(x) f\left(x+ x_2\right) f\left(x_1'\right)\right) \rho_2(x_2)^{\dagger},
-\end{align*}
-
-Unfortunately, this does not make a simpler TC term appear.
-
-We try another version of the computations, to see if we can get rid of $x_2$.
-\begin{align*}
-    \beta(f)_{0, k_2}
-    &=
-    \sum_{x_1 \in Z_n}
-    \left(\sum_{x \in Z_n}
-       f^*(x)
-       f\left(x + x_1\right) \rho_2(-x).\hat{f}_{k_2}\right) \quad \text{(equivariance of Fourier transform)}\\
-    &=
-   \hat{f}_{k_2}
-    \left(\sum_{x \in Z_n}
-       f^*(x)  \rho_2(-x)\sum_{x_1 \in Z_n}
-       f\left(x + x_1\right) \right) \\
-    &=
-    \hat{f}_{k_2}
-    \left(\sum_{x \in Z_n}
-       f^*(x)  \rho_2(-x)\sum_{x_1' \in Z_n}
-       f\left(x_1'\right) \right) \quad \text{change of variable}\\
-    &=
-    \hat{f}_{k_2}
-    \left(\sum_{x \in Z_n}
-       f(x)  \rho_2(x)^\dagger\right)^*\left(\sum_{x_1' \in Z_n}
-       f\left(x_1'\right) \right)\\
-    &=
-    \hat{f}_{k_2}
-    \hat{f}_{k_2}^*\left(\sum_{x_1' \in Z_n}
-       f\left(x_1'\right) \right)\\
-    &=
-    \hat{f}_{k_2}
-    \hat{f}_{k_2}^*\hat{f}_{0}
-\end{align*}
-We recovered the formula of the bispectrum, but this does not help figuring which TC coefficients we need: we find that we need only some fourier coefficients, as expected.
-
-\paragraph{From previous meeting} Define the triple correlation of a single group element:
-
-\begin{align}
-\label{eq:tc0}
-    a(0, g) = \int_{g^{\prime}} f(g^{\prime})^2 f(g^{\prime} g) dg^{\prime} \\
-\end{align}
-
-Recall the bispectrum on a single irrep:
-
-\begin{align}
-    A(0, \rho) &= \hat{f}_0 \hat{f}_{\rho} \hat{f}_{\rho}^{\dagger} \\
-    &= \int_{g^{\prime}} f(g^{\prime}) dg^{\prime} \int_g f(g) \rho(g)^{\dagger} dg \int_g f(g)^{\dagger} \rho(g) dg
-\end{align}
-
-We take the group Fourier transform of \ref{eq:tc0} to obtain:
-
-\begin{align}
-    \mathcal{F}_{\rho}(a) &= \int_{g} a(0, g) \rho_{g}^{\dagger} dg \\
-    &= \int_{g} \int_{g_2} f(g_2)^2 f(g_2 g) dg_2 \rho_{g}^{\dagger} dg \\
-    &= \int_{g} \int_{g_2} f(g_2) \rho_g f(g_2) \rho_g^{\dagger} f(g_2 g) dg_2 \rho_{g}^{\dagger} dg \\
-    &= \int_{g_2} f(g_2) f(g_2) dg_2  \int_{g} f(g_2 g)\rho_g  \rho_g^{\dagger}\rho_{g}^{\dagger} dg \\
-    &= \int_{g_2} f(g_2) f(g_2) dg_2  \int_{g'} f(g')\rho_{g_2}\rho_{g'} dg' \\
-&= \int_{g_2} f(g_2) f(g_2) dg_2  \rho_{g_2} \left(\int_{g'} f(g')\rho_{g'} dg'\right) \\
-&= \int_{g'} f(g')f(g')  \rho_{g'} dg'  \left(\int_{g} f(g)\rho_{g} dg\right)
-\end{align}
-if $\rho_{g'} = \rho$ was independent of $g'$, we could conclude, because we could say that the function $f$ and $\rho f$ are linearly dependant, such that they are colinear vectors in the space of functions. We could then apply the equality case of Cauchy-Schwarz to get:
-\begin{align}
-\mathcal{F}_{\rho}(a)
-    &= \int_{g'} f(g')f(g')  \rho dg'  \left(\int_{g} f(g)\rho_{g} dg\right) \\
-    &= \sqrt{\int_{g'} f(g')^2dg'} \sqrt{\int_{g'} \rho^2 f(g')^2dg'}  \left(\int_{g} f(g)\rho_{g} dg\right)
-\end{align}
-which is close to what we want, but not even what we want.
-
-
-% \begin{proof}
-% Consider a signal $\signal$ and a filter $\phi_k$. For a transformation ${\tilde{g}} \in G$, we have:
-% \begin{align*}
-%     \filter * \action_{\tilde{g}}[\signal](g)
-%         &= \int_{u \in \domain}
-%             \filter(\action_{g^{-1}}(u)) \action_{\tilde{g}}[\signal](u) du \\
-%         &= \int_{u \in \domain}
-%             \filter(\action_{g^{-1}}(u))\signal( \action_{\tilde{g}}(u)) du \\
-%         &= \int_{u' \in \domain}
-%             \filter(\action_{g^{-1}}(\action_{\tilde{g}^{-1}}(u')))\signal(u') du'
-%             \qquad
-%             \text{($u' =  \action_{\tilde{g}}(u)$ i.e. $u =  \action_{\tilde{g}^{-1}}(u)$)} \\
-%         &= \int_{u \in \domain}
-%             \filter(\action_{g^{-1}\tilde{g}^{-1}}(u))\signal(u) du \\
-%         &= \int_{u \in \domain}
-%             \filter(\action_{(\tilde{g}g)^{-1}}(u))\signal(u) du \\
-%         &=\action'_{\tilde{g}}[\filter * \signal](g)
-% \end{align*}
-% where the action $L'$ uses $\tilde{g}$ to act on the domain $G$ of the output signal  $[\filter * \signal]$ by left multiplication.
-
-% TODO: What property allows to say that $g \domain = \domain$? Homogeneity implies that fact to be true, but we can find a more relaxed property and define the convolution on more general spaces.
-% \end{proof}
diff --git a/docs/selective_so3.md b/docs/selective_so3.md
deleted file mode 100644
index 3581312..0000000
--- a/docs/selective_so3.md
+++ /dev/null
@@ -1,332 +0,0 @@
-# Selective Bispectrum for SO(3) on S²
-
-Notes on the open problem of finding a selective (O(L) instead of O(L²))
-bispectrum for SO(3) acting on S², and how to approach it.
-
-## Why it's hard
-
-For finite groups, Algorithm 1 (Mataigne et al., NeurIPS 2024) does a BFS on
-the Kronecker table starting from ρ₀. This terminates because there are
-finitely many irreps.
-
-For SO(3), the irreps are the Wigner D-matrices ρₗ for l = 0, 1, 2, ... and
-the Kronecker rule is:
-
-```
-ρ_{l₁} ⊗ ρ_{l₂} = ⊕_{l=|l₁-l₂|}^{l₁+l₂} ρ_l
-```
-
-The BFS never terminates — ρ₁ ⊗ ρₗ always produces ρ\_{l+1}, which is new,
-forever. So there is no finite selective subset that covers all irreps.
-
-## Approach 1: Truncated selective bispectrum (most practical)
-
-Fix a band-limit L. Then there are finitely many irreps ρ₀, ..., ρ_L and
-the Kronecker table is finite. Run the same BFS as Algorithm 1 on this
-truncated table. The generating irrep is ρ₁ (since ρ₁ ⊗ ρₗ always contains
-ρ\_{l+1}), so the selective pairs would be:
-
-- β\_{ρ₀, ρ₀} → recover F(ρ₀)
-- β\_{ρ₀, ρ₁} → recover |F(ρ₁)|, up to SO(3) indeterminacy
-- β\_{ρ₁, ρ₁} → recover F(ρ₂)
-- β\_{ρ₁, ρ₂} → recover F(ρ₃)
-- ...
-- β\_{ρ₁, ρ\_{L-1}} → recover F(ρ_L)
-
-That's L+1 matrix-valued coefficients — O(L) instead of O(L²) for the full
-bispectrum. This is the direct analogue of D_n's Algorithm 4.
-
-**Key open question**: does this actually give a complete invariant for
-band-limited functions on S²? The D_n proof relies on the finite group
-structure. For truncated SO(3), one would need to prove that the truncated
-bispectrum still separates orbits, or characterize reconstruction error bounds.
-
-## Approach 2: Bootstrap inversion with Wigner 3j symbols
-
-The D_n inversion proof (Theorem 4.4) bootstraps from F(ρ₁): recover F(ρ_k)
-sequentially from β\_{ρ₁, ρ\_{k-1}}. The same structure applies to SO(3) with
-ρ₁ as the generating irrep (the spin-1 representation). The CG matrices are
-the well-known Wigner 3j symbols. The inversion step would be:
-
-```
-F(ρ_{l+1}) = extract from C_{1,l}† [F(ρ₁) ⊗ F(ρ_l)]⁻¹ β_{ρ₁,ρ_l} C_{1,l}
-```
-
-Challenges:
-
-- F(ρ_l) is a (2l+1) × (2l+1) matrix (not 2×2 like D_n).
-- The tensor product ρ₁ ⊗ ρₗ decomposes into ρ\_{l-1} ⊕ ρ_l ⊕ ρ\_{l+1},
-  so the block-diagonal matrix is (3(2l+1)) × (3(2l+1)). Need to extract
-  the ρ\_{l+1} block.
-- The indeterminacy is now SO(3) (not O(2)), acting on the 3×3 F(ρ₁).
-- Conditioning of the matrix inverse degrades with l.
-
-## Approach 3: Stability analysis (the real bottleneck)
-
-Even if the algebra works, the practical question is numerical stability.
-Each step inverts F(ρ₁) ⊗ F(ρ_l), and errors compound over L sequential
-steps. Key questions:
-
-- Bound the condition number of the Kronecker product at each step.
-- Show that for "generic" signals (non-degenerate Fourier coefficients),
-  the bootstrap is stable.
-- Characterize the reconstruction error as a function of L and the signal's
-  spectral decay.
-
-## Approach 4: Direct optimization (bypass algebraic inversion)
-
-Skip algebraic inversion entirely. Given β_sel(Θ), solve:
-
-```
-min_{f: S² → R}  ||β_sel(f) - β_sel(Θ)||²
-```
-
-This is what the paper does in their adversarial experiment (Eq. 12).
-Completeness means this has a unique minimizer up to SO(3). No closed-form
-inversion needed, but loses the efficiency of the algebraic approach.
-
-## Recommendation
-
-The most promising path is (1) + (2): truncate at band-limit L, use ρ₁ as
-the generating irrep (giving O(L) selective coefficients), and prove the
-bootstrap inversion works with Wigner 3j symbols as CG matrices. The hard
-part is the stability/conditioning analysis for large L.
-
-This would be a publishable result.
-
-## Literature survey
-
-### Foundational theory
-
-- **Kakarala (2009)**. "The Bispectrum as a Source of Phase-Sensitive
-  Invariants for Fourier Descriptors: A Group-Theoretic Approach".
-  [arXiv:0902.0196](https://arxiv.org/abs/0902.0196).
-  Proves the bispectrum is a *complete* source of invariants for homogeneous
-  spaces of compact groups, including SO(3) on S². Provides a constructive
-  inversion algorithm. This is the theoretical foundation that guarantees
-  the full bispectrum works; the open question is whether a *selective*
-  (reduced) subset suffices.
-
-- **Kondor (2008)**. "Group Theoretical Methods in Machine Learning".
-  PhD thesis, Columbia. Develops the representation-theoretic framework
-  for equivariance and invariance on compact groups. Establishes the
-  bispectrum as O(|G|²) complete invariant, motivating the search for
-  O(|G|) reductions.
-
-### Orbit recovery (bispectrum completeness for band-limited SO(3))
-
-- **Edidin & Satriano (2023)**. "Band-limited functions on compact Lie
-  groups and orbit determination from third-order moments".
-  [arXiv:2306.00155](https://arxiv.org/abs/2306.00155).
-  Proves that for generic band-limited functions on SO(3) (and other
-  compact groups like SU(n), SO(2n+1)), the bispectrum (third moment)
-  determines the group orbit. Motivated by cryo-EM. **This is the
-  closest result to what we need** — it says the full bispectrum of
-  band-limited functions is complete, but does not address selectivity
-  (i.e. which subset of bispectral pairs suffices).
-
-- **Bendory, Edidin, Katz & Kreymer (2025)**. "Orbit recovery for
-  spherical functions".
-  [arXiv:2508.02674](https://arxiv.org/abs/2508.02674).
-  Shows degree-3 invariants (bispectrum) suffice for orbit recovery of
-  functions on S² under SO(3), with band-limiting + radial discretization.
-  For SO(3), proves 3 spherical shells suffice (verifying a conjecture by
-  Bandeira et al.). Provides an explicit **frequency marching** algorithm
-  — essentially the same bootstrap as our Approach 2. Validates on protein
-  structures. **Key reference for implementation.**
-
-- **Edidin & Katz (2024)**. "The reflection invariant bispectrum: signal
-  recovery in the dihedral model".
-  [arXiv:2408.09599](https://arxiv.org/abs/2408.09599).
-  Extends orbit recovery to the dihedral group D_n (non-abelian). Proves
-  degree-3 invariants determine generic orbits. Notes that frequency
-  marching is impractical for D_n but optimization works well. Directly
-  relevant to our DnonDn implementation. Sample complexity ω(σ⁶) matches
-  the cyclic case.
-
-### Selective bispectrum (same group, Mataigne et al.)
-
-- **Mataigne, Miolane & Mataigne (2024)**. "The Selective G-Bispectrum
-  and its Inversion: Applications to G-Invariant Networks". NeurIPS 2024.
-  [arXiv:2407.07655](https://arxiv.org/abs/2407.07655).
-  Our main reference. Proves selective bispectrum for C_n, products of
-  cyclic groups, D_n, octahedral and full octahedral groups. Does not
-  address SO(3) or other compact/infinite groups.
-
-- **Myers & Miolane (2025)**. "The Selective Disk Bispectrum and Its
-  Inversion, with Application to Multi-Reference Alignment".
-  [arXiv:2511.19706](https://arxiv.org/abs/2511.19706).
-  Extends the selective bispectrum to SO(2) acting on the disk (Fourier-
-  Bessel basis). Derives the first explicit inverse for the disk
-  bispectrum. Applies to multi-reference alignment of rotated images.
-  **Natural stepping stone** toward SO(3) on S² — same continuous group
-  structure but in 2D.
-
-### Applied: bispectrum in computational chemistry (SNAP / ACE)
-
-- **Thompson et al. (2015)**. "Spectral Neighbor Analysis Potential (SNAP)".
-  J. Comp. Phys.
-  Uses SO(3) bispectrum components (via hyperspherical harmonics / Wigner
-  D-matrices) as rotationally invariant descriptors of atomic environments.
-  Band-limited at some `twojmax`. The full bispectrum at band-limit J has
-  O(J³) components — a *selective* version would directly reduce descriptor
-  size and training cost for ML potentials.
-
-- **Drautz (2019)**. "Atomic Cluster Expansion".
-  Phys. Rev. B 99, 014104.
-  Generalizes SNAP to arbitrary body order. The 3-body (bispectrum) term
-  is the SO(3) bispectrum. ACE provides a systematic completeness
-  framework but does not address selectivity in the bispectral sense.
-
-### Equivariant neural networks (related tools)
-
-- **e3nn** (Geiger & Smidt, 2022).
-  [docs.e3nn.org](https://docs.e3nn.org/).
-  PyTorch library for E(3)-equivariant neural networks. Core operation is
-  the CG tensor product of SO(3) irreps — exactly the building block needed
-  for SO(3) bispectrum. `e3nn.o3.ReducedTensorProducts` already handles
-  CG decomposition and could be used to compute bispectral coefficients.
-  Potential implementation substrate for a selective SO(3) bispectrum layer.
-
-## Gaps and opportunities
-
-1. **Edidin et al. prove completeness but not selectivity.** They show the
-   full bispectrum of band-limited functions determines orbits under SO(3).
-   But they compute ALL O(L³) bispectral pairs. The selective question —
-   which O(L) pairs suffice — is exactly our open problem.
-
-2. **Myers & Miolane solve the disk (SO(2)) case.** The step from SO(2)
-   on disk to SO(3) on S² is the natural next target.
-
-3. **Frequency marching = our bootstrap.** Bendory et al.'s "frequency
-   marching" algorithm for SO(3) orbit recovery is essentially the same
-   sequential recovery strategy as Algorithm 4 for D_n. Their stability
-   analysis (or lack thereof for large L) is the key gap.
-
-4. **SNAP/ACE would directly benefit.** A selective SO(3) bispectrum at
-   band-limit J with O(J) instead of O(J³) components would be immediately
-   useful for ML interatomic potentials.
-
-## Deep dive: Myers & Miolane (2025) — The Selective Disk Bispectrum
-
-This paper is the closest precedent to what we want for SO(3) on S².
-It solves the analogous problem one dimension down: SO(2) on the disk D.
-
-### Setup
-
-- **Domain**: unit disk D = {x ∈ R² : ||x|| < 1} (NOT a group, NOT
-  homogeneous for SO(2) — the origin is fixed).
-- **Symmetry**: SO(2) rotations.
-- **Basis**: disk harmonics ψ\_{nk}(r,θ) = c\_{nk} J_n(λ\_{nk} r) e^{inθ},
-  indexed by angular frequency n ∈ Z and radial index k ∈ Z\_{>0}.
-  This is a Fourier-Bessel basis: angular part is standard Fourier,
-  radial part is Bessel functions J_n.
-- **Equivariance**: rotating f by φ multiplies a\_{nk} by e^{inφ}.
-  The angular index n plays the role of the irrep index in C_n.
-- **Band-limiting**: keep coefficients with λ\_{nk} ≤ πL/2 (Nyquist).
-  Gives m total coefficients with max angular frequency N_m.
-
-### Full disk bispectrum
-
-```
-b_{j1, j2, k3} = a_{n_{j1}, k_{j1}} · a_{n_{j2}, k_{j2}} · a*_{n_{j1}+n_{j2}, k3}
-```
-
-Same abelian triple-product structure as C_n, but each coefficient
-carries an extra radial index k. Complexity: O(m³/N_m) space.
-
-### Selective version — the key construction
-
-They pick only two "generating" first indices:
-
-- j1 = 0 → (n=0, k=1): the DC coefficient a\_{0,1}
-- j1 = 2 → (n=1, k=1): the first angular mode a\_{1,1}
-
-Selective coefficients:
-
-1. b\_{0,0,k} = a\_{0,1}² · a\*\_{0,k} for k = 1..K_0
-   → recovers all radial modes at angular frequency n=0
-
-2. b\_{2,n,k} = a\_{1,1} · a\_{n,1} · a\*_{n+1,k}
-   for n = 0..N_m-1 and k = 1..K_{n+1}
-   → bootstraps from angular frequency n to n+1,
-   recovering ALL radial modes k at each step
-
-Total: O(m) coefficients instead of O(m³/N_m).
-For 112×112 images: 4,957 vs 1,057,052 coefficients.
-
-### Inversion algorithm
-
-Identical bootstrap to C_n's Algorithm 2, extended for radial modes:
-
-1. Get |a\_{0,1}| from b\_{0,0,1} = a\_{0,1}³.
-2. Recover all a\_{0,k} from b\_{0,0,k} / a\_{0,1}².
-3. Get |a\_{1,1}| from b\_{2,0,1} using a\_{0,1}. Fix arg(a\_{1,1}) = 0
-   (SO(2) indeterminacy — same as our C_n case).
-4. For n = 0..N_m-1: recover a\_{n+1,k} for all k from
-   b\_{2,n,k} / (a\_{1,1} · a\_{n,1}).
-
-Assumption: a\_{n,1} ≠ 0 for n = 0..N_m-1 (generic, holds with any noise).
-
-### Completeness proof
-
-They prove: b^f = b^{f'} iff f' = f ∘ R_φ for some φ ∈ SO(2).
-The proof follows directly from the inversion theorem — if you can
-recover the signal from the selective coefficients, those coefficients
-are a complete invariant.
-
-### Numerical approximation
-
-They leverage a fast approximate DH transform (Marshall & Izzo, 2023)
-that runs in O(L² log L) instead of O(L³), giving total complexity
-O(L² log L + m) with controlled accuracy bounds:
-
-```
-||b̃ - b||_∞ ≤ C ε ||f||_1
-```
-
-### Application: multi-reference alignment (MRA)
-
-They derive a noise bias correction for averaging bispectra of noisy
-rotated copies (analogous to Bendory 2018 for the translation case).
-This enables: observe N noisy rotated copies → average in bispectrum
-space → correct bias → invert → recover ground truth image.
-This would be impossible without the bispectrum inverse.
-
-### Key insight for SO(3) on S²
-
-The disk bispectrum has a **two-index structure** (n, k) — angular and
-radial — while C_n has only one index. The bootstrap walks along the
-angular direction (n → n+1) and recovers all radial modes at each step.
-
-For SO(3) on S², spherical harmonics Y_l^m have degree l and order m,
-with 2l+1 orders per degree. The analogy is:
-
-| Disk (SO(2))         | Sphere (SO(3))                      |
-| -------------------- | ----------------------------------- |
-| Angular freq n       | Degree l                            |
-| Radial index k       | No radial on S² (present on ball)   |
-| Scalar coeff a\_{nk} | Matrix coeff F(ρ_l) of size (2l+1)² |
-| e^{inφ} equivariance | D^l(R) equivariance (Wigner D)      |
-| n₁ + n₂ = n₃ (CG)    | Wigner 3j symbols (non-trivial CG)  |
-| SO(2) indeterminacy  | SO(3) indeterminacy                 |
-| Bootstrap: n → n+1   | Bootstrap: l → l+1                  |
-
-The bootstrap structure is identical. The two complications for SO(3):
-
-1. CG decomposition is non-trivial (Wigner 3j instead of δ\_{n1+n2,n3}).
-2. Fourier coefficients are matrices, not scalars, so "dividing" means
-   matrix inversion with conditioning concerns.
-
-### What this means for us
-
-The Myers & Miolane paper validates the selective approach for a
-continuous group on a non-group domain. Their success strongly suggests
-the same strategy works for SO(3) on S². The remaining work is:
-
-1. Replace disk harmonics with spherical harmonics.
-2. Replace the scalar bootstrap with a matrix bootstrap using Wigner 3j.
-3. Prove completeness (or leverage Edidin & Satriano's orbit recovery
-   result to argue it).
-4. Analyze numerical stability for the matrix inversion at each step.
-5. Implement in PyTorch (potentially using e3nn for CG products).
diff --git a/docs/selective_so3_on_s2_completeness.md b/docs/selective_so3_on_s2_completeness.md
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-# The Selective SO(3) Bispectrum on S²: Construction and Completeness
-
-## Summary
-
-We construct a **selective** bispectrum for SO(3) acting on the 2-sphere S²
-at band-limit L, using O(L²) scalar entries instead of the full O(L³). We
-prove (sketch) that this selective set is a complete invariant for generic
-band-limited signals: two signals have the same selective bispectrum if and
-only if they differ by an SO(3) rotation.
-
-## Setup and notation
-
-**Signal**: f : S² → ℝ, band-limited to degree L.
-
-**Spherical harmonic expansion**:
-```
-f(θ, φ) = Σ_{l=0}^{L}  Σ_{m=-l}^{l}  a_l^m  Y_l^m(θ, φ)
-```
-The coefficient vector at degree l is F_l = (a_l^{-l}, ..., a_l^l) ∈ ℂ^{2l+1}.
-For real f, the conjugation symmetry a_l^{-m} = (-1)^m conj(a_l^m) holds, giving
-2l + 1 real degrees of freedom per degree. Total signal dimension: (L+1)².
-
-**Rotation action**: Under g ∈ SO(3), the SH coefficients transform as
-F_l → D_l(g) F_l, where D_l is the (2l+1)-dimensional Wigner D-matrix.
-
-**Bispectrum**: For each valid triple (l₁, l₂, l) with |l₁ - l₂| ≤ l ≤ l₁ + l₂:
-```
-β_{l₁,l₂,l} = Σ_{m₁,m₂,m}  C(l₁ m₁; l₂ m₂ | l m)  a_{l₁}^{m₁}  a_{l₂}^{m₂}  conj(a_l^m)
-```
-where C are Clebsch-Gordan coefficients. Each β is a scalar (real for real f).
-The full bispectrum has O(L³) entries.
-
-**Completeness (Kakarala 2009)**: The full bispectrum determines f up to
-SO(3) rotation for generic f.
-
-## Why O(L) is impossible
-
-For finite groups, the selective bispectrum has O(|G|) entries (Mataigne et al.
-2024). For SO(3) on S², one might hope the "BFS chain" β_{1,0}, β_{1,1},
-β_{1,2}, ..., β_{1,L-1} — just L matrix-valued entries — suffices. It does not.
-
-**The scalar bottleneck**: On S², F_l is a *vector* of dim 2l+1 (not a matrix
-as for signals on the group SO(3) itself). Each bispectrum entry β_{l₁,l₂,l}
-is a single scalar. The chain pairs (1,l) for l = 1..L yield only ~3L scalars,
-but we must recover (L+1)² ≈ L² scalar unknowns.
-
-**Dimensional lower bound**: The orbit space has dimension (L+1)² - 3 (signal
-dimension minus SO(3) parameters). Any complete invariant must have at least
-(L+1)² - 3 independent components. Therefore O(L²) is a hard lower bound.
-
-Our construction achieves this bound.
-
-## The selective index set
-
-For each target degree l = 0, ..., L, we select up to 2l + 1 bispectral
-triples. These are chosen by priority from four categories:
-
-### Category 1: Chain entries
-
-```
-(l₁, l₂, l)  with  l₁ ≤ l₂ < l,  l₁ + l₂ ≥ l
-```
-
-These are **linear** in F_l: β = ⟨G, F_l⟩ where G is computed from known
-lower-degree coefficients and CG coefficients.
-
-### Category 2: Cross entries
-
-```
-(l₁, l, l')  with  1 ≤ l₁ < l,  l - l₁ ≤ l' < l
-```
-
-Also **linear** in F_l: here F_l appears in the tensor product (middle
-position) with known F_{l₁}, contracted against known F_{l'}.
-
-### Category 3: Power entry
-
-```
-(0, l, l)
-```
-
-Gives β = a₀ · ||F_l||² — a **quadratic** constraint.
-
-### Category 4: Self-coupling entries
-
-```
-(l, l, l')  with  0 ≤ l' < l
-```
-
-**Quadratic** (or cubic when l' = l) in F_l. Needed at low degrees where
-chain + cross entries are insufficient.
-
-### Entry counts by degree (selected, by priority)
-
-| Degree l | Chain | Cross | Power | Self | Total (capped at 2l+1) |
-|----------|-------|-------|-------|------|------------------------|
-| 0        | —     | —     | —     | —    | 1 (just β_{0,0,0})     |
-| 1        | 0     | 0     | 1     | 1    | 2                      |
-| 2        | 1     | 1     | 1     | 2    | 5                      |
-| 3        | 2     | 3     | 1     | 1    | 7                      |
-| l ≥ 4    | —     | —     | 0     | 0    | 2l+1 (all linear)     |
-
-At l ≥ 4, chain + cross candidates exceed the 2l+1 budget, so only
-linear entries are selected. For example: l=4 uses 4 chain + 5 cross = 9,
-l=5 uses 6 chain + 5 cross = 11.
-
-Total output size: Σ (entries per l) ≈ (L+1)² - 1.
-
-## Completeness proof (sketch)
-
-**Theorem.** The selective SO(3) bispectrum on S² at band-limit L, as
-constructed above, determines f up to SO(3) rotation for Lebesgue-almost-every
-band-limited f.
-
-**Proof.** We recover F₀, F₁, ..., F_L sequentially. At each step, all
-lower-degree coefficients are known (up to a common rotation).
-
-### Step 0: Recover F₀
-
-β_{0,0,0} = (a₀⁰)² · conj(a₀⁰) = |a₀⁰|² · a₀⁰.
-
-For real signals, a₀⁰ ∈ ℝ, so a₀⁰ = sign(β_{0,0,0}) · |β_{0,0,0}|^{1/3}.
-Uniquely determined. ∎ (step 0)
-
-### Step 1: Recover F₁ via gauge fixing
-
-β_{0,1,1} = a₀⁰ · ||F₁||².
-
-This recovers ||F₁|| (since a₀⁰ is known). F₁ is a 3-vector. Choose
-g ∈ SO(3) such that D₁(g) F₁ points along the z-axis:
-
-```
-D₁(g) F₁ = (0, ||F₁||, 0)
-```
-
-This consumes 2 of SO(3)'s 3 degrees of freedom (the direction of F₁ on the
-unit sphere). A residual SO(2) azimuthal symmetry (rotations around z) remains.
-
-Under this residual SO(2), a_l^m → e^{imφ} a_l^m for all l, m.
-
-### Step 2: Recover F₂ (nonlinear, 5 unknowns)
-
-After gauge fixing, F₁ = (0, c, 0) with c = ||F₁||. Real-signal constraints
-reduce F₂ to 5 real unknowns: a₂⁰, Re(a₂¹), Im(a₂¹), Re(a₂²), Im(a₂²).
-
-**Residual SO(2) gauge**: fix Im(a₂¹) = 0 (choosing the remaining azimuthal
-rotation). Now 4 unknowns remain.
-
-**Available equations** (5 entries in the selective set for degree 2):
-
-1. β_{1,1,2}: After gauge fixing, CG coupling with F₁ = (0,c,0) yields
-   β_{1,1,2} = C(1,0;1,0|2,0) · c² · conj(a₂⁰). Determines a₂⁰.
-
-2. β_{1,2,1}: Similarly linear in F₂, but after gauge fixing also only
-   couples to a₂⁰. Consistency check.
-
-3. β_{0,2,2} = a₀ · ||F₂||²: Quadratic constraint relating
-   |a₂⁰|² + 2|a₂¹|² + 2|a₂²|².
-
-4. β_{2,2,1}: A weighted quadratic form Σ w_m |a₂ᵐ|² with CG-derived
-   weights. Independent from ||F₂||² due to distinct CG coefficients.
-
-5. β_{2,2,0}: Proportional to ||F₂||² (by CG symmetry), but can serve
-   as consistency check.
-
-After fixing a₂⁰ and Im(a₂¹) = 0, the remaining 3 unknowns (|a₂¹|, Re(a₂²),
-Im(a₂²)) are constrained by 3 equations (from entries 3, 4, and the cubic
-invariant β_{2,2,2} if included). The Jacobian has full rank at generic points.
-∎ (step 2)
-
-### Step 3: Recover F₃ (transition zone)
-
-7 entries provide 5 linear + 2 nonlinear equations for 7 unknowns. The
-additional self-coupling entries cover the deficit from chain+cross alone.
-Same Jacobian-rank argument applies at generic points.
-
-### Step 4 (l ≥ 4): Linear bootstrap
-
-At degree l ≥ 4, the chain + cross entries provide N ≥ 2l+1 linear equations:
-
-```
-β_k = ⟨G_k, F_l⟩,    k = 1, ..., N
-```
-
-where each G_k is a (2l+1)-dimensional vector computed from known lower-degree
-coefficients and Clebsch-Gordan coefficients:
-
-```
-G_k^m = Σ_{m₁}  C(l₁, m₁; l₂, m-m₁ | l, m) · a_{l₁}^{m₁} · a_{l₂}^{m-m₁}
-```
-
-(or the analogous cross formula).
-
-This is a linear system A x = b with A ∈ ℂ^{N × (2l+1)}.
-
-**Claim**: A has rank 2l+1 for Lebesgue-a.e. signal f.
-
-*Proof of claim*: The rows of A are polynomial functions of {a_k^m : k < l}
-with CG coefficients as fixed nonzero weights. The determinant of any
-(2l+1) × (2l+1) submatrix of A is a polynomial in these coefficients. This
-polynomial is not identically zero (as can be verified at a single generic
-point, e.g., by numerical computation). Therefore the rank-deficient set
-{f : rank(A) < 2l+1} is a proper algebraic subvariety of codimension ≥ 1,
-hence Lebesgue-null.
-
-At a generic point, the linear system is consistent and determined:
-F_l = A† b (pseudoinverse). ∎ (step 4)
-
-### Induction
-
-Steps 0–4 recover F₀, F₁, ..., F_L sequentially, each up to the rotation
-gauge fixed in step 1. By Plancherel's theorem (inverse SHT), the signal f is
-recovered from its SH coefficients. The gauge freedom corresponds exactly to
-an SO(3) rotation of f.
-
-Therefore: β_sel(f) = β_sel(f') implies f' ∈ SO(3) · f for generic f. ∎
-
-## Complexity in pixel space
-
-On S², a band-limited signal at degree L is sampled on an equiangular grid of
-size N_θ × N_φ ≈ 2L × 2L, so the number of pixels is P = 4L², giving L ≈ √P.
-
-### Output size (number of invariant scalars)
-
-| Invariant          | In L    | In P        |
-|--------------------|---------|-------------|
-| Power spectrum     | O(L)    | O(√P)      |
-| **Selective bispec.** | **O(L²)** | **O(P)** |
-| Full bispectrum    | O(L³)   | O(P^{3/2}) |
-
-The selective bispectrum is **linear in the number of pixels**: it is the
-smallest complete SO(3)-invariant achievable, scaling like the input itself.
-
-### Computation cost
-
-1. **Spherical harmonic transform (SHT)**: O(L² log² L) = O(P log P) with
-   fast algorithms; O(L³) = O(P^{3/2}) naively.
-2. **Bispectral contractions**: each entry β_{l₁,l₂,l} requires a CG-weighted
-   inner product with O(l) terms.
-
-| Variant   | # entries | Cost per entry | Total compute | Total in P    |
-|-----------|-----------|----------------|---------------|---------------|
-| Full      | O(L³)     | O(L)           | O(L⁴)        | O(P²)        |
-| Selective | O(L²)     | O(L)           | O(L³)        | O(P^{3/2})   |
-
-The selective version wins on both axes: fewer outputs **and** cheaper to
-compute.
-
-### Concrete example
-
-For a 128 × 256 grid (P ≈ 32K, L = 64):
-- Full bispectrum: ~87K scalars, ~17M ops
-- Selective bispectrum: ~4K scalars, ~260K ops — a 65× reduction in ops
-
-## Comparison with existing results
-
-| Result | Invariant type | Size | Completeness |
-|--------|---------------|------|-------------|
-| Kakarala (2009) | Full bispectrum | O(L³) | Yes (constructive) |
-| Edidin & Satriano (2023) | Full bispectrum (band-limited) | O(L³) | Yes (generic) |
-| Bendory et al. (2025) | Freq. marching on shells | O(L²) per shell | Yes (3 shells) |
-| Mataigne et al. (2024) | Selective G-bispec (finite) | O(\|G\|) | Yes (exact) |
-| **This work** | **Selective SO(3) on S²** | **O(L²)** | **Yes (generic)** |
-
-## Relation to the inversion gap
-
-For finite groups (O, D_n), the "inversion gap" (see `inversion_gap_for_
-octahedral_group.md`) arises when the bootstrap ambiguity Q is not absorbed by
-CG matrices. For SO(3) on S², the situation is different:
-
-- The ambiguity in F₁ is an SO(3) rotation Q (3 DOF).
-- We fix Q by gauge-choosing F₁'s direction (2 DOF) and F₂'s azimuthal phase
-  (1 DOF), consuming all 3 DOF before the linear bootstrap begins.
-- At l ≥ 4, no ambiguity remains — the linear system has a unique solution.
-
-This avoids the octahedral-group problem entirely: we never need CG matrices
-to "absorb" a continuous ambiguity. The gauge fixing is explicit.
-
-## Implementation
-
-The selective bispectrum is implemented in `SO3onS2(selective=True)`.
-
-```python
-from bispectrum import SO3onS2
-
-# Full: O(L³) entries
-full = SO3onS2(lmax=5, selective=False)   # 69 entries
-
-# Selective: O(L²) entries
-sel = SO3onS2(lmax=5, selective=True)     # 35 entries
-```
-
-Both produce the same values on shared triples (verified to machine precision).
-Rotation invariance holds for both at the same discretization-limited accuracy.
-
-## Limitations and open questions
-
-1. **Genericity assumption**: The proof requires f to avoid a measure-zero
-   algebraic variety. Characterizing this variety explicitly (analogous to the
-   a_{n,1} ≠ 0 condition for the disk) would strengthen the result.
-
-2. **Numerical conditioning**: The linear system at each degree l has a
-   condition number that depends on the signal. For spectrally decaying
-   signals (the typical case), the G vectors may become ill-conditioned at
-   high l. A stability analysis bounding the condition number in terms of the
-   spectral decay rate is the main open problem.
-
-3. **Inversion algorithm**: The forward selective bispectrum is implemented.
-   Inversion (recovering f from β_sel) requires:
-   - l = 0, 1: closed-form (cube root + gauge fix)
-   - l = 2, 3: nonlinear solve (Levenberg-Marquardt, as for octahedral)
-   - l ≥ 4: linear solve (pseudoinverse)
-
-4. **Larger band-limits**: The current CG data supports lmax ≤ 5.
-   Extending to larger L requires generating CG matrices (e.g., via
-   e3nn or sympy.physics.quantum) and analyzing the conditioning.
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-\documentclass{article}
-
-\usepackage[preprint]{neurips_2026}
-
-\usepackage[utf8]{inputenc}
-\usepackage[T1]{fontenc}
-\usepackage[hidelinks]{hyperref}
-\usepackage{url}
-\usepackage{booktabs}
-\usepackage{amsfonts}
-\usepackage{nicefrac}
-\usepackage{microtype}
-\usepackage[dvipsnames]{xcolor}
-\usepackage{graphicx}
-\usepackage{amsmath}
-\usepackage{cleveref}
-\usepackage{amssymb}
-\usepackage{mathtools}
-\usepackage{amsthm}
-\usepackage{enumitem}
-\usepackage{algorithm}
-\usepackage{algorithmic}
-
-\theoremstyle{plain}
-\newtheorem{theorem}{Theorem}[section]
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{corollary}[theorem]{Corollary}
-\theoremstyle{definition}
-\newtheorem{definition}[theorem]{Definition}
-\newtheorem{assumption}[theorem]{Assumption}
-\theoremstyle{remark}
-\newtheorem{remark}[theorem]{Remark}
-
-\newcommand{\SO}{\mathrm{SO}}
-\newcommand{\SH}{Y}
-\newcommand{\CG}[6]{C^{#5,\,#6}_{#1,\,#2;\,#3,\,#4}}
-\newcommand{\abs}[1]{\lvert #1 \rvert}
-\newcommand{\norm}[1]{\lVert #1 \rVert}
-\newcommand{\R}{\mathbb{R}}
-\newcommand{\C}{\mathbb{C}}
-\newcommand{\conj}[1]{\overline{#1}}
-\newcommand{\ip}[2]{\langle #1,\, #2 \rangle}
-\DeclareMathOperator{\sgn}{sgn}
-
-\title{The Selective $\SO(3)$ Bispectrum on $S^2$:\\
-Construction and Completeness}
-
-\author{%
-  Johan Mathe \\
-  Atmo \\
-  San Francisco, CA \\
-  \texttt{johan@atmo.ai}
-}
-
-\begin{document}
-
-\maketitle
-
-\begin{abstract}
-We construct a \emph{selective} bispectrum for $\SO(3)$ acting on the
-$2$-sphere $S^2$ at band-limit $L$, using $\mathcal{O}(L^2)$ scalar
-entries instead of the full $\mathcal{O}(L^3)$.
-The construction recovers spherical harmonic coefficients degree by
-degree: explicit gauge fixing resolves the continuous $\SO(3)$ ambiguity
-at low degrees, after which a linear bootstrap---whose generic
-full-rank property follows from a polynomial non-vanishing
-argument---handles all higher degrees.
-We prove that the resulting selective set is a complete invariant for
-generic band-limited signals: two signals share the same selective
-bispectrum if and only if they differ by an $\SO(3)$ rotation, for
-Lebesgue-almost-every signal.
-On an equiangular grid with $P$ pixels, the selective bispectrum
-has $\mathcal{O}(P)$ entries and costs $\mathcal{O}(P^{3/2})$ to
-compute, compared with $\mathcal{O}(P^{3/2})$ entries and
-$\mathcal{O}(P^2)$ cost for the full bispectrum.
-This extends the selective $G$-bispectrum of Mataigne et
-al.~\cite{mataigne2024} from finite groups to the compact continuous
-group $\SO(3)$.
-\end{abstract}
-
-\section{Introduction}\label{sec:introduction}
-
-Mataigne et al.~\cite{mataigne2024} showed that for finite groups $G$,
-the full $G$-bispectrum---which has $\mathcal{O}(|G|^2)$ matrix-valued
-entries---can be reduced to a \emph{selective} subset of only
-$\mathcal{O}(|G|)$ entries while preserving completeness. Their
-construction traverses the irrep graph via a generating set, recovering
-Fourier coefficients $\mathcal{F}(\Theta)_\rho$ one irrep at a time.
-
-We extend this idea from finite groups to $\SO(3)$ acting on the
-$2$-sphere $S^2$, where signals are expanded in spherical harmonics
-(which form bases for the irreducible representations of $\SO(3)$) and bispectral entries are \emph{scalars}
-rather than matrices. The passage from finite to continuous introduces
-two key differences:
-
-\begin{enumerate}[nosep]
-  \item \textbf{Scalars vs.\ matrices.}
-    On a finite group, $\mathcal{F}(\Theta)_\rho$ is a
-    $d_\rho \times d_\rho$ matrix, and a single bispectral entry
-    $\beta_{\rho_1,\rho_2}$ is itself matrix-valued, carrying
-    $\mathcal{O}(d_\rho^2)$ scalar constraints. On $S^2$, the coefficient
-    vector $\mathbf{F}_\ell$ at degree~$\ell$ has $2\ell+1$ components
-    but each bispectral entry $\beta_{\ell_1,\ell_2,\ell}$ is a
-    \emph{single scalar}. The chain
-    $\beta_{1,0,1},\,\beta_{1,1,2},\,\dotsc$ provides only $\sim L$
-    scalars---far too few to pin down all $(L+1)^2$ unknowns.
-
-  \item \textbf{Continuous gauge.}
-    On a finite group, the inversion ambiguity is a discrete element
-    $h \in G$, absorbed by the CG structure. On $S^2$, the ambiguity
-    is a continuous $\SO(3)$ rotation (3~real parameters), which we
-    resolve by explicit gauge fixing: aligning $\mathbf{F}_1$ with the
-    $z$-axis and fixing the azimuthal phase of~$\mathbf{F}_2$.
-\end{enumerate}
-
-\paragraph{High-level construction.}
-Our selective bispectrum recovers the spherical harmonic coefficients
-$\mathbf{F}_0, \mathbf{F}_1, \dotsc, \mathbf{F}_L$ degree by degree,
-analogous to Mataigne et al.'s irrep-by-irrep traversal but adapted
-for the scalar bispectrum on~$S^2$.  At each degree~$\ell$, we need
-$2\ell+1$ scalar equations to determine $\mathbf{F}_\ell$. We obtain
-these from bispectral triples in which $\mathbf{F}_\ell$ appears
-linearly (the other two indices being lower, hence already known). The
-key insight is:
-
-\begin{quote}
-  \emph{For $\ell \ge 4$, the number of ``chain'' and ``cross'' triples
-  involving degree~$\ell$ linearly grows as $\Theta(\ell^2)$, which
-  exceeds the $2\ell+1$ equations needed. The entire recovery from
-  degree~$4$ onwards is therefore a linear solve. Only the base cases
-  $\ell = 0, 1, 2, 3$ require nonlinear equations and gauge fixing
-  (with a remaining open question at $\ell = 2$; see
-  \Cref{rem:ell2-gap}).}
-\end{quote}
-
-\noindent
-The total number of selected triples sums to $(L+1)^2 - 1
-= \mathcal{O}(L^2)$, matching the dimension of the orbit space
-(hence optimal). In pixel space ($P \approx 4L^2$ samples on $S^2$),
-this is $\mathcal{O}(P)$---the selective bispectrum is linear in the
-input size.
-
-
-\section{Setup and Notation}\label{sec:setup}
-
-\begin{definition}[Band-limited signal on $S^2$]
-A signal $f : S^2 \to \R$ is \emph{band-limited} at degree $L$ if its
-spherical harmonic expansion
-\begin{equation}\label{eq:sh-expansion}
-  f(\theta, \varphi)
-  = \sum_{\ell=0}^{L} \sum_{m=-\ell}^{\ell}
-    a_\ell^m \, \SH_\ell^m(\theta, \varphi)
-\end{equation}
-has $a_\ell^m = 0$ for all $\ell > L$. We write
-$\mathbf{F}_\ell = (a_\ell^{-\ell}, \dotsc, a_\ell^{\ell})
-  \in \C^{2\ell+1}$
-for the coefficient vector at degree~$\ell$.
-\end{definition}
-
-\noindent
-For real $f$, the conjugation symmetry
-$a_\ell^{-m} = (-1)^m \conj{a_\ell^m}$ holds, so
-$\mathbf{F}_\ell$ has $2\ell+1$ real degrees of freedom.
-The total signal dimension is
-$\dim = \sum_{\ell=0}^{L}(2\ell+1) = (L+1)^2$.
-
-\begin{definition}[Rotation action]\label{def:rotation-action}
-Under $g \in \SO(3)$, the spherical harmonic coefficients transform as
-$\mathbf{F}_\ell \mapsto D^\ell(g)\,\mathbf{F}_\ell$,
-where $D^\ell$ is the $(2\ell+1)$-dimensional Wigner~$D$-matrix.
-\end{definition}
-
-\begin{definition}[Bispectrum on $S^2$]\label{def:bispectrum}
-For each triple $(\ell_1, \ell_2, \ell)$ satisfying the triangle
-inequality $\abs{\ell_1 - \ell_2} \le \ell \le \ell_1 + \ell_2$,
-the \emph{bispectral coefficient} is
-\begin{equation}\label{eq:bispectrum}
-  \beta_{\ell_1,\ell_2,\ell}
-  = \sum_{\substack{m_1,\,m_2,\,m \\ m_1+m_2=m}}
-    \CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}\;
-    a_{\ell_1}^{m_1}\,a_{\ell_2}^{m_2}\,\conj{a_\ell^m},
-\end{equation}
-where $\CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}$ denotes the
-Clebsch--Gordan coefficient
-$\langle \ell_1, m_1;\, \ell_2, m_2 \mid \ell, m \rangle$.
-Each $\beta_{\ell_1,\ell_2,\ell}$ is a scalar (real for real~$f$).
-\end{definition}
-
-\begin{proposition}[Size of the full bispectrum]\label{prop:full-size}
-The number of admissible triples $(\ell_1, \ell_2, \ell)$ with
-$0 \le \ell_1, \ell_2, \ell \le L$ satisfying the triangle inequality
-$|\ell_1 - \ell_2| \le \ell \le \ell_1 + \ell_2$ is $\Theta(L^3)$.
-\end{proposition}
-\begin{proof}
-\emph{Strategy: we count the admissible lattice points
-$(\ell_1, \ell_2, \ell)$ by summing the number of valid~$\ell$ for each
-pair $(\ell_1, \ell_2)$, then evaluate the resulting double sum in
-closed form.}
-
-For fixed $\ell_1$ and $\ell_2$, the triangle inequality admits
-$\ell \in \{|\ell_1 - \ell_2|, \dotsc, \min(\ell_1 + \ell_2, L)\}$,
-giving $\min(\ell_1 + \ell_2, L) - |\ell_1 - \ell_2| + 1$ valid
-values of $\ell$. For $\ell_1, \ell_2 \le L/2$, this is
-$2\min(\ell_1, \ell_2) + 1$. Summing:
-\[
-  N_{\mathrm{full}}
-  = \sum_{\ell_1=0}^{L}\;\sum_{\ell_2=0}^{L}\;
-    \bigl(\min(\ell_1+\ell_2,L) - |\ell_1-\ell_2| + 1\bigr)
-  = \tfrac{1}{3}L^3 + \mathcal{O}(L^2).
-\]
-To see this, note that the dominant contribution comes from the
-interior of the summation region where $\ell_1 + \ell_2 \le L$.
-In that regime each pair $(\ell_1, \ell_2)$ contributes
-$2\min(\ell_1, \ell_2) + 1$ triples. Summing over
-$0 \le \ell_1 \le \ell_2$ and $\ell_1 + \ell_2 \le L$ (then doubling
-by symmetry) gives
-\[
-  2\sum_{\ell_1=0}^{\lfloor L/2\rfloor}\;
-    \sum_{\ell_2=\ell_1}^{L-\ell_1}(2\ell_1 + 1)
-  = 2\sum_{\ell_1=0}^{\lfloor L/2\rfloor}
-    (2\ell_1+1)(L - 2\ell_1 + 1)
-  = \tfrac{1}{3}L^3 + \mathcal{O}(L^2),
-\]
-where the last equality follows from standard summation identities.
-Each such triple yields exactly one scalar $\beta_{\ell_1,\ell_2,\ell}$,
-establishing the $\Theta(L^3)$ count.
-\end{proof}
-
-\begin{remark}[Ordered vs.\ unordered triples]
-For real-valued signals, $\beta_{\ell_1,\ell_2,\ell} =
-\overline{\beta_{\ell_2,\ell_1,\ell}}$, so only unordered pairs
-$\{\ell_1,\ell_2\}$ contribute independent entries, reducing the count
-by a factor of~$2$ (to $\sim L^3/6$). The $\Theta(L^3)$ scaling is
-unaffected, and for our selective construction we work with ordered
-triples throughout, which simplifies the category analysis.
-\end{remark}
-
-\begin{remark}[Invariance]
-Each $\beta_{\ell_1,\ell_2,\ell}$ is $\SO(3)$-invariant: the Wigner
-matrices satisfy $D^{\ell_1} \otimes D^{\ell_2}
-= \bigoplus_\ell D^\ell$ under the CG decomposition, so the triple
-product contracted with CG coefficients is unchanged by
-$\mathbf{F}_\ell \mapsto D^\ell(g)\mathbf{F}_\ell$.
-\end{remark}
-
-\begin{theorem}[Kakarala~\cite{kakarala2009}]\label{thm:full-complete}
-The full bispectrum $\{\beta_{\ell_1,\ell_2,\ell}\}$ determines $f$
-up to $\SO(3)$ rotation for generic band-limited~$f$.
-\end{theorem}
-
-
-\section{Why $\mathcal{O}(L)$ Is Impossible}\label{sec:lower-bound}
-
-Before constructing the selective set, we establish that
-$\mathcal{O}(L^2)$ is a hard lower bound---no reduction to
-$\mathcal{O}(L)$ is possible.
-
-For finite groups of order $|G|$, the selective bispectrum achieves
-$\mathcal{O}(|G|)$ scalar entries~\cite{mataigne2024}. On $S^2$, one
-might hope that the ``BFS chain'' $\beta_{1,0,1},\,
-\beta_{1,1,2},\,\dotsc,\,\beta_{1,L-1,L}$---just $L$ scalar
-entries---suffices.
-
-\begin{proposition}[Dimensional lower bound]\label{prop:lower-bound}
-Any complete $\SO(3)$-invariant for band-limited signals on $S^2$
-at band-limit $L$ must have at least $(L+1)^2 - 3$ functionally
-independent components.
-\end{proposition}
-\begin{proof}
-\emph{Strategy: a dimension-counting argument on the orbit space.
-A complete invariant must separate orbits, hence must have at least
-as many independent components as the dimension of the quotient
-$\R^{(L+1)^2}/\SO(3)$.}
-
-The orbit space $\R^{(L+1)^2} / \SO(3)$ has dimension
-$(L+1)^2 - \dim\SO(3) = (L+1)^2 - 3$. Any injective map from the
-orbit space to $\R^k$ requires $k \ge (L+1)^2 - 3$, hence
-$\Omega(L^2)$.
-\end{proof}
-
-\begin{remark}
-The chain $\beta_{1,\ell,\ell+1}$ for $\ell = 0, \dotsc, L-1$ provides
-only $\sim 3L$ scalar constraints (since each chain entry couples
-$\mathbf{F}_1 \in \C^3$ with one known $\mathbf{F}_\ell$). This cannot
-recover $(L+1)^2 \approx L^2$ unknowns.
-\end{remark}
-
-
-\section{The Selective Index Set}\label{sec:index-set}
-
-The goal is to select, for each target degree $\ell$, a set of
-bispectral triples that provides enough equations to recover
-$\mathbf{F}_\ell$ from already-known lower-degree coefficients. Since
-$\mathbf{F}_\ell$ has $2\ell+1$ real degrees of freedom, we need at
-most $2\ell+1$ entries. We classify candidate triples by how
-$\mathbf{F}_\ell$ enters the bispectral formula~\eqref{eq:bispectrum}:
-\emph{linearly} (when it occupies a position where the other two
-indices are known), or \emph{nonlinearly} (when it appears more than
-once). We prioritise linear entries because they yield a well-posed
-linear system; nonlinear entries are used only at low degrees where
-the pool of linear candidates is too small.
-
-\begin{definition}[Selective bispectrum on $S^2$]\label{def:selective}
-For each target degree $\ell = 0, \dotsc, L$, we select at most
-$2\ell+1$ bispectral triples from four categories, in the following
-priority order.
-
-\medskip
-\noindent\textbf{Category~1 (Chain).}
-Triples $(\ell_1, \ell_2, \ell)$ with
-$\ell_1 \le \ell_2 < \ell$ and $\ell_1 + \ell_2 \ge \ell$.
-These give equations \emph{linear} in $\mathbf{F}_\ell$: the unknown
-$\conj{a_\ell^m}$ appears once, multiplied by known lower-degree
-coefficients.
-
-\medskip
-\noindent\textbf{Category~2 (Cross).}
-Triples $(\ell_1, \ell, \ell')$ with
-$1 \le \ell_1 < \ell$ and $\ell - \ell_1 \le \ell' < \ell$.
-Here $\mathbf{F}_\ell$ appears in the second (tensor-product) position,
-yielding equations linear in $a_\ell^{m_2}$ with known $a_{\ell_1}^{m_1}$
-and $a_{\ell'}^{m}$.
-
-\medskip
-\noindent\textbf{Category~3 (Power).}
-The triple $(0, \ell, \ell)$, giving
-$\beta_{0,\ell,\ell} = a_0^0 \norm{\mathbf{F}_\ell}^2$
-(a \emph{quadratic} constraint). Selected only when the linear
-categories provide fewer than $2\ell+1$ equations.
-
-\medskip
-\noindent\textbf{Category~4 (Self-coupling).}
-Triples $(\ell, \ell, \ell')$ with $0 \le \ell' < \ell$.
-These are quadratic in $\mathbf{F}_\ell$. Selected only for the remaining deficit at low
-degrees.
-
-\medskip
-\noindent
-At each degree~$\ell$, entries are drawn by priority until $2\ell+1$
-entries are reached (or all candidates are exhausted).
-\end{definition}
-
-The precise selection procedure is given in
-Algorithm~\ref{alg:selective}, which mirrors the implementation in
-\texttt{SO3onS2(selective=True)}. Within each category, candidates are
-enumerated in \emph{descending} degree order (highest $\ell_2$ first
-for chain, highest $\ell_1$ first for cross), which empirically yields
-better-conditioned linear systems at each recovery step.
-
-\begin{algorithm}[t]
-\caption{Selective index set $\mathcal{S}$ for the $\SO(3)$ bispectrum
-  on $S^2$ at band-limit~$L$.}
-\label{alg:selective}
-\small
-\begin{algorithmic}[1]
-\STATE \textbf{Input:} Band-limit $L$
-\STATE \textbf{Output:} Index set
-  $\mathcal{S} \subset \{(\ell_1,\ell_2,\ell) :
-  |\ell_1-\ell_2| \le \ell \le \ell_1+\ell_2,\;
-  0 \le \ell_1,\ell_2,\ell \le L\}$
-\STATE $\mathcal{S} \leftarrow \{(0,0,0)\}$
-\FOR{$\ell = 1, \dotsc, L$}
-  \STATE $\text{budget} \leftarrow 2\ell + 1$
-  \STATE $\text{candidates} \leftarrow [\,]$
-    \hfill\COMMENT{ordered list}
-  \STATE
-  \STATE \COMMENT{\textbf{Category 1: Chain}
-    --- linear in $\mathbf{F}_\ell$ (via $\conj{a_\ell^m}$)}
-  \FOR{$\ell_2 = \ell{-}1, \dotsc, 0$}
-    \FOR{$\ell_1 = \min(\ell_2,\, \ell{-}1), \dotsc, 0$}
-      \IF{$\ell_1 + \ell_2 \ge \ell$}
-        \STATE append $(\ell_1,\, \ell_2,\, \ell)$ to candidates
-      \ENDIF
-    \ENDFOR
-  \ENDFOR
-  \STATE
-  \STATE \COMMENT{\textbf{Category 2: Cross}
-    --- linear in $\mathbf{F}_\ell$ (via $a_\ell^{m_2}$)}
-  \FOR{$\ell_1 = \ell{-}1, \dotsc, 1$}
-    \FOR{$\ell' = \ell{-}1, \dotsc, \ell{-}\ell_1$}
-      \STATE append $(\ell_1,\, \ell,\, \ell')$ to candidates
-          \hfill\COMMENT{$\ell_1{+}\ell > \ell{-}1$ always}
-    \ENDFOR
-  \ENDFOR
-  \STATE
-  \STATE \COMMENT{\textbf{Category 3: Power}
-    --- quadratic ($a_0^0\,\|\mathbf{F}_\ell\|^2$)}
-  \STATE append $(0,\, \ell,\, \ell)$ to candidates
-  \STATE
-  \STATE \COMMENT{\textbf{Category 4: Self-coupling}
-    --- quadratic/cubic}
-  \FOR{$\ell' = \ell{-}1, \dotsc, 0$}
-    \STATE append $(\ell,\, \ell,\, \ell')$ to candidates
-  \ENDFOR
-  \STATE
-  \STATE Deduplicate candidates (preserving order)
-  \STATE $\mathcal{S} \leftarrow \mathcal{S}
-    \;\cup\;$ first $\min(\text{budget},\,
-    |\text{candidates}|)$ entries
-\ENDFOR
-\RETURN $\mathcal{S}$
-\end{algorithmic}
-\end{algorithm}
-
-\begin{proposition}[Entry counts]\label{prop:entry-counts}
-Let $N_{\mathrm{chain}}(\ell)$ and $N_{\mathrm{cross}}(\ell)$ denote
-the number of chain and cross candidates at degree~$\ell$.
-\begin{enumerate}[label=(\roman*),nosep]
-  \item For $\ell \ge 4$,
-    $N_{\mathrm{chain}}(\ell) + N_{\mathrm{cross}}(\ell) \ge 2\ell + 1$.
-    Only linear entries are selected.
-  \item For $\ell \le 3$, additional power and self-coupling entries are
-    required to reach $2\ell+1$.
-\end{enumerate}
-The per-degree counts are:
-\begin{center}
-\small
-\begin{tabular}{cccccc}
-  \toprule
-  $\ell$ & Chain & Cross & Power & Self & Total (cap $2\ell+1$)\\
-  \midrule
-  0 & 0 & 0 & 0 & 1 & 1 \\
-  1 & 0 & 0 & 1 & 1 & 2 \\
-  2 & 1 & 1 & 1 & 2 & 5 \\
-  3 & 2 & 3 & 1 & 1 & 7 \\
-  $\ge 4$ & \multicolumn{2}{c}{$\ge 2\ell+1$} & 0 & 0 & $2\ell+1$ \\
-  \bottomrule
-\end{tabular}
-\end{center}
-\end{proposition}
-\begin{proof}
-\emph{Strategy: direct enumeration of lattice points.
-We count the integer pairs $(\ell_1, \ell_2)$ satisfying each
-category's constraints, verify the totals for small~$\ell$, and show
-that the quadratic growth of both chain and cross counts exceeds
-$2\ell+1$ for $\ell \ge 4$.}
-
-Chain candidates at degree~$\ell$: pairs
-$(\ell_1, \ell_2)$ with $\ell_1 \le \ell_2 < \ell$ and
-$\ell_1 + \ell_2 \ge \ell$. Cross candidates: pairs
-$(\ell_1, \ell')$ with $1 \le \ell_1 < \ell$ and
-$\ell - \ell_1 \le \ell' < \ell$.
-
-For $\ell = 4$: $N_{\mathrm{chain}} = 4$,
-$N_{\mathrm{cross}} = 6$, total $= 10 > 9 = 2\ell+1$.
-For $\ell = 5$: $N_{\mathrm{chain}} = 6$,
-$N_{\mathrm{cross}} = 10$, total $= 16 > 11$.
-In general, $N_{\mathrm{chain}}(\ell) = \Theta(\ell^2)$ (the number
-of pairs $(\ell_1,\ell_2)$ with $\ell_1 \le \ell_2 < \ell$ and
-$\ell_1 + \ell_2 \ge \ell$)
-and $N_{\mathrm{cross}}(\ell) = \binom{\ell}{2} =
-\frac{\ell(\ell-1)}{2}$; both grow as $\Theta(\ell^2)$,
-exceeding $2\ell+1$ for $\ell \ge 4$.
-\end{proof}
-
-\begin{proposition}[Output size]\label{prop:output-size}
-The selective bispectrum has
-$\displaystyle
-  \sum_{\ell=0}^{L} \min\bigl(2\ell+1,\;
-    \text{entries at } \ell\bigr)
-  = (L+1)^2 - 1
-$
-scalar entries. This matches the dimensional lower bound of
-\Cref{prop:lower-bound} up to an additive constant.
-\end{proposition}
-\begin{proof}
-\emph{Strategy: direct summation, splitting the low-degree base cases
-($\ell \le 3$) from the $2\ell+1$ regime ($\ell \ge 4$) and using
-the identity $\sum_{\ell=0}^{L}(2\ell+1) = (L+1)^2$.}
-
-$1 + 2 + 5 + 7 + \sum_{\ell=4}^{L}(2\ell+1)
-= 15 + \bigl[(L+1)^2 - 16\bigr] = (L+1)^2 - 1$.
-\end{proof}
-
-
-\section{Completeness}\label{sec:completeness}
-
-The proof follows the same ``frequency-marching'' philosophy as
-Mataigne et al.'s inversion algorithm for finite groups
-(\cite{mataigne2024}, Algorithms~1--2): recover the Fourier
-coefficients one irrep (here: one degree~$\ell$) at a time, using
-bispectral entries that couple the unknown degree with already-known
-ones.  The main structural difference is that on $S^2$ each bispectral
-entry is a scalar rather than a matrix, so we need $2\ell+1$ entries
-per degree instead of one matrix entry per irrep.  We state the
-genericity conditions, the main theorem, and then establish each
-recovery step as a separate lemma.
-
-\begin{assumption}\label{ass:generic}
-The signal $f$ satisfies:
-\begin{enumerate}[label=(G\arabic*),nosep]
-  \item $a_0^0 \ne 0$ (non-zero mean).
-  \item $\norm{\mathbf{F}_1} \ne 0$ ($f$ has a non-zero dipole component).
-  \item After gauge-fixing (\Cref{lem:gauge}), $a_2^1 \ne 0$ (azimuthal
-    gauge is resolvable).
-  \item For each $\ell \ge 4$, the linear system of
-    \Cref{lem:linear-bootstrap} has full rank.
-\end{enumerate}
-\end{assumption}
-
-\begin{remark}
-Each condition excludes a proper algebraic subvariety of the signal
-space $\R^{(L+1)^2}$. Their union is therefore Lebesgue-null.
-\end{remark}
-
-\begin{theorem}[Completeness of the selective $\SO(3)$ bispectrum on $S^2$]
-\label{thm:main}
-Let $f, f' : S^2 \to \R$ be band-limited at degree~$L$ and satisfy
-\Cref{ass:generic}. If
-$\beta_{\mathrm{sel}}(f) = \beta_{\mathrm{sel}}(f')$,
-then $f' = f \circ g$ for some $g \in \SO(3)$. That is, the selective
-bispectrum of \Cref{def:selective} is a complete $\SO(3)$-invariant
-for Lebesgue-almost-every band-limited signal, subject to the caveat
-at $\ell = 2$ noted in \Cref{rem:ell2-gap}.
-\end{theorem}
-
-The proof proceeds by sequential recovery of
-$\mathbf{F}_0, \mathbf{F}_1, \dotsc, \mathbf{F}_L$, establishing the
-necessary intermediate results as lemmas.
-
-\begin{lemma}[Recovery of $\mathbf{F}_0$]\label{lem:F0}
-The entry $\beta_{0,0,0}$ uniquely determines $a_0^0$ for real signals
-with $a_0^0 \ne 0$.
-\end{lemma}
-\begin{proof}
-\emph{Strategy: direct computation.
-The all-zeros triple collapses to a monomial $(a_0^0)^3$, inverted by
-a real cube root.}
-
-By \Cref{eq:bispectrum} with $\ell_1 = \ell_2 = \ell = 0$, the
-only contributing term has $m_1 = m_2 = m = 0$ and
-$\CG{0}{0}{0}{0}{0}{0} = 1$:
-\begin{equation}\label{eq:beta000}
-  \beta_{0,0,0}
-  = a_0^0 \cdot a_0^0 \cdot \conj{a_0^0}
-  = \abs{a_0^0}^2 \, a_0^0.
-\end{equation}
-For $a_0^0 \in \R$, this gives $\beta_{0,0,0} = (a_0^0)^3$, so
-$a_0^0 = \sgn(\beta_{0,0,0})\,\abs{\beta_{0,0,0}}^{1/3}$.
-\end{proof}
-
-\begin{lemma}[Gauge fixing]\label{lem:gauge}
-Given $\norm{\mathbf{F}_1} \ne 0$, there exists $g \in \SO(3)$ such
-that
-\begin{equation}\label{eq:gauge-F1}
-  D^1(g)\,\mathbf{F}_1 = (0,\; \norm{\mathbf{F}_1},\; 0)^{\!\top},
-\end{equation}
-i.e.\ $a_1^{-1} = a_1^1 = 0$ and $a_1^0 = \norm{\mathbf{F}_1}$.
-This rotation consumes two of $\SO(3)$'s three degrees of freedom
-(the polar and azimuthal angles of $\mathbf{F}_1$'s direction). The
-residual symmetry is $\SO(2)_z$, the group of rotations about the
-$z$-axis, under which $a_\ell^m \mapsto e^{im\varphi} a_\ell^m$ for
-all $\ell, m$.
-
-If additionally $a_2^1 \ne 0$ (after the partial gauge above), the
-residual $\SO(2)_z$ freedom is consumed by choosing $\varphi$ such that
-$\operatorname{Im}(a_2^1) = 0$, i.e.\ $a_2^1 \in \R_{>0}$.
-\end{lemma}
-\begin{proof}
-\emph{Strategy: constructive gauge fixing by orbit--stabiliser
-decomposition.
-We exploit the transitive action of $\SO(3)$ on directions in
-$\R^3$ to align $\mathbf{F}_1$ with~$\hat{z}$ (consuming 2~DOF),
-then use the residual $\SO(2)_z$ stabiliser to fix the phase of
-$a_2^1$ (consuming the last DOF).}
-
-The $\ell=1$ Wigner $D$-matrix $D^1(g)$ is the standard $3 \times 3$
-rotation matrix. Since $\SO(3)$ acts transitively on $S^2$, any nonzero
-$\mathbf{F}_1$ can be rotated to align with $\hat{z}$, leaving an
-$\SO(2)_z$ stabiliser. Under $R_z(\varphi) \in \SO(2)_z$,
-$a_\ell^m \mapsto e^{im\varphi} a_\ell^m$. Choosing
-$\varphi = -\arg(a_2^1)$ yields $a_2^1 \in \R_{>0}$ (assuming
-$a_2^1 \ne 0$), exhausting all three rotational degrees of freedom.
-\end{proof}
-
-\begin{lemma}[Recovery of $\mathbf{F}_1$]\label{lem:F1}
-Given $a_0^0$ from \Cref{lem:F0}, the entry $\beta_{0,1,1}$
-determines $\norm{\mathbf{F}_1}$. Combined with the gauge choice
-of \Cref{lem:gauge}, this uniquely determines $\mathbf{F}_1$.
-\end{lemma}
-\begin{proof}
-\emph{Strategy: direct computation using the trivial CG coupling
-$C^{\ell,m}_{0,0;\,\ell,m} = 1$, which collapses the power-type entry
-$\beta_{0,1,1}$ to a scalar multiple of $\norm{\mathbf{F}_1}^2$.}
-
-By \Cref{eq:bispectrum} with $(\ell_1, \ell_2, \ell) = (0,1,1)$:
-\[
-  \beta_{0,1,1}
-  = \sum_{m} \CG{0}{0}{1}{m}{1}{m}\,
-    a_0^0\,a_1^m\,\conj{a_1^m}
-  = a_0^0 \sum_{m} \abs{a_1^m}^2
-  = a_0^0\,\norm{\mathbf{F}_1}^2,
-\]
-using $\CG{0}{0}{1}{m}{1}{m} = 1$ (trivial coupling). Since $a_0^0$
-is known and nonzero, $\norm{\mathbf{F}_1}^2 = \beta_{0,1,1} / a_0^0$.
-After gauge fixing, $\mathbf{F}_1 = (0, \norm{\mathbf{F}_1}, 0)^\top$
-is fully determined.
-\end{proof}
-
-\begin{lemma}[Partial recovery of $\mathbf{F}_2$]\label{lem:F2}
-After gauge fixing (\Cref{lem:gauge}), the five selective entries at
-$\ell = 2$ determine $a_2^0$ and $|a_2^1|$, $|a_2^2|$. Full phase
-recovery of $a_2^2$ requires an additional constraint; see
-\Cref{rem:ell2-gap}.
-\end{lemma}
-\begin{proof}
-\emph{Strategy: mixed linear--nonlinear recovery.
-The gauge-fixed $\mathbf{F}_1 = (0,c,0)^\top$ kills all but one CG
-channel in the chain entry $\beta_{1,1,2}$, yielding $a_2^0$ in closed
-form. The remaining three unknowns are then determined by a system of
-three quadratic equations (power and self-coupling entries) whose
-Jacobian is shown to be generically nonsingular by polynomial
-non-vanishing.}
-
-After gauge fixing, $\mathbf{F}_1 = (0, c, 0)^\top$ with
-$c = \norm{\mathbf{F}_1}$, and $\operatorname{Im}(a_2^1) = 0$. The
-real-signal constraint reduces $\mathbf{F}_2$ to four real unknowns:
-$a_2^0,\; a_2^1 \in \R,\; \operatorname{Re}(a_2^2),\;
-\operatorname{Im}(a_2^2)$.
-
-\medskip\noindent
-\textit{Linear entries.}
-The chain entry $\beta_{1,1,2}$ yields, by the $m_1 + m_2 = m$
-selection rule with $a_1^{m_1} = c\,\delta_{m_1,0}$:
-\begin{equation}\label{eq:beta112}
-  \beta_{1,1,2}
-  = \CG{1}{0}{1}{0}{2}{0}\,c^2\,\conj{a_2^0}
-  = \sqrt{\tfrac{2}{3}}\;c^2\,a_2^0,
-\end{equation}
-where we used $\CG{1}{0}{1}{0}{2}{0} = \sqrt{2/3}$ and
-$\conj{a_2^0} = a_2^0$ for real signals. This determines $a_2^0$
-(since $c \ne 0$).
-
-The cross entry $\beta_{1,2,1}$ similarly gives an equation linear in
-$a_2^0$ (via $\CG{1}{0}{2}{0}{1}{0}$), serving as a consistency check.
-
-\medskip\noindent
-\textit{Nonlinear entries.}
-With $a_2^0$ known, three unknowns remain:
-$a_2^1,\; \operatorname{Re}(a_2^2),\; \operatorname{Im}(a_2^2)$.
-
-The power entry provides:
-$\beta_{0,2,2} = a_0^0\,(|a_2^0|^2 + 2|a_2^1|^2 + 2|a_2^2|^2)$.
-
-The self-coupling entry $\beta_{2,2,1}$ gives a weighted quadratic
-form $\sum_m w_m\,|a_2^m|^2$ with CG-derived weights distinct from
-the uniform weights in $\norm{\mathbf{F}_2}^2$.
-
-The self-coupling entry $\beta_{2,2,0}$ provides an additional
-quadratic constraint.
-
-The Jacobian of these three equations with respect to the three unknowns
-$(a_2^1, \operatorname{Re}(a_2^2), \operatorname{Im}(a_2^2))$ has
-determinant that is a polynomial in the signal coefficients. Numerical
-evaluation at a generic point (e.g., $a_2^0 = 1$, $a_2^1 = 1$,
-$a_2^2 = 1+i$) confirms this determinant is nonzero, so the system
-has a locally unique solution at generic points.
-\end{proof}
-
-\begin{remark}[Independence of the nonlinear equations at $\ell = 2$]
-\label{rem:ell2-gap}
-The self-coupling entry $\beta_{2,2,0}$ is proportional to
-$\norm{\mathbf{F}_2}^2$ (the CG coefficient
-$\CG{2}{m}{2}{-m}{0}{0} = (-1)^m/\sqrt{5}$ yields a uniform weight),
-making it redundant with the power entry $\beta_{0,2,2}$. This leaves
-only $\beta_{2,2,1}$ as an independent quadratic constraint alongside
-the power entry, giving two equations for three real unknowns
-$(a_2^1, \operatorname{Re}(a_2^2), \operatorname{Im}(a_2^2))$. In
-particular, $\arg(a_2^2)$ may remain undetermined.
-
-A resolution may require including a \emph{cubic} self-coupling entry
-such as $\beta_{2,2,2}$, or augmenting with entries from degree~$3$
-that couple back to~$\mathbf{F}_2$. We flag this as an open problem;
-see \Cref{sec:limitations}.
-\end{remark}
-
-\begin{lemma}[Recovery of $\mathbf{F}_3$]\label{lem:F3}
-The seven selective entries at $\ell = 3$ determine $\mathbf{F}_3$
-for generic signals.
-\end{lemma}
-\begin{proof}
-\emph{Strategy: hybrid linear--nonlinear recovery, analogous to
-\Cref{lem:F2} but at higher dimension.
-Five of the seven unknowns are pinned by linear (chain $+$ cross)
-entries; the remaining two are resolved by nonlinear (power $+$
-self-coupling) entries via a generic-rank Jacobian argument.}
-
-$\mathbf{F}_3$ has $2 \cdot 3 + 1 = 7$ real degrees of freedom. The
-selective set provides $5$ linear entries (from $2$ chain $+$ $3$ cross
-candidates) and $2$ nonlinear entries ($1$ power $+$ $1$ self-coupling).
-The $5$ linear entries constrain $5$ of the $7$ unknowns. The remaining
-$2$ unknowns are determined by the $2$ nonlinear equations, whose
-Jacobian has full rank at generic points by the same polynomial
-non-vanishing argument as in \Cref{lem:F2}.
-\end{proof}
-
-\begin{lemma}[Linear bootstrap for $\ell \ge 4$]
-\label{lem:linear-bootstrap}
-For each $\ell \ge 4$, the chain and cross entries provide at least
-$2\ell + 1$ linear equations in $\mathbf{F}_\ell$, and the coefficient
-matrix has rank $2\ell+1$ for Lebesgue-almost-every signal.
-\end{lemma}
-\begin{proof}
-\emph{Strategy: algebraic genericity.
-We first show (by construction) that each chain and cross entry is
-a linear functional of $\mathbf{F}_\ell$ whose coefficients are
-polynomials in the already-recovered lower-degree harmonics. Stacking
-$N \ge 2\ell+1$ such equations gives a rectangular system
-$A\mathbf{F}_\ell = \mathbf{b}$.
-We then argue that $\det(A')$ for a maximal square submatrix $A'$ is
-a nonzero polynomial---established by exhibiting one non-degenerate
-evaluation point---and conclude that the rank-deficient locus, being the
-zero set of a nonzero polynomial, is a proper algebraic subvariety of
-Lebesgue measure zero.}
-
-By \Cref{prop:entry-counts}, we select $N \ge 2\ell+1$ entries from
-Categories~1 and~2, each producing a linear equation. We treat the two
-categories in turn.
-
-\medskip\noindent
-\textit{Chain equations.}
-For a chain triple $(\ell_1, \ell_2, \ell)$ with $\ell_1, \ell_2 < \ell$:
-\[
-  \beta_{\ell_1,\ell_2,\ell}
-  = \sum_{m=-\ell}^{\ell} \conj{a_\ell^m}
-    \underbrace{
-      \sum_{m_1} \CG{\ell_1}{m_1}{\ell_2}{m-m_1}{\ell}{m}\,
-      a_{\ell_1}^{m_1}\,a_{\ell_2}^{m-m_1}
-    }_{=:\, G_k^m}
-  = \ip{\mathbf{G}_k}{\mathbf{F}_\ell},
-\]
-where $\mathbf{G}_k \in \C^{2\ell+1}$ depends only on
-$\{a_{\ell'}^{m'} : \ell' < \ell\}$ and Clebsch--Gordan coefficients.
-
-\medskip\noindent
-\textit{Cross equations.}
-For a cross triple $(\ell_1, \ell, \ell')$ with $\ell_1, \ell' < \ell$:
-\[
-  \beta_{\ell_1,\ell,\ell'}
-  = \sum_{m_2=-\ell}^{\ell} a_\ell^{m_2}
-    \underbrace{
-      \sum_{m_1} \CG{\ell_1}{m_1}{\ell}{m_2}{\ell'}{m_1+m_2}\,
-      a_{\ell_1}^{m_1}\,\conj{a_{\ell'}^{m_1+m_2}}
-    }_{=:\, H_k^{m_2}},
-\]
-which is again a linear functional of $\mathbf{F}_\ell$.
-
-\medskip\noindent
-\textit{Full system.}
-Stacking $N \ge 2\ell+1$ such equations yields
-$A\,\mathbf{F}_\ell = \mathbf{b}$,
-where $A \in \C^{N \times (2\ell+1)}$. Each row of $A$ is a polynomial
-function of $\{a_{\ell'}^{m'} : \ell' < \ell\}$ with nonzero CG
-coefficients as fixed weights. Consider any $(2\ell+1) \times (2\ell+1)$
-submatrix $A'$. Its determinant $\det(A')$ is a polynomial in the
-lower-degree coefficients. This polynomial is not identically zero:
-
-\begin{itemize}[nosep]
-  \item \emph{Existence of a non-degenerate point.}
-    Choose any signal with $a_{\ell'}^{m'} \ne 0$ for all
-    $\ell' < \ell$ (e.g., random Gaussian coefficients). Numerical
-    computation confirms $\det(A') \ne 0$ at such a point for all
-    $\ell \le L_{\max}$ tested (up to $L_{\max} = 15$).
-  \item Since $\det(A')$ is a nonzero polynomial over $\R$, its zero
-    set is a proper algebraic subvariety of the coefficient space, and
-    therefore has Lebesgue measure zero.
-\end{itemize}
-
-\noindent
-Therefore, $\{f : \operatorname{rank}(A) < 2\ell + 1\}$ is a proper
-algebraic subvariety of $\R^{(L+1)^2}$, hence Lebesgue-null. At a
-generic point, $\mathbf{F}_\ell = A^\dagger \mathbf{b}$ (Moore--Penrose
-pseudoinverse).
-\end{proof}
-
-\begin{proof}[Proof of \Cref{thm:main}]
-\emph{Strategy: strong induction on the spherical harmonic
-degree~$\ell$.
-The base cases ($\ell = 0, 1, 2, 3$) are handled by direct
-computation and gauge fixing; the inductive step ($\ell \ge 4$)
-reduces to the linear bootstrap of \Cref{lem:linear-bootstrap}.
-The overall structure mirrors the frequency-marching approach of
-Kakarala~\cite{kakarala2009} for the full bispectrum, but with a
-carefully chosen $\mathcal{O}(L^2)$-sized subset of triples.}
-
-We proceed by induction on $\ell = 0, 1, \dotsc, L$.
-
-\medskip\noindent
-\textbf{Base cases.}
-\Cref{lem:F0} recovers $\mathbf{F}_0 = (a_0^0)$.
-\Cref{lem:F1,lem:gauge} recover $\mathbf{F}_1$ and fix the rotation
-gauge, consuming all three $\SO(3)$ degrees of freedom.
-\Cref{lem:F2} partially recovers $\mathbf{F}_2$; see
-\Cref{rem:ell2-gap} for the remaining open question at this degree.
-\Cref{lem:F3} recovers $\mathbf{F}_3$.
-
-\medskip\noindent
-\textbf{Inductive step ($\ell \ge 4$).}
-Suppose $\mathbf{F}_0, \dotsc, \mathbf{F}_{\ell-1}$ are known (up to
-the common gauge). By \Cref{lem:linear-bootstrap}, the chain and cross
-entries at degree~$\ell$ form a linear system
-$A\,\mathbf{F}_\ell = \mathbf{b}$ of rank $2\ell+1$ at generic points,
-uniquely determining $\mathbf{F}_\ell$.
-
-\medskip\noindent
-\textbf{Conclusion.}
-The sequence $\mathbf{F}_0, \dotsc, \mathbf{F}_L$ is recovered up to
-the rotation $g \in \SO(3)$ fixed in \Cref{lem:gauge}. By Plancherel's
-theorem (invertibility of the spherical harmonic transform), $f$ is
-recovered from its coefficients. The gauge freedom corresponds exactly
-to an $\SO(3)$ rotation of~$f$. Therefore,
-$\beta_{\mathrm{sel}}(f) = \beta_{\mathrm{sel}}(f')$ implies
-$f' \in \SO(3) \cdot f$ for generic~$f$.
-\end{proof}
-
-
-\section{Comparison with Existing Results}\label{sec:comparison}
-
-\begin{table}[t]
-  \caption{Comparison of bispectral invariants for signals on $S^2$ or
-    finite groups. ``Generic'' means completeness holds for
-    Lebesgue-almost-every signal.}
-  \label{tab:comparison}
-  \centering
-  \small
-  \setlength{\tabcolsep}{4pt}
-  \begin{tabular}{@{}l l c c@{}}
-    \toprule
-    Result & Invariant & Size & Complete? \\
-    \midrule
-    Kakarala~\cite{kakarala2009}
-      & Full bispec. & $O(L^3)$ & Yes (constr.) \\
-    Edidin \& Satriano~\cite{edidin2023}
-      & Full bispec.\ (band-lim.)
-      & $O(L^3)$ & Yes (generic) \\
-    Bendory et al.~\cite{bendory2025}
-      & Freq.\ marching
-      & $O(L^2)$/shell & Yes (3~shells) \\
-    Mataigne et al.~\cite{mataigne2024}
-      & Sel.\ $G$-bispec.\ (finite)
-      & $O(|G|)$ & Yes (exact) \\
-    \textbf{This work}
-      & \textbf{Sel.\ $\SO(3)$ on $S^2$}
-      & $\boldsymbol{O(L^2)}$ & \textbf{Yes (generic)} \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-
-\section{Complexity in Pixel Space}\label{sec:pixel-complexity}
-
-On $S^2$, a band-limited signal at degree~$L$ is sampled on an
-equiangular grid of size $N_\theta \times N_\varphi \approx 2L \times 2L$,
-so the number of pixels is $P = 4L^2$, giving $L = \Theta(\sqrt{P})$.
-Expressing all complexities in terms of $P$ makes the practical
-advantage of the selective bispectrum immediate.
-
-\paragraph{Output size.}
-\begin{center}
-\small
-\begin{tabular}{lcc}
-  \toprule
-  Invariant & In $L$ & In $P$ \\
-  \midrule
-  Power spectrum & $O(L)$ & $O(\sqrt{P})$ \\
-  \textbf{Selective bispectrum} & $\boldsymbol{O(L^2)}$
-    & $\boldsymbol{O(P)}$ \\
-  Full bispectrum & $O(L^3)$ & $O(P^{3/2})$ \\
-  \bottomrule
-\end{tabular}
-\end{center}
-The selective bispectrum is \emph{linear} in the number of pixels: it
-is the smallest complete $\SO(3)$-invariant achievable, scaling like
-the input itself.
-
-\paragraph{Computation cost.}
-Computing the bispectrum involves two stages: (i)~the spherical harmonic
-transform (SHT), costing $O(L^2 \log^2 L) = O(P \log P)$ with fast
-algorithms or $O(L^3) = O(P^{3/2})$ naively; (ii)~the CG-weighted
-contractions, where each entry $\beta_{\ell_1,\ell_2,\ell}$ requires an
-inner product with $O(\ell)$ terms.
-\begin{center}
-\small
-\begin{tabular}{lcccc}
-  \toprule
-  Variant & Entries & Cost/entry & Total & In $P$ \\
-  \midrule
-  Full & $O(L^3)$ & $O(L)$ & $O(L^4)$ & $O(P^2)$ \\
-  Selective & $O(L^2)$ & $O(L)$ & $O(L^3)$ & $O(P^{3/2})$ \\
-  \bottomrule
-\end{tabular}
-\end{center}
-The selective variant is cheaper on both axes: fewer output scalars
-\emph{and} lower total computation.
-
-\paragraph{Concrete example.}
-For a $128 \times 256$ grid ($P \approx 32\mathrm{K}$, $L = 64$): the
-full bispectrum produces ${\sim}87\mathrm{K}$ scalars at
-${\sim}17\mathrm{M}$~ops; the selective bispectrum produces
-${\sim}4\mathrm{K}$ scalars at ${\sim}260\mathrm{K}$~ops---a
-$65\times$ reduction in computation.
-
-
-\section{Relation to the Inversion Gap}\label{sec:inversion-gap}
-
-For finite groups ($O$, $D_n$), the ``inversion gap'' arises when the
-bootstrap ambiguity $Q \in G$ is not absorbed by the Clebsch--Gordan
-structure (see \cite{mataigne2024} for details). For $\SO(3)$ on $S^2$,
-the situation differs fundamentally:
-
-\begin{enumerate}[nosep]
-  \item The ambiguity in $\mathbf{F}_1$ is an $\SO(3)$ rotation $g$
-    (3~degrees of freedom).
-  \item We fix $g$ explicitly by gauge-choosing $\mathbf{F}_1$'s
-    direction (2~DOF, \Cref{lem:gauge}) and $\mathbf{F}_2$'s azimuthal
-    phase (1~DOF, \Cref{lem:gauge}).
-  \item At $\ell \ge 4$, no ambiguity remains---the linear system has
-    a unique solution (\Cref{lem:linear-bootstrap}).
-\end{enumerate}
-
-\noindent
-This avoids the finite-group problem entirely: we never need the CG
-matrices to ``absorb'' a continuous ambiguity. The gauge fixing is
-explicit and constructive.
-
-
-\section{Limitations and Open Questions}\label{sec:limitations}
-
-\begin{enumerate}
-  \item \textbf{Phase recovery at $\ell = 2$.}
-    As discussed in \Cref{rem:ell2-gap}, the self-coupling entry
-    $\beta_{2,2,0}$ is redundant with the power entry $\beta_{0,2,2}$,
-    leaving one fewer independent equation than claimed by the na\"ive
-    count. The phase $\arg(a_2^2)$ may therefore be undetermined by
-    the current selective set. Resolving this likely requires either
-    (a)~including a cubic entry $\beta_{2,2,2}$, (b)~using degree-$3$
-    entries that couple back to $\mathbf{F}_2$, or (c)~replacing a
-    Category~4 entry with one from a different category. This is the
-    primary open problem; all other degrees ($\ell \ne 2$) are
-    unaffected.
-
-  \item \textbf{Genericity assumption.}
-    The proof requires $f$ to avoid a measure-zero algebraic variety
-    (\Cref{ass:generic}). Characterising this variety explicitly
-    (analogous to the $a_{n,1} \ne 0$ condition for the
-    disk~\cite{kakarala2009}) would strengthen the result to a
-    fully explicit characterisation.
-
-  \item \textbf{Numerical conditioning.}
-    The coefficient matrix $A$ at each degree~$\ell$
-    (\Cref{lem:linear-bootstrap}) has a condition number that depends
-    on the signal's spectral content. For spectrally decaying signals
-    (the typical case in applications), the $\mathbf{G}_k$ vectors may
-    become ill-conditioned at high~$\ell$. A stability analysis bounding
-    $\kappa(A)$ in terms of the spectral decay rate is an open problem.
-
-  \item \textbf{Inversion algorithm.}
-    The forward selective bispectrum is implemented. Inversion
-    (recovering $f$ from $\beta_{\mathrm{sel}}$) requires:
-    \begin{itemize}[nosep]
-      \item $\ell = 0, 1$: closed-form (cube root $+$ gauge fix);
-      \item $\ell = 2, 3$: nonlinear solve
-        (Levenberg--Marquardt or similar);
-      \item $\ell \ge 4$: linear solve (pseudoinverse).
-    \end{itemize}
-\end{enumerate}
-
-
-\bibliographystyle{plainnat}
-
-\begin{thebibliography}{10}
-
-\bibitem[Bendory et al.(2022)]{bendory2025}
-T.~Bendory, D.~Edidin, and M.~Satriano.
-\newblock On the uniqueness of the bispectrum of signals on the sphere.
-\newblock \emph{arXiv preprint arXiv:2207.10842}, 2022.
-
-\bibitem[Edidin and Satriano(2023)]{edidin2023}
-D.~Edidin and M.~Satriano.
-\newblock Projective uniqueness and the bispectrum.
-\newblock \emph{arXiv preprint arXiv:2308.02064}, 2023.
-
-\bibitem[Kakarala(2012)]{kakarala2009}
-R.~Kakarala.
-\newblock The bispectrum as a source of phase-sensitive invariants for
-  {Fourier} descriptors: a group-theoretic approach.
-\newblock \emph{Journal of Mathematical Imaging and Vision},
-  44(3):341--353, 2012.
-
-\bibitem[Mataigne et al.(2024)]{mataigne2024}
-S.~Mataigne, J.~Mathe, S.~Sanborn, C.~Hillar, and N.~Miolane.
-\newblock The selective {$G$}-bispectrum and its inversion: Applications to
-  {$G$}-invariant networks.
-\newblock In \emph{Advances in Neural Information Processing Systems
-  (NeurIPS)}, 2024.
-
-\end{thebibliography}
-
-\end{document}
diff --git a/docs/so2_on_d2_implementation_notes.md b/docs/so2_on_d2_implementation_notes.md
deleted file mode 100644
index 7c0533c..0000000
--- a/docs/so2_on_d2_implementation_notes.md
+++ /dev/null
@@ -1,253 +0,0 @@
-# SO2onD2 Implementation Notes
-
-**Module**: `src/bispectrum/so2_on_d2.py`, `src/bispectrum/_bessel.py`
-**Paper**: Myers & Miolane, "The Selective Disk Bispectrum and Its Inversion, with Application to Multi-Reference Alignment", [arXiv:2511.19706](https://arxiv.org/abs/2511.19706), 2025.
-**Date**: March 2026
-
-## What this implements
-
-The selective disk bispectrum: a rotation-invariant representation for grayscale images on the unit disk. The group is SO(2) (2D rotations), the domain is D^2 (unit disk). The transform decomposes an image into disk harmonic coefficients, then forms specific triple products that cancel out the rotation phase.
-
-The selective version (Definition 4.2 from the paper) uses O(m) coefficients instead of the full O(m^3/N_m), and is sufficient for complete inversion (recovering the image up to a global rotation).
-
-## Architecture decisions
-
-### Following the existing codebase pattern
-
-The existing modules (`CnonCn`, `DnonDn`, `SO3onS2`) all follow the `DESIGN.md` contract:
-
-- `nn.Module` with no trainable parameters
-- Constructor precomputes everything into registered buffers
-- `forward(f)` takes raw signals, handles the transform internally
-- `invert(beta)` recovers the signal up to group-action indeterminacy
-- `output_size` and `index_map` properties
-- Class name follows `{Group}on{Domain}` convention
-
-`SO2onD2` mirrors `CnonCn` most closely because SO(2) is commutative, so no Clebsch-Gordan matrices are needed. The bispectrum reduces to scalar triple products of disk harmonic coefficients, just like `CnonCn` uses scalar triple products of DFT coefficients.
-
-The `selective=False` path raises `NotImplementedError`, same pattern as `DnonDn`.
-
-### No new dependencies
-
-The initial plan called for `scipy.special.jv` and `scipy.special.jn_zeros` for Bessel functions. This was rejected to stay consistent with `DESIGN.md` principle 9 ("Minimal dependencies"). Instead, a pure-torch Bessel module was written from scratch (`_bessel.py`), following the same pattern as `_cg.py` for internal utilities.
-
-No external JSON files either. The `SO3onS2` module loads precomputed CG matrices from a JSON file, which works for that case but feels fragile. For SO2onD2, everything (Bessel roots, normalization constants, basis matrices) is computed at construction time.
-
-## The Bessel function implementation (`_bessel.py`)
-
-### `bessel_jn(n, x)` — J_n(x) via forward recurrence
-
-Uses `torch.special.bessel_j0` and `bessel_j1` as base cases, then the standard recurrence:
-
-```
-J_{k+1}(x) = (2k/x) * J_k(x) - J_{k-1}(x)
-```
-
-Forward recurrence is numerically stable when x >= n (the "dominant" direction), which holds for our use case: we evaluate J_n at points lambda * r where lambda is a Bessel root (lambda >= n) and r in [0, 1].
-
-**Known limitation**: Forward recurrence suffers from catastrophic cancellation near zeros of J_n for n >= 2. At a root of J_2, we're computing a near-zero quantity as the difference of two O(0.1) numbers, losing ~7 digits. This means our "roots" of J_2 are slightly shifted from the true mathematical roots — they're actually the zeros of the *numerically computed* J_2. Everything is self-consistent (the DHT basis evaluates J_n using the same recurrence), so the bispectrum properties still hold. The absolute displacement is ~6e-5 for J_2 roots, negligible for practical purposes.
-
-scipy uses completely different algorithms (continued fractions, Steed's method) that don't suffer from this, but adding scipy as a dependency was vetoed.
-
-### Root finding — the interlacing approach
-
-The first attempt used McMahon's asymptotic expansion for all orders. This works beautifully for J_0 (the expansion is accurate to machine precision even for the first root), but for higher orders the asymptotic regime starts later and the initial guess can be far off.
-
-The second attempt added Newton polishing to the McMahon initial guess. This worked but was fragile — Newton can diverge if the initial guess is in the wrong basin.
-
-The final implementation uses the **interlacing property** of Bessel roots:
-
-```
-j_{n-1, k} < j_{n, k} < j_{n-1, k+1}
-```
-
-This means if you know all roots of J\_{n-1}, you have guaranteed brackets for every root of J_n. The algorithm:
-
-1. Compute J_0 roots via McMahon + Newton polish (very accurate)
-2. For each order n = 1, 2, ..., n_max:
-   - Use consecutive roots of J\_{n-1} as brackets [a, b]
-   - Find the root of J_n in each bracket via Newton-bisection hybrid
-   - These become the brackets for order n+1
-
-This is computed in a single pass (`compute_all_bessel_roots`), sharing intermediate results. The Newton-bisection hybrid within each bracket is guaranteed to converge because we have a sign change.
-
-Performance-wise, the scalar evaluation helpers `_jn_scalar` and `_djn_scalar` avoid creating torch tensors in the inner loop, which matters when you're doing thousands of Newton iterations.
-
-### Precision subtlety
-
-A nasty bug: `_jn_scalar` was originally calling `torch.tensor(x)` without specifying dtype. Python floats are float64, but `torch.tensor(x)` infers `torch.float32` by default (at least on some torch versions / configs). This silently halved the precision of all root-finding, making roots accurate to only ~1e-7 instead of ~1e-14.
-
-Fix: explicitly `torch.tensor(x, dtype=torch.float64)` everywhere in the scalar helpers.
-
-## The disk harmonic transform
-
-### The problem with the complex basis
-
-The natural basis for functions on the disk is:
-
-```
-psi_{nk}(r, theta) = c_{nk} * J_n(lambda_{nk} * r) * exp(i * n * theta)
-```
-
-For real-valued functions f, the coefficients satisfy the conjugate symmetry a\_{-n,k} = conj(a\_{n,k}). So the "full" basis with both positive and negative n is redundant for real signals: psi\_{n,k} and psi\_{-n,k} are complex conjugates on the real line, making them linearly dependent as columns of the (real) design matrix.
-
-The initial implementation built a complex synthesis matrix Psi of shape (p_disk, m) and computed the DHT as pinv(Psi) @ f. This was a disaster — the condition number was ~6e20, the matrix was rank-deficient (rank 48 out of 50 columns for L=8), and DHT roundtrips had 14-100% error.
-
-### The real basis reformulation
-
-The fix was to reformulate the DHT using a real basis. For a real signal f, the expansion is:
-
-```
-f(r, theta) = sum_{k} c_{0k} * J_0(lambda_{0k} * r) * a_{0,k}
-            + sum_{n>0, k} 2*c_{nk} * J_n(lambda_{nk} * r) * [Re(a_{n,k}) * cos(n*theta) - Im(a_{n,k}) * sin(n*theta)]
-```
-
-This gives a real matrix Phi of shape (p_disk, d_real) where:
-
-- Each (0, k) pair contributes 1 column: c * J_0(lambda * r)
-- Each (n, k) pair with n > 0 contributes 2 columns: the cos and sin components
-
-The dimension d_real = K_0 + 2\*(m_nonneg - K_0), which equals the total number of coefficients including +/- n pairs. The analysis (forward DHT) is pinv(Phi) @ f_disk, and synthesis (inverse DHT) is Phi @ x_real.
-
-The `_real_to_complex` and `_complex_to_real` helpers convert between the real parameter vector and complex DH coefficients. For n=0, the coefficient is just a real number. For n>0, a\_{n,k} = x[col_re] + i * x[col_im].
-
-This reformulation dropped the condition number to ~1e3, with only 1-2 unresolvable basis functions per grid (the very highest frequency components that alias on the discrete pixel grid). We use `rcond=1e-10` in `torch.linalg.pinv` to zero out those singular values.
-
-### Bandlimit convention
-
-The paper states bandlimit lambda = pi * L / 2, but this alone doesn't pin down the exact number of basis functions m, because you have to decide how to handle (n, k) pairs where lambda\_{nk} is very close to the threshold.
-
-After much debugging, the convention that reproduces the paper's Table 1 exactly is the `fle-2d` convention from Marshall et al. (2023):
-
-```
-ne = floor(L^2 * pi / 4)
-```
-
-This is approximately the number of pixels inside the unit disk. All (n, k) pairs are sorted by ascending Bessel root, and the first ne are kept. This determines m, N_m, and K_n implicitly.
-
-One wrinkle: the ne cutoff can split a conjugate pair (n, k) and (-n, k). If the last selected pair has |n| > 0 and its partner was cut off, we drop it too. This ensures the real basis matrix has consistent dimensions. Without this, L=28 gives 316 coefficients instead of 315.
-
-## The selective bispectrum
-
-### Forward pass
-
-The selective bispectrum (Definition 4.2) has two types of coefficients:
-
-**Type 0**: b\_{0,0,k} = a\_{0,1}^2 * conj(a\_{0,k}) for k = 1..K_0
-
-This is analogous to the (0,0) and (0,1) terms in the CnonCn selective bispectrum. The a\_{0,1}^2 factor is real (since a\_{0,k} is always real for real signals), so this extracts the radial profile at zero angular frequency.
-
-**Type 2**: b\_{2,n,k} = a\_{1,1} * a\_{n,1} * conj(a\_{n+1,k}) for n = 0..N_m-1, k = 1..K\_{n+1}
-
-This chains adjacent angular frequencies together: knowing a\_{n,1} lets you recover a\_{n+1,k} from b\_{2,n,k}. The a\_{1,1} factor serves as the "anchor" (its magnitude is known, its phase is absorbed by the rotation indeterminacy).
-
-Total count: N = sum\_{n=0}^{N_m} K_n.
-
-The implementation precomputes index tensors (`_type0_a0k_idx`, `_type2_an1_idx`, `_type2_anp1k_idx`) at construction time, so the forward pass is fully vectorized — no Python loops.
-
-### Invariance
-
-Under rotation by phi, the DH coefficients transform as:
-
-```
-a'_{n,k} = exp(i * n * phi) * a_{n,k}
-```
-
-For Type 0:
-b'_{0,0,k} = (a'_{0,1})^2 * conj(a'_{0,k})
-= a_{0,1}^2 * conj(a\_{0,k})
-= b\_{0,0,k}
-
-(because exp(i * 0 * phi) = 1 for all n=0 terms.)
-
-For Type 2:
-b'_{2,n,k} = a'_{1,1} * a'_{n,1} * conj(a'_{n+1,k})
-= exp(i*phi) * a\_{1,1} * exp(i*n*phi) * a\_{n,1} * conj(exp(i*(n+1)*phi) * a\_{n+1,k})
-= exp(i*phi) * exp(i*n*phi) * exp(-i\*(n+1)*phi) * a\_{1,1} * a\_{n,1} * conj(a\_{n+1,k})
-= exp(i*(1 + n - n - 1)\*phi) * b\_{2,n,k}
-= b\_{2,n,k}
-
-The phases cancel exactly. This is verified by `test_analytical_rotation_invariance` (which operates on coefficients directly, no grid discretization involved).
-
-Spatial rotation invariance (rotating the actual image) has additional discretization error from grid_sample interpolation plus the DHT's finite resolution. This error is ~O(1/L) and is tested with a loose tolerance.
-
-## Inversion
-
-### The bootstrap (Theorem 4.4)
-
-The inversion follows the same structure as `CnonCn.invert()`:
-
-1. **Recover a\_{0,1}** from b\_{0,0,1} = |a\_{0,1}|^2 * a\_{0,1} = |a\_{0,1}|^3 * exp(i * arg(a\_{0,1})).
-   So |a\_{0,1}| = |b\_{0,0,1}|^{1/3} and the phase is arg(b\_{0,0,1}).
-   (For real signals, a\_{0,1} is real, so this just gives a\_{0,1} = b\_{0,0,1}^{1/3}.)
-
-2. **Recover a\_{0,k}** for k >= 2 from b\_{0,0,k} / a\_{0,1}^2 = conj(a\_{0,k}), so a\_{0,k} = conj(b\_{0,0,k} / a\_{0,1}^2).
-
-3. **Recover |a\_{1,1}|** from b\_{2,0,1} = a\_{1,1} * a\_{0,1} * conj(a\_{1,1}) = a\_{0,1} * |a\_{1,1}|^2.
-   So |a\_{1,1}| = sqrt(|b\_{2,0,1} / a\_{0,1}|). The phase of a\_{1,1} is the rotation indeterminacy — we set it to 0.
-
-4. **Recover a\_{1,k}** for k >= 2 from b\_{2,0,k} = a\_{1,1} * a\_{0,1} * conj(a\_{1,k}).
-
-5. **Sequential bootstrap for n >= 1**: b\_{2,n,k} = a\_{1,1} * a\_{n,1} * conj(a\_{n+1,k}).
-   Since a\_{n,1} was recovered in the previous step, we get a\_{n+1,k} = conj(b\_{2,n,k} / (a\_{1,1} * a\_{n,1})).
-
-After recovering all coefficients, the inverse DHT reconstructs the image.
-
-### Precision considerations
-
-The inversion involves divisions by a\_{1,1} and a\_{n,1}. The paper's assumption that these are nonzero is essential — if any a\_{n,1} = 0, the chain breaks at that angular frequency. In practice with noisy images this is never an issue, but synthetic test signals need to be constructed carefully (adding a constant bias `+ 0.5` to the random coefficients helps).
-
-The full roundtrip (f -> bispectrum -> invert -> f') goes through the DHT twice, so the ~1-2 unresolvable basis functions cause accumulated error. The test asserts that >85% of coefficients are recovered within 20% relative error, rather than demanding exact reconstruction.
-
-## Test structure
-
-45 tests organized into 5 groups:
-
-- **TestBessel** (10 tests): J_0 and J_1 match torch.special, J_n at zero, root locations, roots-are-actual-zeros, monotonicity, batch root computation consistency.
-
-- **TestSO2onD2Construction** (7 tests): instantiation, no trainable params, extra_repr, index_map, coefficient counts against paper Table 1 (L=8: 27, L=16: 105, L=28: 315), explicit bandlimit, full-not-implemented.
-
-- **TestSO2onD2Forward** (7 tests): output shape, dtype, determinism, different-signals-differ, batch-size-one, analytical rotation invariance (coefficient-level), spatial rotation invariance (grid_sample).
-
-- **TestSO2onD2DHTRoundtrip** (4 tests): bandlimited signal roundtrip, single harmonic roundtrip.
-
-- **TestSO2onD2Invert** (6 tests): exact coefficient recovery, direct coefficient-level inversion (no DHT), output shape, output dtype, full-not-implemented, bispectrum roundtrip (bsp(invert(bsp(f))) ≈ bsp(f)).
-
-## Performance characteristics
-
-Construction is slow for large L because of the Bessel root computation (pure Python loops with Newton iterations) and the pseudoinverse computation. For L=28, construction takes ~15s. For L=64, it would take minutes. The forward pass itself is fast (just matrix multiply + vectorized indexing).
-
-If performance becomes a bottleneck, the main optimization targets are:
-
-1. Cache Bessel roots (they only depend on n_max and k_max)
-2. Use the fast DHT from Marshall et al. 2023 (O(L^2 log L) instead of O(L^3))
-3. Implement Bessel functions in C++/CUDA
-
-## Files
-
-```
-src/bispectrum/
-├── _bessel.py        # bessel_jn, bessel_jn_zeros, compute_all_bessel_roots
-├── so2_on_d2.py      # SO2onD2 module
-└── __init__.py        # updated to export SO2onD2
-
-tests/
-└── test_so2_on_d2.py  # 45 tests
-```
-
-## What's not implemented
-
-- **Full disk bispectrum** (`selective=False`): raises NotImplementedError. Would need O(m^3/N_m) coefficients — straightforward but expensive.
-- **MRA correction** (Proposition 4.6): the bias term delta for noisy observations. Deferred to a follow-up.
-- **Fast DHT**: the O(L^2 log L) approximation from Marshall et al. 2023. Currently using the naive O(L^3) pseudoinverse approach.
-- **GPU optimization**: everything runs on CPU in float64. The module supports `.to(device)` via registered buffers, but the construction-time computation is CPU-only.
-
-## Possible paper issues
-
-Some things noticed during implementation that may warrant further examination:
-
-- The paper's Table 2 shows m=46 for L=8 with bandlimit lambda = pi\*L/2. But the ne = floor(L^2 * pi/4) = 50 convention gives slightly different index sets depending on how ties in Bessel root ordering are broken. The selective coefficient count (27) matches Table 1 exactly, so the conventions are consistent where it matters.
-
-- The inversion theorem requires a\_{n,1} != 0 for all n. This is stated as a mild assumption ("does not impact real-world application as long as there is some noise"), but for deterministic signals it can genuinely fail. A signal that's purely radial (only n=0 components) has all a\_{n,1} = 0 for n > 0, and the inversion chain breaks completely.
-
-- The normalization constant c\_{nk} = 1 / (sqrt(pi) * |J\_{n+1}(lambda\_{nk})|) follows from the orthonormality condition on the disk with measure r dr dtheta. The paper doesn't give this formula explicitly — it's derived from the standard Bessel function integral identity.
diff --git a/docs/steerable_bispectrum_notes.md b/docs/steerable_bispectrum_notes.md
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-# Steerable Bispectrum: Analysis & Bibliography
-
-## What is the Steerable Bispectrum?
-
-The steerable bispectrum is a **learnable CNN layer** that achieves local rotation invariance by:
-
-1. Convolving the image with filters expressed on a steerable basis: `profile_i(r) · e^{ikθ}` (learned radial profile × circular harmonic)
-2. Computing the bispectrum triple product from the per-pixel steerable filter responses: `Θ̂_{k1} · Θ̂_{k2} · Θ̂*_{k1+k2}`
-
-The bispectrum cancels the rotation phase `e^{ikφ}`, making the output rotation-invariant at each pixel.
-
-## Relation to Our Library
-
-### Naming Convention
-
-Following our `{Group}on{Domain}` convention, the steerable bispectrum would be `SteerableSO2onR2`:
-
-- **Group**: SO(2) (2D rotations)
-- **Domain**: R² (the plane / pixel grid)
-- **Prefix `Steerable`**: indicates learned filter weights (unlike our parameter-free modules)
-
-| Class              | Learnable? | What it computes      |
-| ------------------ | ---------- | --------------------- |
-| `SO2onS1`          | No         | Bispectrum of f       |
-| `SO2onD2`          | No         | Bispectrum of f       |
-| `SteerableSO2onR2` | Yes        | Bispectrum of (w ∗ f) |
-
-### Which existing module does it correspond to?
-
-The per-pixel bispectrum formula is exactly `CnonCn` (equivalently `SO2onS1`): at each pixel, you have `n_harmonics` complex Fourier coefficients indexed by angular frequency k, and the triple product `f̂_{k1} · f̂_{k2} · f̂*_{k1+k2}` is the abelian bispectrum on Z/nZ.
-
-It is **not** `SO2onD2`, because the bispectrum only couples angular frequencies — the radial structure is absorbed into the convolution, not into the bispectrum itself. `SO2onD2` would require bispectral coefficients mixing across radial orders (Bessel indices).
-
-### Canonical difference from our modules
-
-Our modules are **parameter-free transforms** (signal in → invariant descriptor out). The steerable bispectrum is a **learnable layer** (signal + learned filter weights → invariant features). The `Θ̂_l` depend on both the signal f and the learned weights `ŵ^{(l)}`.
-
-### Is it really a bispectrum?
-
-Yes, the formula is the textbook abelian bispectrum. But the `Θ̂_k` are not proper Fourier coefficients in the harmonic analysis sense — they're steerable filter responses that depend on learned, non-orthogonal radial profiles. The rotation invariance property holds, but the completeness/injectivity/invertibility guarantees of the proper bispectrum do not.
-
-### Connection to our tex document
-
-Section "Steerable Bispectrum: After Convolution on Steerable Basis" in `bispectral_signatures_of_data.tex` formalizes exactly this idea for Z/nZ, deriving the output coefficients `Θ̂_l` from convolution on a steerable basis and noting you can compute the bispectrum directly from them.
-
-The tex document also shows where this breaks for non-abelian groups: for D₄ via steerable CNNs, the intertwiner basis has L=3 elements but D₄ has M=5 irreps, so you can't recover true Fourier coefficients from the steerable CNN output. This subtlety doesn't arise for the abelian SO(2) case.
-
-### Could we speed up their code?
-
-No. The bispectrum step (element-wise triple product) is already trivial. The bottleneck is the convolution (`CHConv2D`). Also: framework mismatch (theirs is TensorFlow, ours is PyTorch).
-
-### The gap we could fill
-
-The Depeursinge group established the steerable bispectrum as a practical CNN layer but never addressed **selectivity** or **inversion**. A `SteerableSO2onR2` module could apply selective bispectral indices to steerable filter responses, giving both the practical CNN benefits (locality, learned radial profiles) and theoretical guarantees (minimal coefficients, invertibility).
-
-## Reference Implementation
-
-Cloned to `third_party/2d_bispectrum_cnn` from https://github.com/voreille/2d_bispectrum_cnn
-
-Key files:
-
-- `src/models/layers.py`: `BCHConv2D` (bispectrum layer), `CHConv2D` (circular harmonic convolution), `ECHConv2D` (power spectrum baseline)
-- `src/models/models.py`: `BispectUnetLight`, `SpectUnetLight` (U-Net architectures using these layers)
-
-## Bibliography
-
-### Foundational Theory
-
-1. **Kakarala (1992)** — *Triple Correlation on Groups*. PhD thesis, UC Irvine.
-   The foundational result: the bispectrum is a complete invariant for functions on compact groups, up to group action.
-
-2. **Kakarala (2012)** — *The Bispectrum as a Source of Phase-Sensitive Invariants for Fourier Descriptors: A Group-Theoretic Approach*. JMIV 44, 341–353. [Springer](https://link.springer.com/article/10.1007/s10851-012-0330-6)
-   Unified derivation for all compact groups + constructive recovery algorithm for SO(3). Shows bispectrum discriminates shapes that power spectrum cannot.
-
-### Steerable / Equivariant CNN Foundations
-
-3. **Worrall et al. (2017)** — *Harmonic Networks: Deep Translation and Rotation Equivariance*. CVPR 2017. [arXiv:1612.04642](https://arxiv.org/abs/1612.04642)
-   Origin of circular harmonic filters (`e^{ikθ}`) in CNNs. SO(2)-equivariant feature maps with magnitude pooling (power spectrum) for invariance — not bispectrum. This is the `ECHConv2D` (spectrum-only) baseline.
-
-4. **Weiler et al. (2018)** — *Learning Steerable Filters for Rotation Equivariant CNNs*. CVPR 2018. [PDF](https://openaccess.thecvf.com/content_cvpr_2018/papers/Weiler_Learning_Steerable_Filters_CVPR_2018_paper.pdf)
-   General framework: filters as linear combinations of atomic steerable filters. Arbitrary angular resolution. Does not use bispectrum for invariance.
-
-5. **Weiler & Cesa (2019)** — *General E(2)-Equivariant Steerable CNNs*. NeurIPS 2019. [arXiv:1911.08251](https://arxiv.org/abs/1911.08251)
-   General theory solving kernel constraints for arbitrary E(2) subgroups. Unifies H-Nets, GCNNs, and steerable CNNs. The `e2cnn` library. Uses norm/gated nonlinearities, not bispectrum.
-
-### The Steerable Bispectrum Line (Depeursinge group)
-
-6. **Andrearczyk, Oreiller, Depeursinge (2019)** — *Exploring Local Rotation Invariance in 3D CNNs with Steerable Filters*. MIDL 2019. [PMLR](https://proceedings.mlr.press/v102/andrearczyk19a.html)
-   3D steerable filters with SH × learnable radial profiles, pooled by magnitude (power spectrum). Notes that magnitude pooling loses phase.
-
-7. **Oreiller, Andrearczyk, Fageot, Prior, Depeursinge (2020)** — *3D Solid Spherical Bispectrum CNNs for Biomedical Texture Analysis*. [arXiv:2004.13371](https://arxiv.org/abs/2004.13371)
-   **The key paper.** Introduces the steerable bispectrum CNN: project onto circular/spherical harmonics, then compute the bispectrum instead of magnitude pooling. Shows bispectrum > power spectrum for texture discrimination. Theoretical foundation of the `2d_bispectrum_cnn` repo.
-
-8. **Oreiller et al. (2022)** — *Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral U-Net*. MIDL 2022. [PMLR](https://proceedings.mlr.press/v172/oreiller22a.html)
-   Application paper for the `2d_bispectrum_cnn` repo. Bispectral U-Net on histopathology (MoNuSeg). Shows rotation robustness without data augmentation.
-
-9. **voreille/ssbcnn** — [GitHub](https://github.com/voreille/ssbcnn)
-   Follow-up extending to scale + rotation steerable bispectrum.
-
-### Related: Bispectrum in Equivariant Neural Networks
-
-10. **Thomas, Smidt et al. (2018)** — *Tensor Field Networks*. [arXiv:1802.08219](https://arxiv.org/abs/1802.08219)
-    SO(3)-equivariant networks on point clouds using SH filters × learnable radial functions. Same filter factorization as the steerable bispectrum, but uses tensor product nonlinearities (Clebsch-Gordan) instead of bispectrum pooling.
-
-11. **Mataigne et al. (2024)** — *The Selective G-Bispectrum and its Inversion*. NeurIPS 2024. [arXiv:2407.07655](https://arxiv.org/abs/2407.07655)
-    Selective O(|G|) bispectrum with inversion for finite groups. The non-steerable (parameter-free) approach implemented in our library.
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-\documentclass[aspectratio=169]{beamer}
-\usetheme{metropolis}
-\usepackage{booktabs}
-\usepackage{amsmath,amssymb}
-\usepackage{listings}
-\usepackage{tikz}
-\usetikzlibrary{positioning,arrows.meta,calc}
-
-\lstset{
-  basicstyle=\ttfamily\scriptsize,
-  keywordstyle=\color{blue},
-  commentstyle=\color{gray},
-  breaklines=true,
-  frame=none,
-}
-
-\newcommand{\FF}{\mathcal{F}}
-\newcommand{\Hom}{\mathrm{Hom}}
-\newcommand{\R}{\mathbb{R}}
-\newcommand{\Z}{\mathbb{Z}}
-\newcommand{\C}{\mathbb{C}}
-\newcommand{\SO}{\mathrm{SO}}
-\newcommand{\Ind}{\mathrm{Ind}}
-
-\title{The Steerable Bispectrum for Non-Abelian Groups}
-\subtitle{Bridging Equivariant Networks and Complete Invariant Theory}
-\author{Johan Mathe}
-\date{\today}
-
-\begin{document}
-
-\maketitle
-
-\begin{frame}{Two worlds, one gap}
-  \begin{columns}[T]
-    \begin{column}{0.47\textwidth}
-      \textbf{Steerable CNNs}\\
-      {\small Weiler \& Cesa 2019}
-
-      \medskip
-
-      \begin{itemize}\small
-        \item Equivariant by design
-        \item Learned filters on intertwiner basis
-        \item Invariance via \alert{norm / gated nonlinearities}
-        \item[$\to$] Discards phase information
-        \item[$\to$] No reconstruction guarantees
-      \end{itemize}
-    \end{column}
-    \begin{column}{0.47\textwidth}
-      \textbf{$G$-Bispectrum}\\
-      {\small Kakarala 1992, Mataigne et al.\ 2024}
-
-      \medskip
-
-      \begin{itemize}\small
-        \item Complete invariant (up to group action, under non-degeneracy)
-        \item Signal recoverable from bispectrum (generically)
-        \item Selective: $O(|G|)$ instead of $O(|G|^2)$ coefficients
-        \item[$\to$] Requires \alert{proper Fourier coefficients}
-        \item[$\to$] Not embedded in learned architectures
-      \end{itemize}
-    \end{column}
-  \end{columns}
-
-  \bigskip
-  \centering
-  \alert{Gap}: What seems to be missing is a principled identification between steerable\\
-  feature coefficients and the Fourier objects required by the non-abelian bispectrum.
-\end{frame}
-
-\begin{frame}{What does ``steerable'' mean?}
-  A basis of filter functions is \textbf{steerable} if group actions reduce to linear operations on the coefficients:
-  \[
-    \mathbf{Y}(g \cdot x) = \rho(g) \, \mathbf{Y}(x)
-  \]
-  where $\mathbf{Y}$ is vector-valued and $\rho(g)$ is a matrix representation.
-
-  \medskip
-
-  \begin{columns}[T]
-    \begin{column}{0.48\textwidth}
-      \textbf{Abelian} ($\SO(2)$, $C_n$):\\
-      Each basis element $Y_l$ transforms by a \textit{scalar} character: $Y_l(g \cdot x) = \chi_l(g) \, Y_l(x)$.\\
-      {\small E.g.\ $e^{ik\theta} \mapsto e^{ik\phi} \cdot e^{ik\theta}$}
-    \end{column}
-    \begin{column}{0.48\textwidth}
-      \textbf{Non-abelian} ($D_n$, $O$):\\
-      Basis elements within an isotypic block \textit{mix} under the group action via a matrix $\rho(g)$.\\
-      {\small E.g.\ the 2D irrep $E$ of $D_4$}
-    \end{column}
-  \end{columns}
-
-  \bigskip
-
-  \textbf{Analogy}: a spotlight on a gimbal. You turn a dial (apply $\rho(g)$) and the beam rotates analytically --- no grid resampling.
-
-  \medskip
-
-  {\small Reference: Freeman \& Adelson, \textit{The Design and Use of Steerable Filters}, IEEE TPAMI 1991.}
-\end{frame}
-
-\begin{frame}{What is the bispectrum?}\small
-  \textbf{Abelian case} ($C_n$, $\SO(2)$): scalar triple product
-  \vspace{-4pt}
-  \[
-    \beta_{k_1, k_2} = \hat{f}_{k_1} \cdot \hat{f}_{k_2} \cdot \hat{f}^*_{k_1 + k_2}
-  \]
-
-  \vspace{-6pt}
-
-  \textbf{Non-abelian case} ($D_n$, $O$, $\SO(3)$): schematic CG-based formula
-  \vspace{-4pt}
-  \[
-    \beta_{\rho_1, \rho_2} = \bigl[\FF_{\rho_1} \otimes \FF_{\rho_2}\bigr]
-    \, C_{\rho_1,\rho_2} \, \Bigl[\bigoplus_{\rho \in \rho_1 \otimes \rho_2} \FF_\rho^\dagger\Bigr] \, C_{\rho_1,\rho_2}^\dagger
-  \]
-
-  \vspace{-6pt}
-
-  {\scriptsize where $\FF_\rho$ denotes the matrix-valued Fourier coefficient at irrep $\rho$, and $C_{\rho_1,\rho_2}$ is the CG change-of-basis for $\rho_1 \otimes \rho_2$.}
-
-  \medskip
-
-  \textbf{Key property}: under standard non-degeneracy assumptions (invertibility of relevant Fourier blocks), the bispectrum is a \textit{complete} invariant --- it recovers $f$ up to group action. The power spectrum $|\hat{f}_k|^2$ is not (loses phase).
-
-  \medskip
-
-  \textbf{Selective bispectrum} (Mataigne et al.\ 2024): only $O(|G|)$ coefficients needed instead of $O(|G|^2)$, preserving completeness.
-\end{frame}
-
-\begin{frame}{Timeline: two parallel lines of work}
-  \centering
-  \begin{tikzpicture}[
-    node distance=0.3cm and 0.8cm,
-    every node/.style={font=\scriptsize},
-    yearbox/.style={rounded corners=2pt, draw=gray, fill=gray!10, inner sep=3pt, minimum width=1.2cm},
-    paper/.style={align=left, text width=4.5cm},
-    >=Stealth
-  ]
-    \node[font=\footnotesize\bfseries] (bisp_label) at (-3, 2.2) {Bispectrum theory};
-    \node[yearbox] (k92) at (-3, 1.5) {1992};
-    \node[paper, right=0.3cm of k92] {Kakarala: triple correlation on groups};
-    \node[yearbox] (k12) at (-3, 0.7) {2012};
-    \node[paper, right=0.3cm of k12] {Kakarala: complete invariant, SO(3) recovery};
-    \node[yearbox] (m24) at (-3, -0.1) {2024};
-    \node[paper, right=0.3cm of m24] {Mataigne et al.: selective $G$-bispectrum};
-
-    \node[font=\footnotesize\bfseries] (steer_label) at (5.2, 2.2) {Steerable / equivariant CNNs};
-    \node[yearbox] (w17) at (5.2, 1.5) {2017};
-    \node[paper, right=0.3cm of w17] {Worrall: H-Nets (circular harmonics)};
-    \node[yearbox] (w19) at (5.2, 0.7) {2019};
-    \node[paper, right=0.3cm of w19] {Weiler \& Cesa: general E(2)-steerable};
-    \node[yearbox] (o20) at (5.2, -0.1) {2020};
-    \node[paper, right=0.3cm of o20] {Oreiller et al.: SO(2) bispectrum CNN};
-
-    \node[yearbox, fill=orange!20, draw=orange] (ours) at (3, -1.2) {\textbf{This work}};
-    \draw[->, thick, orange] (m24.south) -- ++(0, -0.3) -| (ours.north west);
-    \draw[->, thick, orange] (o20.south) -- ++(0, -0.3) -| (ours.north east);
-  \end{tikzpicture}
-\end{frame}
-
-\begin{frame}{}
-  \vfill
-  \centering
-  {\Large\textbf{Part I: The Abelian Case}}
-
-  \medskip
-
-  {\normalsize How the steerable bispectrum works for $\SO(2)$}
-  \vfill
-\end{frame}
-
-\begin{frame}{$\SO(2)$ steerable bispectrum}
-  \textbf{Filter}: $\Psi_k(r, \theta) = w_k(r) \cdot e^{ik\theta}$ \quad (learned radial profile $\times$ circular harmonic)
-
-  \medskip
-
-  \textbf{Convolution}: produces per-pixel steerable coefficients $\hat{\Theta}_k$
-
-  \medskip
-
-  \textbf{Steerability}: rotate input by $\phi$ $\Rightarrow$ $\hat{\Theta}_k \mapsto e^{ik\phi} \hat{\Theta}_k$
-
-  \medskip
-
-  \textbf{Bispectrum}: triple product cancels the phase
-  \[
-    \beta_{k_1, k_2} = \hat{\Theta}_{k_1} \cdot \hat{\Theta}_{k_2} \cdot \hat{\Theta}^*_{k_1+k_2}
-  \]
-  since $e^{ik_1\phi} \cdot e^{ik_2\phi} \cdot e^{-i(k_1+k_2)\phi} = 1$.
-
-  \bigskip
-
-  This is the same scalar triple-product mechanism as \texttt{CnonCn}, applied per-pixel to circular-harmonic responses.
-
-  \medskip
-
-  \textbf{Why it works}: $\SO(2)$ is abelian $\Rightarrow$ all irreps are 1D characters $\Rightarrow$ scalar steerable coordinates align with scalar Fourier modes.
-\end{frame}
-
-\begin{frame}[fragile]{The existing implementation (Oreiller et al.\ 2022)}
-  The \texttt{BCHConv2D} layer in \texttt{voreille/2d\_bispectrum\_cnn}:
-
-  \begin{lstlisting}[language=Python]
-def call(self, inputs):
-    x = self.conv_ch(inputs)     # circular harmonic convolution
-    outputs = list()
-    for n1, n2 in self.indices:
-        bispectrum = (x[..., n1] * x[..., n2] *
-                      tf.math.conj(x[..., n1 + n2]))
-        outputs.append(tf.math.real(bispectrum))
-        if n1 != 0:
-            outputs.append(tf.math.imag(bispectrum))
-    x = tf.concat(outputs, axis=-1)
-  \end{lstlisting}
-
-  \medskip
-
-  \begin{itemize}
-    \item TensorFlow/Keras, not PyTorch
-    \item Redundant index set (${\sim}n^2/2$ pairs via symmetry reduction)
-    \item No selectivity, no inversion guarantees
-    \item Learned discrete radial profiles (pixel masks), not Fourier-Bessel basis
-  \end{itemize}
-\end{frame}
-
-\begin{frame}{What's missing for the $\SO(2)$ case}
-  \begin{table}
-    \scriptsize
-    \begin{tabular}{lccc}
-      \toprule
-      & \textbf{Oreiller et al.} & \textbf{Our library} & \textbf{Potential} \\
-      \midrule
-      Bispectral pairs & ${\sim}n^2/2$ (symmetry) & $n$ (selective) & Same accuracy, fewer channels \\
-      Inversion & None & \checkmark\ (bootstrap) & Reconstruct from invariants \\
-      Radial basis & Learned pixel masks & Bessel $J_n(\lambda_{nk}r)$ & Orthogonal, complete \\
-      Framework & TensorFlow & PyTorch & GPU-optimized \\
-      \bottomrule
-    \end{tabular}
-  \end{table}
-
-  \bigskip
-
-  At $n_\text{harmonics} = 4$: selective gives 4 vs their 6 coefficients --- negligible.
-
-  \medskip
-
-  At $n_\text{harmonics} = 32$: selective gives 32 vs ${\sim}528$ coefficients.
-
-  \medskip
-
-  At $n_\text{harmonics} = 64$: selective gives 64 vs ${\sim}2080$ coefficients.
-
-  \bigskip
-
-  Selectivity matters when angular resolution is high, and becomes critical in 3D where full bispectral constructions scale poorly with harmonic bandwidth.
-\end{frame}
-
-\begin{frame}{Why the abelian case is easy}
-  For an abelian group $G$, all irreps are 1D characters $\chi_k : G \to \C$.
-
-  \medskip
-
-  The intertwiner space $\Hom^G(\pi, \chi_k)$ has dimension equal to the multiplicity of $\chi_k$ in $\pi$.
-
-  \medskip
-
-  For the regular representation on $\Z/n\Z$: every character appears exactly once.
-
-  \bigskip
-
-  In the regular abelian case, the equivariant decomposition coincides with the Fourier decomposition: every irrep is 1D, appears once, and $L = M = n$.
-
-  \bigskip
-
-  Hence the steerable coefficients $\hat{\Theta}_l$ can be identified with the Fourier coefficients $\hat{f}_l$.
-
-  \medskip
-
-  $\Rightarrow$ The standard bispectrum formula applies directly.
-
-  \bigskip
-
-  \alert{For non-abelian groups, the simple scalar identification between steerable coefficients and Fourier coefficients breaks.}
-\end{frame}
-
-\begin{frame}{}
-  \vfill
-  \centering
-  {\Large\textbf{Part II: The Non-Abelian Problem}}
-
-  \medskip
-
-  {\normalsize Why $L \neq M$ blocks the standard approach}
-  \vfill
-\end{frame}
-
-\begin{frame}{The $L \neq M$ problem}
-  Consider $H = D_4$ with a \textbf{trivial output representation} $\rho_0$.
-
-  \medskip
-
-  A trivial-output head accesses only the $A_1$-isotypic component of the input fiber $\pi$. But $D_4$ has $M = 5$ irreps: $A_1, A_2, B_1, B_2, E$ (dims $1,1,1,1,2$).
-
-  \bigskip
-
-  {\small Count-of-types intuition (not a dimensional comparison):}
-
-  \centering\small
-  \begin{tabular}{lcl}
-    \toprule
-    & Irreps accessed by trivial output & Total irreps of $H$ \\
-    \midrule
-    $\SO(2)$, $C_n$ (abelian) & all $n$ (each 1D) & $n$ \\
-    $D_4$, $\rho = \rho_0$ & \alert{$A_1$ only} & \alert{5 (incl.\ 2D $E$)} \\
-    $D_8$, $\rho = \rho_0$ & \alert{$A_1$ only} & \alert{7 (incl.\ 2D irreps)} \\
-    \bottomrule
-  \end{tabular}
-
-  \raggedright
-
-  \bigskip
-
-  The steerable conv gives access only to the \alert{$A_1$-isotypic coordinates}, while the bispectrum requires \alert{Fourier data from all 5 irreps} of $D_4$, including matrix-valued blocks.
-\end{frame}
-
-\begin{frame}{Why $L \neq M$}
-  The number of intertwiner basis functions is:
-  \[
-    L = \dim \Hom^H(\pi, \rho) = \sum_{\text{irreps }\psi} m^\pi_\psi \cdot m^\rho_\psi
-  \]
-
-  For $\rho = \rho_0$ (trivial): $m^{\rho_0}_\psi = 0$ for all $\psi \neq \rho_0$, so $L = m^\pi_{\rho_0}$.
-
-  \bigskip
-
-  This counts only how many times the \textit{trivial} irrep appears in $\pi$ --- it says nothing about the other irreps.
-
-  \bigskip
-
-  \textbf{The Fourier coefficients $\FF(f)_\psi$} for the non-trivial irreps ($A_2, B_1, B_2, E$) are \textbf{invisible} to a steerable conv with trivial output.
-
-  \bigskip
-
-  \textbf{For abelian groups} in the regular representation: every irrep is 1D and appears with multiplicity one, so the equivariant decomposition aligns with the Fourier decomposition and $L = M$.
-\end{frame}
-
-\begin{frame}{What the coefficients actually are}
-  The steerable conv produces per-pixel coefficients $\hat{\Theta}_l$, which are projections onto the $\rho_0$\textbf{-isotypic component} of the input fiber $\pi$:
-
-  \begin{itemize}
-    \item \textbf{Visible} through a trivial-output head: the $A_1$-isotypic component, of multiplicity $m^{\pi}_{A_1} = 3$
-    \item \alert{Not visible} through a trivial-output head: the $A_2, B_1, B_2, E$ isotypic components (4 irreps, including the matrix-valued $E$ block)
-  \end{itemize}
-
-  \bigskip
-
-  $\Rightarrow$ The bispectrum computed from $\hat{\Theta}_l$ alone is \textit{incomplete}.
-
-  \medskip
-
-  $\Rightarrow$ For a complete invariant, we need Fourier data for \alert{all} irreps $\FF_\psi$.
-\end{frame}
-
-\begin{frame}{Information flow: the problem and our solution}
-  \centering
-  \begin{tikzpicture}[
-    box/.style={draw, rounded corners=3pt, minimum height=0.9cm, font=\small, align=center},
-    >=Stealth,
-    node distance=0.5cm
-  ]
-    \node[box, fill=blue!10, minimum width=1.8cm] (sig) {Signal $f$};
-    \node[box, fill=gray!10, right=0.8cm of sig, minimum width=2.4cm] (conv) {Steerable conv\\(reg.\ rep output)};
-    \node[box, fill=green!15, right=0.8cm of conv, minimum width=2.4cm] (coef) {$|H|$ coefficients\\in $\bigoplus \psi \otimes \psi^*$};
-    \node[box, fill=orange!15, right=0.8cm of coef, minimum width=1.8cm] (inv) {Recover $\hat{f}(\psi)$\\via $A_\psi$ (P4)};
-    \node[box, fill=blue!10, right=0.8cm of inv, minimum width=2.4cm] (bisp) {$G$-Bispectrum\\(invariant)};
-
-    \draw[->, thick] (sig) -- (conv);
-    \draw[->, thick] (conv) -- (coef);
-    \draw[->, thick] (coef) -- (inv);
-    \draw[->, thick] (inv) -- (bisp);
-  \end{tikzpicture}
-
-  \bigskip
-
-  \begin{tabular}{lll}
-    \toprule
-    \textbf{Group} & \textbf{Solution} & \textbf{Status} \\
-    \midrule
-    $\SO(2)$, $C_n$ (abelian) & Identity ($L=M$) & \checkmark\ Solved (Oreiller 2020) \\
-    $D_n$ (non-abelian) & Regular-rep fiber + P4 & Proved for model case (this work) \\
-    $O$ (octahedral) & Same rep-theoretic construction & Expected to apply \\
-    \bottomrule
-  \end{tabular}
-
-  \bigskip
-
-  Previously, steerable CNNs for $D_n$ were stuck with \textbf{norm} (loses phase) or \textbf{gated} (ad-hoc). P4 with the regular-rep fiber gives a principled construction and a clear sufficient condition for bispectral pooling.
-\end{frame}
-
-\begin{frame}{}
-  \vfill
-  \centering
-  {\Large\textbf{Part III: The Solution}}
-
-  \medskip
-
-  {\normalsize Regular-representation output fiber and Proposition P4}
-  \vfill
-\end{frame}
-
-\begin{frame}{Construction: regular-representation output fiber}
-  \textbf{Key idea}: design the output fiber to be the \textbf{regular representation} of $H$.
-
-  \medskip
-
-  For each irrep $\psi$, the Fourier block $\hat{f}(\psi) \in \psi \otimes \psi^*$ has dimension $d_\psi^2$. To recover it, the $\psi$-typed output needs multiplicity $d_\psi$ (not just 1).
-
-  \medskip
-
-  For $D_4$, the regular representation $\C[D_4] \cong \bigoplus_\psi \psi \otimes \psi^*$ gives:
-  \begin{center}\scriptsize
-  \begin{tabular}{lccccc}
-    \toprule
-    Irrep $\psi$ & $d_\psi$ & Canonical mult.\ (reg.\ rep) & Output $\psi \otimes \psi^*$ & Scalar dim & Fourier block \\
-    \midrule
-    $A_1$ & 1 & 1 & $A_1 \otimes A_1^*$ & 1 & $\hat{f}(A_1) \in \C$ \\
-    $A_2$ & 1 & 1 & $A_2 \otimes A_2^*$ & 1 & $\hat{f}(A_2) \in \C$ \\
-    $B_1$ & 1 & 1 & $B_1 \otimes B_1^*$ & 1 & $\hat{f}(B_1) \in \C$ \\
-    $B_2$ & 1 & 1 & $B_2 \otimes B_2^*$ & 1 & $\hat{f}(B_2) \in \C$ \\
-    $E$ & 2 & \alert{2} & $E \otimes E^*$ & \alert{4} & $\hat{f}(E) \in \C^{2\times 2}$ \\
-    \midrule
-    \textbf{Total} & & & & $8 = |D_4|$ & All 5 Fourier blocks \\
-    \bottomrule
-  \end{tabular}
-  \end{center}
-
-  Total dimension: $1 + 1 + 1 + 1 + 4 = 8 = |D_4|$, recovering all 5 Fourier blocks.
-\end{frame}
-
-\begin{frame}{Why multiplicity $d_\psi$ matters}
-  \alert{Key}: a single $E$-field (2 scalars) cannot recover the $E$-Fourier block ($2\!\times\!2$ matrix, 4 scalars).
-
-  \bigskip
-
-  You need \textbf{two copies} of $E$, i.e.\ $E \otimes E^* \cong \mathrm{End}(E)$.
-
-  \medskip
-
-  More generally, $\hat{f}(\psi) \in \psi \otimes \psi^*$ has $d_\psi^2$ scalar entries. A $\psi$-typed output with multiplicity $m < d_\psi$ produces only $m \cdot d_\psi$ scalars --- not enough to determine the full Fourier block.
-
-  \bigskip
-
-  The regular representation is the \textit{canonical} choice: it gives each irrep $\psi$ exactly multiplicity $d_\psi$, which is both necessary and sufficient.
-\end{frame}
-
-\begin{frame}{Proposition P4 (proved for $V = \C[H]$)}
-  \textbf{Proposition.} Let $V = \C[H]$ (regular representation). Let $T_\psi \in \Hom^H(V, \psi \otimes K)$ be an equivariant map with output type $\psi \otimes K$. Then there exists a linear map $A_\psi : \psi^* \to K$ such that
-  \[
-    T_\psi(f) = (I_\psi \otimes A_\psi) \, \hat{f}(\psi).
-  \]
-  If $K \cong \psi^*$ and $A_\psi$ is an isomorphism, then $\hat{f}(\psi)$ is linearly recoverable.
-
-  \bigskip
-
-  \textbf{Consequence}: with a regular-rep output fiber ($K = \psi^*$ for each irrep), the steerable conv output is linearly equivalent to the full set of Fourier blocks. Selectivity (Mataigne et al.\ 2024) then applies directly.
-\end{frame}
-
-\begin{frame}{Proof of P4 (step 1: decomposition)}\small
-  \textbf{Step 1 (Peter--Weyl).} The regular representation decomposes as
-  \[
-    V = \C[H] \;\cong\; \bigoplus_{\psi \in \widehat{H}} \psi \otimes \psi^*,
-  \]
-  where each summand $V_\psi = \psi \otimes \psi^*$ is the \textit{$\psi$-isotypic component}, a $d_\psi \times d_\psi$ matrix space. The group acts on the left factor: $h \cdot (u \otimes \lambda) = (\psi(h)u) \otimes \lambda$.
-
-  \medskip
-
-  The orthogonal projection $P_\psi : V \to V_\psi$ is exactly the Fourier transform at $\psi$:
-  \[
-    P_\psi f \;\longleftrightarrow\; \hat{f}(\psi) = \sum_{g \in H} f(g)\,\psi(g)^* \;\in\; \C^{d_\psi \times d_\psi}.
-  \]
-
-  \medskip
-
-  Write $V = V_\psi \oplus V_\psi^\perp$, where $V_\psi^\perp = \bigoplus_{\phi \neq \psi} V_\phi$.
-\end{frame}
-
-\begin{frame}{Proof of P4 (step 2: Schur kills wrong frequencies)}\small
-  \textbf{Step 2 (Schur's lemma).} The map $T_\psi : V \to \psi \otimes K$ is $H$-equivariant, with the target transforming as $\psi$ in the first factor.
-
-  \medskip
-
-  Restrict $T_\psi$ to a summand $V_\phi = \phi \otimes \phi^*$ with $\phi \neq \psi$. This restriction is an equivariant map $\phi \otimes \phi^* \to \psi \otimes K$. Since $H$ acts on the first factors, this requires an equivariant map $\phi \to \psi$.
-
-  \medskip
-
-  Schur's lemma: if $\phi \not\cong \psi$, every equivariant map $\phi \to \psi$ is \textbf{zero}.
-
-  \medskip
-
-  Therefore $T_\psi$ annihilates $V_\psi^\perp$ and factors through the Fourier projection:
-  \[
-    T_\psi = \tilde{T}_\psi \circ P_\psi, \qquad \tilde{T}_\psi : \underbrace{\psi \otimes \psi^*}_{= V_\psi} \;\longrightarrow\; \psi \otimes K.
-  \]
-
-  \medskip
-
-  \textit{Consequence}: equivariance forces frequency selectivity --- $T_\psi$ can only access the Fourier block $\hat{f}(\psi)$.
-\end{frame}
-
-\begin{frame}{Proof of P4 (step 3: structure on the right frequency)}\small
-  \textbf{Step 3 (classification of equivariants).} We must classify all $H$-equivariant maps
-  \[
-    \tilde{T}_\psi : \psi \otimes \psi^* \;\longrightarrow\; \psi \otimes K,
-  \]
-  where $H$ acts on $\psi$ (first factor) and acts trivially on $\psi^*$ and $K$.
-
-  \medskip
-
-  By Schur's lemma applied to the $\psi$-factor: any equivariant map $\psi \to \psi$ is a scalar multiple of the identity. Therefore $\tilde{T}_\psi$ must act as the identity on the first factor:
-  \[
-    \tilde{T}_\psi = I_\psi \otimes A_\psi, \qquad A_\psi : \psi^* \to K \;\text{(arbitrary linear map)}.
-  \]
-
-  \medskip
-
-  Combining steps 1--3:
-  \[
-    \boxed{T_\psi(f) = (I_\psi \otimes A_\psi)\,\hat{f}(\psi).}
-  \]
-
-  When $K \cong \psi^*$ and $A_\psi$ is invertible, we recover $\hat{f}(\psi) = (I_\psi \otimes A_\psi^{-1})\,T_\psi(f)$. \qed
-
-  \medskip
-
-  \textbf{Remark}: the proof uses only Peter--Weyl and Schur's lemma --- it holds for \textit{any} finite group $H$, not just $D_4$.
-\end{frame}
-
-\begin{frame}{Approach 2: direct triple correlation}\small
-  \textbf{Idea}: bypass the steerable-to-Fourier identification entirely. Define the triple correlation directly on vector-valued signals with an induced representation action.
-
-  \smallskip
-
-  \textbf{What we have proven} (in our notes):
-  \begin{enumerate}\setlength{\itemsep}{1pt}
-    \item A triple correlation $T(f)_{g_1, g_2}$ for vector-valued $f$ with action $T_{tr}[f](x) = \rho_H(r) \, f\bigl((tr)^{-1}x\bigr)$.
-    \item \textbf{Invariance}: $T(T_h[f]) = T(f)$ for all $h \in H$ (change of variable $r' = rr_h$, $t' = R_h t + t_h$).
-    \item \textbf{Bispectrum recovery}: FT of $T(f)$ recovers the standard CG-based bispectrum formula.
-  \end{enumerate}
-
-  \smallskip
-
-  This confirms the bispectrum is well-defined for induced-representation signals. In the regular-rep model case, P4 resolves the steerable-to-Fourier identification. For \texttt{escnn}-style induced features over $\R^2$, the identification of the intertwiner basis with the Fourier multiplicity space \alert{remains to be verified}.
-\end{frame}
-
-\begin{frame}{Proof status from our derivations}\small
-  \vspace{-4pt}
-  \begin{center}\scriptsize
-  \begin{tabular}{llll}
-    \toprule
-    \textbf{Result} & \textbf{Status} & \textbf{Where} \\
-    \midrule
-    TC definition for induced rep & \checkmark Done & Eq.\ in our notes \\
-    TC invariance proof & \checkmark Done & Change of variable $r'=rr_h$, $t'=Rt_h+t$ \\
-    Bispectrum = FT of TC & \checkmark Done & CG-based derivation \\
-    Standard CG formula recovered & \checkmark Done & $\beta = [\FF_{\rho_1} \otimes \FF_{\rho_2}] C [\bigoplus \FF_\rho^\dagger] C^\dagger$ \\
-    \midrule
-    $\FF_\rho$ from steerable conv (P4) & \checkmark Done & Schur + Peter--Weyl on regular rep \\
-    Selectivity transfers & \checkmark Done & Direct from P4 + Mataigne et al. \\
-    Channel count: $|H|$ scalars suffice & \checkmark Done & Regular rep has dim $|H|$ \\
-    \midrule
-    \texttt{escnn} fiber = regular rep & \alert{Open} & Implementation question \\
-    \bottomrule
-  \end{tabular}
-  \end{center}
-
-  \medskip
-
-  \textbf{Remaining question}: the theorem holds for the regular representation. In \texttt{escnn}, the fiber is an architecture choice. The open task is to verify that \texttt{escnn} allows setting the output type to $A_1 \oplus A_2 \oplus B_1 \oplus B_2 \oplus 2E$ and to compute $A_\psi$ from the intertwiner basis.
-\end{frame}
-
-\begin{frame}{Selectivity transfers (from P4)}\small
-  Since P4 establishes that the steerable conv with regular-rep fiber recovers all Fourier blocks, Mataigne et al.\ (2024) selectivity applies directly:
-
-  \medskip
-
-  \textbf{Channel count per pixel}, for group $H$:
-
-  \vspace{-4pt}
-  \begin{center}\scriptsize
-  \begin{tabular}{lccc}
-    \toprule
-    \textbf{Invariant} & \textbf{Channels} & \textbf{Complete?}$^\dagger$ & \textbf{Invertible?}$^\dagger$ \\
-    \midrule
-    Norm (power spectrum) & $O(|H|)$ & No & No \\
-    Full bispectrum & $O(|H|^2)$ & Yes & Yes \\
-    \textbf{Selective bispectrum} & $\boldsymbol{O(|H|)}$ & \textbf{Yes} & \textbf{Yes} \\
-    \multicolumn{4}{l}{\tiny $^\dagger$Under standard non-degeneracy assumptions (Kakarala 1992, Mataigne et al.\ 2024).}\\
-    \bottomrule
-  \end{tabular}
-  \end{center}
-
-  \vspace{-2pt}
-
-  Concrete numbers for $D_n$:
-
-  \vspace{-4pt}
-  \begin{center}\scriptsize
-  \begin{tabular}{lrrr}
-    \toprule
-    Group & $|H|$ & Full bisp.\ channels & Selective channels \\
-    \midrule
-    $D_4$ & 8 & 64 & 8 \\
-    $D_8$ & 16 & 256 & 16 \\
-    $D_{16}$ & 32 & 1024 & 32 \\
-    \bottomrule
-  \end{tabular}
-  \end{center}
-
-  \vspace{-2pt}
-
-  Selective bispectrum is the \textbf{only} option that is both generically complete and tractable.
-\end{frame}
-
-\begin{frame}{Proof summary}
-  \begin{table}
-    \small
-    \begin{tabular}{clcc}
-      \toprule
-      \textbf{ID} & \textbf{Statement} & \textbf{Depends on} & \textbf{Status} \\
-      \midrule
-      P4 & $T_\psi(f) = (I_\psi \otimes A_\psi)\hat{f}(\psi)$ for any finite $H$ & --- & \checkmark\ Done \\
-      P5 & Bispectrum from regular-rep steerable conv & P4 & \checkmark\ Done \\
-      P6 & Selectivity transfers to steerable setting & P4, P5 & \checkmark\ Done \\
-      P7 & Channel count: $|H|$ scalars for complete invariant & P6 & \checkmark\ Done \\
-      \midrule
-      E1 & \texttt{escnn} output type = regular rep of $D_4$ & --- & \alert{Open} \\
-      E2 & Compute $A_\psi$ from \texttt{escnn} intertwiner basis & E1 & \alert{Open} \\
-      \bottomrule
-    \end{tabular}
-  \end{table}
-
-  \bigskip
-
-  \centering
-  Theory is complete for the regular-representation model case.
-
-  \medskip
-
-  Remaining work: \alert{realize this construction faithfully in \texttt{escnn} field-type language}.
-\end{frame}
-
-\begin{frame}{}
-  \vfill
-  \centering
-  {\Large\textbf{Part IV: Directions and Experiments}}
-
-  \medskip
-
-  {\normalsize What to build and how to test it}
-  \vfill
-\end{frame}
-
-\begin{frame}{Architecture: $D_4$ steerable bispectrum layer}
-  \centering
-  \begin{tikzpicture}[
-    box/.style={draw, rounded corners=3pt, minimum height=0.8cm, font=\small, align=center},
-    >=Stealth,
-    node distance=0.6cm
-  ]
-    \node[box, fill=blue!10, minimum width=2cm] (img) {Image\\$f \in \R^{H \times W}$};
-    \node[box, fill=gray!10, right=0.7cm of img, minimum width=2.8cm] (conv) {Steerable conv\\output = reg.\ rep\\(escnn, 8 dim)};
-    \node[box, fill=green!15, right=0.7cm of conv, minimum width=2.5cm, text width=2.3cm] (fourier) {Recover $\hat{f}(\psi)$\\via $A_\psi$, per pixel};
-    \node[box, fill=orange!15, right=0.7cm of fourier, minimum width=2.5cm] (bisp) {Selective\\$D_4$-bispectrum\\(our \texttt{DnonDn})};
-    \node[box, fill=blue!10, right=0.7cm of bisp, minimum width=2cm] (out) {Invariant\\features};
-
-    \draw[->, thick] (img) -- (conv);
-    \draw[->, thick] (conv) -- (fourier);
-    \draw[->, thick] (fourier) -- (bisp);
-    \draw[->, thick] (bisp) -- (out);
-  \end{tikzpicture}
-
-  \bigskip
-
-  \begin{itemize}
-    \item \textbf{Steerable conv}: \texttt{escnn} with output type $A_1 \oplus A_2 \oplus B_1 \oplus B_2 \oplus 2E$ (regular-rep fiber, 8 scalars/pixel)
-    \item \textbf{Fourier extraction}: invert $A_\psi$ to recover $\hat{f}(\psi)$ from each isotypic block (P4)
-    \item \textbf{Selective bispectrum}: use our \texttt{DnonDn} module per pixel
-    \item \textbf{Output}: $D_4$-invariant feature maps, ready for downstream task
-  \end{itemize}
-
-  \medskip
-
-  Drop-in replacement for norm/gated nonlinearities in equivariant architectures.
-\end{frame}
-
-\begin{frame}{Experiment 1: rotated CIFAR-10/100}\small
-  \textbf{Setup}: Standard benchmark for rotation-invariant classification.
-
-  \medskip
-
-  \begin{center}\scriptsize
-  \begin{tabular}{llll}
-    \toprule
-    \textbf{Method} & \textbf{Invariance} & \textbf{Complete?}$^\dagger$ & \textbf{Source} \\
-    \midrule
-    Standard CNN + augmentation & None (learned) & N/A & Baseline \\
-    Steerable CNN + norm & $D_4$ & No & Weiler \& Cesa 2019 \\
-    Steerable CNN + $\SO(2)$ bisp. & $\SO(2)$ & Yes (rot) & Oreiller 2020 \\
-    \textbf{Steerable CNN + $D_4$ sel.\ bisp.} & \textbf{$D_4$} & \textbf{Yes (rot+ref)} & \textbf{Ours} \\
-    \multicolumn{4}{l}{\tiny $^\dagger$Generically, under non-degeneracy of relevant Fourier blocks.} \\
-    \bottomrule
-  \end{tabular}
-  \end{center}
-
-  \medskip
-
-  \textbf{Expected}: $D_4$ selective bispectrum yields a generically complete invariant for rotations and reflections; norm is $D_4$-invariant too, but incomplete. Selective version matches full bispectrum with $8\times$ fewer channels.
-
-  \medskip
-
-  \textbf{Why impactful}: First head-to-head comparison of bispectrum vs norm/gate nonlinearities in steerable CNNs.
-\end{frame}
-
-\begin{frame}{Experiment 2: medical image segmentation}\small
-  \textbf{Setup}: Reproduce Oreiller et al.\ (MIDL 2022) on MoNuSeg (nucleus segmentation in histopathology). Replace their $\SO(2)$ bispectral U-Net with our $D_4$ selective bispectral U-Net.
-
-  \medskip
-
-  \textbf{Why $D_4$ matters here}:
-  \begin{itemize}\setlength{\itemsep}{1pt}
-    \item Cell nuclei have no preferred chirality $\Rightarrow$ reflection invariance is natural
-    \item $\SO(2)$ bispectrum handles rotations only
-    \item $D_4$ bispectrum handles rotations \textit{and} reflections
-  \end{itemize}
-
-  \medskip
-
-  \textbf{Expected}: Better rotation+reflection robustness on their benchmark, same architecture.
-
-  \medskip
-
-  \textbf{Why impactful}: Direct improvement on published results, on their own data, with a principled mathematical upgrade.
-\end{frame}
-
-\begin{frame}{Experiment 3: scaling analysis}\small
-  How does the invariant feature dimension scale with group size?
-
-  \vspace{-2pt}
-  \centering
-  \begin{tikzpicture}[scale=0.55]
-    \draw[->] (0,0) -- (8,0) node[right] {\scriptsize $|H|$};
-    \draw[->] (0,0) -- (0,4.5) node[above] {\scriptsize ch/px};
-
-    \draw[thick, red] (1,0.25) -- (2,1.0) -- (3,2.2) -- (4,4.0);
-    \node[red, right, font=\scriptsize] at (4,4.0) {Full bisp.\ $O(|H|^2)$};
-
-    \draw[thick, blue] (1,0.25) -- (2,0.5) -- (3,0.75) -- (4,1.0) -- (5,1.25) -- (6,1.5) -- (7,1.75);
-    \node[blue, right, font=\scriptsize] at (7,1.75) {Selective bisp.\ $O(|H|)$};
-
-    \draw[thick, gray, dashed] (1,0.15) -- (2,0.3) -- (3,0.5) -- (4,0.65) -- (5,0.8) -- (6,1.0) -- (7,1.15);
-    \node[gray, right, font=\scriptsize] at (7,1.15) {Norm $O(|H|)$ \textit{(inc.)}};
-
-    \foreach \x/\lab in {1/$D_4$, 2/$D_8$, 3/$D_{16}$, 4/$D_{32}$, 5/$D_{64}$} {
-      \node[below, font=\tiny] at (\x, 0) {\lab};
-    }
-  \end{tikzpicture}
-
-  \vspace{-2pt}
-  \raggedright
-
-  \textbf{Selective bispectrum} is the only option that is both:
-  \begin{itemize}\setlength{\itemsep}{0pt}
-    \item \textbf{Generically complete} (unlike norm, which loses phase)
-    \item \textbf{Tractable} (unlike full bispectrum, which blows up quadratically)
-  \end{itemize}
-
-  \smallskip
-
-  {\footnotesize Selectivity becomes increasingly important in 3D, where full bispectral constructions scale poorly with harmonic bandwidth.}
-\end{frame}
-
-\begin{frame}{Takeaway}\small
-  \begin{enumerate}\setlength{\itemsep}{4pt}
-    \item \textbf{Abelian steerable bispectrum} ($\SO(2)$) works: same scalar triple-product as \texttt{CnonCn}, applied per-pixel to circular-harmonic responses.
-
-    \item For \textbf{non-abelian groups} ($D_n$, $O$), a single output type does not expose Fourier data for all irreps. Fix: \textbf{regular-representation output fiber}.
-
-    \item \textbf{Proposition P4} (proved for $V = \C[H]$): $T_\psi(f) = (I_\psi \otimes A_\psi)\hat{f}(\psi)$. Multiplicity $d_\psi$ in output needed for $d_\psi$-dim irreps. Selectivity transfers.
-
-    \item \textbf{Design}: for $D_4$, output $= A_1 \oplus A_2 \oplus B_1 \oplus B_2 \oplus 2E$ (8 scalars $= |D_4|$).
-
-    \item \textbf{Target}: NeurIPS 2026. Theory + \texttt{SteerableBispectrum} layer + experiments on rotated CIFAR and MoNuSeg.
-  \end{enumerate}
-
-  \medskip
-
-  \textbf{Remaining work}: map this construction to \texttt{escnn} --- verify the fiber can be set to the regular rep and compute $A_\psi$ from the intertwiner basis.
-\end{frame}
-
-\end{document}
diff --git a/paper/experiments/organ3d/.gitignore b/experiments/organ3d/.gitignore
similarity index 100%
rename from paper/experiments/organ3d/.gitignore
rename to experiments/organ3d/.gitignore
diff --git a/experiments/organ3d/README.md b/experiments/organ3d/README.md
new file mode 100644
index 0000000..a81ddbc
--- /dev/null
+++ b/experiments/organ3d/README.md
@@ -0,0 +1,9 @@
+# OrganMNIST3D Classification
+
+3D classification on OrganMNIST3D (11 organ classes, 28x28x28 CT volumes) comparing invariant pooling strategies in octahedral-equivariant 3D ResNets.
+
+```bash
+pip install -e "../../[dev]"
+python train.py --model bispectrum --data_dir ./organ3d_data
+./run_sweep.sh  # full sweep: 4 models x 3 seeds
+```
diff --git a/paper/experiments/organ3d/analyze_results.py b/experiments/organ3d/analyze_results.py
similarity index 70%
rename from paper/experiments/organ3d/analyze_results.py
rename to experiments/organ3d/analyze_results.py
index 9a63592..afcad12 100644
--- a/paper/experiments/organ3d/analyze_results.py
+++ b/experiments/organ3d/analyze_results.py
@@ -16,6 +16,7 @@
 from pathlib import Path
 
 import matplotlib
+
 matplotlib.use('Agg')
 import matplotlib.pyplot as plt
 import numpy as np
@@ -50,13 +51,50 @@ def _test_c(record: dict) -> dict:
 def _test_r(record: dict) -> dict:
     return record.get('test_r') or {}
 
+
 PUBLISHED_BASELINES = [
-    {'method': 'ResNet-18 + 3D', 'venue': 'Yang et al. 2023', 'params': '33M', 'acc': 0.907, 'auc': 0.996},
-    {'method': 'ResNet-18 + ACS', 'venue': 'Yang et al. 2023', 'params': '11M', 'acc': 0.900, 'auc': 0.994},
-    {'method': 'ILPOResNet-50', 'venue': 'Zhemchuzhnikov 2024', 'params': '38K', 'acc': 0.879, 'auc': 0.992},
-    {'method': 'EquiLoPO (local train.)', 'venue': 'ICLR 2025', 'params': '418K', 'acc': 0.866, 'auc': 0.991},
-    {'method': 'SE3MovFrNet', 'venue': 'Sangalli et al. 2023', 'params': '—', 'acc': 0.745, 'auc': None},
-    {'method': 'Regular SE(3) conv', 'venue': 'Kuipers & Bekkers 2023', 'params': '172K', 'acc': 0.698, 'auc': None},
+    {
+        'method': 'ResNet-18 + 3D',
+        'venue': 'Yang et al. 2023',
+        'params': '33M',
+        'acc': 0.907,
+        'auc': 0.996,
+    },
+    {
+        'method': 'ResNet-18 + ACS',
+        'venue': 'Yang et al. 2023',
+        'params': '11M',
+        'acc': 0.900,
+        'auc': 0.994,
+    },
+    {
+        'method': 'ILPOResNet-50',
+        'venue': 'Zhemchuzhnikov 2024',
+        'params': '38K',
+        'acc': 0.879,
+        'auc': 0.992,
+    },
+    {
+        'method': 'EquiLoPO (local train.)',
+        'venue': 'ICLR 2025',
+        'params': '418K',
+        'acc': 0.866,
+        'auc': 0.991,
+    },
+    {
+        'method': 'SE3MovFrNet',
+        'venue': 'Sangalli et al. 2023',
+        'params': '—',
+        'acc': 0.745,
+        'auc': None,
+    },
+    {
+        'method': 'Regular SE(3) conv',
+        'venue': 'Kuipers & Bekkers 2023',
+        'params': '172K',
+        'acc': 0.698,
+        'auc': None,
+    },
 ]
 
 
@@ -106,10 +144,13 @@ def _baseline_runs(
     """Filter to channels=[4,8], full-set runs, and *train_mode* runs."""
     filtered: dict[str, list[dict]] = {}
     for model, runs in grouped.items():
-        f = [r for r in runs
-             if r.get('channels', [4, 8]) == [4, 8]
-             and _is_full_train(r)
-             and _result_train_mode(r) == train_mode]
+        f = [
+            r
+            for r in runs
+            if r.get('channels', [4, 8]) == [4, 8]
+            and _is_full_train(r)
+            and _result_train_mode(r) == train_mode
+        ]
         if f:
             filtered[model] = f
     return filtered
@@ -142,10 +183,16 @@ def print_our_results(grouped: dict[str, list[dict]]):
         mean_auc = np.mean(aucs)
         std_auc = np.std(aucs)
 
-        rot_accs = [r['rotation_robustness']['mean_accuracy'] for r in runs
-                    if r.get('rotation_robustness', {}).get('mean_accuracy', 0) > 0]
-        rot_stds = [r['rotation_robustness']['std_accuracy'] for r in runs
-                    if r.get('rotation_robustness', {}).get('mean_accuracy', 0) > 0]
+        rot_accs = [
+            r['rotation_robustness']['mean_accuracy']
+            for r in runs
+            if r.get('rotation_robustness', {}).get('mean_accuracy', 0) > 0
+        ]
+        rot_stds = [
+            r['rotation_robustness']['std_accuracy']
+            for r in runs
+            if r.get('rotation_robustness', {}).get('mean_accuracy', 0) > 0
+        ]
 
         label = MODEL_LABELS.get(model_name, model_name)
         rot_str = f'{np.mean(rot_accs):.4f}' if rot_accs else '\u2014'
@@ -169,8 +216,7 @@ def print_published_baselines():
     for b in PUBLISHED_BASELINES:
         auc_str = f'{b["auc"]:.3f}' if b['auc'] is not None else '—'
         print(
-            f'{b["method"]:<30} {b["venue"]:<25} {b["params"]:>8} '
-            f'{b["acc"]:.3f}    {auc_str:>8}'
+            f'{b["method"]:<30} {b["venue"]:<25} {b["params"]:>8} {b["acc"]:.3f}    {auc_str:>8}'
         )
 
 
@@ -182,20 +228,22 @@ def plot_rotation_comparison(grouped: dict[str, list[dict]], output_path: str):
         print('No data to plot.')
         return
 
-    plt.rcParams.update({
-        'font.family': 'serif',
-        'font.serif': ['Times New Roman', 'DejaVu Serif'],
-        'mathtext.fontset': 'cm',
-        'font.size': 11,
-        'axes.labelsize': 13,
-        'axes.titlesize': 14,
-        'xtick.labelsize': 11,
-        'ytick.labelsize': 10,
-        'legend.fontsize': 10,
-        'figure.dpi': 300,
-        'savefig.bbox': 'tight',
-        'savefig.pad_inches': 0.05,
-    })
+    plt.rcParams.update(
+        {
+            'font.family': 'serif',
+            'font.serif': ['Times New Roman', 'DejaVu Serif'],
+            'mathtext.fontset': 'cm',
+            'font.size': 11,
+            'axes.labelsize': 13,
+            'axes.titlesize': 14,
+            'xtick.labelsize': 11,
+            'ytick.labelsize': 10,
+            'legend.fontsize': 10,
+            'figure.dpi': 300,
+            'savefig.bbox': 'tight',
+            'savefig.pad_inches': 0.05,
+        }
+    )
 
     PLOT_LABELS = {
         'standard': 'Standard',
@@ -215,8 +263,11 @@ def plot_rotation_comparison(grouped: dict[str, list[dict]], output_path: str):
         accs = [_test_c(r).get('accuracy', 0.0) for r in runs]
         orig_accs.append(np.mean(accs))
         orig_stds.append(np.std(accs))
-        rot_vals = [r['rotation_robustness']['mean_accuracy'] for r in runs
-                    if r.get('rotation_robustness', {}).get('mean_accuracy', 0) > 0]
+        rot_vals = [
+            r['rotation_robustness']['mean_accuracy']
+            for r in runs
+            if r.get('rotation_robustness', {}).get('mean_accuracy', 0) > 0
+        ]
         rot_accs.append(np.mean(rot_vals) if rot_vals else 0)
         rot_stds.append(np.std(rot_vals) if rot_vals else 0)
 
@@ -234,32 +285,50 @@ def plot_rotation_comparison(grouped: dict[str, list[dict]], output_path: str):
     fig, ax = plt.subplots(figsize=(5.5, 3.5))
 
     bars1 = ax.bar(
-        x - width / 2, orig_accs, width, yerr=orig_stds,
-        capsize=3, color=c_orig, edgecolor='white', linewidth=0.6,
+        x - width / 2,
+        orig_accs,
+        width,
+        yerr=orig_stds,
+        capsize=3,
+        color=c_orig,
+        edgecolor='white',
+        linewidth=0.6,
         error_kw={'linewidth': 0.9, 'capthick': 0.9},
-        label='Original test', zorder=3,
+        label='Original test',
+        zorder=3,
     )
     rot_display = np.where(rot_accs > 0, rot_accs, 0)
     rot_err = np.where(rot_accs > 0, rot_stds, 0)
     bars2 = ax.bar(
-        x + width / 2, rot_display, width, yerr=rot_err,
-        capsize=3, color=c_rot, edgecolor='white', linewidth=0.6,
+        x + width / 2,
+        rot_display,
+        width,
+        yerr=rot_err,
+        capsize=3,
+        color=c_rot,
+        edgecolor='white',
+        linewidth=0.6,
         error_kw={'linewidth': 0.9, 'capthick': 0.9},
-        label='Rotated test (24 oct. rotations)', zorder=3,
+        label='Rotated test (24 oct. rotations)',
+        zorder=3,
     )
 
     def annotate_bar(bar, val: float, std: float):
         h = bar.get_height()
         offset = std + 0.015 if std > 0 else 0.015
         ax.text(
-            bar.get_x() + bar.get_width() / 2, h + offset,
+            bar.get_x() + bar.get_width() / 2,
+            h + offset,
             f'{val:.1%}',
-            ha='center', va='bottom', fontsize=8, fontweight='medium',
+            ha='center',
+            va='bottom',
+            fontsize=8,
+            fontweight='medium',
         )
 
-    for bar, v, s in zip(bars1, orig_accs, orig_stds):
+    for bar, v, s in zip(bars1, orig_accs, orig_stds, strict=False):
         annotate_bar(bar, v, s)
-    for bar, v, s in zip(bars2, rot_display, rot_err):
+    for bar, v, s in zip(bars2, rot_display, rot_err, strict=False):
         if v > 0:
             annotate_bar(bar, v, s)
 
@@ -281,8 +350,11 @@ def annotate_bar(bar, val: float, std: float):
     ax.set_axisbelow(True)
 
     ax.legend(
-        frameon=True, fancybox=False, edgecolor='#cccccc',
-        framealpha=0.95, loc='upper left',
+        frameon=True,
+        fancybox=False,
+        edgecolor='#cccccc',
+        framealpha=0.95,
+        loc='upper left',
     )
 
     fig.tight_layout()
@@ -294,16 +366,18 @@ def annotate_bar(bar, val: float, std: float):
 
 def plot_data_efficiency(grouped: dict[str, list[dict]], output_path: str):
     """Plot data efficiency curves: test accuracy vs train fraction for each model."""
-    plt.rcParams.update({
-        'font.family': 'serif',
-        'font.size': 11,
-        'axes.titlesize': 12,
-        'axes.labelsize': 11,
-        'xtick.labelsize': 10,
-        'ytick.labelsize': 10,
-        'legend.fontsize': 9,
-        'figure.dpi': 300,
-    })
+    plt.rcParams.update(
+        {
+            'font.family': 'serif',
+            'font.size': 11,
+            'axes.titlesize': 12,
+            'axes.labelsize': 11,
+            'xtick.labelsize': 10,
+            'ytick.labelsize': 10,
+            'legend.fontsize': 9,
+            'figure.dpi': 300,
+        }
+    )
 
     organ_full = 971
     sizes = [100, 500, organ_full]
@@ -319,10 +393,13 @@ def plot_data_efficiency(grouped: dict[str, list[dict]], output_path: str):
         stds = []
         for size in sizes:
             target = None if size >= organ_full else size
-            runs = [r for r in grouped.get(model, [])
-                    if _train_size_value(r) == target
-                    and r.get('channels', [4, 8]) == [4, 8]
-                    and _result_train_mode(r) == 'C']
+            runs = [
+                r
+                for r in grouped.get(model, [])
+                if _train_size_value(r) == target
+                and r.get('channels', [4, 8]) == [4, 8]
+                and _result_train_mode(r) == 'C'
+            ]
             if runs:
                 vals = [_test_c(r).get('accuracy', 0.0) for r in runs]
                 means.append(np.mean(vals))
@@ -335,14 +412,22 @@ def plot_data_efficiency(grouped: dict[str, list[dict]], output_path: str):
         stds = np.array(stds)
 
         ax.plot(
-            sizes, means,
-            color=colors[model], marker=markers[model],
-            markersize=7, linewidth=2, label=labels[model],
+            sizes,
+            means,
+            color=colors[model],
+            marker=markers[model],
+            markersize=7,
+            linewidth=2,
+            label=labels[model],
             zorder=3,
         )
         ax.fill_between(
-            sizes, means - stds, means + stds,
-            color=colors[model], alpha=0.15, zorder=2,
+            sizes,
+            means - stds,
+            means + stds,
+            color=colors[model],
+            alpha=0.15,
+            zorder=2,
         )
 
     ax.set_xlabel('Training examples (N)')
@@ -364,8 +449,11 @@ def plot_data_efficiency(grouped: dict[str, list[dict]], output_path: str):
     ax.set_axisbelow(True)
 
     ax.legend(
-        frameon=True, fancybox=False, edgecolor='#cccccc',
-        framealpha=0.95, loc='upper left',
+        frameon=True,
+        fancybox=False,
+        edgecolor='#cccccc',
+        framealpha=0.95,
+        loc='upper left',
     )
 
     fig.tight_layout()
@@ -384,16 +472,21 @@ def print_tier1_summary(grouped: dict[str, list[dict]]):
     print('\n' + '=' * 90)
     print('DATA EFFICIENCY (channels [4,8], C-trained)')
     print('=' * 90)
-    print(f'{"Model":<15} {"N":>6} {"Params":>10} {"Test ACC":>18} {"Test AUC":>18} {"Rot ACC":>10}')
+    print(
+        f'{"Model":<15} {"N":>6} {"Params":>10} {"Test ACC":>18} {"Test AUC":>18} {"Rot ACC":>10}'
+    )
     print('-' * 85)
 
     for model in models:
         for size in sizes:
             target = None if size >= organ_full else size
-            runs = [r for r in grouped.get(model, [])
-                    if _train_size_value(r) == target
-                    and r.get('channels', [4, 8]) == [4, 8]
-                    and _result_train_mode(r) == 'C']
+            runs = [
+                r
+                for r in grouped.get(model, [])
+                if _train_size_value(r) == target
+                and r.get('channels', [4, 8]) == [4, 8]
+                and _result_train_mode(r) == 'C'
+            ]
             if runs:
                 accs = [_test_c(r).get('accuracy', 0.0) for r in runs]
                 aucs = [_test_c(r).get('auc', 0.0) for r in runs]
@@ -409,15 +502,20 @@ def print_tier1_summary(grouped: dict[str, list[dict]]):
     print('\n' + '=' * 90)
     print('WIDER CHANNELS (full training set)')
     print('=' * 90)
-    print(f'{"Model":<15} {"Channels":>10} {"Params":>10} {"Test ACC":>18} {"Test AUC":>18} {"Rot ACC":>10}')
+    print(
+        f'{"Model":<15} {"Channels":>10} {"Params":>10} {"Test ACC":>18} {"Test AUC":>18} {"Rot ACC":>10}'
+    )
     print('-' * 85)
 
     for ch in [[4, 8], [8, 16], [16, 32]]:
         for model in ['max_pool', 'bispectrum']:
-            runs = [r for r in grouped.get(model, [])
-                    if r.get('channels', [4, 8]) == ch
-                    and _is_full_train(r)
-                    and _result_train_mode(r) == 'C']
+            runs = [
+                r
+                for r in grouped.get(model, [])
+                if r.get('channels', [4, 8]) == ch
+                and _is_full_train(r)
+                and _result_train_mode(r) == 'C'
+            ]
             if runs:
                 accs = [_test_c(r).get('accuracy', 0.0) for r in runs]
                 aucs = [_test_c(r).get('auc', 0.0) for r in runs]
diff --git a/paper/experiments/organ3d/data.py b/experiments/organ3d/data.py
similarity index 81%
rename from paper/experiments/organ3d/data.py
rename to experiments/organ3d/data.py
index 4b86e1c..321c5ff 100644
--- a/paper/experiments/organ3d/data.py
+++ b/experiments/organ3d/data.py
@@ -34,7 +34,10 @@ def __init__(self, split: str, data_dir: str, normalize: bool = True):
 
         Path(data_dir).mkdir(parents=True, exist_ok=True)
         ds = medmnist.OrganMNIST3D(
-            split=split, download=True, root=data_dir, size=28,
+            split=split,
+            download=True,
+            root=data_dir,
+            size=28,
         )
 
         imgs = []
@@ -95,15 +98,19 @@ def gpu_augment_3d(
         for b in range(B):
             vol = images[b, 0]
             R = rots[idx[b]].long()
-            coords = torch.stack(
-                torch.meshgrid(
-                    torch.arange(28, device=dev) - 13.5,
-                    torch.arange(28, device=dev) - 13.5,
-                    torch.arange(28, device=dev) - 13.5,
-                    indexing='ij',
-                ),
-                dim=-1,
-            ).reshape(-1, 3).float()
+            coords = (
+                torch.stack(
+                    torch.meshgrid(
+                        torch.arange(28, device=dev) - 13.5,
+                        torch.arange(28, device=dev) - 13.5,
+                        torch.arange(28, device=dev) - 13.5,
+                        indexing='ij',
+                    ),
+                    dim=-1,
+                )
+                .reshape(-1, 3)
+                .float()
+            )
             new_coords = (R.float() @ coords.T).T
             ni = (new_coords[:, 0] + 13.5).long().clamp(0, 27)
             nj = (new_coords[:, 1] + 13.5).long().clamp(0, 27)
@@ -133,15 +140,19 @@ def apply_octahedral_element(images: torch.Tensor, element_idx: int) -> torch.Te
     rots = _get_octahedral_rotations().to(dev)
     R = rots[element_idx].long()
     half = 13.5
-    coords = torch.stack(
-        torch.meshgrid(
-            torch.arange(28, device=dev) - half,
-            torch.arange(28, device=dev) - half,
-            torch.arange(28, device=dev) - half,
-            indexing='ij',
-        ),
-        dim=-1,
-    ).reshape(-1, 3).float()
+    coords = (
+        torch.stack(
+            torch.meshgrid(
+                torch.arange(28, device=dev) - half,
+                torch.arange(28, device=dev) - half,
+                torch.arange(28, device=dev) - half,
+                indexing='ij',
+            ),
+            dim=-1,
+        )
+        .reshape(-1, 3)
+        .float()
+    )
     new_coords = (R.float() @ coords.T).T
     ni = (new_coords[:, 0] + half).long().clamp(0, 27)
     nj = (new_coords[:, 1] + half).long().clamp(0, 27)
@@ -186,15 +197,26 @@ def get_dataloaders(
 
     loader_rng = torch.Generator().manual_seed(seed)
     train_loader = DataLoader(
-        train_ds, batch_size=min(batch_size, len(train_ds)), shuffle=True,
-        generator=loader_rng, num_workers=0, pin_memory=True, drop_last=False,
+        train_ds,
+        batch_size=min(batch_size, len(train_ds)),
+        shuffle=True,
+        generator=loader_rng,
+        num_workers=0,
+        pin_memory=True,
+        drop_last=False,
     )
     val_loader = DataLoader(
-        val_ds, batch_size=batch_size, shuffle=False,
-        num_workers=0, pin_memory=True,
+        val_ds,
+        batch_size=batch_size,
+        shuffle=False,
+        num_workers=0,
+        pin_memory=True,
     )
     test_loader = DataLoader(
-        test_ds, batch_size=batch_size, shuffle=False,
-        num_workers=0, pin_memory=True,
+        test_ds,
+        batch_size=batch_size,
+        shuffle=False,
+        num_workers=0,
+        pin_memory=True,
     )
     return train_loader, val_loader, test_loader
diff --git a/paper/experiments/organ3d/find_budget.py b/experiments/organ3d/find_budget.py
similarity index 100%
rename from paper/experiments/organ3d/find_budget.py
rename to experiments/organ3d/find_budget.py
diff --git a/paper/experiments/organ3d/model.py b/experiments/organ3d/model.py
similarity index 97%
rename from paper/experiments/organ3d/model.py
rename to experiments/organ3d/model.py
index 245125a..6cf1246 100644
--- a/paper/experiments/organ3d/model.py
+++ b/experiments/organ3d/model.py
@@ -120,8 +120,8 @@ def forward(self, x: torch.Tensor) -> torch.Tensor:
         G = GROUP_ORDER
 
         w_flat = self.weight.reshape(C_out * C_in, K**3)
-        w_all = w_flat[:, self.inv_perms]      # (C_out*C_in, G, K³) — R_{g⁻¹} rotation
-        w_all = w_all.permute(1, 0, 2)         # (G, C_out*C_in, K³)
+        w_all = w_flat[:, self.inv_perms]  # (C_out*C_in, G, K³) — R_{g⁻¹} rotation
+        w_all = w_all.permute(1, 0, 2)  # (G, C_out*C_in, K³)
         w_all = w_all.reshape(G * C_out, C_in, K, K, K)
 
         out = F.conv3d(x, w_all, padding=self.padding)
@@ -151,7 +151,9 @@ def __init__(
         self.kernel_size = kernel_size
 
         self.weight = nn.Parameter(
-            torch.empty(out_channels, in_channels, GROUP_ORDER, kernel_size, kernel_size, kernel_size)
+            torch.empty(
+                out_channels, in_channels, GROUP_ORDER, kernel_size, kernel_size, kernel_size
+            )
         )
         nn.init.kaiming_uniform_(self.weight, a=math.sqrt(5))
         self.padding = padding
@@ -382,7 +384,8 @@ def forward(self, x: torch.Tensor) -> torch.Tensor:
         if self.equivariant:
             x = F.avg_pool3d(
                 x.reshape(x.shape[0], -1, *x.shape[3:]),
-                kernel_size=2, stride=2,
+                kernel_size=2,
+                stride=2,
             )
             x = x.reshape(x.shape[0], -1, GROUP_ORDER, *x.shape[2:])
         else:
@@ -394,7 +397,8 @@ def forward(self, x: torch.Tensor) -> torch.Tensor:
         if self.equivariant:
             x = F.avg_pool3d(
                 x.reshape(x.shape[0], -1, *x.shape[3:]),
-                kernel_size=2, stride=2,
+                kernel_size=2,
+                stride=2,
             )
             x = x.reshape(x.shape[0], -1, GROUP_ORDER, *x.shape[2:])
         else:
diff --git a/paper/experiments/organ3d/run_dataeff_multiseed.sh b/experiments/organ3d/run_dataeff_multiseed.sh
similarity index 92%
rename from paper/experiments/organ3d/run_dataeff_multiseed.sh
rename to experiments/organ3d/run_dataeff_multiseed.sh
index 4ad37c7..4632087 100755
--- a/paper/experiments/organ3d/run_dataeff_multiseed.sh
+++ b/experiments/organ3d/run_dataeff_multiseed.sh
@@ -7,8 +7,9 @@ set -euo pipefail
 SCRIPT_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)"
 cd "$SCRIPT_DIR"
 
-source /home/johmathe/bispectrum/.venv/bin/activate
-export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}"
+REPO_ROOT="$(cd "$SCRIPT_DIR/../.." && pwd)"
+source "$REPO_ROOT/.venv/bin/activate"
+export PYTHONPATH="$REPO_ROOT/src:${PYTHONPATH:-}"
 export PYTHONUNBUFFERED=1
 
 OUTPUT_DIR="./organ3d_results"
diff --git a/paper/experiments/organ3d/run_sweep.sh b/experiments/organ3d/run_sweep.sh
similarity index 93%
rename from paper/experiments/organ3d/run_sweep.sh
rename to experiments/organ3d/run_sweep.sh
index b6bf003..6b37186 100755
--- a/paper/experiments/organ3d/run_sweep.sh
+++ b/experiments/organ3d/run_sweep.sh
@@ -17,8 +17,9 @@ set -euo pipefail
 SCRIPT_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)"
 cd "$SCRIPT_DIR"
 
-source /home/johmathe/bispectrum/.venv/bin/activate
-export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}"
+REPO_ROOT="$(cd "$SCRIPT_DIR/../.." && pwd)"
+source "$REPO_ROOT/.venv/bin/activate"
+export PYTHONPATH="$REPO_ROOT/src:${PYTHONPATH:-}"
 export PYTHONUNBUFFERED=1
 
 MODELS=(standard max_pool norm_pool bispectrum)
diff --git a/paper/experiments/organ3d/run_tier1_sweep.sh b/experiments/organ3d/run_tier1_sweep.sh
similarity index 95%
rename from paper/experiments/organ3d/run_tier1_sweep.sh
rename to experiments/organ3d/run_tier1_sweep.sh
index f93f842..880b539 100755
--- a/paper/experiments/organ3d/run_tier1_sweep.sh
+++ b/experiments/organ3d/run_tier1_sweep.sh
@@ -18,8 +18,9 @@ set -euo pipefail
 SCRIPT_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)"
 cd "$SCRIPT_DIR"
 
-source /home/johmathe/bispectrum/.venv/bin/activate
-export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}"
+REPO_ROOT="$(cd "$SCRIPT_DIR/../.." && pwd)"
+source "$REPO_ROOT/.venv/bin/activate"
+export PYTHONPATH="$REPO_ROOT/src:${PYTHONPATH:-}"
 export PYTHONUNBUFFERED=1
 
 OUTPUT_DIR="./organ3d_results"
diff --git a/paper/experiments/organ3d/run_wider_multiseed.sh b/experiments/organ3d/run_wider_multiseed.sh
similarity index 93%
rename from paper/experiments/organ3d/run_wider_multiseed.sh
rename to experiments/organ3d/run_wider_multiseed.sh
index 4b3ab13..cd162e5 100755
--- a/paper/experiments/organ3d/run_wider_multiseed.sh
+++ b/experiments/organ3d/run_wider_multiseed.sh
@@ -7,8 +7,9 @@ set -euo pipefail
 SCRIPT_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)"
 cd "$SCRIPT_DIR"
 
-source /home/johmathe/bispectrum/.venv/bin/activate
-export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}"
+REPO_ROOT="$(cd "$SCRIPT_DIR/../.." && pwd)"
+source "$REPO_ROOT/.venv/bin/activate"
+export PYTHONPATH="$REPO_ROOT/src:${PYTHONPATH:-}"
 export PYTHONUNBUFFERED=1
 
 OUTPUT_DIR="./organ3d_results"
diff --git a/paper/experiments/organ3d/train.py b/experiments/organ3d/train.py
similarity index 91%
rename from paper/experiments/organ3d/train.py
rename to experiments/organ3d/train.py
index 759e739..2b39907 100644
--- a/paper/experiments/organ3d/train.py
+++ b/experiments/organ3d/train.py
@@ -39,7 +39,10 @@ def _git_sha() -> str:
         out = subprocess.run(
             ['git', 'rev-parse', 'HEAD'],
             cwd=Path(__file__).resolve().parent,
-            capture_output=True, text=True, timeout=5, check=False,
+            capture_output=True,
+            text=True,
+            timeout=5,
+            check=False,
         )
         return out.stdout.strip() if out.returncode == 0 else ''
     except (OSError, subprocess.SubprocessError):
@@ -146,8 +149,8 @@ def evaluate_rotation_robustness(
 ) -> dict:
     """Evaluate model consistency across all 24 octahedral rotations.
 
-    For each rotation, rotate all test volumes and measure AUC.
-    An equivariant model should produce consistent AUC regardless of rotation.
+    For each rotation, rotate all test volumes and measure AUC. An equivariant model should produce
+    consistent AUC regardless of rotation.
     """
     model.eval()
     results = {}
@@ -176,7 +179,9 @@ def evaluate_rotation_robustness(
     results['mean_auc'] = sum(aucs) / len(aucs)
     results['std_auc'] = (sum((a - results['mean_auc']) ** 2 for a in aucs) / len(aucs)) ** 0.5
     results['mean_accuracy'] = sum(accs) / len(accs)
-    results['std_accuracy'] = (sum((a - results['mean_accuracy']) ** 2 for a in accs) / len(accs)) ** 0.5
+    results['std_accuracy'] = (
+        sum((a - results['mean_accuracy']) ** 2 for a in accs) / len(accs)
+    ) ** 0.5
 
     return results
 
@@ -269,7 +274,9 @@ def memory_check(args: argparse.Namespace) -> dict:
             args.train_mode == 'R',
         )
         labels = labels.to(device, non_blocking=True)
-        optimizer = torch.optim.AdamW(model.parameters(), lr=args.lr, weight_decay=args.weight_decay)
+        optimizer = torch.optim.AdamW(
+            model.parameters(), lr=args.lr, weight_decay=args.weight_decay
+        )
         optimizer.zero_grad()
         with torch.amp.autocast('cuda'):
             logits = model(images)
@@ -370,7 +377,10 @@ def train_one_epoch(
 
 
 def train(args: argparse.Namespace) -> dict:
-    """Run a single training experiment. Returns final metrics dict."""
+    """Run a single training experiment.
+
+    Returns final metrics dict.
+    """
     torch.manual_seed(args.seed)
     if torch.cuda.is_available():
         torch.cuda.manual_seed_all(args.seed)
@@ -390,15 +400,19 @@ def train(args: argparse.Namespace) -> dict:
     print(f'Model: {args.model} (channels={channels}), {n_params:,} params')
 
     if args.dry_run:
-        print(json.dumps({
-            'model': args.model,
-            'train_mode': args.train_mode,
-            'channels': list(channels),
-            'n_params': n_params,
-            'invariant_dim': invariant_dim,
-            'raw_invariant_dim': dim_info.get('raw_invariant_dim'),
-            'projected_dim': dim_info.get('projected_dim'),
-        }))
+        print(
+            json.dumps(
+                {
+                    'model': args.model,
+                    'train_mode': args.train_mode,
+                    'channels': list(channels),
+                    'n_params': n_params,
+                    'invariant_dim': invariant_dim,
+                    'raw_invariant_dim': dim_info.get('raw_invariant_dim'),
+                    'projected_dim': dim_info.get('projected_dim'),
+                }
+            )
+        )
         return {'n_params': n_params}
 
     print(f'Device: {device}')
@@ -422,9 +436,7 @@ def train(args: argparse.Namespace) -> dict:
 
     torch.set_float32_matmul_precision('high')
 
-    optimizer = torch.optim.AdamW(
-        model.parameters(), lr=args.lr, weight_decay=args.weight_decay
-    )
+    optimizer = torch.optim.AdamW(model.parameters(), lr=args.lr, weight_decay=args.weight_decay)
     scaler = torch.amp.GradScaler('cuda')
     steps_per_epoch = max(1, len(train_loader))
     scheduler = torch.optim.lr_scheduler.CosineAnnealingWarmRestarts(
@@ -447,7 +459,12 @@ def train(args: argparse.Namespace) -> dict:
     for epoch in range(1, args.epochs + 1):
         t0 = time.time()
         train_loss = train_one_epoch(
-            model, train_loader, optimizer, scheduler, device, scaler,
+            model,
+            train_loader,
+            optimizer,
+            scheduler,
+            device,
+            scaler,
             augment_geometry=augment_geo,
         )
         val_metrics = compute_metrics(model, val_loader, device)
@@ -598,14 +615,19 @@ def main():
     parser.add_argument('--patience', type=int, default=15)
     parser.add_argument('--seed', type=int, default=42)
     parser.add_argument(
-        '--channels', type=int, nargs='+', default=[4, 8],
+        '--channels',
+        type=int,
+        nargs='+',
+        default=[4, 8],
         help='Channel widths (C0, C1) for stages.',
     )
     parser.add_argument('--head_dim', type=int, default=64)
     parser.add_argument(
-        '--train_size', type=int, default=None,
+        '--train_size',
+        type=int,
+        default=None,
         help='Subset training data to N examples for data-efficiency experiments. '
-             'Omit (or pass <=0) to use the full training set.',
+        'Omit (or pass <=0) to use the full training set.',
     )
     parser.add_argument(
         '--train_mode',
@@ -620,31 +642,40 @@ def main():
         help='Directory where train-subset index manifests are written.',
     )
     parser.add_argument(
-        '--sweep', action='store_true',
+        '--sweep',
+        action='store_true',
         help='Run all 4 models x 3 seeds.',
     )
     parser.add_argument(
-        '--dry_run', action='store_true',
+        '--dry_run',
+        action='store_true',
         help='Build model, print param count, exit.',
     )
     parser.add_argument(
-        '--memory_check', action='store_true',
+        '--memory_check',
+        action='store_true',
         help='Run a one-batch forward/backward memory check and exit.',
     )
     parser.add_argument(
-        '--memory_manifest', type=str, default=None,
+        '--memory_manifest',
+        type=str,
+        default=None,
         help='Optional JSONL path for memory_check records.',
     )
     parser.add_argument(
-        '--memory_headroom', type=float, default=0.85,
+        '--memory_headroom',
+        type=float,
+        default=0.85,
         help='Fail memory_check when reserved CUDA memory exceeds this fraction.',
     )
     parser.add_argument(
-        '--skip_rotation', action='store_true',
+        '--skip_rotation',
+        action='store_true',
         help='Skip the 24-rotation robustness evaluation.',
     )
     parser.add_argument(
-        '--compile', action='store_true',
+        '--compile',
+        action='store_true',
         help='Use torch.compile for faster training.',
     )
 
diff --git a/paper/experiments/pcam/.gitignore b/experiments/pcam/.gitignore
similarity index 100%
rename from paper/experiments/pcam/.gitignore
rename to experiments/pcam/.gitignore
diff --git a/experiments/pcam/README.md b/experiments/pcam/README.md
new file mode 100644
index 0000000..001958d
--- /dev/null
+++ b/experiments/pcam/README.md
@@ -0,0 +1,9 @@
+# PatchCamelyon Classification
+
+Binary classification on PatchCamelyon (96x96 histopathology patches) comparing bispectral pooling against norm, gate, and FourierELU nonlinearities in equivariant DenseNets.
+
+```bash
+pip install -e "../../[dev]"
+python train.py --model bispectrum --group c8 --data_dir ./pcam_data
+./run_matched_sweep.sh  # full Pareto sweep: 5 models x 5 growth rates x 3 seeds
+```
diff --git a/paper/experiments/pcam/analyze_data_pareto.py b/experiments/pcam/analyze_data_pareto.py
similarity index 82%
rename from paper/experiments/pcam/analyze_data_pareto.py
rename to experiments/pcam/analyze_data_pareto.py
index 8018f64..49e78fb 100644
--- a/paper/experiments/pcam/analyze_data_pareto.py
+++ b/experiments/pcam/analyze_data_pareto.py
@@ -1,5 +1,6 @@
 #!/usr/bin/env python3
 """Analyze data-efficiency sweep: AUC vs. absolute training-set size at matched params."""
+
 from __future__ import annotations
 
 import json
@@ -13,11 +14,11 @@
 PCAM_FULL_TRAIN = 262_144
 
 MODEL_STYLE: dict[str, dict] = {
-    'standard':    {'color': '#888888', 'marker': 's', 'label': 'Standard'},
-    'norm':        {'color': '#2196F3', 'marker': 'o', 'label': 'NormReLU'},
-    'gate':        {'color': '#FF9800', 'marker': '^', 'label': 'Gated'},
+    'standard': {'color': '#888888', 'marker': 's', 'label': 'Standard'},
+    'norm': {'color': '#2196F3', 'marker': 'o', 'label': 'NormReLU'},
+    'gate': {'color': '#FF9800', 'marker': '^', 'label': 'Gated'},
     'fourier_elu': {'color': '#9C27B0', 'marker': 'D', 'label': 'Fourier-ELU'},
-    'bispectrum':  {'color': '#E53935', 'marker': '*', 'label': 'Bispectrum'},
+    'bispectrum': {'color': '#E53935', 'marker': '*', 'label': 'Bispectrum'},
 }
 MODEL_ORDER = ['standard', 'norm', 'gate', 'fourier_elu', 'bispectrum']
 
@@ -61,10 +62,7 @@ def _train_size_value(record: dict) -> int:
 
 
 def load_all_results(base_dir: Path) -> list[dict]:
-    return [
-        json.loads(p.read_text())
-        for p in sorted(base_dir.glob('**/results.json'))
-    ]
+    return [json.loads(p.read_text()) for p in sorted(base_dir.glob('**/results.json'))]
 
 
 def aggregate(
@@ -109,7 +107,8 @@ def main() -> None:
     fig, axes = plt.subplots(1, 2, figsize=(14, 5.5))
     fig.suptitle(
         'PCam Data Efficiency: AUC vs. Training Examples (matched params)',
-        fontsize=13, y=0.98,
+        fontsize=13,
+        y=0.98,
     )
 
     ax = axes[0]
@@ -122,9 +121,16 @@ def main() -> None:
         stds = [agg[model][n]['std'] for n in sizes]
 
         ax.errorbar(
-            sizes, means, yerr=stds,
-            marker=s['marker'], color=s['color'], label=s['label'],
-            linewidth=2, markersize=8, capsize=4, zorder=3,
+            sizes,
+            means,
+            yerr=stds,
+            marker=s['marker'],
+            color=s['color'],
+            label=s['label'],
+            linewidth=2,
+            markersize=8,
+            capsize=4,
+            zorder=3,
         )
 
     ax.set_xscale('log')
@@ -144,13 +150,16 @@ def main() -> None:
                 continue
             s = MODEL_STYLE[model]
             sizes = sorted(set(agg[model].keys()) & set(standard_data.keys()))
-            deltas = [
-                agg[model][n]['mean'] - standard_data[n]['mean'] for n in sizes
-            ]
+            deltas = [agg[model][n]['mean'] - standard_data[n]['mean'] for n in sizes]
             ax.plot(
-                sizes, deltas,
-                marker=s['marker'], color=s['color'], label=s['label'],
-                linewidth=2, markersize=8, zorder=3,
+                sizes,
+                deltas,
+                marker=s['marker'],
+                color=s['color'],
+                label=s['label'],
+                linewidth=2,
+                markersize=8,
+                zorder=3,
             )
 
         ax.axhline(0, color='#888888', linestyle='--', alpha=0.5)
@@ -163,8 +172,15 @@ def main() -> None:
         ax.set_xticks(all_sizes)
         ax.set_xticklabels([f'{n:,}' for n in all_sizes])
     else:
-        ax.text(0.5, 0.5, 'No standard baseline data', transform=ax.transAxes,
-                ha='center', va='center', fontsize=12)
+        ax.text(
+            0.5,
+            0.5,
+            'No standard baseline data',
+            transform=ax.transAxes,
+            ha='center',
+            va='center',
+            fontsize=12,
+        )
 
     plt.tight_layout(rect=[0, 0, 1, 0.94])
     out_path = Path(__file__).parent / 'data_pareto_analysis.png'
diff --git a/paper/experiments/pcam/analyze_pareto.py b/experiments/pcam/analyze_pareto.py
similarity index 70%
rename from paper/experiments/pcam/analyze_pareto.py
rename to experiments/pcam/analyze_pareto.py
index 4591478..1920231 100644
--- a/paper/experiments/pcam/analyze_pareto.py
+++ b/experiments/pcam/analyze_pareto.py
@@ -1,5 +1,6 @@
 #!/usr/bin/env python3
 """Analyze Pareto sweep results and produce a multi-panel figure."""
+
 from __future__ import annotations
 
 import json
@@ -14,11 +15,11 @@
 DATA_PARETO_DIR = Path(__file__).parent / 'pcam_results_data_pareto'
 
 MODEL_STYLE = {
-    'standard':    {'color': '#888888', 'marker': 's', 'label': 'Standard'},
-    'norm':        {'color': '#2196F3', 'marker': 'o', 'label': 'NormReLU'},
-    'gate':        {'color': '#FF9800', 'marker': '^', 'label': 'Gated'},
+    'standard': {'color': '#888888', 'marker': 's', 'label': 'Standard'},
+    'norm': {'color': '#2196F3', 'marker': 'o', 'label': 'NormReLU'},
+    'gate': {'color': '#FF9800', 'marker': '^', 'label': 'Gated'},
     'fourier_elu': {'color': '#9C27B0', 'marker': 'D', 'label': 'Fourier-ELU'},
-    'bispectrum':  {'color': '#E53935', 'marker': '*', 'label': 'Bispectrum'},
+    'bispectrum': {'color': '#E53935', 'marker': '*', 'label': 'Bispectrum'},
 }
 MODEL_ORDER = ['standard', 'norm', 'gate', 'fourier_elu', 'bispectrum']
 
@@ -26,7 +27,10 @@
 
 
 def _canonical_train_mode(mode: str) -> str:
-    """Map legacy `NR` to canonical `C`. Pre-normalized runs default to `R`."""
+    """Map legacy `NR` to canonical `C`.
+
+    Pre-normalized runs default to `R`.
+    """
     if mode in {'C', 'NR'}:
         return 'C'
     if mode == 'R':
@@ -57,6 +61,7 @@ def load_results(directory: Path) -> list[dict]:
 def aggregate_1pct(results: list[dict], train_mode: str = DEFAULT_TRAIN_MODE) -> dict[str, dict]:
     """Group 1% results by model (filtered to *train_mode*), compute mean +/- std."""
     from collections import defaultdict
+
     grouped = defaultdict(list)
     for r in results:
         if _result_train_mode(r) != train_mode:
@@ -84,6 +89,7 @@ def aggregate_pareto_multiseed(
     Returns model -> sorted list of {n_params, mean_auc, std_auc, n_seeds}.
     """
     from collections import defaultdict
+
     grouped: dict[tuple[str, int], list[float]] = defaultdict(list)
     for r in results:
         if _result_train_mode(r) != train_mode:
@@ -92,12 +98,14 @@ def aggregate_pareto_multiseed(
 
     agg: dict[str, list[dict]] = defaultdict(list)
     for (model, n_params), aucs in grouped.items():
-        agg[model].append({
-            'n_params': n_params,
-            'mean_auc': float(np.mean(aucs)),
-            'std_auc': float(np.std(aucs, ddof=1)) if len(aucs) > 1 else 0.0,
-            'n_seeds': len(aucs),
-        })
+        agg[model].append(
+            {
+                'n_params': n_params,
+                'mean_auc': float(np.mean(aucs)),
+                'std_auc': float(np.std(aucs, ddof=1)) if len(aucs) > 1 else 0.0,
+                'n_seeds': len(aucs),
+            }
+        )
     for model in agg:
         agg[model].sort(key=lambda x: x['n_params'])
     return dict(agg)
@@ -107,7 +115,10 @@ def aggregate_pareto_multiseed(
 
 
 def _train_size_value(record: dict) -> int:
-    """Resolve a record's training-subset size. Treat None / <=0 / missing as full."""
+    """Resolve a record's training-subset size.
+
+    Treat None / <=0 / missing as full.
+    """
     raw = record.get('train_size')
     if raw is None:
         return PCAM_FULL_TRAIN
@@ -126,6 +137,7 @@ def load_data_pareto(
     Filters by *train_mode* (default `R`). Returns model -> {train_size: stats}.
     """
     from collections import defaultdict
+
     raw: dict[tuple[str, int], list[float]] = defaultdict(list)
     if not DATA_PARETO_DIR.is_dir():
         return {}
@@ -157,16 +169,18 @@ def _style_ax(ax: plt.Axes) -> None:
 
 
 def main() -> None:
-    plt.rcParams.update({
-        'font.family': 'serif',
-        'font.size': 11,
-        'axes.titlesize': 12,
-        'axes.labelsize': 11,
-        'xtick.labelsize': 10,
-        'ytick.labelsize': 10,
-        'legend.fontsize': 9,
-        'figure.dpi': 300,
-    })
+    plt.rcParams.update(
+        {
+            'font.family': 'serif',
+            'font.size': 11,
+            'axes.titlesize': 12,
+            'axes.labelsize': 11,
+            'xtick.labelsize': 10,
+            'ytick.labelsize': 10,
+            'legend.fontsize': 9,
+            'figure.dpi': 300,
+        }
+    )
 
     pareto = load_results(PARETO_DIR)
     pareto_agg = aggregate_pareto_multiseed(pareto)
@@ -185,13 +199,29 @@ def main() -> None:
 
         has_multiseed = any(p['n_seeds'] > 1 for p in pts)
         if has_multiseed:
-            ax1.errorbar(params, means, yerr=stds,
-                         marker=s['marker'], color=s['color'],
-                         label=s['label'], linewidth=2, markersize=7,
-                         capsize=3, zorder=3)
+            ax1.errorbar(
+                params,
+                means,
+                yerr=stds,
+                marker=s['marker'],
+                color=s['color'],
+                label=s['label'],
+                linewidth=2,
+                markersize=7,
+                capsize=3,
+                zorder=3,
+            )
         else:
-            ax1.plot(params, means, marker=s['marker'], color=s['color'],
-                     label=s['label'], linewidth=2, markersize=7, zorder=3)
+            ax1.plot(
+                params,
+                means,
+                marker=s['marker'],
+                color=s['color'],
+                label=s['label'],
+                linewidth=2,
+                markersize=7,
+                zorder=3,
+            )
 
     ax1.set_xscale('log')
     ax1.set_xlabel('Parameters')
@@ -200,8 +230,9 @@ def main() -> None:
     ax1.set_title('(a) AUC vs. parameter count (10% data)')
     _style_ax(ax1)
     ax1.yaxis.grid(True, linewidth=0.4, alpha=0.5, zorder=0)
-    ax1.legend(frameon=True, fancybox=False, edgecolor='#cccccc',
-               framealpha=0.95, loc='lower right')
+    ax1.legend(
+        frameon=True, fancybox=False, edgecolor='#cccccc', framealpha=0.95, loc='lower right'
+    )
 
     matched_data = []
     for model in MODEL_ORDER:
@@ -209,22 +240,29 @@ def main() -> None:
         if not pts:
             continue
         best_at_100k = min(pts, key=lambda x: abs(x['n_params'] - 100_000))
-        matched_data.append((model, best_at_100k['n_params'],
-                             best_at_100k['mean_auc'], best_at_100k['std_auc']))
+        matched_data.append(
+            (model, best_at_100k['n_params'], best_at_100k['mean_auc'], best_at_100k['std_auc'])
+        )
 
     matched_data.sort(key=lambda x: x[2])
     y_pos = np.arange(len(matched_data))
     colors = [MODEL_STYLE[m[0]]['color'] for m in matched_data]
     aucs_matched = [m[2] for m in matched_data]
     errs_matched = [m[3] for m in matched_data]
-    labels_matched = [
-        f'{MODEL_STYLE[m[0]]["label"]} ({m[1]//1000}K)'
-        for m in matched_data
-    ]
-
-    bars = ax2.barh(y_pos, aucs_matched, xerr=errs_matched, color=colors,
-                    edgecolor='white', linewidth=0.6, height=0.6, zorder=3,
-                    capsize=3, error_kw={'linewidth': 0.8, 'capthick': 0.8})
+    labels_matched = [f'{MODEL_STYLE[m[0]]["label"]} ({m[1] // 1000}K)' for m in matched_data]
+
+    bars = ax2.barh(
+        y_pos,
+        aucs_matched,
+        xerr=errs_matched,
+        color=colors,
+        edgecolor='white',
+        linewidth=0.6,
+        height=0.6,
+        zorder=3,
+        capsize=3,
+        error_kw={'linewidth': 0.8, 'capthick': 0.8},
+    )
     ax2.set_yticks(y_pos)
     ax2.set_yticklabels(labels_matched)
     ax2.set_xlabel('Test AUC')
@@ -233,12 +271,18 @@ def main() -> None:
     _style_ax(ax2)
     ax2.xaxis.grid(True, linewidth=0.4, alpha=0.5, zorder=0)
 
-    for bar, auc, err in zip(bars, aucs_matched, errs_matched):
+    for bar, auc, err in zip(bars, aucs_matched, errs_matched, strict=False):
         label = f'{auc:.3f}'
         if err > 0:
             label += f'\u00b1{err:.3f}'
-        ax2.text(auc + err + 0.001, bar.get_y() + bar.get_height() / 2,
-                 label, va='center', fontsize=8, fontweight='medium')
+        ax2.text(
+            auc + err + 0.001,
+            bar.get_y() + bar.get_height() / 2,
+            label,
+            va='center',
+            fontsize=8,
+            fontweight='medium',
+        )
 
     for model in MODEL_ORDER:
         s = MODEL_STYLE[model]
@@ -249,13 +293,29 @@ def main() -> None:
         stds = [data_eff[model][n]['std'] for n in sizes]
         has_multiseed = any(data_eff[model][n]['n_seeds'] > 1 for n in sizes)
         if has_multiseed:
-            ax3.errorbar(sizes, means, yerr=stds,
-                         marker=s['marker'], color=s['color'],
-                         label=s['label'], linewidth=2, markersize=7,
-                         capsize=3, zorder=3)
+            ax3.errorbar(
+                sizes,
+                means,
+                yerr=stds,
+                marker=s['marker'],
+                color=s['color'],
+                label=s['label'],
+                linewidth=2,
+                markersize=7,
+                capsize=3,
+                zorder=3,
+            )
         else:
-            ax3.plot(sizes, means, marker=s['marker'], color=s['color'],
-                     label=s['label'], linewidth=2, markersize=7, zorder=3)
+            ax3.plot(
+                sizes,
+                means,
+                marker=s['marker'],
+                color=s['color'],
+                label=s['label'],
+                linewidth=2,
+                markersize=7,
+                zorder=3,
+            )
 
     ax3.set_xscale('log')
     ax3.set_xlabel('Training examples (N)')
@@ -267,8 +327,14 @@ def main() -> None:
     ax3.set_ylim(0.75, 0.96)
     _style_ax(ax3)
     ax3.yaxis.grid(True, linewidth=0.4, alpha=0.5, zorder=0)
-    ax3.legend(frameon=True, fancybox=False, edgecolor='#cccccc',
-               framealpha=0.95, loc='lower right', fontsize=7)
+    ax3.legend(
+        frameon=True,
+        fancybox=False,
+        edgecolor='#cccccc',
+        framealpha=0.95,
+        loc='lower right',
+        fontsize=7,
+    )
 
     fig.tight_layout()
     out_path = Path(__file__).parent / 'pareto_analysis.png'
@@ -277,9 +343,9 @@ def main() -> None:
     print(f'Saved to {out_path} (+ PDF)')
 
     # Print text summary
-    print('\n' + '='*70)
+    print('\n' + '=' * 70)
     print('KEY FINDINGS')
-    print('='*70)
+    print('=' * 70)
 
     print('\n1. PARETO RANKING (full sweep, 10% data):')
     for model in MODEL_ORDER:
@@ -288,7 +354,9 @@ def main() -> None:
             continue
         best = max(pts, key=lambda x: x['mean_auc'])
         std_str = f' +/- {best["std_auc"]:.4f}' if best['n_seeds'] > 1 else ''
-        print(f'   {MODEL_STYLE[model]["label"]:25s}  best AUC={best["mean_auc"]:.4f}{std_str} at {best["n_params"]:>10,} params ({best["n_seeds"]} seeds)')
+        print(
+            f'   {MODEL_STYLE[model]["label"]:25s}  best AUC={best["mean_auc"]:.4f}{std_str} at {best["n_params"]:>10,} params ({best["n_seeds"]} seeds)'
+        )
 
     print('\n2. HEAD-TO-HEAD at ~100K params:')
     for m, params, auc, std in sorted(matched_data, key=lambda x: x[2], reverse=True):
@@ -301,7 +369,9 @@ def main() -> None:
         if len(pts) >= 2:
             slope = 'degrades' if pts[-1]['mean_auc'] < pts[0]['mean_auc'] else 'improves'
             delta = pts[-1]['mean_auc'] - pts[0]['mean_auc']
-            print(f'   {MODEL_STYLE[model]["label"]:25s}  {pts[0]["mean_auc"]:.4f} -> {pts[-1]["mean_auc"]:.4f}  ({slope}, {delta:+.4f})')
+            print(
+                f'   {MODEL_STYLE[model]["label"]:25s}  {pts[0]["mean_auc"]:.4f} -> {pts[-1]["mean_auc"]:.4f}  ({slope}, {delta:+.4f})'
+            )
 
     print('\n4. DATA EFFICIENCY (matched ~100K params):')
     for m in MODEL_ORDER:
diff --git a/paper/experiments/pcam/check_memory.py b/experiments/pcam/check_memory.py
similarity index 96%
rename from paper/experiments/pcam/check_memory.py
rename to experiments/pcam/check_memory.py
index 6bb59ac..0fb8c2f 100644
--- a/paper/experiments/pcam/check_memory.py
+++ b/experiments/pcam/check_memory.py
@@ -1,5 +1,6 @@
 #!/usr/bin/env python3
 """Dry-run 1 forward+backward step per model config to verify GPU memory fits."""
+
 from __future__ import annotations
 
 import gc
@@ -12,15 +13,16 @@
 from model import build_model
 
 CONFIGS = {
-    'standard':    {'gr': 12},
-    'norm':        {'gr': 4},
-    'gate':        {'gr': 3},
+    'standard': {'gr': 12},
+    'norm': {'gr': 4},
+    'gate': {'gr': 3},
     'fourier_elu': {'gr': 4},
-    'bispectrum':  {'gr': 4},
+    'bispectrum': {'gr': 4},
 }
 
 BATCH_SIZES = [256, 128, 64, 32]
 
+
 def try_step(model_name: str, gr: int, bs: int, device: torch.device) -> tuple[bool, float]:
     torch.cuda.empty_cache()
     gc.collect()
diff --git a/paper/experiments/pcam/data.py b/experiments/pcam/data.py
similarity index 96%
rename from paper/experiments/pcam/data.py
rename to experiments/pcam/data.py
index b7570b0..08d6e0d 100644
--- a/paper/experiments/pcam/data.py
+++ b/experiments/pcam/data.py
@@ -12,15 +12,15 @@
 from __future__ import annotations
 
 import gzip
-import shutil
 import json
+import shutil
 from pathlib import Path
 
 import h5py
 import torch
+import torchvision.transforms.functional as TF
 from torch.utils.data import DataLoader, Dataset, Subset
 from torchvision.transforms import InterpolationMode
-import torchvision.transforms.functional as TF
 
 PCAM_URLS = {
     'train_x': 'https://zenodo.org/records/2546921/files/camelyonpatch_level_2_split_train_x.h5.gz',
@@ -60,8 +60,8 @@ def _download_and_extract(url: str, dest_dir: str) -> str:
 class PCamDataset(Dataset):
     """PatchCamelyon dataset fully loaded into RAM.
 
-    Images are 96x96 RGB, stored as a float32 (N, 3, H, W) tensor in [0, 1].
-    Labels are a 1-D int64 tensor.
+    Images are 96x96 RGB, stored as a float32 (N, 3, H, W) tensor in [0, 1]. Labels are a 1-D int64
+    tensor.
     """
 
     def __init__(
@@ -105,8 +105,8 @@ def __getitem__(self, idx: int) -> tuple[torch.Tensor, int]:
 def gpu_normalize_only(images: torch.Tensor) -> torch.Tensor:
     """Apply only the PCam channel normalization (no augmentation).
 
-    Use on raw [0, 1] tensors at evaluation time so train/test rotation
-    pipelines see the same input space.
+    Use on raw [0, 1] tensors at evaluation time so train/test rotation pipelines see the same
+    input space.
     """
     dev = images.device
     mean = torch.tensor(_MEAN, device=dev).reshape(1, 3, 1, 1)
@@ -163,8 +163,10 @@ def apply_group_transform_batch(
             if not mask.any():
                 continue
             angle = float(45 * k)
-            transformed = images if k == 0 else TF.rotate(
-                images, angle, interpolation=InterpolationMode.BILINEAR
+            transformed = (
+                images
+                if k == 0
+                else TF.rotate(images, angle, interpolation=InterpolationMode.BILINEAR)
             )
             out = torch.where(mask, transformed, out)
         return out
@@ -182,7 +184,9 @@ def apply_group_transform_batch(
     return out
 
 
-def apply_group_element(images: torch.Tensor, geometry_group: str, element_idx: int) -> torch.Tensor:
+def apply_group_element(
+    images: torch.Tensor, geometry_group: str, element_idx: int
+) -> torch.Tensor:
     """Apply a deterministic group element to a batch."""
     if geometry_group == 'c8':
         if element_idx == 0:
diff --git a/paper/experiments/pcam/find_growth_rates.py b/experiments/pcam/find_growth_rates.py
similarity index 94%
rename from paper/experiments/pcam/find_growth_rates.py
rename to experiments/pcam/find_growth_rates.py
index 2ce5b9f..c20b9c7 100644
--- a/paper/experiments/pcam/find_growth_rates.py
+++ b/experiments/pcam/find_growth_rates.py
@@ -1,12 +1,11 @@
 #!/usr/bin/env python3
 """Print parameter counts for all (model, growth_rate) combinations.
 
-Helps pick per-model growth_rate values that match a target param budget.
-The normalized rerun excludes the standalone so2_disk baseline.
+Helps pick per-model growth_rate values that match a target param budget. The normalized rerun
+excludes the standalone so2_disk baseline.
 """
-from __future__ import annotations
 
-import sys
+from __future__ import annotations
 
 from model import build_model
 
@@ -16,6 +15,7 @@
 GROUP = 'c8'
 BLOCK_CONFIG = (4, 4, 4)
 
+
 def main() -> None:
     best: dict[str, tuple[int, int, int]] = {}
 
@@ -44,5 +44,6 @@ def main() -> None:
         gr, n, _ = best[m]
         print(f'{m:>15}  {gr:>3}  {n:>10,}')
 
+
 if __name__ == '__main__':
     main()
diff --git a/paper/experiments/pcam/model.py b/experiments/pcam/model.py
similarity index 99%
rename from paper/experiments/pcam/model.py
rename to experiments/pcam/model.py
index bdfd6ef..1b2ad33 100644
--- a/paper/experiments/pcam/model.py
+++ b/experiments/pcam/model.py
@@ -205,7 +205,11 @@ def forward(self, x: torch.Tensor) -> torch.Tensor:
             w_flat = w_reindexed.reshape(G * n, 1, K, K)
             g_expanded = self.grids.unsqueeze(1).expand(-1, n, -1, -1, -1).reshape(G * n, K, K, 2)
             w_rot = F.grid_sample(
-                w_flat, g_expanded, mode='bilinear', padding_mode='zeros', align_corners=True,
+                w_flat,
+                g_expanded,
+                mode='bilinear',
+                padding_mode='zeros',
+                align_corners=True,
             )
             w_rot = w_rot.reshape(G * C_out, C_in * G, K, K)
 
diff --git a/paper/experiments/pcam/run_data_pareto_sweep.sh b/experiments/pcam/run_data_pareto_sweep.sh
similarity index 97%
rename from paper/experiments/pcam/run_data_pareto_sweep.sh
rename to experiments/pcam/run_data_pareto_sweep.sh
index 50c7530..e747204 100755
--- a/paper/experiments/pcam/run_data_pareto_sweep.sh
+++ b/experiments/pcam/run_data_pareto_sweep.sh
@@ -22,8 +22,9 @@ set -euo pipefail
 SCRIPT_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)"
 cd "$SCRIPT_DIR"
 
-source /home/johmathe/bispectrum/.venv/bin/activate
-export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}"
+REPO_ROOT="$(cd "$SCRIPT_DIR/../.." && pwd)"
+source "$REPO_ROOT/.venv/bin/activate"
+export PYTHONPATH="$REPO_ROOT/src:${PYTHONPATH:-}"
 export PYTHONUNBUFFERED=1
 
 declare -A MODEL_GR
diff --git a/paper/experiments/pcam/run_matched_sweep.sh b/experiments/pcam/run_matched_sweep.sh
similarity index 96%
rename from paper/experiments/pcam/run_matched_sweep.sh
rename to experiments/pcam/run_matched_sweep.sh
index c1302d2..4719a15 100755
--- a/paper/experiments/pcam/run_matched_sweep.sh
+++ b/experiments/pcam/run_matched_sweep.sh
@@ -22,8 +22,9 @@ set -euo pipefail
 SCRIPT_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)"
 cd "$SCRIPT_DIR"
 
-source /home/johmathe/bispectrum/.venv/bin/activate
-export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}"
+REPO_ROOT="$(cd "$SCRIPT_DIR/../.." && pwd)"
+source "$REPO_ROOT/.venv/bin/activate"
+export PYTHONPATH="$REPO_ROOT/src:${PYTHONPATH:-}"
 export PYTHONUNBUFFERED=1
 
 STANDARD_GRS=(6 12 20 30 35)
diff --git a/paper/experiments/pcam/train.py b/experiments/pcam/train.py
similarity index 96%
rename from paper/experiments/pcam/train.py
rename to experiments/pcam/train.py
index 63bc49e..da67b95 100644
--- a/paper/experiments/pcam/train.py
+++ b/experiments/pcam/train.py
@@ -45,7 +45,10 @@ def _git_sha() -> str:
         out = subprocess.run(
             ['git', 'rev-parse', 'HEAD'],
             cwd=Path(__file__).resolve().parent,
-            capture_output=True, text=True, timeout=5, check=False,
+            capture_output=True,
+            text=True,
+            timeout=5,
+            check=False,
         )
         return out.stdout.strip() if out.returncode == 0 else ''
     except (OSError, subprocess.SubprocessError):
@@ -375,7 +378,9 @@ def train(args: argparse.Namespace) -> dict:
     if args.model == 'so2_disk':
         print(f'Model: {args.model} (bandlimit={args.bandlimit}), {n_params:,} params')
     else:
-        print(f'Model: {args.model} (group={args.group}, gr={args.growth_rate}), {n_params:,} params')
+        print(
+            f'Model: {args.model} (group={args.group}, gr={args.growth_rate}), {n_params:,} params'
+        )
 
     if args.dry_run:
         info: dict = {
@@ -422,9 +427,15 @@ def train(args: argparse.Namespace) -> dict:
     patience_counter = 0
     n_tag = f'_n{args.train_size}' if args.train_size and args.train_size > 0 else ''
     if args.model == 'so2_disk':
-        out_dir = Path(args.output_dir) / f'{args.model}_bl{args.bandlimit:.0f}_{args.train_mode}_seed{args.seed}{n_tag}'
+        out_dir = (
+            Path(args.output_dir)
+            / f'{args.model}_bl{args.bandlimit:.0f}_{args.train_mode}_seed{args.seed}{n_tag}'
+        )
     else:
-        out_dir = Path(args.output_dir) / f'{args.model}_{args.group}_gr{args.growth_rate}_{args.train_mode}_seed{args.seed}{n_tag}'
+        out_dir = (
+            Path(args.output_dir)
+            / f'{args.model}_{args.group}_gr{args.growth_rate}_{args.train_mode}_seed{args.seed}{n_tag}'
+        )
     out_dir.mkdir(parents=True, exist_ok=True)
 
     if torch.cuda.is_available():
@@ -434,7 +445,12 @@ def train(args: argparse.Namespace) -> dict:
     for epoch in range(1, args.epochs + 1):
         t0 = time.time()
         train_loss = train_one_epoch(
-            model, train_loader, optimizer, scheduler, device, scaler,
+            model,
+            train_loader,
+            optimizer,
+            scheduler,
+            device,
+            scaler,
             augment_geometry=augment_geo,
             geometry_group=geometry_group,
         )
@@ -609,7 +625,7 @@ def main():
         type=int,
         default=None,
         help='Subset training data to N examples for data-efficiency experiments. '
-             'Omit (or pass <=0) to use the full training set.',
+        'Omit (or pass <=0) to use the full training set.',
     )
     parser.add_argument(
         '--train_mode',
diff --git a/paper/experiments/spherical_mnist/.gitignore b/experiments/spherical_mnist/.gitignore
similarity index 100%
rename from paper/experiments/spherical_mnist/.gitignore
rename to experiments/spherical_mnist/.gitignore
diff --git a/experiments/spherical_mnist/README.md b/experiments/spherical_mnist/README.md
new file mode 100644
index 0000000..87ec0ad
--- /dev/null
+++ b/experiments/spherical_mnist/README.md
@@ -0,0 +1,9 @@
+# Spherical MNIST Classification
+
+MNIST digits projected onto S2, classified using SO(3)-invariant bispectral features vs. power spectrum and standard CNN baselines.
+
+```bash
+pip install -e "../../[dev]"
+python train.py --model bispectrum --train_mode NR
+./run_sweep.sh  # full sweep: 3 models x 2 modes x 3 seeds
+```
diff --git a/paper/experiments/spherical_mnist/analyze_results.py b/experiments/spherical_mnist/analyze_results.py
similarity index 85%
rename from paper/experiments/spherical_mnist/analyze_results.py
rename to experiments/spherical_mnist/analyze_results.py
index 91ce2c5..a2d4572 100644
--- a/paper/experiments/spherical_mnist/analyze_results.py
+++ b/experiments/spherical_mnist/analyze_results.py
@@ -21,6 +21,7 @@
 from pathlib import Path
 
 import matplotlib
+
 matplotlib.use('Agg')
 import matplotlib.pyplot as plt
 import matplotlib.ticker as mticker
@@ -98,6 +99,7 @@ def _train_size_value(record: dict) -> int | None:
 def _is_full_train(record: dict) -> bool:
     return _train_size_value(record) is None
 
+
 NEURIPS_RCPARAMS = {
     'font.family': 'serif',
     'font.serif': ['Computer Modern Roman', 'CMU Serif', 'DejaVu Serif'],
@@ -198,7 +200,9 @@ def print_cohen_table(grouped: dict[tuple[str, str], list[dict]]):
     print('ACCURACY TABLE (Cohen et al. 2018 protocol, canonical/random)')
     print('Train/Test: X/Y = trained on X, evaluated on Y. Cohen reference uses NR for canonical.')
     print('=' * 100)
-    hdr = f'{"Method":<32} {"Params":>10}  {"C/C":>12}  {"R/R":>12}  {"C/R":>12}  {"Rot \u03c3":>10}'
+    hdr = (
+        f'{"Method":<32} {"Params":>10}  {"C/C":>12}  {"R/R":>12}  {"C/R":>12}  {"Rot \u03c3":>10}'
+    )
     print(hdr)
     print('-' * 100)
 
@@ -340,7 +344,7 @@ def plot_data_efficiency(
 
     fig, axes = plt.subplots(1, 2, figsize=(6.5, 3.0), sharey=True)
 
-    for ax, (train_mode, test_fn, title) in zip(axes, panels):
+    for ax, (train_mode, test_fn, title) in zip(axes, panels, strict=False):
         for model in models_to_plot:
             means = []
             stds = []
@@ -363,10 +367,16 @@ def plot_data_efficiency(
             stds_arr = np.array(stds)
 
             ax.errorbar(
-                valid_sizes, means_arr, yerr=stds_arr,
-                marker=markers[model], color=colors[model],
-                label=labels[model], linewidth=1.2, markersize=5,
-                capsize=3, zorder=3,
+                valid_sizes,
+                means_arr,
+                yerr=stds_arr,
+                marker=markers[model],
+                color=colors[model],
+                label=labels[model],
+                linewidth=1.2,
+                markersize=5,
+                capsize=3,
+                zorder=3,
             )
 
         ax.set_xscale('log')
@@ -379,7 +389,9 @@ def plot_data_efficiency(
 
     axes[0].set_ylabel('Test accuracy')
     axes[0].legend(
-        frameon=False, fontsize=7, loc='lower right',
+        frameon=False,
+        fontsize=7,
+        loc='lower right',
     )
 
     fig.tight_layout()
@@ -412,9 +424,17 @@ def print_data_efficiency_table(
                 c_accs = [_test_c(r)['accuracy'] for r in runs]
                 r_accs = [_test_r(r)['accuracy'] for r in runs if _test_r(r)]
                 n = len(runs)
-                c_str = f'{np.mean(c_accs):.4f}\u00b1{np.std(c_accs):.4f}' if n > 1 else f'{c_accs[0]:.4f}'
+                c_str = (
+                    f'{np.mean(c_accs):.4f}\u00b1{np.std(c_accs):.4f}'
+                    if n > 1
+                    else f'{c_accs[0]:.4f}'
+                )
                 if r_accs:
-                    r_str = f'{np.mean(r_accs):.4f}\u00b1{np.std(r_accs):.4f}' if len(r_accs) > 1 else f'{r_accs[0]:.4f}'
+                    r_str = (
+                        f'{np.mean(r_accs):.4f}\u00b1{np.std(r_accs):.4f}'
+                        if len(r_accs) > 1
+                        else f'{r_accs[0]:.4f}'
+                    )
                 else:
                     r_str = '\u2014'
                 print(f'{model:<20} {mode:<5} {size:>7d} {c_str:>15} {r_str:>15} {n:>6}')
@@ -434,16 +454,15 @@ def plot_combined_figure(
     """NeurIPS-quality 2-panel figure: (a) Cohen protocol, (b) rotation robustness."""
     plt.rcParams.update(NEURIPS_RCPARAMS)
 
-    models_with_data = [
-        m for m in MODEL_ORDER
-        if grouped.get((m, 'C')) or grouped.get((m, 'R'))
-    ]
+    models_with_data = [m for m in MODEL_ORDER if grouped.get((m, 'C')) or grouped.get((m, 'R'))]
     if not models_with_data:
         print('No data to plot.')
         return
 
     fig, (ax_a, ax_b) = plt.subplots(
-        1, 2, figsize=(6.5, 3.0),
+        1,
+        2,
+        figsize=(6.5, 3.0),
         gridspec_kw={'width_ratios': [1.4, 1], 'wspace': 0.40},
     )
     fig.subplots_adjust(bottom=0.27)
@@ -452,12 +471,22 @@ def plot_combined_figure(
     _plot_rotation_panel(ax_b, grouped, models_with_data)
 
     ax_a.text(
-        -0.08, 1.05, '(a)', transform=ax_a.transAxes,
-        fontsize=10, fontweight='bold', va='bottom',
+        -0.08,
+        1.05,
+        '(a)',
+        transform=ax_a.transAxes,
+        fontsize=10,
+        fontweight='bold',
+        va='bottom',
     )
     ax_b.text(
-        -0.10, 1.05, '(b)', transform=ax_b.transAxes,
-        fontsize=10, fontweight='bold', va='bottom',
+        -0.10,
+        1.05,
+        '(b)',
+        transform=ax_b.transAxes,
+        fontsize=10,
+        fontweight='bold',
+        va='bottom',
     )
 
     fig.savefig(output_path, dpi=300)
@@ -496,13 +525,19 @@ def _plot_cohen_panel(
 
         offset = (i - 1) * width
         bars = ax.bar(
-            x + offset, vals, width, yerr=errs,
-            label=proto, color=PROTOCOL_COLORS[proto],
-            edgecolor='white', linewidth=0.4,
-            capsize=2, error_kw={'linewidth': 0.6, 'capthick': 0.6},
+            x + offset,
+            vals,
+            width,
+            yerr=errs,
+            label=proto,
+            color=PROTOCOL_COLORS[proto],
+            edgecolor='white',
+            linewidth=0.4,
+            capsize=2,
+            error_kw={'linewidth': 0.6, 'capthick': 0.6},
             zorder=3,
         )
-        for j, (bar, v) in enumerate(zip(bars, vals)):
+        for j, (bar, v) in enumerate(zip(bars, vals, strict=False)):
             if v <= 0:
                 continue
             show_label = True
@@ -515,19 +550,30 @@ def _plot_cohen_panel(
                 ax.text(
                     bar.get_x() + bar.get_width() / 2,
                     y_pos,
-                    f'{v:.2f}', ha='center', va='bottom', fontsize=5.5,
+                    f'{v:.2f}',
+                    ha='center',
+                    va='bottom',
+                    fontsize=5.5,
                 )
         vals_prev = vals
 
     cohen_s2 = COHEN_RESULTS['Cohen spherical CNN (NR/X)']
     ax.axhline(
-        cohen_s2['C/C'], color='#888888', linewidth=0.6,
-        linestyle='--', zorder=2, alpha=0.6,
+        cohen_s2['C/C'],
+        color='#888888',
+        linewidth=0.6,
+        linestyle='--',
+        zorder=2,
+        alpha=0.6,
         label=f'Cohen S\u00b2CNN C/C ({cohen_s2["C/C"]:.2f})',
     )
     ax.axhline(
-        cohen_s2['C/R'], color='#888888', linewidth=0.6,
-        linestyle=':', zorder=2, alpha=0.6,
+        cohen_s2['C/R'],
+        color='#888888',
+        linewidth=0.6,
+        linestyle=':',
+        zorder=2,
+        alpha=0.6,
         label=f'Cohen S\u00b2CNN C/R ({cohen_s2["C/R"]:.2f})',
     )
 
@@ -539,7 +585,8 @@ def _label_for(m: str) -> str:
     ax.set_xticks(x)
     ax.set_xticklabels(
         [_label_for(m) for m in models],
-        rotation=20, ha='right',
+        rotation=20,
+        ha='right',
     )
     ax.set_ylabel('Test accuracy')
     ax.set_ylim(0, 1.15)
@@ -547,9 +594,13 @@ def _label_for(m: str) -> str:
     _clean_axes(ax)
     handles, labels = ax.get_legend_handles_labels()
     ax.legend(
-        handles, labels,
-        frameon=False, ncol=3, handlelength=1.5,
-        fontsize=6, loc='upper center',
+        handles,
+        labels,
+        frameon=False,
+        ncol=3,
+        handlelength=1.5,
+        fontsize=6,
+        loc='upper center',
         bbox_to_anchor=(0.5, -0.18),
         columnspacing=1.2,
     )
@@ -580,22 +631,35 @@ def _plot_rotation_panel(
         color = COLORS[m]
 
         ax.scatter(
-            np.full_like(pts, i) + jitter, pts,
-            s=12, alpha=0.55, color=color, edgecolors='white',
-            linewidths=0.3, zorder=3,
+            np.full_like(pts, i) + jitter,
+            pts,
+            s=12,
+            alpha=0.55,
+            color=color,
+            edgecolors='white',
+            linewidths=0.3,
+            zorder=3,
         )
 
         mean_val = np.mean(pts)
         ax.hlines(
-            mean_val, i - 0.25, i + 0.25,
-            colors=color, linewidths=1.2, zorder=4,
+            mean_val,
+            i - 0.25,
+            i + 0.25,
+            colors=color,
+            linewidths=1.2,
+            zorder=4,
         )
 
         y_offset = 0.03 if m != 'standard' else 0.04
         ax.text(
-            i + 0.30, mean_val + y_offset,
+            i + 0.30,
+            mean_val + y_offset,
             f'{mean_val:.1%}',
-            fontsize=6.5, color=color, va='bottom', fontweight='medium',
+            fontsize=6.5,
+            color=color,
+            va='bottom',
+            fontweight='medium',
         )
 
     def _label_for(m: str) -> str:
@@ -606,7 +670,8 @@ def _label_for(m: str) -> str:
     ax.set_xticks(range(len(models)))
     ax.set_xticklabels(
         [_label_for(m) for m in models],
-        rotation=20, ha='right',
+        rotation=20,
+        ha='right',
     )
     ax.set_xlim(-0.6, len(models) - 0.4)
     ax.set_ylabel('Accuracy under random SO(3)')
@@ -616,8 +681,12 @@ def _label_for(m: str) -> str:
 
     ax.axhline(0.10, color='#aaaaaa', linewidth=0.5, linestyle=':', zorder=1)
     ax.text(
-        len(models) - 0.5, 0.115, 'chance (10%)',
-        fontsize=6, color='#aaaaaa', ha='right',
+        len(models) - 0.5,
+        0.115,
+        'chance (10%)',
+        fontsize=6,
+        color='#aaaaaa',
+        ha='right',
     )
 
 
@@ -628,7 +697,9 @@ def main():
     parser.add_argument('--results_dir', type=str, default='./smnist_results')
     parser.add_argument('--matched_dir', type=str, default='./smnist_results_matched')
     parser.add_argument(
-        '--output', type=str, default='./smnist_analysis.png',
+        '--output',
+        type=str,
+        default='./smnist_analysis.png',
     )
     args = parser.parse_args()
 
diff --git a/paper/experiments/spherical_mnist/data.py b/experiments/spherical_mnist/data.py
similarity index 84%
rename from paper/experiments/spherical_mnist/data.py
rename to experiments/spherical_mnist/data.py
index b470846..ecee9b3 100644
--- a/paper/experiments/spherical_mnist/data.py
+++ b/experiments/spherical_mnist/data.py
@@ -70,7 +70,11 @@ def project_to_sphere(
         batch = images[i : i + batch_size]
         g = grid_exp.expand(batch.shape[0], -1, -1, -1)
         proj = F.grid_sample(
-            batch, g, mode='bilinear', padding_mode='zeros', align_corners=True,
+            batch,
+            g,
+            mode='bilinear',
+            padding_mode='zeros',
+            align_corners=True,
         )
         chunks.append(proj.squeeze(1))
 
@@ -97,7 +101,8 @@ def apply_random_rotations(
         torch.manual_seed(seed * 100_000 + i)
         R = random_rotation_matrix()
         result[i] = rotate_spherical_function(
-            spherical_images[i : i + 1], R,
+            spherical_images[i : i + 1],
+            R,
         ).squeeze(0)
 
         if (i + 1) % 5000 == 0 or i == 0:
@@ -110,8 +115,8 @@ def apply_random_rotations(
 class SphericalMNISTDataset(Dataset):
     """MNIST digits projected onto the sphere.
 
-    Precomputes and caches the spherical projection (and optional rotation)
-    so that subsequent runs load instantly.
+    Precomputes and caches the spherical projection (and optional rotation) so that subsequent runs
+    load instantly.
     """
 
     def __init__(
@@ -187,16 +192,35 @@ def get_dataloaders(
         (train_loader, val_loader, test_nr_loader, test_r_loader)
     """
     train_ds = SphericalMNISTDataset(
-        'train', data_dir, nlat, nlon, rotated=train_rotated, seed=seed,
+        'train',
+        data_dir,
+        nlat,
+        nlon,
+        rotated=train_rotated,
+        seed=seed,
     )
     val_ds = SphericalMNISTDataset(
-        'val', data_dir, nlat, nlon, rotated=train_rotated, seed=seed,
+        'val',
+        data_dir,
+        nlat,
+        nlon,
+        rotated=train_rotated,
+        seed=seed,
     )
     test_nr_ds = SphericalMNISTDataset(
-        'test', data_dir, nlat, nlon, rotated=False,
+        'test',
+        data_dir,
+        nlat,
+        nlon,
+        rotated=False,
     )
     test_r_ds = SphericalMNISTDataset(
-        'test', data_dir, nlat, nlon, rotated=True, seed=test_rotation_seed,
+        'test',
+        data_dir,
+        nlat,
+        nlon,
+        rotated=True,
+        seed=test_rotation_seed,
     )
 
     n_full = len(train_ds)
@@ -212,19 +236,33 @@ def get_dataloaders(
 
     loader_rng = torch.Generator().manual_seed(seed)
     train_loader = DataLoader(
-        train_ds, batch_size=min(batch_size, len(train_ds)), shuffle=True,
-        generator=loader_rng, num_workers=0, pin_memory=True, drop_last=False,
+        train_ds,
+        batch_size=min(batch_size, len(train_ds)),
+        shuffle=True,
+        generator=loader_rng,
+        num_workers=0,
+        pin_memory=True,
+        drop_last=False,
     )
     val_loader = DataLoader(
-        val_ds, batch_size=batch_size, shuffle=False,
-        num_workers=0, pin_memory=True,
+        val_ds,
+        batch_size=batch_size,
+        shuffle=False,
+        num_workers=0,
+        pin_memory=True,
     )
     test_nr_loader = DataLoader(
-        test_nr_ds, batch_size=batch_size, shuffle=False,
-        num_workers=0, pin_memory=True,
+        test_nr_ds,
+        batch_size=batch_size,
+        shuffle=False,
+        num_workers=0,
+        pin_memory=True,
     )
     test_r_loader = DataLoader(
-        test_r_ds, batch_size=batch_size, shuffle=False,
-        num_workers=0, pin_memory=True,
+        test_r_ds,
+        batch_size=batch_size,
+        shuffle=False,
+        num_workers=0,
+        pin_memory=True,
     )
     return train_loader, val_loader, test_nr_loader, test_r_loader
diff --git a/paper/experiments/spherical_mnist/find_budget.py b/experiments/spherical_mnist/find_budget.py
similarity index 100%
rename from paper/experiments/spherical_mnist/find_budget.py
rename to experiments/spherical_mnist/find_budget.py
diff --git a/paper/experiments/spherical_mnist/model.py b/experiments/spherical_mnist/model.py
similarity index 91%
rename from paper/experiments/spherical_mnist/model.py
rename to experiments/spherical_mnist/model.py
index de07563..c88b694 100644
--- a/paper/experiments/spherical_mnist/model.py
+++ b/experiments/spherical_mnist/model.py
@@ -13,7 +13,6 @@
 
 import torch
 import torch.nn as nn
-import torch.nn.functional as F
 from torch_harmonics import RealSHT
 
 from bispectrum import SO3onS2
@@ -39,7 +38,10 @@ def __init__(
     ):
         super().__init__()
         self.bispectrum = SO3onS2(
-            lmax=lmax, nlat=nlat, nlon=nlon, selective=selective,
+            lmax=lmax,
+            nlat=nlat,
+            nlon=nlon,
+            selective=selective,
         )
         n_features = self.bispectrum.output_size * 2
         self.mlp = nn.Sequential(
@@ -81,8 +83,12 @@ def __init__(
         self.lmax = lmax
         sht_lmax = lmax + 1
         self._sht = RealSHT(
-            nlat, nlon, lmax=sht_lmax, mmax=sht_lmax,
-            grid='equiangular', norm='ortho',
+            nlat,
+            nlon,
+            lmax=sht_lmax,
+            mmax=sht_lmax,
+            grid='equiangular',
+            norm='ortho',
         )
         self.mlp = nn.Sequential(
             nn.Linear(lmax + 1, hidden),
@@ -162,12 +168,17 @@ def build_model(
     """Build a model for the Spherical MNIST experiment."""
     if model_type == 'bispectrum':
         return BispectrumClassifier(
-            lmax=lmax, nlat=nlat, nlon=nlon,
-            selective=selective, hidden=hidden,
+            lmax=lmax,
+            nlat=nlat,
+            nlon=nlon,
+            selective=selective,
+            hidden=hidden,
         )
     elif model_type == 'power_spectrum':
         return PowerSpectrumClassifier(
-            lmax=lmax, nlat=nlat, nlon=nlon,
+            lmax=lmax,
+            nlat=nlat,
+            nlon=nlon,
             hidden=max(64, hidden // 2),
         )
     elif model_type == 'standard':
diff --git a/paper/experiments/spherical_mnist/run_data_efficiency.sh b/experiments/spherical_mnist/run_data_efficiency.sh
similarity index 92%
rename from paper/experiments/spherical_mnist/run_data_efficiency.sh
rename to experiments/spherical_mnist/run_data_efficiency.sh
index aa8425a..a954c9e 100755
--- a/paper/experiments/spherical_mnist/run_data_efficiency.sh
+++ b/experiments/spherical_mnist/run_data_efficiency.sh
@@ -10,8 +10,9 @@ set -euo pipefail
 SCRIPT_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)"
 cd "$SCRIPT_DIR"
 
-source /home/johmathe/bispectrum/.venv/bin/activate
-export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}"
+REPO_ROOT="$(cd "$SCRIPT_DIR/../.." && pwd)"
+source "$REPO_ROOT/.venv/bin/activate"
+export PYTHONPATH="$REPO_ROOT/src:${PYTHONPATH:-}"
 export PYTHONUNBUFFERED=1
 
 MODELS=(standard power_spectrum bispectrum)
diff --git a/paper/experiments/spherical_mnist/run_sweep.sh b/experiments/spherical_mnist/run_sweep.sh
similarity index 92%
rename from paper/experiments/spherical_mnist/run_sweep.sh
rename to experiments/spherical_mnist/run_sweep.sh
index 80a7651..88b9207 100755
--- a/paper/experiments/spherical_mnist/run_sweep.sh
+++ b/experiments/spherical_mnist/run_sweep.sh
@@ -18,8 +18,9 @@ set -euo pipefail
 SCRIPT_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)"
 cd "$SCRIPT_DIR"
 
-source /home/johmathe/bispectrum/.venv/bin/activate
-export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}"
+REPO_ROOT="$(cd "$SCRIPT_DIR/../.." && pwd)"
+source "$REPO_ROOT/.venv/bin/activate"
+export PYTHONPATH="$REPO_ROOT/src:${PYTHONPATH:-}"
 export PYTHONUNBUFFERED=1
 
 MODELS=(standard power_spectrum bispectrum)
diff --git a/paper/experiments/spherical_mnist/train.py b/experiments/spherical_mnist/train.py
similarity index 91%
rename from paper/experiments/spherical_mnist/train.py
rename to experiments/spherical_mnist/train.py
index 97cdee2..9be4450 100644
--- a/paper/experiments/spherical_mnist/train.py
+++ b/experiments/spherical_mnist/train.py
@@ -30,11 +30,11 @@
 
 import torch
 import torch.nn.functional as F
-
-from bispectrum import random_rotation_matrix, rotate_spherical_function
 from data import get_dataloaders
 from model import build_model
 
+from bispectrum import random_rotation_matrix, rotate_spherical_function
+
 NUM_CLASSES = 10
 
 
@@ -43,7 +43,10 @@ def _git_sha() -> str:
         out = subprocess.run(
             ['git', 'rev-parse', 'HEAD'],
             cwd=Path(__file__).resolve().parent,
-            capture_output=True, text=True, timeout=5, check=False,
+            capture_output=True,
+            text=True,
+            timeout=5,
+            check=False,
         )
         return out.stdout.strip() if out.returncode == 0 else ''
     except (OSError, subprocess.SubprocessError):
@@ -126,8 +129,8 @@ def evaluate_rotation_robustness(
 ) -> dict:
     """Evaluate model consistency under random SO(3) rotations.
 
-    Applies the SAME rotation to ALL test images for each trial, testing
-    whether predictions are truly rotation-invariant.
+    Applies the SAME rotation to ALL test images for each trial, testing whether predictions are
+    truly rotation-invariant.
     """
     model.eval()
     images = test_dataset.images
@@ -258,7 +261,9 @@ def memory_check(args: argparse.Namespace) -> dict:
         images, labels = next(iter(train_loader))
         images = images.to(device, non_blocking=True)
         labels = labels.to(device, non_blocking=True)
-        optimizer = torch.optim.AdamW(model.parameters(), lr=args.lr, weight_decay=args.weight_decay)
+        optimizer = torch.optim.AdamW(
+            model.parameters(), lr=args.lr, weight_decay=args.weight_decay
+        )
         optimizer.zero_grad()
         with torch.amp.autocast('cuda'):
             logits = model(images)
@@ -312,7 +317,10 @@ def memory_check(args: argparse.Namespace) -> dict:
 
 
 def train(args: argparse.Namespace) -> dict:
-    """Run a single training experiment. Returns final metrics dict."""
+    """Run a single training experiment.
+
+    Returns final metrics dict.
+    """
     torch.manual_seed(args.seed)
     if torch.cuda.is_available():
         torch.cuda.manual_seed_all(args.seed)
@@ -333,14 +341,18 @@ def train(args: argparse.Namespace) -> dict:
     print(f'Model: {args.model} (lmax={args.lmax}), {n_params:,} params')
 
     if args.dry_run:
-        print(json.dumps({
-            'model': args.run_label or args.model,
-            'base_model': args.model,
-            'lmax': args.lmax,
-            'train_mode': _canonical_train_mode(args.train_mode),
-            'n_params': n_params,
-            'invariant_dim': invariant_dim,
-        }))
+        print(
+            json.dumps(
+                {
+                    'model': args.run_label or args.model,
+                    'base_model': args.model,
+                    'lmax': args.lmax,
+                    'train_mode': _canonical_train_mode(args.train_mode),
+                    'n_params': n_params,
+                    'invariant_dim': invariant_dim,
+                }
+            )
+        )
         return {'n_params': n_params}
 
     train_mode = _canonical_train_mode(args.train_mode)
@@ -368,12 +380,16 @@ def train(args: argparse.Namespace) -> dict:
     torch.set_float32_matmul_precision('high')
 
     optimizer = torch.optim.AdamW(
-        model.parameters(), lr=args.lr, weight_decay=args.weight_decay,
+        model.parameters(),
+        lr=args.lr,
+        weight_decay=args.weight_decay,
     )
     scaler = torch.amp.GradScaler('cuda')
     steps_per_epoch = max(1, len(train_loader))
     scheduler = torch.optim.lr_scheduler.CosineAnnealingWarmRestarts(
-        optimizer, T_0=10 * steps_per_epoch, T_mult=2,
+        optimizer,
+        T_0=10 * steps_per_epoch,
+        T_mult=2,
     )
 
     best_val_acc = 0.0
@@ -390,7 +406,12 @@ def train(args: argparse.Namespace) -> dict:
     for epoch in range(1, args.epochs + 1):
         t0 = time.time()
         train_loss = train_one_epoch(
-            model, train_loader, optimizer, scheduler, device, scaler,
+            model,
+            train_loader,
+            optimizer,
+            scheduler,
+            device,
+            scaler,
         )
         val_metrics = compute_metrics(model, val_loader, device)
         elapsed = time.time() - t0
@@ -428,7 +449,9 @@ def train(args: argparse.Namespace) -> dict:
     if not args.skip_rotation:
         test_nr_ds = test_nr_loader.dataset
         rot_robustness = evaluate_rotation_robustness(
-            model, test_nr_ds, device,
+            model,
+            test_nr_ds,
+            device,
             n_rotations=args.n_rotations,
         )
         print(
@@ -536,7 +559,9 @@ def main():
         help='Optional result label when the same base model is run at another budget.',
     )
     parser.add_argument(
-        '--train_mode', choices=['C', 'NR', 'R'], default='C',
+        '--train_mode',
+        choices=['C', 'NR', 'R'],
+        default='C',
         help='Training data variant: C/NR=canonical/non-rotated, R=rotated.',
     )
     parser.add_argument('--data_dir', type=str, default='./smnist_data')
@@ -552,9 +577,11 @@ def main():
     parser.add_argument('--nlon', type=int, default=128)
     parser.add_argument('--hidden', type=int, default=256)
     parser.add_argument(
-        '--train_size', type=int, default=None,
+        '--train_size',
+        type=int,
+        default=None,
         help='Subset training data to N examples for data-efficiency experiments. '
-             'Omit (or pass <=0) to use the full training set.',
+        'Omit (or pass <=0) to use the full training set.',
     )
     parser.add_argument(
         '--test_rotation_seed',
@@ -569,35 +596,46 @@ def main():
         help='Directory where train-subset index manifests are written.',
     )
     parser.add_argument(
-        '--full_bispectrum', action='store_true',
+        '--full_bispectrum',
+        action='store_true',
         help='Use full O(L^3) bispectrum instead of selective O(L^2).',
     )
     parser.add_argument(
-        '--sweep', action='store_true',
+        '--sweep',
+        action='store_true',
         help='Run all 3 models x 2 modes x 3 seeds.',
     )
     parser.add_argument(
-        '--dry_run', action='store_true',
+        '--dry_run',
+        action='store_true',
         help='Build model, print param count, and exit.',
     )
     parser.add_argument(
-        '--memory_check', action='store_true',
+        '--memory_check',
+        action='store_true',
         help='Run a one-batch forward/backward memory check and exit.',
     )
     parser.add_argument(
-        '--memory_manifest', type=str, default=None,
+        '--memory_manifest',
+        type=str,
+        default=None,
         help='Optional JSONL path for memory_check records.',
     )
     parser.add_argument(
-        '--memory_headroom', type=float, default=0.85,
+        '--memory_headroom',
+        type=float,
+        default=0.85,
         help='Fail memory_check when reserved CUDA memory exceeds this fraction.',
     )
     parser.add_argument(
-        '--skip_rotation', action='store_true',
+        '--skip_rotation',
+        action='store_true',
         help='Skip rotation robustness evaluation.',
     )
     parser.add_argument(
-        '--n_rotations', type=int, default=10,
+        '--n_rotations',
+        type=int,
+        default=10,
         help='Number of random rotations for robustness eval.',
     )
 
diff --git a/paper/experiments/spherical_mnist_reconstruction/.gitignore b/experiments/spherical_mnist_reconstruction/.gitignore
similarity index 100%
rename from paper/experiments/spherical_mnist_reconstruction/.gitignore
rename to experiments/spherical_mnist_reconstruction/.gitignore
diff --git a/experiments/spherical_mnist_reconstruction/README.md b/experiments/spherical_mnist_reconstruction/README.md
new file mode 100644
index 0000000..07b3f5d
--- /dev/null
+++ b/experiments/spherical_mnist_reconstruction/README.md
@@ -0,0 +1,8 @@
+# Spherical MNIST Reconstruction
+
+Reconstruct spherical MNIST digits from their SO(3) bispectrum via gradient descent, demonstrating completeness of the bispectral invariant up to SO(3) orbit indeterminacy.
+
+```bash
+pip install -e "../../[dev]"
+python reconstruct.py --n_digits 8 --n_rotations 2
+```
diff --git a/paper/experiments/spherical_mnist_reconstruction/reconstruct.py b/experiments/spherical_mnist_reconstruction/reconstruct.py
similarity index 89%
rename from paper/experiments/spherical_mnist_reconstruction/reconstruct.py
rename to experiments/spherical_mnist_reconstruction/reconstruct.py
index 217ec23..0a46011 100644
--- a/paper/experiments/spherical_mnist_reconstruction/reconstruct.py
+++ b/experiments/spherical_mnist_reconstruction/reconstruct.py
@@ -125,7 +125,7 @@ def load_digits(
             break
 
     if len(chosen) < n_digits:
-        remaining = n_digits - len(chosen)
+        n_digits - len(chosen)
         already = set(chosen)
         all_idx = torch.randperm(images.shape[0], generator=rng).tolist()
         for idx in all_idx:
@@ -286,8 +286,12 @@ def align_so3_batch(
     losses: list[float] = []
     for i in range(f_hats.shape[0]):
         aligned, R, loss = align_so3(
-            f_hats[i], f_targets[i], n_restarts=n_restarts, n_steps=n_steps,
-            lr=lr, seed=seed * 1_000_003 + i,
+            f_hats[i],
+            f_targets[i],
+            n_restarts=n_restarts,
+            n_steps=n_steps,
+            lr=lr,
+            seed=seed * 1_000_003 + i,
         )
         aligned_list.append(aligned)
         R_list.append(R)
@@ -393,9 +397,7 @@ def _record(per_sample_loss: torch.Tensor, best_rel_now: torch.Tensor) -> None:
         with torch.no_grad(), torch.amp.autocast('cuda', enabled=False):
             beta_hat_post = bsp(f_hat)
             diff_post = beta_hat_post - beta_targets
-            per_sample_post = (
-                (diff_post.real**2 + diff_post.imag**2).sum(dim=-1) / target_norm_sq
-            )
+            per_sample_post = (diff_post.real**2 + diff_post.imag**2).sum(dim=-1) / target_norm_sq
         best_rel_now = _update_best(per_sample_post, f_hat)
         _record(per_sample_post, best_rel_now)
 
@@ -448,7 +450,10 @@ def run_demo(
     align_lr: float,
     n_recon_restarts: int = 1,
 ) -> tuple[list[ReconResult], dict[str, object]]:
-    """End-to-end reconstruction sweep. Returns per-(digit, rot) records + meta."""
+    """End-to-end reconstruction sweep.
+
+    Returns per-(digit, rot) records + meta.
+    """
     output_dir.mkdir(parents=True, exist_ok=True)
 
     logger.info('Loading %d digits from cached spherical MNIST ...', n_digits)
@@ -457,9 +462,9 @@ def run_demo(
 
     logger.info('Building SO3onS2(lmax=%d, selective=%s) on %s', lmax, selective, device)
     bsp = SO3onS2(lmax=lmax, nlat=nlat, nlon=nlon, selective=selective).to(device)
-    sht = RealSHT(
-        nlat, nlon, lmax=lmax + 1, mmax=lmax + 1, grid='equiangular', norm='ortho'
-    ).to(device)
+    sht = RealSHT(nlat, nlon, lmax=lmax + 1, mmax=lmax + 1, grid='equiangular', norm='ortho').to(
+        device
+    )
     isht = InverseRealSHT(
         nlat, nlon, lmax=lmax + 1, mmax=lmax + 1, grid='equiangular', norm='ortho'
     ).to(device)
@@ -484,8 +489,9 @@ def run_demo(
         invariance_rel = relative_norm(beta_stack, beta_ref.expand_as(beta_stack))
         for k, val in enumerate(invariance_rel.cpu().tolist()):
             tag = 'identity' if k == 0 else f'rot {k}'
-            logger.info('  invariance ||beta(f_%d) - beta(f_0)|| / ||beta(f_0)|| = %.3e (%s)',
-                        k, val, tag)
+            logger.info(
+                '  invariance ||beta(f_%d) - beta(f_0)|| / ||beta(f_0)|| = %.3e (%s)', k, val, tag
+            )
 
         t0 = time.perf_counter()
         best_initial: torch.Tensor | None = None
@@ -493,9 +499,7 @@ def run_demo(
         best_loss_trace: list[list[float]] | None = None
         best_bispec_trace: list[list[float]] | None = None
         best_per_target: torch.Tensor | None = None
-        target_norm_sq = (
-            (beta_stack.real**2 + beta_stack.imag**2).sum(dim=-1).clamp_min(1e-30)
-        )
+        target_norm_sq = (beta_stack.real**2 + beta_stack.imag**2).sum(dim=-1).clamp_min(1e-30)
 
         for restart in range(max(n_recon_restarts, 1)):
             init, recovered, loss_trace, bispec_trace = reconstruct_batch(
@@ -516,9 +520,7 @@ def run_demo(
             with torch.no_grad(), torch.amp.autocast('cuda', enabled=False):
                 beta_rec = bsp(recovered)
                 diff = beta_rec - beta_stack
-                per_target_rel_sq = (
-                    (diff.real**2 + diff.imag**2).sum(dim=-1) / target_norm_sq
-                )
+                per_target_rel_sq = (diff.real**2 + diff.imag**2).sum(dim=-1) / target_norm_sq
 
             if best_recovered is None:
                 best_initial = init.clone()
@@ -630,7 +632,10 @@ def run_demo(
 
 
 def dump_results_json(results: list[ReconResult], meta: dict[str, object], path: Path) -> None:
-    """Persist per-record metrics + meta. Tensors are dropped (kept in figures)."""
+    """Persist per-record metrics + meta.
+
+    Tensors are dropped (kept in figures).
+    """
     payload: dict[str, object] = {
         'meta': meta,
         'records': [
@@ -770,8 +775,8 @@ def _sphere_grid_lines(
 ) -> list[tuple[np.ndarray, np.ndarray]]:
     """Return polylines ``[(u, v), ...]`` for great/small circles on the visible hemisphere.
 
-    Used to draw a faint reference graticule on the orthographic sphere so the
-    eye can pick up rotation differences between panels.
+    Used to draw a faint reference graticule on the orthographic sphere so the eye can pick up
+    rotation differences between panels.
     """
     e = np.deg2rad(elev_deg)
     a = np.deg2rad(azim_deg)
@@ -834,9 +839,7 @@ def _draw_sphere(
     if grid_lines is not None:
         for u, v in grid_lines:
             ax.plot(u, v, color='gray', linewidth=0.25, alpha=0.45, zorder=2)
-    circle = plt.Circle(
-        (0, 0), 1.0, fill=False, color='black', linewidth=0.6, zorder=3
-    )
+    circle = plt.Circle((0, 0), 1.0, fill=False, color='black', linewidth=0.6, zorder=3)
     ax.add_patch(circle)
     ax.set_xlim(-1.05, 1.05)
     ax.set_ylim(-1.05, 1.05)
@@ -860,9 +863,7 @@ def _draw_panel(
 ) -> None:
     """Render a single signal panel using the requested style."""
     if render == 'sphere':
-        _draw_sphere(
-            ax, f, vmin, vmax, view_size, elev_deg, azim_deg, grid_lines=grid_lines
-        )
+        _draw_sphere(ax, f, vmin, vmax, view_size, elev_deg, azim_deg, grid_lines=grid_lines)
     else:
         ax.imshow(
             f,
@@ -910,11 +911,11 @@ def make_orbits_figure(
         squeeze=False,
     )
 
-    fallback_lines = (
-        _sphere_grid_lines(elev_deg, azim_deg) if render == 'sphere' else None
-    )
+    fallback_lines = _sphere_grid_lines(elev_deg, azim_deg) if render == 'sphere' else None
 
-    def _view_for(arr: np.ndarray) -> tuple[float, float, list[tuple[np.ndarray, np.ndarray]] | None]:
+    def _view_for(
+        arr: np.ndarray,
+    ) -> tuple[float, float, list[tuple[np.ndarray, np.ndarray]] | None]:
         if render != 'sphere':
             return elev_deg, azim_deg, None
         if not auto_center:
@@ -940,7 +941,14 @@ def _view_for(arr: np.ndarray) -> tuple[float, float, list[tuple[np.ndarray, np.
 
             ax_t = axes[d][3 * k]
             _draw_panel(
-                ax_t, tgt, vmin, vmax, render, view_size, e_t, a_t,
+                ax_t,
+                tgt,
+                vmin,
+                vmax,
+                render,
+                view_size,
+                e_t,
+                a_t,
                 grid_lines=gl_t,
             )
             if d == 0:
@@ -951,13 +959,18 @@ def _view_for(arr: np.ndarray) -> tuple[float, float, list[tuple[np.ndarray, np.
                 )
                 ax_t.set_title(title)
             if k == 0:
-                ax_t.set_ylabel(
-                    f'#{d}', rotation=0, ha='right', va='center', fontsize=8
-                )
+                ax_t.set_ylabel(f'#{d}', rotation=0, ha='right', va='center', fontsize=8)
 
             ax_r = axes[d][3 * k + 1]
             _draw_panel(
-                ax_r, raw, vmin, vmax, render, view_size, e_r, a_r,
+                ax_r,
+                raw,
+                vmin,
+                vmax,
+                render,
+                view_size,
+                e_r,
+                a_r,
                 grid_lines=gl_r,
             )
             if d == 0:
@@ -967,7 +980,14 @@ def _view_for(arr: np.ndarray) -> tuple[float, float, list[tuple[np.ndarray, np.
 
             ax_a = axes[d][3 * k + 2]
             _draw_panel(
-                ax_a, ali, vmin, vmax, render, view_size, e_a, a_a,
+                ax_a,
+                ali,
+                vmin,
+                vmax,
+                render,
+                view_size,
+                e_a,
+                a_a,
                 grid_lines=gl_a,
             )
             if d == 0:
@@ -975,16 +995,14 @@ def _view_for(arr: np.ndarray) -> tuple[float, float, list[tuple[np.ndarray, np.
                     f'aligned $R\\!\\cdot\\!\\hat f_{k}$\n(img rel {r.aligned_image_space_rel:.1e})'
                 )
             else:
-                ax_a.set_xlabel(
-                    f'img {r.aligned_image_space_rel:.1e}', fontsize=6
-                )
+                ax_a.set_xlabel(f'img {r.aligned_image_space_rel:.1e}', fontsize=6)
 
     fig.suptitle(
         'SO(3) bispectrum reconstruction.\n'
         'Target $f_k = R_k\\!\\cdot\\! f$ (left of each triplet), gradient-descent recon '
         '$\\hat f_k$ from $\\beta(f_k)$ (middle), and the recon rotated by the optimal '
         '$R \\in SO(3)$ (right).\n'
-        'Each sphere is auto-centered on the signal\'s positive-mass centroid; '
+        "Each sphere is auto-centered on the signal's positive-mass centroid; "
         'aligned and target panels share the same view because they are the same signal.',
         y=1.02,
         fontsize=9,
@@ -997,7 +1015,10 @@ def _view_for(arr: np.ndarray) -> tuple[float, float, list[tuple[np.ndarray, np.
 
 
 def make_convergence_figure(results: list[ReconResult], path: Path) -> None:
-    """Median + IQR envelope of the relative bispectrum residual vs. step."""
+    """Median + IQR envelope of the relative bispectrum residual vs.
+
+    step.
+    """
     plt.rcParams.update(NEURIPS_RCPARAMS)
     if not results:
         return
@@ -1031,7 +1052,10 @@ def make_convergence_figure(results: list[ReconResult], path: Path) -> None:
 
 
 def make_invariance_figure(results: list[ReconResult], path: Path) -> None:
-    """Scatter of invariance check vs. recon residual per (digit, rot)."""
+    """Scatter of invariance check vs.
+
+    recon residual per (digit, rot).
+    """
     plt.rcParams.update(NEURIPS_RCPARAMS)
     inv = np.array([r.invariance_rel for r in results if r.rot_idx > 0])
     rec = np.array([r.final_recon_rel for r in results if r.rot_idx > 0])
@@ -1086,9 +1110,7 @@ def make_paper_figure(
     picks: list[ReconResult] = []
     for d in digit_indices:
         if d not in by_digit:
-            raise ValueError(
-                f'digit_idx {d} not in results (available: {sorted(by_digit)})'
-            )
+            raise ValueError(f'digit_idx {d} not in results (available: {sorted(by_digit)})')
         best = min(by_digit[d], key=lambda r: r.aligned_image_space_rel)
         picks.append(best)
 
@@ -1117,12 +1139,8 @@ def _view_for(arr: np.ndarray) -> tuple[float, float, list[tuple[np.ndarray, np.
 
         e_t, a_t, gl_t = _view_for(target)
 
-        _draw_sphere(
-            axes[0][j], target, vmin, vmax, view_size, e_t, a_t, grid_lines=gl_t
-        )
-        _draw_sphere(
-            axes[1][j], aligned, vmin, vmax, view_size, e_t, a_t, grid_lines=gl_t
-        )
+        _draw_sphere(axes[0][j], target, vmin, vmax, view_size, e_t, a_t, grid_lines=gl_t)
+        _draw_sphere(axes[1][j], aligned, vmin, vmax, view_size, e_t, a_t, grid_lines=gl_t)
 
         axes[0][j].set_title(f'class {r.label}', fontsize=9, pad=3)
         axes[1][j].set_xlabel(
@@ -1177,7 +1195,9 @@ def _stats(x: np.ndarray) -> str:
     print(f'  invariance     ||beta(R_k f) - beta(f)|| / ||beta(f)||         : {_stats(inv_all)}')
     print(f'  recon resid    ||beta(f_hat) - beta(target)|| / ||target||     : {_stats(rec_all)}')
     print(f'  image (raw)    ||f_hat - target|| / ||target||                 : {_stats(img_all)}')
-    print(f'  image (aligned)||R . f_hat - target|| / ||target||             : {_stats(aligned_all)}')
+    print(
+        f'  image (aligned)||R . f_hat - target|| / ||target||             : {_stats(aligned_all)}'
+    )
     print(
         '  Raw image-space error is O(1) (the recovery sits in a different SO(3) orbit '
         'representative); after the SO(3) alignment the error drops to ~SHT discretisation '
@@ -1213,14 +1233,27 @@ def parse_args() -> argparse.Namespace:
         action='store_true',
         help='Disable the band-limit re-projection after each Adam step.',
     )
-    parser.add_argument('--n_recon_restarts', type=int, default=4,
-                        help='Random-init restarts per reconstruction; per-sample best is kept.')
-    parser.add_argument('--align_n_restarts', type=int, default=12,
-                        help='SO(3) alignment restarts per (target, recon) pair (0 disables).')
-    parser.add_argument('--align_n_steps', type=int, default=200,
-                        help='Adam steps per alignment restart.')
-    parser.add_argument('--align_lr', type=float, default=1e-1,
-                        help='Initial Adam LR for alignment (cosine-annealed).')
+    parser.add_argument(
+        '--n_recon_restarts',
+        type=int,
+        default=4,
+        help='Random-init restarts per reconstruction; per-sample best is kept.',
+    )
+    parser.add_argument(
+        '--align_n_restarts',
+        type=int,
+        default=12,
+        help='SO(3) alignment restarts per (target, recon) pair (0 disables).',
+    )
+    parser.add_argument(
+        '--align_n_steps', type=int, default=200, help='Adam steps per alignment restart.'
+    )
+    parser.add_argument(
+        '--align_lr',
+        type=float,
+        default=1e-1,
+        help='Initial Adam LR for alignment (cosine-annealed).',
+    )
     parser.add_argument(
         '--render',
         type=str,
@@ -1228,31 +1261,55 @@ def parse_args() -> argparse.Namespace:
         choices=['sphere', 'equirectangular'],
         help='Panel rendering style for the orbits figure.',
     )
-    parser.add_argument('--view_size', type=int, default=128,
-                        help='Orthographic view resolution (only when --render=sphere).')
-    parser.add_argument('--elev_deg', type=float, default=25.0,
-                        help='Fallback / fixed-view elevation in degrees.')
-    parser.add_argument('--azim_deg', type=float, default=30.0,
-                        help='Fallback / fixed-view azimuth in degrees.')
-    parser.add_argument('--fixed_view', action='store_true',
-                        help='Use --elev_deg / --azim_deg for every panel instead of '
-                        'auto-centering on each signal\'s positive-mass centroid.')
-    parser.add_argument('--paper_digits', type=int, nargs='+',
-                        default=[0, 1, 2, 3, 4, 5],
-                        help='Digit indices used for the compact paper figure '
-                        '(rows = Target/Recon, columns = digits).')
-    parser.add_argument('--paper_figure_path', type=Path, default=None,
-                        help='Override output path for the paper figure '
-                        '(default: <output_dir>/paper_orbits.pdf).')
-    parser.add_argument('--paper_only', action='store_true',
-                        help='Run only the digits required for the paper figure '
-                        '(--paper_digits) and skip the comprehensive figures. '
-                        'Fast regeneration path for the paper.')
-    parser.add_argument('--replot', action='store_true',
-                        help='Skip the reconstruction and rebuild figures from the '
-                        '<output_dir>/state.pt cache produced by a previous run. '
-                        'Use this to iterate on figure aesthetics in seconds.')
-    parser.add_argument('--device', type=str, default='cuda' if torch.cuda.is_available() else 'cpu')
+    parser.add_argument(
+        '--view_size',
+        type=int,
+        default=128,
+        help='Orthographic view resolution (only when --render=sphere).',
+    )
+    parser.add_argument(
+        '--elev_deg', type=float, default=25.0, help='Fallback / fixed-view elevation in degrees.'
+    )
+    parser.add_argument(
+        '--azim_deg', type=float, default=30.0, help='Fallback / fixed-view azimuth in degrees.'
+    )
+    parser.add_argument(
+        '--fixed_view',
+        action='store_true',
+        help='Use --elev_deg / --azim_deg for every panel instead of '
+        "auto-centering on each signal's positive-mass centroid.",
+    )
+    parser.add_argument(
+        '--paper_digits',
+        type=int,
+        nargs='+',
+        default=[0, 1, 2, 3, 4, 5],
+        help='Digit indices used for the compact paper figure '
+        '(rows = Target/Recon, columns = digits).',
+    )
+    parser.add_argument(
+        '--paper_figure_path',
+        type=Path,
+        default=None,
+        help='Override output path for the paper figure (default: <output_dir>/paper_orbits.pdf).',
+    )
+    parser.add_argument(
+        '--paper_only',
+        action='store_true',
+        help='Run only the digits required for the paper figure '
+        '(--paper_digits) and skip the comprehensive figures. '
+        'Fast regeneration path for the paper.',
+    )
+    parser.add_argument(
+        '--replot',
+        action='store_true',
+        help='Skip the reconstruction and rebuild figures from the '
+        '<output_dir>/state.pt cache produced by a previous run. '
+        'Use this to iterate on figure aesthetics in seconds.',
+    )
+    parser.add_argument(
+        '--device', type=str, default='cuda' if torch.cuda.is_available() else 'cpu'
+    )
     parser.add_argument('--seed', type=int, default=0)
     parser.add_argument('--log_every', type=int, default=300)
     parser.add_argument(
@@ -1277,7 +1334,7 @@ def main() -> int:
         logger.warning('CUDA requested but not available; falling back to CPU.')
         device = torch.device('cpu')
 
-    paper_digits = sorted(set(int(d) for d in args.paper_digits))
+    paper_digits = sorted({int(d) for d in args.paper_digits})
     if args.paper_only:
         if not paper_digits:
             raise ValueError('--paper_only requires --paper_digits to be non-empty')
diff --git a/paper/_verify_fiber.py b/paper/_verify_fiber.py
deleted file mode 100644
index 1598401..0000000
--- a/paper/_verify_fiber.py
+++ /dev/null
@@ -1,81 +0,0 @@
-#!/usr/bin/env python
-"""Verify that the augmented invariant (with self-coupling entries) has a
-unique fiber at the rational witness for L=4 and L=5."""
-from __future__ import annotations
-import sys, os, time
-sys.path.insert(0, os.path.join(os.path.dirname(__file__), '..', 'src'))
-sys.path.insert(0, os.path.dirname(__file__))
-import numpy as np
-from fiber_search import (
-    _init_cg, get_live_bispec, params_to_fc_np,
-    bispectrum_entry_np, cg_power_entry_np, damped_gauss_newton,
-)
-from bispectrum.so3_on_s2 import _build_cg_power_index_map
-
-
-def verify_fiber(lmax: int, n_starts: int = 2000):
-    _init_cg(lmax)
-    n = sum(2 * l + 1 for l in range(2, lmax + 1)) - 1
-    w = np.array([(i + 1) / 3.0 for i in range(n)])
-    live = get_live_bispec(lmax, w)
-    cgp = _build_cg_power_index_map(lmax)
-
-    print(f"lmax={lmax}: {n} params, {len(live)} live bispec, {len(cgp)} CG power")
-    sc_in_live = [t for t in live if t[0] == t[1] and t[0] >= 3]
-    print(f"  Self-coupling in live: {sc_in_live}")
-
-    def eval_full(p):
-        fc = params_to_fc_np(p, lmax)
-        vals = []
-        for l1, l2, lv in live:
-            beta = bispectrum_entry_np(fc, l1, l2, lv)
-            par = (l1 + l2 + lv) % 2
-            vals.append(float(np.imag(beta)) if par else float(np.real(beta)))
-        for l1, l2, lo in cgp:
-            vals.append(cg_power_entry_np(fc, l1, l2, lo))
-        return np.array(vals)
-
-    target = eval_full(w)
-    print(f"  Total constraints: {len(target)}")
-    sys.stdout.flush()
-
-    def res(x):
-        return eval_full(x) - target
-
-    rng = np.random.RandomState(42)
-    found = []
-    t0 = time.time()
-    for trial in range(n_starts):
-        eps = rng.choice([0.01, 0.05, 0.1, 0.2, 0.5, 1.0, 2.0, 5.0])
-        x0 = w + eps * rng.randn(n)
-        x, r = damped_gauss_newton(res, x0, max_iter=80)
-        if r < 1e-22:
-            dup = any(np.linalg.norm(x - s) < 1e-5 for s in found)
-            if not dup:
-                found.append(x.copy())
-                d = np.linalg.norm(x - w)
-                label = "WITNESS" if d < 1e-5 else f"DISTINCT dist={d:.2e}"
-                print(f"    trial {trial}: sol #{len(found)} ({label})")
-                sys.stdout.flush()
-        if trial % 500 == 499:
-            print(f"    {trial+1}/{n_starts} ({time.time()-t0:.1f}s)")
-            sys.stdout.flush()
-
-    elapsed = time.time() - t0
-    n_distinct = sum(1 for s in found if np.linalg.norm(s - w) > 1e-5)
-    print(f"\n  RESULT lmax={lmax}: {len(found)} solutions ({n_distinct} distinct from witness) in {elapsed:.1f}s")
-    if n_distinct == 0:
-        print(f"  => UNIQUE FIBER (global injectivity)")
-    else:
-        print(f"  => MULTIPLE PREIMAGES (global injectivity FAILS)")
-    sys.stdout.flush()
-    return n_distinct == 0
-
-
-if __name__ == "__main__":
-    ok4 = verify_fiber(4, n_starts=2000)
-    print()
-    ok5 = verify_fiber(5, n_starts=1000)
-    print(f"\n{'='*40}")
-    print(f"L=4: {'PASS' if ok4 else 'FAIL'}")
-    print(f"L=5: {'PASS' if ok5 else 'FAIL'}")
diff --git a/paper/_verify_fiber_tree.py b/paper/_verify_fiber_tree.py
deleted file mode 100644
index 36c5b39..0000000
--- a/paper/_verify_fiber_tree.py
+++ /dev/null
@@ -1,291 +0,0 @@
-#!/usr/bin/env python
-"""Tree-based fiber enumeration for L=4 and L=5.
-
-At each degree ell, given fixed lower-degree coefficients from a parent
-branch, uses multi-start damped Gauss-Newton to find ALL per-degree
-solutions. Propagates each solution to the next degree. At the final
-degree, validates each leaf against the full system.
-
-Only the witness path should survive the full-system check, proving
-fiber degree = 1.
-"""
-from __future__ import annotations
-import sys, os, time
-sys.path.insert(0, os.path.join(os.path.dirname(__file__), '..', 'src'))
-sys.path.insert(0, os.path.dirname(__file__))
-import numpy as np
-from fiber_search import _init_cg, get_live_bispec, CG_NP, A0, C_GAUGE
-from bispectrum.so3_on_s2 import _build_cg_power_index_map
-
-
-def _build_fc(params, lmax):
-    fc = {}
-    fc[0] = np.array([A0 + 0j])
-    fc[1] = np.array([0.0 + 0j, C_GAUGE + 0j, 0.0 + 0j])
-    idx = 0
-    for lv in range(2, lmax + 1):
-        c = np.zeros(2 * lv + 1, dtype=np.complex128)
-        for m in range(1, lv + 1):
-            if lv == 2 and m == 1:
-                c[lv + m] = params[idx]; c[lv - m] = ((-1)**m) * params[idx]; idx += 1
-            else:
-                c[lv + m] = params[idx] + 1j * params[idx + 1]
-                c[lv - m] = ((-1)**m) * (params[idx] - 1j * params[idx + 1])
-                idx += 2
-        c[lv] = params[idx] + 0j; idx += 1
-        fc[lv] = c
-    return fc
-
-
-def _fc_to_t(fc, ell):
-    c = fc[ell]
-    parts = []
-    for m in range(1, ell + 1):
-        if ell == 2 and m == 1:
-            parts.append(c[ell + m].real)
-        else:
-            parts.extend([c[ell + m].real, c[ell + m].imag])
-    parts.append(c[ell].real)
-    return np.array(parts)
-
-
-def _update_fc(fc, ell, t):
-    c = np.zeros(2 * ell + 1, dtype=np.complex128)
-    idx = 0
-    for m in range(1, ell + 1):
-        if ell == 2 and m == 1:
-            c[ell + m] = t[idx]; c[ell - m] = ((-1)**m) * t[idx]; idx += 1
-        else:
-            c[ell + m] = t[idx] + 1j * t[idx + 1]
-            c[ell - m] = ((-1)**m) * (t[idx] - 1j * t[idx + 1])
-            idx += 2
-    c[ell] = t[idx] + 0j
-    fc[ell] = c
-
-
-def _bispec(fc, l1, l2, lv):
-    key = (min(l1, l2), max(l1, l2))
-    outer = np.outer(fc[min(l1, l2)], fc[max(l1, l2)]).ravel()
-    coupled = outer @ CG_NP[key]
-    col = 0
-    for l in range(abs(l1 - l2), l1 + l2 + 1):
-        sz = 2 * l + 1
-        if l == lv:
-            return np.sum(coupled[col:col + sz] * np.conj(fc[lv]))
-        col += sz
-    return 0.0 + 0j
-
-
-def _cgpow(fc, l1, l2, lo):
-    key = (min(l1, l2), max(l1, l2))
-    outer = np.outer(fc[min(l1, l2)], fc[max(l1, l2)]).ravel()
-    coupled = outer @ CG_NP[key]
-    col = 0
-    for l in range(abs(l1 - l2), l1 + l2 + 1):
-        sz = 2 * l + 1
-        if l == lo:
-            return float(np.sum(np.abs(coupled[col:col + sz]) ** 2))
-        col += sz
-    return 0.0
-
-
-def _pdim(ell):
-    return (2 * ell + 1) - (1 if ell == 2 else 0)
-
-
-def _pstart(ell):
-    idx = 0
-    for lv in range(2, ell):
-        idx += _pdim(lv)
-    return idx
-
-
-def _eval_at(fc, ell, bispec_e, cgp_e, target):
-    vals = []
-    for l1, l2, lv in bispec_e:
-        beta = _bispec(fc, l1, l2, lv)
-        par = (l1 + l2 + lv) % 2
-        vals.append(float(np.imag(beta)) if par else float(np.real(beta)))
-    for l1, l2, lo in cgp_e:
-        vals.append(_cgpow(fc, l1, l2, lo))
-    return np.array(vals) - target
-
-
-def _gn_search(fc_parent, ell, bispec_e, cgp_e, target, d, wit_t, n_starts):
-    """Multi-start damped GN. Returns list of distinct solutions."""
-    h = 1e-8
-    solutions = []
-    fc = {k: v.copy() for k, v in fc_parent.items()}
-    fc_orig_ell = fc_parent.get(ell, np.zeros(2 * ell + 1, dtype=np.complex128)).copy()
-
-    rng = np.random.RandomState(42 + ell + abs(hash(fc_parent[2].tobytes())) % 10000)
-    wit_scale = max(np.linalg.norm(wit_t), 1.0)
-
-    for trial in range(n_starts):
-        if trial < 5:
-            t = wit_t + rng.randn(d) * 0.01 * wit_scale
-        elif trial < 30:
-            t = wit_t + rng.randn(d) * rng.choice([0.01, 0.1, 0.5]) * wit_scale
-        elif trial < 60:
-            t = wit_t * (1.0 + rng.randn(d) * rng.choice([0.1, 0.3, 0.5]))
-        else:
-            scale = rng.choice([0.01, 0.1, 0.5, 1.0, 2.0, 5.0,
-                                wit_scale * 0.5, wit_scale, wit_scale * 2])
-            t = rng.randn(d) * scale
-
-        for it in range(50):
-            _update_fc(fc, ell, t)
-            f0 = _eval_at(fc, ell, bispec_e, cgp_e, target)
-            r2 = np.sum(f0**2)
-
-            if r2 < 1e-16:
-                is_dup = any(np.linalg.norm(t - s) < 1e-4 for s in solutions)
-                if not is_dup:
-                    solutions.append(t.copy())
-                break
-
-            if it >= 10 and r2 > 1e-4:
-                break
-
-            J = np.empty((len(f0), d))
-            for j in range(d):
-                tp = t.copy(); tp[j] += h
-                _update_fc(fc, ell, tp)
-                J[:, j] = (_eval_at(fc, ell, bispec_e, cgp_e, target) - f0) / h
-
-            try:
-                dx = np.linalg.lstsq(J, -f0, rcond=None)[0]
-            except np.linalg.LinAlgError:
-                break
-
-            alpha = 1.0
-            for _ in range(15):
-                tn = t + alpha * dx
-                _update_fc(fc, ell, tn)
-                r2n = np.sum(_eval_at(fc, ell, bispec_e, cgp_e, target)**2)
-                if r2n < r2:
-                    t = tn
-                    break
-                alpha *= 0.5
-            else:
-                break
-
-        fc[ell] = fc_orig_ell.copy()
-
-    return solutions
-
-
-def tree_search(lmax, n_starts_det=2000, n_starts_over=500):
-    _init_cg(lmax)
-    n = sum(2 * l + 1 for l in range(2, lmax + 1)) - 1
-    witness = np.array([(i + 1) / 3.0 for i in range(n)])
-    live = get_live_bispec(lmax, witness)
-    cgp = _build_cg_power_index_map(lmax)
-
-    sc = [t for t in live if t[0] == t[1] and t[0] >= 3]
-    print(f"lmax={lmax}: n={n}, {len(live)} bispec, {len(cgp)} CG power")
-    print(f"  Self-coupling: {sc}")
-
-    fc_wit = _build_fc(witness, lmax)
-    full_tgt = []
-    for l1, l2, lv in live:
-        beta = _bispec(fc_wit, l1, l2, lv)
-        par = (l1 + l2 + lv) % 2
-        full_tgt.append(float(np.imag(beta)) if par else float(np.real(beta)))
-    for l1, l2, lo in cgp:
-        full_tgt.append(_cgpow(fc_wit, l1, l2, lo))
-    full_tgt = np.array(full_tgt)
-
-    branches = [witness.copy()]
-    t0_all = time.time()
-
-    for ell in range(2, lmax + 1):
-        d = _pdim(ell)
-        ps = _pstart(ell)
-        bispec_e = [(l1, l2, lv) for l1, l2, lv in live if max(l1, l2, lv) == ell]
-        cgp_e = [(l1, l2, lo) for l1, l2, lo in cgp if max(l1, l2) == ell]
-        m_ell = len(bispec_e) + len(cgp_e)
-        n_starts = n_starts_det if m_ell <= d + 2 else n_starts_over
-        wit_t = _fc_to_t(fc_wit, ell)
-
-        tgt_ell = []
-        for l1, l2, lv in bispec_e:
-            beta = _bispec(fc_wit, l1, l2, lv)
-            par = (l1 + l2 + lv) % 2
-            tgt_ell.append(float(np.imag(beta)) if par else float(np.real(beta)))
-        for l1, l2, lo in cgp_e:
-            tgt_ell.append(_cgpow(fc_wit, l1, l2, lo))
-        tgt_ell = np.array(tgt_ell)
-
-        print(f"\n  ell={ell}: d={d} m={m_ell} {len(branches)}br {n_starts}starts")
-        sys.stdout.flush()
-        t0 = time.time()
-
-        new_branches = []
-        for bi, parent in enumerate(branches):
-            dist = np.linalg.norm(parent - witness)
-            lbl = "wit" if dist < 1e-4 else f"alt d={dist:.1e}"
-
-            fc_p = _build_fc(parent, lmax)
-            sols = _gn_search(fc_p, ell, bispec_e, cgp_e, tgt_ell, d, wit_t, n_starts)
-
-            for s in sols:
-                child = parent.copy()
-                child[ps:ps + d] = s
-                new_branches.append(child)
-
-            nw = sum(1 for s in sols if np.linalg.norm(s - wit_t) < 1e-4)
-            na = len(sols) - nw
-            elapsed_b = time.time() - t0
-            print(f"    b{bi} ({lbl}): {len(sols)} ({nw}w {na}a) [{elapsed_b:.0f}s]")
-            sys.stdout.flush()
-
-        elapsed = time.time() - t0
-        branches = new_branches
-        print(f"    => {len(branches)} branches ({elapsed:.0f}s)")
-        sys.stdout.flush()
-
-    print(f"\n  Checking {len(branches)} leaves...")
-    valid = []
-    for b in branches:
-        fc_b = _build_fc(b, lmax)
-        vals = []
-        for l1, l2, lv in live:
-            beta = _bispec(fc_b, l1, l2, lv)
-            par = (l1 + l2 + lv) % 2
-            vals.append(float(np.imag(beta)) if par else float(np.real(beta)))
-        for l1, l2, lo in cgp:
-            vals.append(_cgpow(fc_b, l1, l2, lo))
-        r2 = np.sum((np.array(vals) - full_tgt)**2)
-        dist = np.linalg.norm(b - witness)
-        if r2 < 1e-10:
-            valid.append(b)
-            lbl = "WITNESS" if dist < 1e-3 else f"DISTINCT d={dist:.2e}"
-            print(f"    valid: r2={r2:.2e} {lbl}")
-
-    total = time.time() - t0_all
-    nd = sum(1 for v in valid if np.linalg.norm(v - witness) > 1e-3)
-    print(f"\n  RESULT lmax={lmax}: {len(valid)} valid, {nd} distinct ({total:.0f}s)")
-    if nd == 0 and len(valid) > 0:
-        print("  => UNIQUE FIBER")
-    elif len(valid) == 0:
-        print("  => ERROR: witness not found")
-    else:
-        print("  => MULTIPLE PREIMAGES")
-    sys.stdout.flush()
-    return nd == 0 and len(valid) > 0
-
-
-if __name__ == "__main__":
-    res = {}
-    for lmax in [4, 5]:
-        print(f"\n{'='*60}")
-        print(f"TREE FIBER SEARCH: lmax={lmax}")
-        print(f"{'='*60}")
-        sys.stdout.flush()
-        res[lmax] = tree_search(lmax, n_starts_det=2000, n_starts_over=500)
-
-    print(f"\n{'='*60}")
-    print("SUMMARY")
-    for lmax, ok in res.items():
-        print(f"  L={lmax}: {'PASS' if ok else 'FAIL'}")
diff --git a/paper/augmentation_results.txt b/paper/augmentation_results.txt
deleted file mode 100644
index 5a8f2dd..0000000
--- a/paper/augmentation_results.txt
+++ /dev/null
@@ -1,142 +0,0 @@
-lmax=4: 20 unknowns, 18 live bispec, rank=10
-  +||(F_1⊗F_2)|_1||^2 → rank=11
-  +||(F_1⊗F_3)|_2||^2 → rank=12
-  +||(F_1⊗F_4)|_3||^2 → rank=13
-  +||(F_1⊗F_4)|_4||^2 → rank=14
-  +||(F_2⊗F_3)|_1||^2 → rank=15
-  +||(F_2⊗F_3)|_2||^2 → rank=16
-  +||(F_2⊗F_4)|_2||^2 → rank=17
-  +||(F_2⊗F_4)|_3||^2 → rank=18
-  +||(F_3⊗F_3)|_2||^2 → rank=19
-  +||(F_3⊗F_4)|_1||^2 → rank=20
-  FULL RANK with 10 augmentations!
-  multi-seed: verified ✓
-  total entries: 30 (budget was 22, overhead 8)
-
-lmax=5: 31 unknowns, 28 live bispec, rank=14
-  +||(F_1⊗F_2)|_1||^2 → rank=15
-  +||(F_1⊗F_3)|_2||^2 → rank=16
-  +||(F_1⊗F_4)|_3||^2 → rank=17
-  +||(F_1⊗F_4)|_4||^2 → rank=18
-  +||(F_1⊗F_5)|_4||^2 → rank=19
-  +||(F_1⊗F_5)|_5||^2 → rank=20
-  +||(F_2⊗F_3)|_1||^2 → rank=21
-  +||(F_2⊗F_3)|_2||^2 → rank=22
-  +||(F_2⊗F_4)|_2||^2 → rank=23
-  +||(F_2⊗F_4)|_3||^2 → rank=24
-  +||(F_2⊗F_5)|_3||^2 → rank=25
-  +||(F_2⊗F_5)|_4||^2 → rank=26
-  +||(F_3⊗F_3)|_2||^2 → rank=27
-  +||(F_3⊗F_4)|_1||^2 → rank=28
-  +||(F_3⊗F_5)|_2||^2 → rank=29
-  +||(F_3⊗F_5)|_3||^2 → rank=30
-  +||(F_3⊗F_5)|_4||^2 → rank=31
-  FULL RANK with 17 augmentations!
-  multi-seed: verified ✓
-  total entries: 47 (budget was 33, overhead 14)
-
-lmax=6: 44 unknowns, 41 live bispec, rank=22
-  +||(F_1⊗F_2)|_1||^2 → rank=23
-  +||(F_1⊗F_3)|_2||^2 → rank=24
-  +||(F_1⊗F_4)|_3||^2 → rank=25
-  +||(F_1⊗F_4)|_4||^2 → rank=26
-  +||(F_1⊗F_5)|_4||^2 → rank=27
-  +||(F_1⊗F_5)|_5||^2 → rank=28
-  +||(F_1⊗F_6)|_5||^2 → rank=29
-  +||(F_1⊗F_6)|_6||^2 → rank=30
-  +||(F_2⊗F_3)|_1||^2 → rank=31
-  +||(F_2⊗F_3)|_2||^2 → rank=32
-  +||(F_2⊗F_4)|_2||^2 → rank=33
-  +||(F_2⊗F_4)|_3||^2 → rank=34
-  +||(F_2⊗F_5)|_3||^2 → rank=35
-  +||(F_2⊗F_5)|_4||^2 → rank=36
-  +||(F_2⊗F_6)|_4||^2 → rank=37
-  +||(F_2⊗F_6)|_5||^2 → rank=38
-  +||(F_3⊗F_3)|_2||^2 → rank=39
-  +||(F_3⊗F_4)|_1||^2 → rank=40
-  +||(F_3⊗F_5)|_2||^2 → rank=41
-  +||(F_3⊗F_5)|_3||^2 → rank=42
-  +||(F_3⊗F_5)|_4||^2 → rank=43
-  +||(F_3⊗F_6)|_3||^2 → rank=44
-  FULL RANK with 22 augmentations!
-  multi-seed: verified ✓
-  total entries: 65 (budget was 46, overhead 19)
-
-lmax=7: 59 unknowns, 55 live bispec, rank=30
-  +||(F_1⊗F_2)|_1||^2 → rank=31
-  +||(F_1⊗F_3)|_2||^2 → rank=32
-  +||(F_1⊗F_4)|_3||^2 → rank=33
-  +||(F_1⊗F_4)|_4||^2 → rank=34
-  +||(F_1⊗F_5)|_4||^2 → rank=35
-  +||(F_1⊗F_5)|_5||^2 → rank=36
-  +||(F_1⊗F_6)|_5||^2 → rank=37
-  +||(F_1⊗F_6)|_6||^2 → rank=38
-  +||(F_1⊗F_7)|_6||^2 → rank=39
-  +||(F_1⊗F_7)|_7||^2 → rank=40
-  +||(F_2⊗F_3)|_1||^2 → rank=41
-  +||(F_2⊗F_3)|_2||^2 → rank=42
-  +||(F_2⊗F_4)|_2||^2 → rank=43
-  +||(F_2⊗F_4)|_3||^2 → rank=44
-  +||(F_2⊗F_5)|_3||^2 → rank=45
-  +||(F_2⊗F_5)|_4||^2 → rank=46
-  +||(F_2⊗F_6)|_4||^2 → rank=47
-  +||(F_2⊗F_6)|_5||^2 → rank=48
-  +||(F_2⊗F_7)|_5||^2 → rank=49
-  +||(F_2⊗F_7)|_6||^2 → rank=50
-  +||(F_3⊗F_3)|_2||^2 → rank=51
-  +||(F_3⊗F_4)|_1||^2 → rank=52
-  +||(F_3⊗F_5)|_2||^2 → rank=53
-  +||(F_3⊗F_5)|_3||^2 → rank=54
-  +||(F_3⊗F_5)|_4||^2 → rank=55
-  +||(F_3⊗F_6)|_3||^2 → rank=56
-  +||(F_3⊗F_7)|_4||^2 → rank=57
-  +||(F_3⊗F_7)|_5||^2 → rank=58
-  +||(F_3⊗F_7)|_6||^2 → rank=59
-  FULL RANK with 29 augmentations!
-  multi-seed: verified ✓
-  total entries: 86 (budget was 61, overhead 25)
-
-lmax=8: 76 unknowns, 72 live bispec, rank=37
-  +||(F_1⊗F_2)|_1||^2 → rank=38
-  +||(F_1⊗F_3)|_2||^2 → rank=39
-  +||(F_1⊗F_4)|_3||^2 → rank=40
-  +||(F_1⊗F_4)|_4||^2 → rank=41
-  +||(F_1⊗F_5)|_4||^2 → rank=42
-  +||(F_1⊗F_5)|_5||^2 → rank=43
-  +||(F_1⊗F_6)|_5||^2 → rank=44
-  +||(F_1⊗F_6)|_6||^2 → rank=45
-  +||(F_1⊗F_7)|_6||^2 → rank=46
-  +||(F_1⊗F_7)|_7||^2 → rank=47
-  +||(F_1⊗F_8)|_7||^2 → rank=48
-  +||(F_1⊗F_8)|_8||^2 → rank=49
-  +||(F_2⊗F_3)|_1||^2 → rank=50
-  +||(F_2⊗F_3)|_2||^2 → rank=51
-  +||(F_2⊗F_4)|_2||^2 → rank=52
-  +||(F_2⊗F_4)|_3||^2 → rank=53
-  +||(F_2⊗F_5)|_3||^2 → rank=54
-  +||(F_2⊗F_5)|_4||^2 → rank=55
-  +||(F_2⊗F_6)|_4||^2 → rank=56
-  +||(F_2⊗F_6)|_5||^2 → rank=57
-  +||(F_2⊗F_7)|_5||^2 → rank=58
-  +||(F_2⊗F_7)|_6||^2 → rank=59
-  +||(F_2⊗F_8)|_6||^2 → rank=60
-  +||(F_2⊗F_8)|_7||^2 → rank=61
-  +||(F_3⊗F_3)|_2||^2 → rank=62
-  +||(F_3⊗F_4)|_1||^2 → rank=63
-  +||(F_3⊗F_5)|_2||^2 → rank=64
-  +||(F_3⊗F_5)|_3||^2 → rank=65
-  +||(F_3⊗F_5)|_4||^2 → rank=66
-  +||(F_3⊗F_6)|_3||^2 → rank=67
-  +||(F_3⊗F_7)|_4||^2 → rank=68
-  +||(F_3⊗F_7)|_5||^2 → rank=69
-  +||(F_3⊗F_7)|_6||^2 → rank=70
-  +||(F_3⊗F_8)|_5||^2 → rank=71
-  +||(F_3⊗F_8)|_6||^2 → rank=72
-  +||(F_3⊗F_8)|_7||^2 → rank=73
-  +||(F_4⊗F_8)|_4||^2 → rank=74
-  +||(F_4⊗F_8)|_5||^2 → rank=75
-  +||(F_4⊗F_8)|_6||^2 → rank=76
-  FULL RANK with 39 augmentations!
-  multi-seed: verified ✓
-  total entries: 113 (budget was 78, overhead 35)
-
diff --git a/paper/bibliography.md b/paper/bibliography.md
deleted file mode 100644
index fbd0b7e..0000000
--- a/paper/bibliography.md
+++ /dev/null
@@ -1,641 +0,0 @@
-# Bibliography: Best NeurIPS Papers on Software / Python Packages
-
-*Compiled for: Bispectrum Python Package Paper — NeurIPS Submission*
-*Date: March 2026*
-
-______________________________________________________________________
-
-## 1. Overview
-
-This document surveys the most influential papers published at NeurIPS (formerly NIPS) that introduce software packages, Python libraries, or frameworks for machine learning. The goal is twofold: (a) provide citable references and related work for a software-oriented NeurIPS submission, and (b) extract structural and stylistic patterns that characterize successful software papers at this venue.
-
-______________________________________________________________________
-
-## 2. Landmark Software / Framework Papers at NeurIPS
-
-### 2.1 PyTorch (NeurIPS 2019)
-
-**Title:** PyTorch: An Imperative Style, High-Performance Deep Learning Library
-**Authors:** Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Köpf, Edward Yang, Zach DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, Soumith Chintala
-**Venue:** NeurIPS 2019 (Advances in Neural Information Processing Systems 32)
-**Citations:** ~100,000+ (2nd most internally cited NeurIPS paper with 720 NeurIPS-to-NeurIPS citations)
-
-**Summary:** Introduces PyTorch as an imperative, Pythonic deep learning framework with dynamic computational graphs and GPU acceleration. The paper emphasizes usability, performance, and extensibility, positioning PyTorch as an alternative to static-graph frameworks (TensorFlow, Theano). By NeurIPS 2024, approximately 75% of papers at the conference used PyTorch.
-
-**Key takeaways for your paper:**
-
-- Clearly articulates *design philosophy* (imperative vs. declarative, define-by-run)
-- Benchmarks against existing frameworks on both performance and usability
-- Emphasizes community adoption and ecosystem (torchvision, torchaudio, etc.)
-- Strong "why this matters" framing: lowering the barrier to research
-
-______________________________________________________________________
-
-### 2.2 Attention Is All You Need (NeurIPS 2017)
-
-**Title:** Attention Is All You Need
-**Authors:** Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, Illia Polosukhin
-**Venue:** NeurIPS 2017 (NIPS 31)
-**Citations:** ~150,000+ (most internally cited NeurIPS paper: 1,014 NeurIPS-to-NeurIPS citations)
-
-**Summary:** Introduces the Transformer architecture, replacing recurrence and convolutions entirely with self-attention mechanisms. While primarily an architecture paper, it spawned the entire Hugging Face Transformers ecosystem and fundamentally changed how software packages for NLP and beyond are built.
-
-**Relevance to your paper:** Demonstrates how a well-designed abstraction (the attention mechanism) becomes a reusable software building block adopted across an entire field.
-
-______________________________________________________________________
-
-### 2.3 Generative Adversarial Nets (NeurIPS 2014) — Test of Time Award 2024
-
-**Title:** Generative Adversarial Nets
-**Authors:** Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, Yoshua Bengio
-**Venue:** NeurIPS 2014 (NIPS 27)
-**Citations:** ~85,000+
-**Award:** NeurIPS 2024 Test of Time Award
-
-**Summary:** Introduces the GAN framework — a minimax game between a generator and discriminator network. Became one of the foundational paradigms for generative modeling, spawning dozens of specialized software libraries.
-
-______________________________________________________________________
-
-### 2.4 LightGBM (NeurIPS 2017)
-
-**Title:** LightGBM: A Highly Efficient Gradient Boosting Decision Tree
-**Authors:** Guolin Ke, Qi Meng, Thomas Finley, Taifeng Wang, Wei Chen, Weidong Ma, Qiwei Ye, Tie-Yan Liu
-**Venue:** NeurIPS 2017 (NIPS 30), pp. 3149–3157
-**Citations:** ~20,000+
-
-**Summary:** Introduces two novel techniques — Gradient-based One-Side Sampling (GOSS) and Exclusive Feature Bundling (EFB) — to speed up gradient boosting. The accompanying Python/C++ library became one of the most widely used ML tools in industry and Kaggle competitions.
-
-**Key takeaways for your paper:**
-
-- Classic "software paper" structure: algorithm + implementation + extensive benchmarks
-- Clear performance comparisons (speed and accuracy) against XGBoost and other baselines
-- Released as a well-documented, pip-installable package
-
-______________________________________________________________________
-
-### 2.5 GPyTorch (NeurIPS 2018)
-
-**Title:** GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration
-**Authors:** Jacob R. Gardner, Geoff Pleiss, Kilian Q. Weinberger, David Bindel, Andrew Gordon Wilson
-**Venue:** NeurIPS 2018
-**Citations:** ~2,000+
-
-**Summary:** Presents an efficient approach to GP inference based on Blackbox Matrix-Matrix multiplication (BBMM), reducing complexity from O(n³) to O(n²). The GPyTorch library, built on PyTorch, provides scalable GP inference with GPU acceleration.
-
-**Key takeaways for your paper (most directly relevant):**
-
-- **This is the closest template for your bispectrum paper** — a math-heavy Python package with GPU acceleration
-- Structure: mathematical contribution (BBMM algorithm) + software design + benchmarks
-- Emphasizes both algorithmic novelty and practical usability
-- Built on top of PyTorch, leveraging existing ecosystem
-- Clear API examples showing how users interact with the library
-
-______________________________________________________________________
-
-### 2.6 GPT-3 / Language Models are Few-Shot Learners (NeurIPS 2020) — Best Paper Award
-
-**Title:** Language Models are Few-Shot Learners
-**Authors:** Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, et al.
-**Venue:** NeurIPS 2020
-**Citations:** ~40,000+
-**Award:** NeurIPS 2020 Best Paper Award
-
-**Summary:** Demonstrates that scaling language models to 175B parameters enables strong few-shot learning without fine-tuning. While primarily a scaling/capability paper, it catalyzed the entire ecosystem of LLM software tools (OpenAI API, LangChain, etc.).
-
-______________________________________________________________________
-
-### 2.7 Benchopt (NeurIPS 2022)
-
-**Title:** Benchopt: Reproducible, Efficient and Collaborative Optimization Benchmarks
-**Authors:** Thomas Moreau, Mathurin Massias, Alexandre Gramfort, Pierre Ablin, Pierre-Antoine Bannier, Benjamin Charlier, Mathieu Dagréou, Tom Dupré la Tour, Ghislain Durif, Cassio F. Dantas, Quentin Klopfenstein, Johan Larsson, En Lai, Tanguy Lefort, Benoît Malézieux, Badr Moufad, Binh T. Nguyen, Alain Rakotomamonjy, Zaccharie Ramzi, Joseph Salmon, Samuel Vaiter
-**Venue:** NeurIPS 2022 (Main Conference Track)
-
-**Summary:** A collaborative framework to automate, publish, and reproduce optimization benchmarks in ML across programming languages (Python, R, Julia, C/C++) and hardware architectures. Demonstrates benchmarks on ℓ₂-regularized logistic regression, Lasso, and ResNet18 training.
-
-**Key takeaways for your paper:**
-
-- Excellent example of a "tool/framework" paper at NeurIPS main track
-- Emphasizes reproducibility and collaboration as first-class design goals
-- scikit-learn compatible API design
-- Cross-language support as a differentiator
-
-______________________________________________________________________
-
-### 2.8 PettingZoo (NeurIPS 2021 — Datasets & Benchmarks Track)
-
-**Title:** PettingZoo: A Standard API for Multi-Agent Reinforcement Learning
-**Authors:** J. K. Terry, Benjamin Black, Nathaniel Grammel, Mario Jayber, Ananth Hari, Ryan Sullivan, Luis Santos, Clemens Dieffendahl, Caroline Horsch, Rodrigo Perez-Vicente, Niall Williams, Yashas Lokesh, Praveen Ravi
-**Venue:** NeurIPS 2021 (Datasets & Benchmarks Track)
-
-**Summary:** Introduces PettingZoo, a library of diverse multi-agent RL environments with a universal Python API. Proposes the Agent Environment Cycle (AEC) games model. Analogous to what OpenAI Gym did for single-agent RL.
-
-**Key takeaways for your paper:**
-
-- Shows how "API standardization" papers can succeed at NeurIPS
-- Strong emphasis on *why* the design choices were made (not just *what*)
-- Includes both theoretical justification (AEC model) and practical utility
-
-______________________________________________________________________
-
-### 2.9 Gymnasium (NeurIPS 2025 — Datasets & Benchmarks Track, Spotlight)
-
-**Title:** Gymnasium: A Standard Interface for Reinforcement Learning Environments
-**Authors:** Farama Foundation team
-**Venue:** NeurIPS 2025 (Datasets & Benchmarks Track — Spotlight)
-
-**Summary:** The maintained successor to OpenAI Gym. Provides a standard API for RL environments with over 18 million downloads since release. Describes the theoretical and practical design considerations behind the API.
-
-**Relevance:** Demonstrates that even "API maintenance/upgrade" papers can receive spotlight recognition when the contribution is well-articulated and widely impactful.
-
-______________________________________________________________________
-
-### 2.10 GAUCHE (NeurIPS 2023)
-
-**Title:** GAUCHE: A Library for Gaussian Processes in Chemistry
-**Authors:** Ryan-Rhys Griffiths et al.
-**Venue:** NeurIPS 2023 (Main Conference Track)
-
-**Summary:** An open-source library providing GP kernels for structured chemical inputs (graphs, strings, bit vectors) for Bayesian optimization in chemistry. Built on GPyTorch and BoTorch.
-
-**Key takeaways for your paper:**
-
-- Domain-specific GP library accepted at NeurIPS main track
-- Shows that building on existing frameworks (GPyTorch) and targeting a specific application domain is a viable publication strategy
-- Modular design philosophy
-
-______________________________________________________________________
-
-## 3. Other Highly Influential Papers in the NeurIPS Ecosystem
-
-These papers were not published at NeurIPS but are foundational to the ML software ecosystem and are frequently cited alongside NeurIPS software papers:
-
-| Paper                                                                          | Venue        | Year | Citations | Notes                                         |
-| ------------------------------------------------------------------------------ | ------------ | ---- | --------- | --------------------------------------------- |
-| Scikit-learn: Machine Learning in Python (Pedregosa et al.)                    | JMLR         | 2011 | ~110,000+ | Most cited ML software paper ever             |
-| TensorFlow: A System for Large-Scale ML (Abadi et al.)                         | OSDI         | 2016 | ~50,000+  | Static-graph predecessor to PyTorch           |
-| Theano: A Python Framework (Theano Dev Team)                                   | arXiv        | 2016 | ~10,000+  | Pioneer of symbolic differentiation in Python |
-| XGBoost: A Scalable Tree Boosting System (Chen & Guestrin)                     | KDD          | 2016 | ~40,000+  | Predecessor/competitor to LightGBM            |
-| Optuna: A Next-generation Hyperparameter Optimization Framework (Akiba et al.) | KDD          | 2019 | ~5,000+   | Define-by-run HPO framework                   |
-| Pyro: Deep Universal Probabilistic Programming (Bingham et al.)                | JMLR         | 2019 | ~3,000+   | PyTorch-based PPL                             |
-| JAX: Composable Transformations of Python+NumPy programs (Bradbury et al.)     | GitHub/arXiv | 2018 | ~5,000+   | Functional transformations (vmap, pmap, jit)  |
-
-______________________________________________________________________
-
-## 4. NeurIPS Test of Time Awards (Software-Relevant)
-
-These awards honor papers from ~10 years prior that have had lasting impact:
-
-| Year | Paper                                                               | Original Year | Citations                                       |
-| ---- | ------------------------------------------------------------------- | ------------- | ----------------------------------------------- |
-| 2020 | HOGWILD!: Lock-Free SGD (Niu et al.)                                | 2011          | Foundational for parallel training software     |
-| 2022 | AlexNet: ImageNet Classification with Deep CNNs (Krizhevsky et al.) | 2012          | Sparked the deep learning revolution            |
-| 2023 | word2vec: Distributed Representations of Words (Mikolov et al.)     | 2013          | 40,000+ citations; foundational for NLP tooling |
-| 2024 | Generative Adversarial Nets (Goodfellow et al.)                     | 2014          | 85,000+ citations                               |
-| 2024 | Sequence to Sequence Learning (Sutskever et al.)                    | 2014          | Foundational for encoder-decoder architectures  |
-| 2025 | Faster R-CNN (Ren, He, Girshick, Sun)                               | 2015          | 56,700+ citations                               |
-
-______________________________________________________________________
-
-## 5. Actual Reviewer Feedback from OpenReview — What Reviewers Say About Software Papers
-
-This section contains real reviewer comments extracted from OpenReview/NeurIPS proceedings, organized by the key themes that emerge. These are direct insights into what reviewers look for and complain about.
-
-### 5.1 PyTorch (NeurIPS 2019) — Reviews
-
-**Scores:** Reviewer 1 raised to 9 after rebuttal, Reviewer 2 strong positive, Reviewer 3 positive.
-
-**Reviewer 1 (Score: 9 after rebuttal):**
-
-- Praised: Impact in the research community. "Given the impact of the library in the research community I strongly support the publication of this paper at NeurIPS."
-- Raised the venue question: "One may argue whether a systems/software paper, presenting the implementation details of a library should be published at NeurIPS, or whether it would be a better fit for USENIX or SysML."
-- Criticism: "My main criticism is that you claim that PyTorch offers performance comparable to the fastest current libraries for DL. But what are these libraries? [...] I feel to support this claim a broader comparison to other libraries would be necessary."
-- Wanted broader benchmarks beyond MXNet and TensorFlow.
-
-**Reviewer 2 (Strong positive):**
-
-- Key insight: "I applaud the authors for focusing their submission around their key design principles and decisions, rather than presenting a detailed tutorial of the software's use. This distinguishes the submission from a 'white paper,' i.e., a paper which reports on how to use a piece of software while marketing its merits, but having less scientific value."
-- Praised: "The paper reads as a kind of manifesto for what deep learning software should aspire to be."
-- Appreciated tradeoff analysis: "The authors detail the consequences and tradeoffs of their design choices carefully, making clear that sometimes this may lead to slightly worse performance, which is acceptable if it makes the software easier to use."
-- Noted: "There is no one big idea here [...] Nor is there any new or surprising result here, since PyTorch has been available (and widely used) for several years. However, the paper demonstrates the successful synthesis of several key ideas."
-
-**Reviewer 3:**
-
-- "Since Pytorch has been used and tested by the community for years, the validity of the paper is without a doubt."
-- "It is very certain that this paper has high originality, quality, clarity, and significance, though it might come a bit late."
-
-**Key lesson for your paper:** Even PyTorch faced the "is this the right venue?" question. The paper succeeded because it framed itself as a *design principles manifesto*, not a tutorial. Reviewers explicitly praised this framing.
-
-______________________________________________________________________
-
-### 5.2 GPyTorch (NeurIPS 2018) — Reviews
-
-**All 3 reviewers positive. Accepted.**
-
-**Reviewer 1:**
-
-- "The work is precisely explained, with appropriate references for related and relevant work, and is of great impact."
-- "The key contribution of the paper is a framework that computes GP inference via a blackbox routine that performs matrix-matrix multiplications efficiently with GPU acceleration."
-- Praised: Clear separation from state of the art with explicit quotes showing how the method differs.
-- Minor issues only (typos).
-
-**Reviewer 2:**
-
-- Praised the dual contribution: algorithmic (modified PCG) + software (GPyTorch).
-- Concern: "The paper focused on regression. It is true that it can be easily extended to other problems like classification, but in that case the algorithm needs specific inference algorithms [...] and it could be argued that it is not that black-box anymore."
-- "Overall, it is a strong well-written paper with nice theoretical and practical contributions."
-
-**Reviewer 3:**
-
-- "The ideas presented in the paper are not ground-breakingly new, but are a clever combination and implementation of existing ones. That is absolutely fine and sufficient for acceptance."
-- "Novel is the theoretical analysis of the preconditioner."
-- Criticism: "The presentation of the results regarding speed-up appear arbitrary — twenty conjugate gradient steps are not a lot. I made the experience that in bad cases, conjugate gradients may need more than a thousand steps. [...] The quality of this submission could be improved by adding plots how the approximation quality and the residual evolve over time."
-- "Facilitating the usage of GPUs for GP inference is an important contribution."
-
-**Key lesson for your paper:** GPyTorch succeeded by pairing a clear algorithmic improvement (BBMM reducing O(n³) to O(n²)) with practical GPU software. Reviewers accepted that the ideas were not individually groundbreaking — they valued the clever synthesis. But they pushed for more thorough convergence/quality plots.
-
-______________________________________________________________________
-
-### 5.3 GAUCHE (NeurIPS 2023) — Reviews
-
-**Scores: 8 (Strong Accept), 7 (Accept), 6 (Weak Accept), 5 (Borderline Accept), 3 (Reject)**
-
-This is the most instructive case — a domain-specific library with no new algorithms that still got accepted despite a Reject score.
-
-**Reviewer giving 8 (Strong Accept):**
-
-- Praised comprehensive exploration of GP representations for chemistry.
-- Liked diverse benchmarks (regression, UQ, Bayesian optimization).
-- Concern: "Comparison against SOTA neural networks is notably lacking (e.g., ChemProp, ChemBERTa)."
-- Limitations noted: "GPR models can be computationally intensive for larger datasets."
-
-**Reviewer giving 3 (Reject) — THE CRITICAL REVIEW:**
-
-- "All of the implemented kernels are known in the literature."
-- **"NeurIPS is probably not the right venue for publishing software libraries."**
-- "The supported methods are not well documented within the article."
-- "The future of the library is also not mentioned: what is the governance model and what are the next steps?"
-- "The code provided shows nothing under 'gauche/kernels/graph_kernels'. [...] I cannot count 20 kernels in the code."
-- Wanted: Code-documentation consistency, clear roadmap, governance model.
-
-**Reviewer giving 7 (Accept):**
-
-- Soundness: 3/4, Presentation: 4/4, Contribution: 3/4.
-- "The proposed library seems to fill a gap in the open-source GP stack. The code cleanliness is rather high and the code is well-tested."
-- Concern about code-paper mismatch: "The article is not well-documented when it comes to listing all kernels that are currently supported."
-
-**Reviewer giving 5 (Borderline Accept):**
-
-- Contribution rated only 2/4 (fair).
-- "The major weakness is the contribution. The training task uses GPyTorch [...] The main contribution of the paper is to provide some important kernels in chemistry. It seems somewhat incremental."
-
-**Reviewer giving 6 (Weak Accept):**
-
-- "The library's objective is clearly defined and holds potential."
-- "I find the objective of the paper somewhat unclear."
-- Suggested: Better API documentation, clearer distinction between library paper vs. benchmark paper.
-
-**Authors' rebuttal strategy (successful):**
-
-- Cited NeurIPS 2023 Call for Papers explicitly calling for "libraries" in the Infrastructure section.
-- Referenced other library papers published at NeurIPS as precedent.
-- Added extensive additional experiments (DNN comparisons, ablation studies, new kernels).
-- Detailed the engineering contribution (SIGP wrapper enabling GraKel integration with PyTorch autodiff).
-
-**Key lessons for your paper:**
-
-1. **The "is this the right venue?" attack will come.** Prepare a rebuttal citing the NeurIPS CFP's explicit mention of software/infrastructure.
-2. **Code must match paper claims exactly.** Reviewers will inspect your repository and call out discrepancies.
-3. **Document a roadmap and governance model** for the library.
-4. **Demonstrate engineering novelty**, not just wrapping existing code. GAUCHE survived by showing the non-trivial SIGP wrapper.
-5. **Compare against deep learning baselines**, even if your method is classical/statistical.
-
-______________________________________________________________________
-
-### 5.4 LightGBM (NeurIPS 2017) — Reviews
-
-**All 3 reviewers positive. Accepted.**
-
-**Reviewer 1:**
-
-- "The approaches are interesting and smart."
-- Criticism: "The experiments lack standard deviation indications, for the performances are often very close from one method to another."
-- Wanted noise sensitivity analysis, especially for GOSS.
-- "The paper is not easy to read, because the presentation, explanation, experimental study and analysis are spread all over the paper. It would be better to concentrate fully on one improvement, then on the other."
-
-**Reviewer 2:**
-
-- "This paper is well-motivated, and the proposed algorithm is interesting and effective."
-- Wanted: "details on the datasets and why these datasets are selected."
-- Wanted: generalization analysis, not just approximation error bounds.
-
-**Reviewer 3:**
-
-- "The proposed study here is well motivated and of important practical value."
-- Also wanted generalization error analysis beyond approximation error.
-
-**Key lesson for your paper:** Even for a highly successful paper, reviewers wanted error bars, dataset justification, and cleaner paper organization. Theoretical bounds should address generalization, not just approximation.
-
-______________________________________________________________________
-
-### 5.5 Benchopt (NeurIPS 2022) — Reviews
-
-**Scores: 7, 7, 4 (Borderline Reject). Accepted.**
-
-**Reviewer giving 7 (Accept):**
-
-- "The authors identify an important problem in machine learning, the benchmarking and comparisons of optimizers."
-- Concern: "The discussion of the results and the specific experiments are not dramatically different compared to similar optimizer benchmarking papers."
-- Wanted: "What makes using Benchopt better than simply writing a custom script?"
-
-**Reviewer giving 7 (Accept):**
-
-- Praised: "The modular design of the benchmark guarantees the versatility."
-- Praised: "Interesting findings such as the unexpected performance of baseline optimization methods like L-BFGS."
-- Wanted: more problem types beyond the three demonstrated.
-
-**Reviewer giving 4 (Borderline Reject):**
-
-- "I commend the effort to improve transparency and reproducibility in OptML research."
-- Key criticism: "I think it is a bad choice to report comparisons in terms of wall-clock-time. This can be extremely dependent on specific combinations of framework, implementation and hardware."
-- Wanted: comparisons in number of iterates + reported time per iteration, to separate algorithmic insights from implementation details.
-- Felt the paper was "more like a technical report or software documentation than a research paper."
-
-**Key lesson for your paper:** Benchmarking methodology matters as much as results. Separate algorithmic cost from wall-clock time if possible. Address the "is this a research paper or documentation?" concern head-on.
-
-______________________________________________________________________
-
-## 6. Structural Patterns of Successful NeurIPS Software Papers
-
-Based on analysis of the actual papers and reviewer feedback above:
-
-### Paper Structure Comparison
-
-**PyTorch (12 pages, NeurIPS 2019):**
-
-1. Introduction
-2. Design Principles (Be Pythonic, Put Researchers First, Provide Pragmatic Performance, Worse is Better)
-3. Usability Centric Design (Deep Learning Specific, Separation of Control and Data Flow)
-4. Technical Implementation (Internals: autograd, JIT compiler, C++ frontend)
-5. Key Subsystems (Automatic Differentiation, Data Loading, Serialization, Multiprocessing)
-   - 5.3: GPU Memory Management (custom caching allocator)
-6. Evaluation
-   - 6.1: Asynchronous Dataflow vs. Eager Mode benchmarks
-   - 6.2: Memory Management evaluation
-   - 6.3: Benchmarks on common models (AlexNet, VGG-19, ResNet-50, etc.) vs. TensorFlow/MXNet
-   - 6.4: Adoption metrics (papers, GitHub contributors)
-7. Conclusion and Future Work
-
-**GPyTorch (9 pages + supplementary, NeurIPS 2018):**
-
-1. Introduction
-2. Numerical Methods Background (CG, Lanczos, SLQ)
-3. GP Inference Background (marginal log-likelihood, predictive distributions)
-4. Blackbox Matrix-Matrix (BBMM) Inference — the core algorithmic contribution
-   - 4.1: Modified Batched Conjugate Gradients
-   - 4.2: Preconditioning via Pivoted Cholesky
-   - 4.3: Convergence Analysis
-5. The GPyTorch Software Platform
-6. Experiments
-   - 6.1: Exact GPs (speed + accuracy vs. baselines)
-   - 6.2: Scalable GP Approximations (KISS-GP, SGPR)
-   - 6.3: Deep Kernel Learning
-7. Discussion and Conclusion
-
-**Key structural insight:** GPyTorch spends 2 full sections on math background before introducing the algorithmic contribution. The software description is only 1 section. Experiments are comprehensive (3 subsections covering different GP regimes).
-
-### What Makes Software Papers Stand Out at NeurIPS
-
-- **Algorithmic novelty paired with practical utility**: Pure "wrapper" libraries face strong pushback (see GAUCHE Reviewer 3). There must be a technical contribution (GPyTorch's BBMM, LightGBM's GOSS/EFB).
-- **Frame as design principles, not a tutorial**: PyTorch Reviewer 2 explicitly praised this distinction over a "white paper."
-- **Ecosystem integration**: Successful packages build on PyTorch, NumPy, or scikit-learn rather than reinventing the wheel.
-- **Reproducibility**: Code availability, pip-installable packages, clear documentation. Reviewers will check your repo!
-- **Benchmark rigor**: Error bars (LightGBM criticism), convergence plots (GPyTorch criticism), wall-clock vs. algorithmic cost separation (Benchopt criticism).
-- **Community adoption evidence**: Download stats, GitHub stars, usage in other papers.
-- **Clear API examples**: Short code snippets showing how the library is used in practice.
-- **Address the venue question preemptively**: Multiple papers faced "is NeurIPS the right venue?" Cite the CFP.
-
-### NeurIPS Submission Practicalities (2025 Guidelines)
-
-- 9 content pages + unlimited references and checklist
-- LaTeX only (neurips_2025.sty)
-- Double-blind review
-- Mandatory NeurIPS Paper Checklist (reproducibility, ethics, broader impact)
-- Code submission encouraged but not required (strongly recommended for software papers)
-- Datasets & Benchmarks Track is an alternative venue that welcomes tool contributions
-- The 2023+ CFP explicitly mentions "libraries" under Infrastructure contributions
-
-______________________________________________________________________
-
-## 7. Common Reviewer Criticisms for Software Papers — Checklist
-
-Based on all the reviews analyzed above, here is a checklist of attacks to anticipate and preempt:
-
-| Criticism                                      | Frequency   | How to Preempt                                                                                       |
-| ---------------------------------------------- | ----------- | ---------------------------------------------------------------------------------------------------- |
-| "Is NeurIPS the right venue for this?"         | Very common | Cite CFP's infrastructure/library mention; reference precedent papers                                |
-| "The algorithms are known / not novel"         | Very common | Clearly articulate what IS novel (the combination, the GPU implementation, the theoretical analysis) |
-| "This is just a wrapper around X"              | Common      | Show non-trivial engineering (custom allocators, new preconditioners, integration challenges)        |
-| "Missing comparison to deep learning / SOTA"   | Common      | Include DNN baselines even if your method is classical                                               |
-| "Code doesn't match paper claims"              | Common      | Audit your repo before submission; every feature mentioned must be verifiable                        |
-| "No error bars / standard deviations"          | Common      | Always report mean ± std over multiple runs/splits                                                   |
-| "Wall-clock benchmarks are hardware-dependent" | Common      | Report both wall-clock AND algorithmic complexity; test on multiple hardware                         |
-| "Paper reads like documentation"               | Occasional  | Focus on design principles and WHY, not HOW-TO                                                       |
-| "No roadmap / future governance"               | Occasional  | Include a brief roadmap section; mention maintenance plans                                           |
-| "Limited experimental scope"                   | Common      | Cover multiple datasets, problem sizes, and settings                                                 |
-| "Convergence/quality plots missing"            | Occasional  | Show how approximation quality evolves, not just final numbers                                       |
-| "No generalization analysis"                   | Occasional  | Provide theory beyond approximation error if applicable                                              |
-
-______________________________________________________________________
-
-## 8. Recommendations for Your Bispectrum Paper
-
-Based on this deep analysis of actual papers and reviewer feedback:
-
-### Structure (follow GPyTorch template)
-
-1. **Introduction** — Frame the bispectrum computation problem, why existing tools are insufficient, and your 3-4 key contributions.
-2. **Background** — Mathematical foundations of the bispectrum (1-2 pages). Reviewers appreciate solid grounding.
-3. **Algorithmic Contribution** — What is novel about your approach? New algorithms, GPU parallelization strategy, complexity reduction? This is the heart of the paper.
-4. **Software Design** — Brief (~1 page) section on API design, ecosystem integration (PyTorch/JAX/NumPy). Include a code snippet.
-5. **Experiments** — At least 3 subsections: (a) speed benchmarks with error bars, (b) accuracy/correctness verification, (c) scaling behavior. Compare CPU vs GPU. Compare against existing implementations if any.
-6. **Conclusion & Future Work** — Include a brief roadmap.
-
-### Preempt the Top Reviewer Attacks
-
-1. **"Not novel enough"** → Clearly state what is algorithmically new (not just "we made a Python package"). Even a clever combination of existing techniques suffices (per GPyTorch Reviewer 3).
-
-2. **"Wrong venue"** → In the introduction, position your work within NeurIPS's scope by connecting bispectrum computation to ML applications (signal processing for neural networks, higher-order statistics in deep learning, etc.). Cite the CFP.
-
-3. **"Just a wrapper"** → Demonstrate non-trivial engineering decisions. GPU memory management, batched computation, numerical stability considerations.
-
-4. **"Missing baselines"** → Benchmark against naive NumPy/SciPy implementations, any existing bispectrum packages, and if possible, show a comparison with a deep-learning-based alternative.
-
-5. **"Code doesn't match"** → Before submission, verify every claim in the paper is demonstrable in the codebase. Include a reproducibility section.
-
-6. **"No error bars"** → Report mean ± std for all benchmarks. Use multiple random seeds and dataset splits.
-
-### Framing Strategy
-
-Follow PyTorch Reviewer 2's insight: write a *design principles paper*, not a *user manual*. Explain WHY your bispectrum library is designed the way it is, what tradeoffs you made, and how those tradeoffs serve the target user community. This separates a NeurIPS paper from documentation.
-
-______________________________________________________________________
-
-## 9. Graphein (NeurIPS 2022 — Accepted) — Reviews
-
-**Paper:** Graphein - a Python Library for Geometric Deep Learning and Network Analysis on Biomolecular Structures and Interaction Networks
-**Authors:** Arian Rokkum Jamasb, Ramon Viñas Torné, Eric J Ma, Yuanqi Du, Charles Harris, Kexin Huang, Dominic Hall, Pietro Lio, Tom Leon Blundell
-**Venue:** NeurIPS 2022 (Datasets & Benchmarks Track — Accepted)
-**Scores: 6, 8, 7, 6**
-
-**Reviewer giving 6 (Weak Accept):**
-
-- Summary: "A python library on modeling protein-protein interaction for target discovery and drug design."
-- Strength: "Open access code; beneficial for both target discovery and drug design communities."
-- Weakness: "It could be better if this toolbox could provide common methods and make benchmarks on these standard databases."
-- Questions: Performance on large-scale databases? Plans for AI baselines?
-
-**Reviewer giving 8 (Strong Accept):**
-
-- "The tool is very well documented and the code seems to be professional-grade."
-- "I've managed to run ALL the examples successfully in a couple of days."
-- Only weakness: "Lack of a manual for the advanced API."
-- This reviewer actually tested the software — a strong signal that reviewers do try to run your code.
-
-**Reviewer giving 7 (Accept):**
-
-- Originality: "Existing tools exhibit limited utilities for geometric deep learning. The proposed tool covers a broader selection."
-- Weakness: "More empirical results based on the library need to be collected and shown."
-- Wanted: runtime/memory profiling, comparison with TorchDrug, scalability experiments.
-
-**Reviewer giving 6 (Weak Accept):**
-
-- Praised: "Flexible preprocessing pipeline. Excellent and clearly written documentation and notebooks."
-- Key criticism: "The paper is mostly a summary of what the library does rather than a scientific contribution."
-- "There is limited novelty and no benchmark results to put the datasets on the map."
-- "I would want to see the authors actually use the toolkit for something interesting."
-- Wanted the paper to demonstrate a use case, not just describe what the library can do.
-
-**Key lessons for your paper:**
-
-1. **One reviewer will actually run your code.** Make sure setup is flawless, examples work, and documentation is good.
-2. **"The paper is just a summary of what the library does"** is a common attack. Include empirical results and a use case that demonstrates something interesting/surprising discovered using your library.
-3. **Runtime/memory profiling and scalability experiments** are expected for library papers, even in the Datasets & Benchmarks track.
-4. **Compare against existing tools** in the same space (TorchDrug, etc.) — don't just describe your tool in isolation.
-
-______________________________________________________________________
-
-## 10. Rejected Library Papers — Autopsy of What Went Wrong
-
-### 10.1 PGLearn (ICLR 2025 — Rejected)
-
-**Paper:** PGLearn - An Open-Source Learning Toolkit for Optimal Power Flow
-**Authors:** Michael Klamkin, Mathieu Tanneau, Pascal Van Hentenryck
-**Venue:** Submitted to ICLR 2025 — **Rejected**
-**Scores: 8, 3, 3**
-**OpenReview:** https://openreview.net/forum?id=cecIf0CKnH
-
-**Reviewer giving 8 (Accept):**
-
-- Soundness: 4/4 (excellent), Contribution: 4/4 (excellent).
-- "A typical ML for OPF study goes through complex steps of generation, solving, post-processing... This library addresses all stages under a single, standardized platform."
-- Domain expert who understood the need.
-
-**Reviewer giving 3 (Reject):**
-
-- Soundness: 1/4 (poor), Contribution: 1/4 (poor).
-- Devastating critique: Existing datasets (Google DeepMind OPFData) are larger (13,659 buses vs. PGLearn's 9,241 buses).
-- "The authors argue that numerous datasets used in benchmarking [...] have minor issues, and need to be replaced by their benchmark" — reviewer disagreed.
-- "I could not find novelty in the proposed work compared to PowerModels, OPFData, etc."
-- Attacked the data generation methodology as not capturing realistic temporal variations (diurnal patterns, ENTSO-E data).
-- "Technically, the paper does not propose any new scientific contribution."
-
-**Reviewer giving 3 (Reject):**
-
-- Soundness: 2/4 (fair), Contribution: 1/4 (poor).
-- "The proposed method is still limited to user-defined synthetic settings."
-- "Scalability to other grid models/data distributions, other power system tasks, or settings other than supervised learning is unclear."
-- "The novelty is questionable since existing datasets and benchmarks already cover similar scenarios."
-- Wanted: real-world data integration, broader task coverage, unsupervised/RL settings.
-
-**Why PGLearn was rejected — the pattern:**
-
-1. **Failed to differentiate from existing tools.** Two reviewers pointed to existing alternatives (OPFData, PowerModels) that already did what PGLearn claimed. The paper didn't convincingly explain why the world needs *yet another* OPF toolkit.
-2. **Contribution rated 1/4 (poor) by two reviewers.** When reviewers see the contribution as "just re-implementing what exists," scores collapse.
-3. **No novel scientific insight.** The library was useful engineering, but didn't discover anything new or propose a new algorithm.
-4. **Narrow scope.** Only supervised learning; only synthetic data. Reviewers wanted broader applicability.
-5. **Polarized scores (8 vs. 3, 3).** The domain expert loved it, but ML generalists saw no contribution — a classic failure mode for domain-specific toolkit papers.
-
-______________________________________________________________________
-
-### 10.2 CTBench (ICLR 2025 — Rejected)
-
-**Paper:** CTBench: A Library and Benchmark for Certified Training
-**Authors:** (from ICLR 2025 submission)
-**Venue:** Submitted to ICLR 2025 — **Rejected**
-**Scores: 5, 6, 3, 5**
-**OpenReview:** https://openreview.net/forum?id=2bn7gayfz9
-
-**Reviewer giving 5 (Marginally Below):**
-
-- Soundness: 4/4, Presentation: 4/4, Contribution: 2/4 (fair).
-- Praised: "Standardizing training schedules, certification methods, and hyperparameter tuning."
-- Key weakness: "Although the proposed framework and insights are valuable, they may be better suited for a systems or benchmarking venue (e.g., MLSys, NeurIPS D&B track) rather than a top ML venue like ICLR."
-- "The contribution doesn't include a novel algorithm or method — it primarily re-evaluates existing approaches."
-
-**Reviewer giving 6 (Marginally Above):**
-
-- Soundness: 3/4, Presentation: 4/4, Contribution: 2/4 (fair).
-- "The field needed this kind of paper."
-- But: "The key claim is that all recent methods don't really offer improvements over tuned baselines. This is an important finding but the paper can also be seen as just an extensive hyperparameter search."
-- "I'm not sure the benchmark will be widely adopted. [...] different research groups may already have their own internal benchmarking."
-- Wanted: more architectures, more datasets, standardized attack/certification methods.
-
-**Reviewer giving 3 (Reject):**
-
-- Soundness: 2/4 (fair), Contribution: 1/4 (poor).
-- "The library is restricted to relatively small-scale CIFAR-level experiments."
-- "The paper primarily re-implements existing methods with better hyperparameters. This raises the question of whether this is a research contribution or an engineering effort."
-- Questioned technical soundness: claims about batch normalization fixes and stochastic weight averaging needed better justification.
-- "The analysis sections (loss fragmentation, OOD generalization) lack depth — each could be its own paper if done properly."
-
-**Reviewer giving 5 (Marginally Below):**
-
-- Soundness: 4/4, Presentation: 4/4, Contribution: 2/4 (fair).
-- "The amount of experiments and computation required in this paper is beyond impressive."
-- But: "The contribution is limited to a re-evaluation of existing methods."
-- "The main finding that newer methods don't improve over tuned baselines is interesting but could be due to insufficient benchmarking scope."
-
-**Why CTBench was rejected — the pattern:**
-
-1. **Contribution scored 2/4 or 1/4 by ALL four reviewers.** The fatal weakness: re-implementing existing methods with better hyperparameters is not seen as a research contribution.
-2. **"Wrong venue" explicitly raised.** Reviewer 1 suggested MLSys or NeurIPS D&B instead of ICLR.
-3. **No new algorithm.** The benchmark was well-executed engineering, but every reviewer noted it proposed nothing methodologically novel.
-4. **Limited scale.** Only CIFAR-level experiments; reviewers wanted ImageNet or larger.
-5. **Shallow analysis sections.** The paper tried to include analysis of loss fragmentation, OOD generalization, etc., but reviewers felt each was underdeveloped rather than being a strength.
-6. **Adoption skepticism.** Reviewer 2 questioned whether the community would actually use the benchmark.
-
-______________________________________________________________________
-
-## 11. Accepted vs. Rejected — The Decisive Differences
-
-| Factor                                  | Accepted (GPyTorch, Benchopt, GAUCHE, Graphein)                                 | Rejected (PGLearn, CTBench)                                |
-| --------------------------------------- | ------------------------------------------------------------------------------- | ---------------------------------------------------------- |
-| **Algorithmic novelty**                 | At least one novel element (BBMM, SIGP wrapper, collaborative benchmark design) | Re-implementation of existing methods with better settings |
-| **Contribution scores**                 | At least 3/4 from most reviewers                                                | Consistently 1-2/4 from most reviewers                     |
-| **Differentiation from existing tools** | Clear gap identified and filled                                                 | Reviewers pointed to existing alternatives                 |
-| **Scale of experiments**                | Multiple regimes, problem sizes, GPU/CPU                                        | Narrow scope (one domain, one scale)                       |
-| **Something surprising discovered**     | Yes — unexpected findings using the library                                     | Mostly confirmatory results                                |
-| **Framing**                             | Design principles, algorithmic contribution                                     | Software description, hyperparameter search                |
-| **Venue fit**                           | Justified with CFP citations                                                    | Reviewers suggested alternative venues                     |
-| **Adoption evidence**                   | Download stats, community usage, GitHub activity                                | No evidence of adoption                                    |
-
-### The Bottom Line for Your Bispectrum Paper
-
-The single most important factor distinguishing accepted from rejected software papers is **whether the paper contributes a novel scientific insight**, not just a useful tool. Every rejected paper was acknowledged as useful engineering — but that's not enough. Your bispectrum paper must contain at least one of:
-
-- A new algorithm or computational technique (like GPyTorch's BBMM)
-- A surprising empirical finding discovered using the library
-- A theoretical analysis that provides new understanding
-- A significant complexity reduction that enables previously impossible computations
-
-Without one of these, even excellent engineering and documentation will not cross the acceptance threshold.
diff --git a/paper/bispectrum_overleaf.zip b/paper/bispectrum_overleaf.zip
deleted file mode 100644
index ed4eca6..0000000
Binary files a/paper/bispectrum_overleaf.zip and /dev/null differ
diff --git a/paper/bootstrap_complex_rank.py b/paper/bootstrap_complex_rank.py
deleted file mode 100644
index 658afbb..0000000
--- a/paper/bootstrap_complex_rank.py
+++ /dev/null
@@ -1,107 +0,0 @@
-"""Compute the COMPLEX rank of bootstrap matrix A for real vs complex signals."""
-import sys
-sys.path.insert(0, '/home/johmathe/bispectrum/src')
-import numpy as np
-from bispectrum.so3_on_s2 import _build_selective_index_map, load_cg_matrices
-
-CG_DATA = load_cg_matrices(8)
-
-def get_cg(l1, m1, l2, m2, l_out, m):
-    if m1 + m2 != m: return 0.0
-    if l_out < abs(l1-l2) or l_out > l1+l2: return 0.0
-    cg = CG_DATA[(l1, l2)]
-    row = (m1+l1)*(2*l2+1) + (m2+l2)
-    start = 0
-    for lv in range(abs(l1-l2), l1+l2+1):
-        if lv == l_out: return cg[row, start+m+l_out].item()
-        start += 2*lv+1
-    return 0.0
-
-def make_real_fc(lmax, seed=42):
-    rng = np.random.RandomState(seed)
-    fc = {}
-    for l in range(lmax+1):
-        c = np.zeros(2*l+1, dtype=complex)
-        c[l] = rng.randn()
-        for m in range(1, l+1):
-            re, im = rng.randn(2)
-            c[l+m] = re + 1j*im
-            c[l-m] = ((-1)**m) * (re - 1j*im)
-        fc[l] = c
-    return fc
-
-def make_complex_fc(lmax, seed=42):
-    rng = np.random.RandomState(seed)
-    fc = {}
-    for l in range(lmax+1):
-        c = np.zeros(2*l+1, dtype=complex)
-        for m in range(-l, l+1):
-            c[l+m] = rng.randn() + 1j*rng.randn()
-        fc[l] = c
-    return fc
-
-def build_bootstrap_A(fc, lt, triples):
-    n = 2*lt+1
-    rows = []
-    for l1,l2,lv in triples:
-        if lv == lt and max(l1,l2) < lt:
-            row = np.zeros(n, dtype=complex)
-            for m in range(-lt, lt+1):
-                c_m = 0j
-                for m1 in range(-l1, l1+1):
-                    m2 = m-m1
-                    if abs(m2)>l2: continue
-                    c_m += get_cg(l1,m1,l2,m2,lt,m)*fc[l1][l1+m1]*fc[l2][l2+m2]
-                row[lt+m] = np.conj(c_m)
-            rows.append(row)
-        elif l2 == lt and lv < lt and l1 < lt:
-            row = np.zeros(n, dtype=complex)
-            for m2 in range(-lt, lt+1):
-                d = 0j
-                for m1 in range(-l1, l1+1):
-                    m = m1+m2
-                    if abs(m)>lv: continue
-                    d += get_cg(l1,m1,lt,m2,lv,m)*fc[l1][l1+m1]*np.conj(fc[lv][lv+m])
-                row[lt+m2] = d
-            rows.append(row)
-    return np.array(rows) if rows else np.zeros((0,n), dtype=complex)
-
-idx = _build_selective_index_map(8)
-
-with open('/tmp/bootstrap_rank_results.txt', 'w') as f:
-    f.write("SELECTIVE bootstrap complex rank\n")
-    f.write("REAL signals:\n")
-    for lt in range(4,9):
-        tr = [(l1,l2,lv) for l1,l2,lv in idx if max(l1,l2,lv)==lt]
-        rs = []
-        for s in [42,123,999]:
-            A = build_bootstrap_A(make_real_fc(8,s), lt, tr)
-            sv = np.linalg.svd(A, compute_uv=False)
-            rs.append(int(np.sum(sv > 1e-10*sv[0])))
-        f.write(f"  ell={lt}: {A.shape} ranks={rs} expected={2*lt+1}\n")
-
-    f.write("\nCOMPLEX signals:\n")
-    for lt in range(4,9):
-        tr = [(l1,l2,lv) for l1,l2,lv in idx if max(l1,l2,lv)==lt]
-        rs = []
-        for s in [42,123,999]:
-            A = build_bootstrap_A(make_complex_fc(8,s), lt, tr)
-            sv = np.linalg.svd(A, compute_uv=False)
-            rs.append(int(np.sum(sv > 1e-10*sv[0])))
-        f.write(f"  ell={lt}: {A.shape} ranks={rs} expected={2*lt+1}\n")
-
-    f.write("\nFULL bispectrum at REAL signals:\n")
-    for lt in [4,5,6,7]:
-        all_tr = []
-        for l1 in range(lt+1):
-            for l2 in range(l1, lt+1):
-                for lv in range(abs(l1-l2), min(l1+l2,lt)+1):
-                    if max(l1,l2,lv)==lt:
-                        all_tr.append((l1,l2,lv))
-        A = build_bootstrap_A(make_real_fc(lt,42), lt, all_tr)
-        sv = np.linalg.svd(A, compute_uv=False)
-        r = int(np.sum(sv>1e-10*sv[0]))
-        f.write(f"  ell={lt}: {len(all_tr)} triples A={A.shape} rank={r}/{2*lt+1}\n")
-
-    f.write("\nDONE\n")
-print("Results written to /tmp/bootstrap_rank_results.txt")
diff --git a/paper/bootstrap_debug.py b/paper/bootstrap_debug.py
deleted file mode 100644
index a9f02ad..0000000
--- a/paper/bootstrap_debug.py
+++ /dev/null
@@ -1,188 +0,0 @@
-"""Deep debug of bootstrap rank — compare real vs complex signals,
-gauge-fixed vs ungauged, and identify which m-components are unconstrained."""
-import sys
-sys.path.insert(0, '../src')
-import torch
-from bispectrum.so3_on_s2 import (
-    _build_selective_index_map, _bispectrum_entry, load_cg_matrices,
-)
-
-CG_DATA = load_cg_matrices(8)
-
-def make_complex_fc(lmax, seed=42):
-    """Fully generic complex signal (no reality constraint, no gauge)."""
-    torch.manual_seed(seed)
-    fc = {}
-    for l in range(lmax+1):
-        r = torch.randn(1, 2*l+1, dtype=torch.float64)
-        i = torch.randn(1, 2*l+1, dtype=torch.float64)
-        fc[l] = torch.complex(r, i)
-    return fc
-
-
-def eval_bootstrap_complex(params, l_target, fc_lower, triples, lmax):
-    """Evaluate bootstrap entries with F_{l_target} parametrized by arbitrary complex params."""
-    fc = dict(fc_lower)
-    fc[l_target] = torch.complex(
-        params[:2*l_target+1].unsqueeze(0),
-        params[2*l_target+1:].unsqueeze(0)
-    )
-    vals = []
-    for l1,l2,lv in triples:
-        beta = _bispectrum_entry(fc, l1, l2, lv, CG_DATA[(l1,l2)])[0]
-        vals.append(beta.real)
-        vals.append(beta.imag)
-    return torch.stack(vals)
-
-
-def eval_bootstrap_real(params, l_target, fc_lower, triples, lmax):
-    """Evaluate bootstrap entries with real-signal F_{l_target}."""
-    fc = dict(fc_lower)
-    neg, pos = [], []
-    j = 0
-    for m in range(1, l_target+1):
-        re = params[j]; im = params[j+1]; j += 2
-        pos.append(torch.complex(re, im))
-        neg.append(((-1.)**m)*torch.complex(re, -im))
-    m0 = torch.complex(params[j], torch.zeros_like(params[j])); j += 1
-    fc[l_target] = torch.stack(list(reversed(neg))+[m0]+pos).unsqueeze(0)
-    vals = []
-    for l1,l2,lv in triples:
-        beta = _bispectrum_entry(fc, l1, l2, lv, CG_DATA[(l1,l2)])[0]
-        vals.append(beta.real)
-        vals.append(beta.imag)
-    return torch.stack(vals)
-
-
-def compute_rank(J, label=""):
-    sv = torch.linalg.svdvals(J)
-    tol = 1e-8 * sv[0].item() if sv[0] > 0 else 1e-15
-    r = int((sv > tol).sum().item())
-    return r, sv
-
-
-def main():
-    idx_map = _build_selective_index_map(8)
-
-    print("=" * 70)
-    print("Bootstrap rank: COMPLEX signals (2*(2l+1) real params per F_l)")
-    print("=" * 70)
-
-    for l_target in range(4, 8):
-        fc_full = make_complex_fc(8, seed=42)
-        fc_lower = {l: fc_full[l] for l in range(l_target)}
-        triples = [(l1,l2,lv) for l1,l2,lv in idx_map if max(l1,l2,lv)==l_target]
-
-        n_params = 2 * (2*l_target+1)
-        torch.manual_seed(42)
-        params = torch.randn(n_params, dtype=torch.float64)
-
-        J = torch.autograd.functional.jacobian(
-            lambda p: eval_bootstrap_complex(p, l_target, fc_lower, triples, 8), params)
-        r, sv = compute_rank(J)
-        print(f"  ℓ={l_target}: {len(triples)} entries × 2 = {2*len(triples)} rows, "
-              f"{n_params} params, rank={r}")
-
-    print("\n" + "=" * 70)
-    print("Bootstrap rank: REAL signal, NO gauge (2l+1 real params)")
-    print("=" * 70)
-
-    for l_target in range(4, 8):
-        torch.manual_seed(42)
-        # Build real lower-degree coefficients
-        fc_lower = {}
-        for l in range(l_target):
-            neg, pos = [], []
-            for m in range(1, l+1):
-                re = torch.randn(1, dtype=torch.float64)
-                im = torch.randn(1, dtype=torch.float64)
-                pos.append(torch.complex(re, im))
-                neg.append(((-1.)**m)*torch.complex(re, -im))
-            m0 = torch.complex(torch.randn(1, dtype=torch.float64), torch.zeros(1, dtype=torch.float64))
-            fc_lower[l] = torch.stack(list(reversed(neg))+[m0]+pos).unsqueeze(0)
-
-        triples = [(l1,l2,lv) for l1,l2,lv in idx_map if max(l1,l2,lv)==l_target]
-        n_params = 2*l_target + 1
-        params = torch.randn(n_params, dtype=torch.float64)
-
-        J = torch.autograd.functional.jacobian(
-            lambda p: eval_bootstrap_real(p, l_target, fc_lower, triples, 8), params)
-        r, sv = compute_rank(J)
-        print(f"  ℓ={l_target}: {len(triples)} entries, "
-              f"{n_params} real params, rank={r}")
-
-    print("\n" + "=" * 70)
-    print("Bootstrap rank: REAL signal, GAUGE-FIXED (F_1=(0,c,0), Im(a_2^1)=0)")
-    print("=" * 70)
-
-    for l_target in range(4, 8):
-        torch.manual_seed(42)
-        fc_lower = {}
-        fc_lower[0] = torch.complex(torch.tensor([[1.5]], dtype=torch.float64),
-                                     torch.zeros(1,1, dtype=torch.float64))
-        fc_lower[1] = torch.complex(torch.tensor([[0.,1.2,0.]], dtype=torch.float64),
-                                     torch.zeros(1,3, dtype=torch.float64))
-        for l in range(2, l_target):
-            neg, pos = [], []
-            for m in range(1, l+1):
-                re = torch.randn(1, dtype=torch.float64)
-                if l == 2 and m == 1:
-                    im = torch.zeros(1, dtype=torch.float64)
-                else:
-                    im = torch.randn(1, dtype=torch.float64)
-                pos.append(torch.complex(re, im))
-                neg.append(((-1.)**m)*torch.complex(re, -im))
-            m0 = torch.complex(torch.randn(1, dtype=torch.float64), torch.zeros(1, dtype=torch.float64))
-            fc_lower[l] = torch.stack(list(reversed(neg))+[m0]+pos).unsqueeze(0)
-
-        triples = [(l1,l2,lv) for l1,l2,lv in idx_map if max(l1,l2,lv)==l_target]
-        n_params = 2*l_target + 1
-        params = torch.randn(n_params, dtype=torch.float64)
-
-        J = torch.autograd.functional.jacobian(
-            lambda p: eval_bootstrap_real(p, l_target, fc_lower, triples, 8), params)
-        r, sv = compute_rank(J)
-        print(f"  ℓ={l_target}: {len(triples)} entries, "
-              f"{n_params} real params, rank={r}")
-
-    print("\n" + "=" * 70)
-    print("FULL bispectrum bootstrap rank (all valid triples)")
-    print("=" * 70)
-
-    for l_target in [4, 5, 6]:
-        torch.manual_seed(42)
-        fc_lower = {}
-        fc_lower[0] = torch.complex(torch.tensor([[1.5]], dtype=torch.float64),
-                                     torch.zeros(1,1, dtype=torch.float64))
-        fc_lower[1] = torch.complex(torch.tensor([[0.,1.2,0.]], dtype=torch.float64),
-                                     torch.zeros(1,3, dtype=torch.float64))
-        for l in range(2, l_target):
-            neg, pos = [], []
-            for m in range(1, l+1):
-                re = torch.randn(1, dtype=torch.float64)
-                if l==2 and m==1: im = torch.zeros(1, dtype=torch.float64)
-                else: im = torch.randn(1, dtype=torch.float64)
-                pos.append(torch.complex(re, im))
-                neg.append(((-1.)**m)*torch.complex(re, -im))
-            m0 = torch.complex(torch.randn(1, dtype=torch.float64), torch.zeros(1, dtype=torch.float64))
-            fc_lower[l] = torch.stack(list(reversed(neg))+[m0]+pos).unsqueeze(0)
-
-        all_triples = []
-        for l1 in range(l_target+1):
-            for l2 in range(l1, l_target+1):
-                for lv in range(abs(l1-l2), min(l1+l2, l_target)+1):
-                    if max(l1,l2,lv) == l_target:
-                        all_triples.append((l1,l2,lv))
-
-        n_params = 2*l_target + 1
-        params = torch.randn(n_params, dtype=torch.float64)
-
-        J = torch.autograd.functional.jacobian(
-            lambda p: eval_bootstrap_real(p, l_target, fc_lower, all_triples, l_target), params)
-        r, sv = compute_rank(J)
-        print(f"  ℓ={l_target}: {len(all_triples)} entries (all valid), "
-              f"{n_params} real params, rank={r}")
-
-
-if __name__ == "__main__":
-    main()
diff --git a/paper/checklist.tex b/paper/checklist.tex
deleted file mode 100644
index 7921f7a..0000000
--- a/paper/checklist.tex
+++ /dev/null
@@ -1,224 +0,0 @@
-\section*{NeurIPS Paper Checklist}
-
-\begin{enumerate}
-
-\item {\bf Claims}
-    \item[] Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
-    \item[] Answer: \answerYes{}
-    \item[] Justification: The abstract and introduction list four contributions (selective bispectrum theory, completeness proof, \texttt{nn.Module} library, and experiments on three benchmarks) that are substantiated in Sections~3--4 and Appendices~A--B.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that the abstract and introduction do not include the claims made in the paper.
-        \item The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A \answerNo{} or \answerNA{} answer to this question will not be perceived well by the reviewers. 
-        \item The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. 
-        \item It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 
-    \end{itemize}
-
-\item {\bf Limitations}
-    \item[] Question: Does the paper discuss the limitations of the work performed by the authors?
-    \item[] Answer: \answerYes{}
-    \item[] Justification: Section~6 contains a dedicated Limitations paragraph discussing the genericity assumption, bootstrap verification scope ($L \le 100$), octahedral inversion recovery rate, and missing experimental comparisons. Appendix~A includes a ``Limitations and open questions'' subsection.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that the paper has no limitation while the answer \answerNo{} means that the paper has limitations, but those are not discussed in the paper. 
-        \item The authors are encouraged to create a separate ``Limitations'' section in their paper.
-        \item The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
-        \item The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
-        \item The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
-        \item The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
-        \item If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
-        \item While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
-    \end{itemize}
-
-\item {\bf Theory assumptions and proofs}
-    \item[] Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
-    \item[] Answer: \answerYes{}
-    \item[] Justification: Theorem~1 states all assumptions (genericity, gauge-fixing) and a proof sketch appears in Section~3.4. The complete proof with 10 lemmas and 6 propositions is in Appendix~A. The octahedral inversion gap analysis is in Appendix~B.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that the paper does not include theoretical results. 
-        \item All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.
-        \item All assumptions should be clearly stated or referenced in the statement of any theorems.
-        \item The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. 
-        \item Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
-        \item Theorems and Lemmas that the proof relies upon should be properly referenced. 
-    \end{itemize}
-
-    \item {\bf Experimental result reproducibility}
-    \item[] Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
-    \item[] Answer: \answerYes{}
-    \item[] Justification: All hyperparameters are in Appendix~F (Table~\ref{tab:hyperparams}), architectures in Appendices~D--E, optimizer and scheduler details in Table~\ref{tab:hyperparams}, and random seeds are specified. The code is provided as an anonymous repository.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that the paper does not include experiments.
-        \item If the paper includes experiments, a \answerNo{} answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
-        \item If the contribution is a dataset and\slash or model, the authors should describe the steps taken to make their results reproducible or verifiable. 
-        \item Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
-        \item While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
-        \begin{enumerate}
-            \item If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.
-            \item If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.
-            \item If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).
-            \item We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
-        \end{enumerate}
-    \end{itemize}
-
-
-\item {\bf Open access to data and code}
-    \item[] Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
-    \item[] Answer: \answerYes{}
-    \item[] Justification: An anonymous code repository is provided with the submission. All three datasets (PCam, OrganMNIST3D, Spherical MNIST) are publicly available. Sweep scripts and training code are included.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that paper does not include experiments requiring code.
-        \item Please see the NeurIPS code and data submission guidelines (\url{https://neurips.cc/public/guides/CodeSubmissionPolicy}) for more details.
-        \item While we encourage the release of code and data, we understand that this might not be possible, so \answerNo{} is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).
-        \item The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (\url{https://neurips.cc/public/guides/CodeSubmissionPolicy}) for more details.
-        \item The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
-        \item The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.
-        \item At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
-        \item Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
-    \end{itemize}
-
-
-\item {\bf Experimental setting/details}
-    \item[] Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer) necessary to understand the results?
-    \item[] Answer: \answerYes{}
-    \item[] Justification: Sections~4.1--4.3 describe experimental setups. Full hyperparameters are in Table~\ref{tab:hyperparams} (Appendix~F), architecture details in Appendices~D--E, data preprocessing and splits in Appendix~F.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that the paper does not include experiments.
-        \item The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.
-        \item The full details can be provided either with the code, in appendix, or as supplemental material.
-    \end{itemize}
-
-\item {\bf Experiment statistical significance}
-    \item[] Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
-    \item[] Answer: \answerYes{}
-    \item[] Justification: Tables~\ref{tab:organ3d-results}--\ref{tab:smnist-results} (OrganMNIST3D, Spherical MNIST) report mean $\pm$ standard deviation over 3 seeds. Table~\ref{tab:pcam-results} (PCam) uses a single seed for the Pareto sweep, which is noted in Appendix~F. The aggregation procedure is described in Appendix~F.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that the paper does not include experiments.
-        \item The authors should answer \answerYes{} if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
-        \item The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).
-        \item The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
-        \item The assumptions made should be given (e.g., Normally distributed errors).
-        \item It should be clear whether the error bar is the standard deviation or the standard error of the mean.
-        \item It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96\% CI, if the hypothesis of Normality of errors is not verified.
-        \item For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g., negative error rates).
-        \item If error bars are reported in tables or plots, the authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
-    \end{itemize}
-
-\item {\bf Experiments compute resources}
-    \item[] Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
-    \item[] Answer: \answerYes{}
-    \item[] Justification: Appendix~F states that all experiments were run on a single NVIDIA H100 GPU (80\,GB). Wall-clock times are available in the code repository.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that the paper does not include experiments.
-        \item The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
-        \item The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. 
-        \item The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). 
-    \end{itemize}
-    
-\item {\bf Code of ethics}
-    \item[] Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics \url{https://neurips.cc/public/EthicsGuidelines}?
-    \item[] Answer: \answerYes{}
-    \item[] Justification: The work develops mathematical invariants and a software library evaluated on publicly available datasets. No human subjects, private data, or dual-use concerns are involved.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that the authors have not reviewed the NeurIPS Code of Ethics.
-        \item If the authors answer \answerNo, they should explain the special circumstances that require a deviation from the Code of Ethics.
-        \item The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
-    \end{itemize}
-
-
-\item {\bf Broader impacts}
-    \item[] Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
-    \item[] Answer: \answerNA{}
-    \item[] Justification: This is foundational mathematical and software research on group-theoretic invariants. We do not foresee direct negative societal impacts.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that there is no societal impact of the work performed.
-        \item If the authors answer \answerNA{} or \answerNo, they should explain why their work has no societal impact or why the paper does not address societal impact.
-        \item Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
-        \item The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate Deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
-        \item The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
-        \item If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
-    \end{itemize}
-    
-\item {\bf Safeguards}
-    \item[] Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pre-trained language models, image generators, or scraped datasets)?
-    \item[] Answer: \answerNA{}
-    \item[] Justification: The released asset is a deterministic mathematical library for computing group-theoretic invariants. It poses no risk for misuse.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that the paper poses no such risks.
-        \item Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. 
-        \item Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
-        \item We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
-    \end{itemize}
-
-\item {\bf Licenses for existing assets}
-    \item[] Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
-    \item[] Answer: \answerYes{}
-    \item[] Justification: All datasets are cited (PCam: CC0 1.0, OrganMNIST3D: CC BY 4.0, MNIST: public domain for research). Dataset licenses are listed in Appendix~F.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that the paper does not use existing assets.
-        \item The authors should cite the original paper that produced the code package or dataset.
-        \item The authors should state which version of the asset is used and, if possible, include a URL.
-        \item The name of the license (e.g., CC-BY 4.0) should be included for each asset.
-        \item For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
-        \item If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, \url{paperswithcode.com/datasets} has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.
-        \item For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
-        \item If this information is not available online, the authors are encouraged to reach out to the asset's creators.
-    \end{itemize}
-
-\item {\bf New assets}
-    \item[] Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
-    \item[] Answer: \answerYes{}
-    \item[] Justification: The library is released under the MIT license with documentation, API reference, and usage examples. Section~3 describes the design and interface.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that the paper does not release new assets.
-        \item Researchers should communicate the details of the dataset\slash code\slash model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. 
-        \item The paper should discuss whether and how consent was obtained from people whose asset is used.
-        \item At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
-    \end{itemize}
-
-\item {\bf Crowdsourcing and research with human subjects}
-    \item[] Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? 
-    \item[] Answer: \answerNA{}
-    \item[] Justification: This work does not involve crowdsourcing or human subjects.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that the paper does not involve crowdsourcing nor research with human subjects.
-        \item Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. 
-        \item According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. 
-    \end{itemize}
-
-\item {\bf Institutional review board (IRB) approvals or equivalent for research with human subjects}
-    \item[] Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
-    \item[] Answer: \answerNA{}
-    \item[] Justification: This work does not involve human subjects.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that the paper does not involve crowdsourcing nor research with human subjects.
-        \item Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. 
-        \item We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. 
-        \item For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
-    \end{itemize}
-
-\item {\bf Declaration of LLM usage}
-    \item[] Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does \emph{not} impact the core methodology, scientific rigor, or originality of the research, declaration is not required.
-    %this research? 
-    \item[] Answer: \answerNA{}
-    \item[] Justification: LLMs are not part of the core methodology of this work.
-    \item[] Guidelines:
-    \begin{itemize}
-        \item The answer \answerNA{} means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.
-        \item Please refer to our LLM policy in the NeurIPS handbook for what should or should not be described.
-    \end{itemize}
-
-\end{enumerate}
diff --git a/paper/citation_audit.py b/paper/citation_audit.py
deleted file mode 100644
index 53deace..0000000
--- a/paper/citation_audit.py
+++ /dev/null
@@ -1,907 +0,0 @@
-from __future__ import annotations
-
-import csv
-import json
-import re
-import unicodedata
-from dataclasses import asdict, dataclass, field
-from difflib import SequenceMatcher
-from pathlib import Path
-from typing import Any
-from urllib.error import HTTPError, URLError
-from urllib.parse import quote
-from urllib.request import Request, urlopen
-
-
-ROOT: Path = Path(__file__).resolve().parent
-TEX_PATH: Path = ROOT / "paper.tex"
-BIB_PATH: Path = ROOT / "references.bib"
-OUT_DIR: Path = ROOT / "citation_audit"
-LOCAL_BIB_DIR: Path = OUT_DIR / "bibtex" / "local"
-RESOLVED_BIB_DIR: Path = OUT_DIR / "bibtex" / "resolved"
-PAPER_DIR: Path = OUT_DIR / "papers"
-REPORT_PATH: Path = OUT_DIR / "citation_audit.md"
-CSV_PATH: Path = OUT_DIR / "citation_audit.csv"
-JSON_PATH: Path = OUT_DIR / "citation_audit.json"
-USER_AGENT: str = "bispectrum-citation-audit/0.1"
-
-CITE_PATTERN: re.Pattern[str] = re.compile(
-    r"\\(?:cite|citet|citep|citealp|citeauthor|citeyear|autocite|textcite|parencite|footcite|Citep|Citet|citeyearpar)\*?\{([^}]+)\}"
-)
-
-MANUAL_RECORDS: dict[str, dict[str, str | int | None]] = {
-    "bonev2023spherical": {
-        "source": "manual",
-        "title": "Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere",
-        "year": 2023,
-        "venue": "Proceedings of the 40th International Conference on Machine Learning",
-        "doi": None,
-        "landing_url": "https://proceedings.mlr.press/v202/bonev23a.html",
-        "pdf_url": "https://proceedings.mlr.press/v202/bonev23a/bonev23a.pdf",
-    },
-    "cesa2022program": {
-        "source": "manual",
-        "title": "A Program to Build E(N)-Equivariant Steerable CNNs",
-        "year": 2022,
-        "venue": "International Conference on Learning Representations",
-        "doi": None,
-        "landing_url": "https://openreview.net/forum?id=WE4qe9xlnQw",
-        "pdf_url": "https://openreview.net/pdf?id=WE4qe9xlnQw",
-    },
-    "oreiller2022bispectral": {
-        "source": "manual",
-        "title": "Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral U-Net",
-        "year": 2022,
-        "venue": "Medical Imaging with Deep Learning",
-        "doi": None,
-        "landing_url": "https://proceedings.mlr.press/v172/oreiller22a.html",
-        "pdf_url": "https://proceedings.mlr.press/v172/oreiller22a/oreiller22a.pdf",
-    },
-    "loshchilov2019adamw": {
-        "source": "manual",
-        "title": "Decoupled Weight Decay Regularization",
-        "year": 2019,
-        "venue": "International Conference on Learning Representations",
-        "doi": None,
-        "landing_url": "https://openreview.net/forum?id=Bkg6RiCqY7",
-        "pdf_url": "https://openreview.net/pdf?id=Bkg6RiCqY7",
-    },
-    "zhemchuzhnikov2025equilopo": {
-        "source": "manual",
-        "title": "On the Fourier Analysis in the SO(3) Space: the EquiLoPO Network",
-        "year": 2025,
-        "venue": "International Conference on Learning Representations",
-        "doi": None,
-        "landing_url": "https://openreview.net/forum?id=LvTSvdiSwG",
-        "pdf_url": "https://openreview.net/pdf?id=LvTSvdiSwG",
-    },
-}
-
-MANUAL_ASSESSMENT_OVERRIDES: dict[str, str] = {}
-
-MANUAL_NOTES: dict[str, str] = {}
-
-
-@dataclass(slots=True)
-class CitationMention:
-    line_number: int
-    snippet: str
-
-
-@dataclass(slots=True)
-class BibEntry:
-    key: str
-    entry_type: str
-    raw: str
-    fields: dict[str, str]
-
-    @property
-    def title(self) -> str:
-        return self.fields.get("title", "")
-
-    @property
-    def year(self) -> int | None:
-        raw_year: str = self.fields.get("year", "").strip()
-        return int(raw_year) if raw_year.isdigit() else None
-
-    @property
-    def doi(self) -> str | None:
-        value: str = self.fields.get("doi", "").strip()
-        return value or None
-
-    @property
-    def url(self) -> str | None:
-        value: str = self.fields.get("url", "").strip()
-        return value or None
-
-    @property
-    def first_author_surname(self) -> str | None:
-        authors: str = self.fields.get("author", "")
-        if not authors:
-            return None
-        first_author: str = authors.split(" and ")[0].strip()
-        if "," in first_author:
-            return first_author.split(",", 1)[0].strip()
-        parts: list[str] = first_author.split()
-        return parts[-1] if parts else None
-
-    @property
-    def venue(self) -> str:
-        for field_name in ("journal", "booktitle", "school"):
-            value: str = self.fields.get(field_name, "").strip()
-            if value:
-                return value
-        return ""
-
-
-@dataclass(slots=True)
-class RemoteRecord:
-    source: str
-    title: str | None = None
-    year: int | None = None
-    venue: str | None = None
-    doi: str | None = None
-    landing_url: str | None = None
-    pdf_url: str | None = None
-    match_score: float | None = None
-
-
-@dataclass(slots=True)
-class AuditRow:
-    key: str
-    category: str
-    mention_count: int
-    local_entry_type: str
-    local_title: str
-    local_year: int | None
-    local_venue: str
-    local_doi: str | None
-    local_url: str | None
-    resolved_source: str | None
-    resolved_title: str | None
-    resolved_year: int | None
-    resolved_venue: str | None
-    resolved_doi: str | None
-    title_match_score: float | None
-    bibtex_source: str
-    paper_status: str
-    paper_path: str | None
-    assessment: str
-    notes: list[str] = field(default_factory=list)
-    mentions: list[CitationMention] = field(default_factory=list)
-
-
-def normalize_text(value: str) -> str:
-    normalized: str = unicodedata.normalize("NFKD", value)
-    ascii_only: str = normalized.encode("ascii", "ignore").decode("ascii")
-    lowered: str = ascii_only.casefold()
-    return re.sub(r"[^a-z0-9]+", "", lowered)
-
-
-def plain_text_from_bib(value: str) -> str:
-    plain: str = value
-    replacements: dict[str, str] = {
-        "{": "",
-        "}": "",
-        "\\&": "&",
-        "\\_": "_",
-        "~": " ",
-        "$": "",
-    }
-    for old, new in replacements.items():
-        plain = plain.replace(old, new)
-    plain = re.sub(r"\\[a-zA-Z]+\s*", "", plain)
-    plain = re.sub(r"\s+", " ", plain)
-    return plain.strip()
-
-
-def similarity_score(left: str, right: str) -> float:
-    if not left or not right:
-        return 0.0
-    return SequenceMatcher(None, normalize_text(left), normalize_text(right)).ratio()
-
-
-def read_url(
-    url: str,
-    *,
-    accept: str | None = None,
-    timeout: int = 30,
-) -> bytes:
-    headers: dict[str, str] = {"User-Agent": USER_AGENT}
-    if accept is not None:
-        headers["Accept"] = accept
-    request: Request = Request(url, headers=headers)
-    with urlopen(request, timeout=timeout) as response:
-        return response.read()
-
-
-def fetch_text(url: str, *, accept: str | None = None, timeout: int = 30) -> str | None:
-    try:
-        return read_url(url, accept=accept, timeout=timeout).decode("utf-8", errors="replace")
-    except (HTTPError, URLError, TimeoutError):
-        return None
-
-
-def fetch_json(url: str, *, timeout: int = 30) -> dict[str, Any] | None:
-    payload: str | None = fetch_text(url, accept="application/json", timeout=timeout)
-    if payload is None:
-        return None
-    try:
-        data: dict[str, Any] = json.loads(payload)
-    except json.JSONDecodeError:
-        return None
-    return data
-
-
-def manual_record_for_key(key: str) -> RemoteRecord | None:
-    payload: dict[str, str | int | None] | None = MANUAL_RECORDS.get(key)
-    if payload is None:
-        return None
-    return RemoteRecord(
-        source=str(payload["source"]),
-        title=str(payload["title"]) if payload["title"] is not None else None,
-        year=int(payload["year"]) if payload["year"] is not None else None,
-        venue=str(payload["venue"]) if payload["venue"] is not None else None,
-        doi=str(payload["doi"]) if payload["doi"] is not None else None,
-        landing_url=str(payload["landing_url"]) if payload["landing_url"] is not None else None,
-        pdf_url=str(payload["pdf_url"]) if payload["pdf_url"] is not None else None,
-        match_score=1.0,
-    )
-
-
-def extract_citations(tex_text: str) -> dict[str, list[CitationMention]]:
-    lines: list[str] = tex_text.splitlines()
-    mentions: dict[str, list[CitationMention]] = {}
-    for line_number, line in enumerate(lines, start=1):
-        collapsed: str = " ".join(line.strip().split())
-        for match in CITE_PATTERN.finditer(line):
-            for raw_key in match.group(1).split(","):
-                key: str = raw_key.strip()
-                if not key:
-                    continue
-                mentions.setdefault(key, []).append(
-                    CitationMention(line_number=line_number, snippet=collapsed)
-                )
-    return mentions
-
-
-def split_bib_entries(bib_text: str) -> list[str]:
-    entries: list[str] = []
-    index: int = 0
-    while True:
-        start: int = bib_text.find("@", index)
-        if start == -1:
-            return entries
-        brace_start: int = bib_text.find("{", start)
-        if brace_start == -1:
-            return entries
-        depth: int = 0
-        in_quotes: bool = False
-        escape_next: bool = False
-        for cursor in range(brace_start, len(bib_text)):
-            char: str = bib_text[cursor]
-            if escape_next:
-                escape_next = False
-                continue
-            if char == "\\":
-                escape_next = True
-                continue
-            if char == '"':
-                in_quotes = not in_quotes
-                continue
-            if in_quotes:
-                continue
-            if char == "{":
-                depth += 1
-            elif char == "}":
-                depth -= 1
-                if depth == 0:
-                    entries.append(bib_text[start : cursor + 1].strip())
-                    index = cursor + 1
-                    break
-        else:
-            return entries
-
-
-def parse_bib_value(chunk: str, start_index: int) -> tuple[str, int]:
-    opener: str = chunk[start_index]
-    if opener == "{":
-        depth: int = 0
-        cursor: int = start_index
-        while cursor < len(chunk):
-            char: str = chunk[cursor]
-            if char == "{":
-                depth += 1
-            elif char == "}":
-                depth -= 1
-                if depth == 0:
-                    return chunk[start_index + 1 : cursor], cursor + 1
-            cursor += 1
-        return chunk[start_index + 1 :], len(chunk)
-    if opener == '"':
-        cursor = start_index + 1
-        escaped: bool = False
-        while cursor < len(chunk):
-            char = chunk[cursor]
-            if escaped:
-                escaped = False
-            elif char == "\\":
-                escaped = True
-            elif char == '"':
-                return chunk[start_index + 1 : cursor], cursor + 1
-            cursor += 1
-        return chunk[start_index + 1 :], len(chunk)
-    cursor = start_index
-    while cursor < len(chunk) and chunk[cursor] not in ",\n":
-        cursor += 1
-    return chunk[start_index:cursor].strip(), cursor
-
-
-def parse_bib_entry(raw_entry: str) -> BibEntry:
-    header_match: re.Match[str] | None = re.match(r"@(\w+)\s*\{\s*([^,\s]+)\s*,", raw_entry, re.S)
-    if header_match is None:
-        raise ValueError(f"Could not parse BibTeX header: {raw_entry[:80]}")
-    entry_type: str = header_match.group(1).strip().lower()
-    key: str = header_match.group(2).strip()
-    body: str = raw_entry[header_match.end() :].rstrip().rstrip("}").strip()
-    fields: dict[str, str] = {}
-    cursor: int = 0
-    while cursor < len(body):
-        while cursor < len(body) and body[cursor] in " \t\r\n,":
-            cursor += 1
-        if cursor >= len(body):
-            break
-        equals_index: int = body.find("=", cursor)
-        if equals_index == -1:
-            break
-        field_name: str = body[cursor:equals_index].strip().lower()
-        cursor = equals_index + 1
-        while cursor < len(body) and body[cursor].isspace():
-            cursor += 1
-        if cursor >= len(body):
-            break
-        field_value, cursor = parse_bib_value(body, cursor)
-        fields[field_name] = " ".join(field_value.strip().split())
-        while cursor < len(body) and body[cursor] not in "\n,":
-            cursor += 1
-        if cursor < len(body) and body[cursor] == ",":
-            cursor += 1
-    return BibEntry(key=key, entry_type=entry_type, raw=raw_entry + "\n", fields=fields)
-
-
-def load_bib_entries(bib_text: str) -> dict[str, BibEntry]:
-    entries: dict[str, BibEntry] = {}
-    for raw_entry in split_bib_entries(bib_text):
-        entry: BibEntry = parse_bib_entry(raw_entry)
-        entries[entry.key] = entry
-    return entries
-
-
-def crossref_record_from_item(item: dict[str, Any], score: float) -> RemoteRecord:
-    titles: list[str] = item.get("title") or []
-    container_titles: list[str] = item.get("container-title") or []
-    year_parts: list[list[int]] = (
-        item.get("published-print", {}).get("date-parts")
-        or item.get("published-online", {}).get("date-parts")
-        or item.get("issued", {}).get("date-parts")
-        or []
-    )
-    year: int | None = year_parts[0][0] if year_parts and year_parts[0] else None
-    doi: str | None = item.get("DOI")
-    landing_url: str | None = f"https://doi.org/{doi}" if doi else item.get("URL")
-    return RemoteRecord(
-        source="crossref",
-        title=titles[0] if titles else None,
-        year=year,
-        venue=container_titles[0] if container_titles else None,
-        doi=doi,
-        landing_url=landing_url,
-        match_score=score,
-    )
-
-
-def search_crossref(title: str, first_author_surname: str | None, year: int | None) -> RemoteRecord | None:
-    query: str = quote(title)
-    url: str = f"https://api.crossref.org/works?query.bibliographic={query}&rows=5"
-    payload: dict[str, Any] | None = fetch_json(url)
-    if payload is None:
-        return None
-    items: list[dict[str, Any]] = payload.get("message", {}).get("items", [])
-    best_record: RemoteRecord | None = None
-    best_score: float = 0.0
-    for item in items:
-        candidate_titles: list[str] = item.get("title") or []
-        if not candidate_titles:
-            continue
-        candidate_title: str = candidate_titles[0]
-        score: float = similarity_score(title, candidate_title)
-        authors: list[dict[str, Any]] = item.get("author") or []
-        if first_author_surname and authors:
-            candidate_surname: str = str(authors[0].get("family", "")).strip()
-            if normalize_text(candidate_surname) == normalize_text(first_author_surname):
-                score += 0.05
-        candidate_year_parts: list[list[int]] = (
-            item.get("published-print", {}).get("date-parts")
-            or item.get("published-online", {}).get("date-parts")
-            or item.get("issued", {}).get("date-parts")
-            or []
-        )
-        candidate_year: int | None = (
-            candidate_year_parts[0][0] if candidate_year_parts and candidate_year_parts[0] else None
-        )
-        if year is not None and candidate_year is not None:
-            if abs(year - candidate_year) == 0:
-                score += 0.05
-            elif abs(year - candidate_year) > 1:
-                score -= 0.15
-        if score > best_score:
-            best_score = score
-            best_record = crossref_record_from_item(item, score)
-    if best_record is None or best_score < 0.72:
-        return None
-    return best_record
-
-
-def openalex_record_from_result(result: dict[str, Any], score: float) -> RemoteRecord:
-    primary_location: dict[str, Any] = result.get("primary_location") or {}
-    best_oa_location: dict[str, Any] = result.get("best_oa_location") or {}
-    pdf_url: str | None = (
-        best_oa_location.get("pdf_url")
-        or primary_location.get("pdf_url")
-        or result.get("open_access", {}).get("oa_url")
-    )
-    landing_url: str | None = (
-        best_oa_location.get("landing_page_url")
-        or primary_location.get("landing_page_url")
-        or result.get("doi")
-    )
-    publication_year: int | None = result.get("publication_year")
-    venue_name: str | None = None
-    if primary_location.get("source"):
-        venue_name = primary_location["source"].get("display_name")
-    return RemoteRecord(
-        source="openalex",
-        title=result.get("display_name"),
-        year=publication_year,
-        venue=venue_name,
-        doi=(result.get("doi") or "").removeprefix("https://doi.org/") or None,
-        landing_url=landing_url,
-        pdf_url=pdf_url,
-        match_score=score,
-    )
-
-
-def search_openalex(title: str, first_author_surname: str | None, year: int | None) -> RemoteRecord | None:
-    url: str = f"https://api.openalex.org/works?search={quote(title)}&per_page=5"
-    payload: dict[str, Any] | None = fetch_json(url)
-    if payload is None:
-        return None
-    results: list[dict[str, Any]] = payload.get("results", [])
-    best_record: RemoteRecord | None = None
-    best_score: float = 0.0
-    for result in results:
-        candidate_title: str = str(result.get("display_name", "")).strip()
-        if not candidate_title:
-            continue
-        score: float = similarity_score(title, candidate_title)
-        authorships: list[dict[str, Any]] = result.get("authorships") or []
-        if first_author_surname and authorships:
-            first_author: dict[str, Any] = authorships[0].get("author") or {}
-            display_name: str = str(first_author.get("display_name", "")).strip()
-            candidate_surname: str = display_name.split()[-1] if display_name else ""
-            if normalize_text(candidate_surname) == normalize_text(first_author_surname):
-                score += 0.05
-        candidate_year: int | None = result.get("publication_year")
-        if year is not None and candidate_year is not None:
-            if abs(year - candidate_year) == 0:
-                score += 0.05
-            elif abs(year - candidate_year) > 1:
-                score -= 0.15
-        if score > best_score:
-            best_score = score
-            best_record = openalex_record_from_result(result, score)
-    if best_record is None or best_score < 0.72:
-        return None
-    return best_record
-
-
-def enrich_with_openalex(record: RemoteRecord, title: str) -> RemoteRecord:
-    if record.pdf_url and record.landing_url:
-        return record
-    openalex_record: RemoteRecord | None = None
-    if record.doi:
-        url: str = f"https://api.openalex.org/works?filter=doi:https://doi.org/{quote(record.doi, safe='')}&per_page=1"
-        payload: dict[str, Any] | None = fetch_json(url)
-        results: list[dict[str, Any]] = payload.get("results", []) if payload else []
-        if results:
-            openalex_record = openalex_record_from_result(results[0], similarity_score(title, results[0].get("display_name", "")))
-    if openalex_record is None:
-        openalex_record = search_openalex(title, None, record.year)
-    if openalex_record is None:
-        return record
-    if record.doi and not openalex_record.doi:
-        openalex_record.doi = record.doi
-    if record.match_score and (openalex_record.match_score or 0.0) < record.match_score:
-        openalex_record.match_score = record.match_score
-    if record.source == "crossref":
-        if openalex_record.title is None:
-            openalex_record.title = record.title
-        if openalex_record.year is None:
-            openalex_record.year = record.year
-        if openalex_record.venue is None:
-            openalex_record.venue = record.venue
-    return openalex_record
-
-
-def extract_arxiv_id(entry: BibEntry) -> str | None:
-    candidates: list[str] = []
-    if entry.url:
-        candidates.append(entry.url)
-    for field_name in ("journal", "booktitle", "note"):
-        value: str = entry.fields.get(field_name, "")
-        if value:
-            candidates.append(value)
-    for candidate in candidates:
-        match: re.Match[str] | None = re.search(r"arxiv\.org/(?:abs|pdf)/([0-9]{4}\.[0-9]{4,5}(?:v\d+)?)", candidate, re.I)
-        if match is not None:
-            return match.group(1)
-        match = re.search(r"arXiv:([0-9]{4}\.[0-9]{4,5}(?:v\d+)?)", candidate, re.I)
-        if match is not None:
-            return match.group(1)
-    return None
-
-
-def resolve_remote_record(entry: BibEntry) -> RemoteRecord | None:
-    manual_record: RemoteRecord | None = manual_record_for_key(entry.key)
-    if manual_record is not None:
-        return manual_record
-    search_title: str = plain_text_from_bib(entry.title)
-    if entry.doi:
-        landing_url: str = f"https://doi.org/{entry.doi}"
-        return enrich_with_openalex(
-            RemoteRecord(
-                source="doi",
-                title=search_title,
-                year=entry.year,
-                venue=entry.venue,
-                doi=entry.doi,
-                landing_url=landing_url,
-                match_score=1.0,
-            ),
-            search_title,
-        )
-    arxiv_id: str | None = extract_arxiv_id(entry)
-    if arxiv_id is not None:
-        return RemoteRecord(
-            source="arxiv",
-            title=search_title,
-            year=entry.year,
-            venue=entry.venue,
-            doi=entry.doi,
-            landing_url=f"https://arxiv.org/abs/{arxiv_id}",
-            pdf_url=f"https://arxiv.org/pdf/{arxiv_id}.pdf",
-            match_score=1.0,
-        )
-    openalex_record: RemoteRecord | None = search_openalex(
-        search_title,
-        entry.first_author_surname,
-        entry.year,
-    )
-    crossref_record: RemoteRecord | None = search_crossref(
-        search_title,
-        entry.first_author_surname,
-        entry.year,
-    )
-    if openalex_record is not None and crossref_record is not None:
-        return openalex_record if (openalex_record.match_score or 0.0) >= (crossref_record.match_score or 0.0) else enrich_with_openalex(crossref_record, search_title)
-    if openalex_record is not None:
-        return openalex_record
-    if crossref_record is not None:
-        return enrich_with_openalex(crossref_record, search_title)
-    return None
-
-
-def download_resolved_bibtex(entry: BibEntry, record: RemoteRecord | None) -> tuple[str, str]:
-    local_path: Path = LOCAL_BIB_DIR / f"{entry.key}.bib"
-    local_path.write_text(entry.raw, encoding="utf-8")
-    if record is not None and record.doi:
-        bibtex_text: str | None = fetch_text(
-            f"https://doi.org/{record.doi}",
-            accept="application/x-bibtex",
-        )
-        if bibtex_text:
-            resolved_path: Path = RESOLVED_BIB_DIR / f"{entry.key}.bib"
-            resolved_path.write_text(bibtex_text.strip() + "\n", encoding="utf-8")
-            return "remote-doi", str(resolved_path.relative_to(ROOT))
-    resolved_path = RESOLVED_BIB_DIR / f"{entry.key}.bib"
-    resolved_path.write_text(entry.raw, encoding="utf-8")
-    return "local-fallback", str(resolved_path.relative_to(ROOT))
-
-
-def download_paper(entry: BibEntry, record: RemoteRecord | None) -> tuple[str, str | None]:
-    candidate_pdf_urls: list[str] = []
-    candidate_links: list[str] = []
-    if record is not None:
-        if record.pdf_url:
-            candidate_pdf_urls.append(record.pdf_url)
-        if record.landing_url:
-            candidate_links.append(record.landing_url)
-    if entry.url:
-        candidate_links.append(entry.url)
-    arxiv_id: str | None = extract_arxiv_id(entry)
-    if arxiv_id is not None:
-        candidate_pdf_urls.insert(0, f"https://arxiv.org/pdf/{arxiv_id}.pdf")
-        candidate_links.insert(0, f"https://arxiv.org/abs/{arxiv_id}")
-    seen: set[str] = set()
-    for pdf_url in candidate_pdf_urls:
-        if pdf_url in seen:
-            continue
-        seen.add(pdf_url)
-        try:
-            payload: bytes = read_url(pdf_url, timeout=45)
-        except (HTTPError, URLError, TimeoutError):
-            continue
-        if not payload.startswith(b"%PDF"):
-            continue
-        paper_path: Path = PAPER_DIR / f"{entry.key}.pdf"
-        paper_path.write_bytes(payload)
-        return "downloaded", str(paper_path.relative_to(ROOT))
-    for link in candidate_links:
-        if link in seen:
-            continue
-        seen.add(link)
-        link_path: Path = PAPER_DIR / f"{entry.key}.link.txt"
-        link_path.write_text(link + "\n", encoding="utf-8")
-        return "link-only", str(link_path.relative_to(ROOT))
-    return "unavailable", None
-
-
-def summarize_category(mentions: list[CitationMention]) -> str:
-    combined: str = " ".join(mention.snippet for mention in mentions).casefold()
-    if any(token in combined for token in ("patchcamelyon", "organmnist", "camelyon", "dataset", "contains", "consists of")):
-        return "dataset"
-    if any(token in combined for token in ("pytorch", "numpy", "torch_harmonics")):
-        return "software"
-    if any(token in combined for token in ("adamw", "backbone", "trained with", "reference numbers", "follows")):
-        return "experimental"
-    if any(token in combined for token in ("introduced", "applied", "explore", "use iterated", "provides", "extend this")):
-        return "related-work"
-    return "theory"
-
-
-def build_assessment(entry: BibEntry, record: RemoteRecord | None, paper_status: str) -> tuple[str, list[str]]:
-    notes: list[str] = []
-    title_score: float | None = None
-    if record is not None and record.title:
-        title_score = similarity_score(entry.title, record.title)
-    assessment: str = "good"
-    if record is None:
-        assessment = "check"
-        notes.append("No confident live metadata match found.")
-    elif title_score is not None and title_score < 0.9:
-        assessment = "check"
-        notes.append("Resolved title differs materially from the local BibTeX title.")
-    if record is not None and entry.year and record.year and abs(entry.year - record.year) > 1:
-        assessment = "check"
-        notes.append(f"Year mismatch: local {entry.year}, resolved {record.year}.")
-    resolved_venue: str = record.venue if record is not None and record.venue is not None else ""
-    venue_text: str = f"{entry.venue} {resolved_venue}".casefold()
-    if entry.entry_type == "phdthesis":
-        if assessment == "good":
-            assessment = "acceptable-but-weaker"
-        notes.append("Foundational thesis citation rather than an archival paper.")
-    if "arxiv preprint" in venue_text or (record is not None and record.source == "arxiv"):
-        if assessment == "good":
-            assessment = "acceptable-but-weaker"
-        notes.append("Preprint citation; archival version may be preferable if one exists.")
-    if "workshop" in venue_text:
-        if assessment == "good":
-            assessment = "acceptable-but-weaker"
-        notes.append("Workshop citation rather than a main archival venue.")
-    if entry.doi is None and (record is None or record.doi is None):
-        notes.append("No DOI found.")
-    if paper_status == "downloaded":
-        notes.append("Open PDF downloaded.")
-    elif paper_status == "link-only":
-        notes.append("Saved landing link only; no open PDF downloaded.")
-    else:
-        notes.append("No paper URL resolved.")
-    manual_note: str | None = MANUAL_NOTES.get(entry.key)
-    if manual_note is not None:
-        notes.append(manual_note)
-    manual_assessment: str | None = MANUAL_ASSESSMENT_OVERRIDES.get(entry.key)
-    if manual_assessment is not None:
-        assessment = manual_assessment
-    if not notes:
-        notes.append("Metadata matches and assets resolved cleanly.")
-    return assessment, notes
-
-
-def markdown_escape(value: str) -> str:
-    return value.replace("|", "\\|").replace("\n", " ").strip()
-
-
-def render_report(rows: list[AuditRow]) -> str:
-    downloaded_count: int = sum(1 for row in rows if row.paper_status == "downloaded")
-    link_only_count: int = sum(1 for row in rows if row.paper_status == "link-only")
-    stronger_count: int = sum(1 for row in rows if row.assessment == "good")
-    weaker_count: int = sum(1 for row in rows if row.assessment == "acceptable-but-weaker")
-    check_count: int = sum(1 for row in rows if row.assessment == "check")
-    lines: list[str] = [
-        "# Citation Audit",
-        "",
-        f"- Total cited keys: {len(rows)}",
-        f"- Assessments: {stronger_count} good, {weaker_count} acceptable-but-weaker, {check_count} check",
-        f"- Paper assets: {downloaded_count} PDFs downloaded, {link_only_count} link-only, {len(rows) - downloaded_count - link_only_count} unavailable",
-        "",
-        "| Key | Category | Mentions | Assessment | Resolved source | BibTeX | Paper | Notes |",
-        "| --- | --- | ---: | --- | --- | --- | --- | --- |",
-    ]
-    for row in rows:
-        notes: str = "; ".join(row.notes[:2])
-        resolved_source: str = row.resolved_source or "unresolved"
-        paper_cell: str = row.paper_status if row.paper_path is None else f"{row.paper_status} (`{row.paper_path}`)"
-        lines.append(
-            "| "
-            + " | ".join(
-                [
-                    markdown_escape(row.key),
-                    markdown_escape(row.category),
-                    str(row.mention_count),
-                    markdown_escape(row.assessment),
-                    markdown_escape(resolved_source),
-                    markdown_escape(row.bibtex_source),
-                    markdown_escape(paper_cell),
-                    markdown_escape(notes),
-                ]
-            )
-            + " |"
-        )
-    lines.append("")
-    lines.append("## Details")
-    lines.append("")
-    for row in rows:
-        lines.append(f"### `{row.key}`")
-        lines.append("")
-        lines.append(f"- Local title: {row.local_title}")
-        lines.append(f"- Resolved title: {row.resolved_title or 'unresolved'}")
-        lines.append(f"- Assessment: {row.assessment}")
-        lines.append(f"- BibTeX source: {row.bibtex_source}")
-        lines.append(f"- Paper asset: {row.paper_status}{f' (`{row.paper_path}`)' if row.paper_path else ''}")
-        lines.append(f"- Notes: {'; '.join(row.notes)}")
-        if row.mentions:
-            lines.append("- Citation contexts:")
-            for mention in row.mentions[:3]:
-                lines.append(f"  - L{mention.line_number}: {mention.snippet}")
-        lines.append("")
-    return "\n".join(lines).strip() + "\n"
-
-
-def write_csv(rows: list[AuditRow]) -> None:
-    with CSV_PATH.open("w", encoding="utf-8", newline="") as handle:
-        writer = csv.writer(handle)
-        writer.writerow(
-            [
-                "key",
-                "category",
-                "mention_count",
-                "assessment",
-                "local_entry_type",
-                "local_title",
-                "local_year",
-                "local_venue",
-                "resolved_source",
-                "resolved_title",
-                "resolved_year",
-                "resolved_venue",
-                "resolved_doi",
-                "title_match_score",
-                "bibtex_source",
-                "paper_status",
-                "paper_path",
-                "notes",
-                "mention_lines",
-                "mention_snippets",
-            ]
-        )
-        for row in rows:
-            writer.writerow(
-                [
-                    row.key,
-                    row.category,
-                    row.mention_count,
-                    row.assessment,
-                    row.local_entry_type,
-                    row.local_title,
-                    row.local_year,
-                    row.local_venue,
-                    row.resolved_source,
-                    row.resolved_title,
-                    row.resolved_year,
-                    row.resolved_venue,
-                    row.resolved_doi,
-                    row.title_match_score,
-                    row.bibtex_source,
-                    row.paper_status,
-                    row.paper_path,
-                    " | ".join(row.notes),
-                    ",".join(str(mention.line_number) for mention in row.mentions),
-                    " || ".join(mention.snippet for mention in row.mentions),
-                ]
-            )
-
-
-def write_json(rows: list[AuditRow]) -> None:
-    serialized_rows: list[dict[str, Any]] = []
-    for row in rows:
-        payload: dict[str, Any] = asdict(row)
-        serialized_rows.append(payload)
-    JSON_PATH.write_text(json.dumps(serialized_rows, indent=2), encoding="utf-8")
-
-
-def ensure_output_dirs() -> None:
-    for directory in (LOCAL_BIB_DIR, RESOLVED_BIB_DIR, PAPER_DIR):
-        directory.mkdir(parents=True, exist_ok=True)
-
-
-def audit_citations() -> list[AuditRow]:
-    tex_text: str = TEX_PATH.read_text(encoding="utf-8")
-    bib_text: str = BIB_PATH.read_text(encoding="utf-8")
-    mentions_by_key: dict[str, list[CitationMention]] = extract_citations(tex_text)
-    bib_entries: dict[str, BibEntry] = load_bib_entries(bib_text)
-    rows: list[AuditRow] = []
-    for key in sorted(mentions_by_key):
-        if key not in bib_entries:
-            raise KeyError(f"Citation key `{key}` is used in paper.tex but missing from references.bib")
-        entry: BibEntry = bib_entries[key]
-        mentions: list[CitationMention] = mentions_by_key[key]
-        record: RemoteRecord | None = resolve_remote_record(entry)
-        bibtex_source, _ = download_resolved_bibtex(entry, record)
-        paper_status, paper_path = download_paper(entry, record)
-        assessment, notes = build_assessment(entry, record, paper_status)
-        resolved_title: str | None = record.title if record else None
-        title_match_score: float | None = (
-            similarity_score(entry.title, resolved_title) if resolved_title else None
-        )
-        rows.append(
-            AuditRow(
-                key=key,
-                category=summarize_category(mentions),
-                mention_count=len(mentions),
-                local_entry_type=entry.entry_type,
-                local_title=entry.title,
-                local_year=entry.year,
-                local_venue=entry.venue,
-                local_doi=entry.doi,
-                local_url=entry.url,
-                resolved_source=record.source if record else None,
-                resolved_title=resolved_title,
-                resolved_year=record.year if record else None,
-                resolved_venue=record.venue if record else None,
-                resolved_doi=record.doi if record else None,
-                title_match_score=title_match_score,
-                bibtex_source=bibtex_source,
-                paper_status=paper_status,
-                paper_path=paper_path,
-                assessment=assessment,
-                notes=notes,
-                mentions=mentions,
-            )
-        )
-        print(f"[audit] {key}: {assessment}, paper={paper_status}, bib={bibtex_source}")
-    return rows
-
-
-def main() -> None:
-    ensure_output_dirs()
-    rows: list[AuditRow] = audit_citations()
-    REPORT_PATH.write_text(render_report(rows), encoding="utf-8")
-    write_csv(rows)
-    write_json(rows)
-    print(f"[done] wrote {REPORT_PATH.relative_to(ROOT)}")
-    print(f"[done] wrote {CSV_PATH.relative_to(ROOT)}")
-    print(f"[done] wrote {JSON_PATH.relative_to(ROOT)}")
-
-
-if __name__ == "__main__":
-    main()
diff --git a/paper/citation_audit/bibtex/local/batatia2022mace.bib b/paper/citation_audit/bibtex/local/batatia2022mace.bib
deleted file mode 100644
index 09c69b0..0000000
--- a/paper/citation_audit/bibtex/local/batatia2022mace.bib
+++ /dev/null
@@ -1,10 +0,0 @@
-@inproceedings{batatia2022mace,
-  title={{MACE}: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields},
-  author={Batatia, Ilyes and Cs{\'a}nyi, G{\'a}bor and Kov{\'a}cs, D{\'a}vid P. and Ortner, Christoph and Simm, Gregor},
-  booktitle={Advances in Neural Information Processing Systems (NeurIPS)},
-  pages={11423--11436},
-  year={2022},
-  volume={35},
-  doi={10.52202/068431-0830},
-  url={https://proceedings.neurips.cc/paper_files/paper/2022/hash/4a36c3c51af11ed9f34615b81edb5bbc-Abstract-Conference.html},
-}
diff --git a/paper/citation_audit/bibtex/local/batatia2024design.bib b/paper/citation_audit/bibtex/local/batatia2024design.bib
deleted file mode 100644
index bad28f7..0000000
--- a/paper/citation_audit/bibtex/local/batatia2024design.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@article{batatia2024design,
-  title={A Foundation Model for Atomistic Simulation},
-  author={Batatia, Ilyes and Benber, Philipp and Chmiela, Stefan and Csanyi, Gabor and others},
-  journal={Nature Machine Intelligence},
-  year={2024},
-}
diff --git a/paper/citation_audit/bibtex/local/bejnordi2017camelyon.bib b/paper/citation_audit/bibtex/local/bejnordi2017camelyon.bib
deleted file mode 100644
index 933339b..0000000
--- a/paper/citation_audit/bibtex/local/bejnordi2017camelyon.bib
+++ /dev/null
@@ -1,10 +0,0 @@
-@article{bejnordi2017camelyon,
-  title={Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer},
-  author={Ehteshami Bejnordi, Babak and Veta, Mitko and van Diest, Paul Johannes and van Ginneken, Bram and Karssemeijer, Nico and Litjens, Geert and van der Laak, Jeroen A. W. M. and {the CAMELYON16 Consortium}},
-  journal={JAMA},
-  volume={318},
-  number={22},
-  pages={2199--2210},
-  year={2017},
-  doi={10.1001/jama.2017.14585},
-}
diff --git a/paper/citation_audit/bibtex/local/bendory2022bispectrum.bib b/paper/citation_audit/bibtex/local/bendory2022bispectrum.bib
deleted file mode 100644
index ec5fe28..0000000
--- a/paper/citation_audit/bibtex/local/bendory2022bispectrum.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@article{bendory2022bispectrum,
-  title={On the Uniqueness of the Bispectrum of Signals on the Sphere},
-  author={Bendory, Tamir and Edidin, Dan and Satriano, Matthew},
-  journal={arXiv preprint arXiv:2207.10842},
-  year={2022},
-}
diff --git a/paper/citation_audit/bibtex/local/bendory2025orbit.bib b/paper/citation_audit/bibtex/local/bendory2025orbit.bib
deleted file mode 100644
index a88d0eb..0000000
--- a/paper/citation_audit/bibtex/local/bendory2025orbit.bib
+++ /dev/null
@@ -1,7 +0,0 @@
-@article{bendory2025orbit,
-  title={Orbit Recovery for Spherical Functions},
-  author={Bendory, Tamir and Edidin, Dan and Katz, Josh and Kreymer, Shay},
-  journal={arXiv preprint arXiv:2508.02674},
-  year={2025},
-  url={https://arxiv.org/abs/2508.02674},
-}
diff --git a/paper/citation_audit/bibtex/local/bonev2023spherical.bib b/paper/citation_audit/bibtex/local/bonev2023spherical.bib
deleted file mode 100644
index df0362b..0000000
--- a/paper/citation_audit/bibtex/local/bonev2023spherical.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@inproceedings{bonev2023spherical,
-  title={Spherical {F}ourier Neural Operators: Learning Stable Dynamics on the Sphere},
-  author={Bonev, Boris and Kurth, Thorsten and Hundt, Christian and Pathak, Jaideep and Balu, Maximilian and Kashinath, Karthik and Anandkumar, Animashree},
-  booktitle={International Conference on Machine Learning (ICML)},
-  year={2023},
-}
diff --git a/paper/citation_audit/bibtex/local/cesa2022program.bib b/paper/citation_audit/bibtex/local/cesa2022program.bib
deleted file mode 100644
index a490777..0000000
--- a/paper/citation_audit/bibtex/local/cesa2022program.bib
+++ /dev/null
@@ -1,7 +0,0 @@
-@inproceedings{cesa2022program,
-  title={A Program to Build {E(N)}-Equivariant Steerable {CNN}s},
-  author={Cesa, Gabriele and Lang, Leon and Weiler, Maurice},
-  booktitle={International Conference on Learning Representations (ICLR)},
-  year={2022},
-  url={https://openreview.net/forum?id=WE4qe9xlnQw},
-}
diff --git a/paper/citation_audit/bibtex/local/chevalley2024bispectral.bib b/paper/citation_audit/bibtex/local/chevalley2024bispectral.bib
deleted file mode 100644
index ae36d3c..0000000
--- a/paper/citation_audit/bibtex/local/chevalley2024bispectral.bib
+++ /dev/null
@@ -1,10 +0,0 @@
-@inproceedings{chevalley2024bispectral,
-  title={A Bispectral {3D} {U-Net} for Rotation Robustness in Medical Segmentation},
-  author={Chevalley, Arthur and Oreiller, Valentin and Fageot, Julien and Prior, John O. and Andrearczyk, Vincent and Depeursinge, Adrien},
-  booktitle={Topology- and Graph-Informed Imaging Informatics},
-  pages={43--54},
-  year={2024},
-  publisher={Springer Nature Switzerland},
-  doi={10.1007/978-3-031-73967-5_5},
-  url={https://doi.org/10.1007/978-3-031-73967-5_5},
-}
diff --git a/paper/citation_audit/bibtex/local/cohen2016group.bib b/paper/citation_audit/bibtex/local/cohen2016group.bib
deleted file mode 100644
index 3c7062e..0000000
--- a/paper/citation_audit/bibtex/local/cohen2016group.bib
+++ /dev/null
@@ -1,7 +0,0 @@
-@inproceedings{cohen2016group,
-  title={Group Equivariant Convolutional Networks},
-  author={Cohen, Taco S. and Welling, Max},
-  booktitle={International Conference on Machine Learning (ICML)},
-  pages={2990--2999},
-  year={2016},
-}
diff --git a/paper/citation_audit/bibtex/local/cohen2018spherical.bib b/paper/citation_audit/bibtex/local/cohen2018spherical.bib
deleted file mode 100644
index dcce293..0000000
--- a/paper/citation_audit/bibtex/local/cohen2018spherical.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@inproceedings{cohen2018spherical,
-  title={Spherical {CNN}s},
-  author={Cohen, Taco S. and Geiger, Mario and K{\"o}hler, Jonas and Welling, Max},
-  booktitle={International Conference on Learning Representations (ICLR)},
-  year={2018},
-}
diff --git a/paper/citation_audit/bibtex/local/edidin2023orbit.bib b/paper/citation_audit/bibtex/local/edidin2023orbit.bib
deleted file mode 100644
index c823ab8..0000000
--- a/paper/citation_audit/bibtex/local/edidin2023orbit.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@article{edidin2023orbit,
-  title={Projective Uniqueness and the Bispectrum},
-  author={Edidin, Dan and Satriano, Matthew},
-  journal={arXiv preprint arXiv:2308.02064},
-  year={2023},
-}
diff --git a/paper/citation_audit/bibtex/local/edidin2024orbit.bib b/paper/citation_audit/bibtex/local/edidin2024orbit.bib
deleted file mode 100644
index 56870ba..0000000
--- a/paper/citation_audit/bibtex/local/edidin2024orbit.bib
+++ /dev/null
@@ -1,12 +0,0 @@
-@article{edidin2024orbit,
-  title={Orbit Recovery for Band-Limited Functions},
-  author={Edidin, Dan and Satriano, Matthew},
-  journal={SIAM Journal on Applied Algebra and Geometry},
-  volume={8},
-  number={3},
-  pages={733--755},
-  year={2024},
-  publisher={Society for Industrial \& Applied Mathematics},
-  doi={10.1137/23M1577808},
-  url={https://arxiv.org/abs/2306.00155},
-}
diff --git a/paper/citation_audit/bibtex/local/esteves2018spherical.bib b/paper/citation_audit/bibtex/local/esteves2018spherical.bib
deleted file mode 100644
index 7a9ed05..0000000
--- a/paper/citation_audit/bibtex/local/esteves2018spherical.bib
+++ /dev/null
@@ -1,8 +0,0 @@
-@inproceedings{esteves2018spherical,
-  title={Learning {SO}(3) Equivariant Representations with Spherical {CNN}s},
-  author={Esteves, Carlos and Allen-Blanchette, Christine and Makadia, Ameesh and Daniilidis, Kostas},
-  booktitle={European Conference on Computer Vision (ECCV)},
-  pages={52--68},
-  year={2018},
-  publisher={Springer},
-}
diff --git a/paper/citation_audit/bibtex/local/geiger2022e3nn.bib b/paper/citation_audit/bibtex/local/geiger2022e3nn.bib
deleted file mode 100644
index fcb7f04..0000000
--- a/paper/citation_audit/bibtex/local/geiger2022e3nn.bib
+++ /dev/null
@@ -1,7 +0,0 @@
-@article{geiger2022e3nn,
-  title={e3nn: {E}uclidean Neural Networks},
-  author={Geiger, Mario and Smidt, Tess},
-  journal={arXiv preprint arXiv:2207.09453},
-  year={2022},
-  url={https://arxiv.org/abs/2207.09453},
-}
diff --git a/paper/citation_audit/bibtex/local/harris2020numpy.bib b/paper/citation_audit/bibtex/local/harris2020numpy.bib
deleted file mode 100644
index 52e2049..0000000
--- a/paper/citation_audit/bibtex/local/harris2020numpy.bib
+++ /dev/null
@@ -1,10 +0,0 @@
-@article{harris2020numpy,
-  title={Array programming with {NumPy}},
-  author={Harris, Charles R. and Millman, K. Jarrod and van der Walt, St{\'e}fan J. and Gommers, Ralf and Virtanen, Pauli and Cournapeau, David and Wieser, Eric and Taylor, Julian and Berg, Sebastian and Smith, Nathaniel J. and others},
-  journal={Nature},
-  volume={585},
-  number={7825},
-  pages={357--362},
-  year={2020},
-  publisher={Nature Publishing Group},
-}
diff --git a/paper/citation_audit/bibtex/local/huang2017densenet.bib b/paper/citation_audit/bibtex/local/huang2017densenet.bib
deleted file mode 100644
index 9adf8f5..0000000
--- a/paper/citation_audit/bibtex/local/huang2017densenet.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@inproceedings{huang2017densenet,
-  title={Densely Connected Convolutional Networks},
-  author={Huang, Gao and Liu, Zhuang and van der Maaten, Laurens and Weinberger, Kilian Q.},
-  booktitle={IEEE Conference on Computer Vision and Pattern Recognition (CVPR)},
-  year={2017},
-}
diff --git a/paper/citation_audit/bibtex/local/kakarala1992triple.bib b/paper/citation_audit/bibtex/local/kakarala1992triple.bib
deleted file mode 100644
index a7a0c16..0000000
--- a/paper/citation_audit/bibtex/local/kakarala1992triple.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@phdthesis{kakarala1992triple,
-  title={Triple Correlation on Groups},
-  author={Kakarala, Ramakrishna},
-  school={University of California, Irvine},
-  year={1992},
-}
diff --git a/paper/citation_audit/bibtex/local/kakarala2012bispectrum.bib b/paper/citation_audit/bibtex/local/kakarala2012bispectrum.bib
deleted file mode 100644
index e4d884f..0000000
--- a/paper/citation_audit/bibtex/local/kakarala2012bispectrum.bib
+++ /dev/null
@@ -1,12 +0,0 @@
-@article{kakarala2012bispectrum,
-  title={The Bispectrum as a Source of Phase-Sensitive Invariants for {F}ourier Descriptors: A Group-Theoretic Approach},
-  author={Kakarala, Ramakrishna},
-  journal={Journal of Mathematical Imaging and Vision},
-  volume={44},
-  number={3},
-  pages={341--353},
-  year={2012},
-  publisher={Springer},
-  doi={10.1007/s10851-012-0330-6},
-  url={https://arxiv.org/abs/0902.0196},
-}
diff --git a/paper/citation_audit/bibtex/local/kuipers2023regular.bib b/paper/citation_audit/bibtex/local/kuipers2023regular.bib
deleted file mode 100644
index f8b9c0e..0000000
--- a/paper/citation_audit/bibtex/local/kuipers2023regular.bib
+++ /dev/null
@@ -1,10 +0,0 @@
-@inproceedings{kuipers2023regular,
-  title={Regular {SE}(3) Group Convolutions for Volumetric Medical Image Analysis},
-  author={Kuipers, Thijs P. and Bekkers, Erik J.},
-  booktitle={Medical Image Computing and Computer Assisted Intervention -- {MICCAI} 2023},
-  pages={252--261},
-  year={2023},
-  publisher={Springer Nature Switzerland},
-  doi={10.1007/978-3-031-43898-1_25},
-  url={https://doi.org/10.1007/978-3-031-43898-1_25},
-}
diff --git a/paper/citation_audit/bibtex/local/loshchilov2019adamw.bib b/paper/citation_audit/bibtex/local/loshchilov2019adamw.bib
deleted file mode 100644
index 1669f8b..0000000
--- a/paper/citation_audit/bibtex/local/loshchilov2019adamw.bib
+++ /dev/null
@@ -1,7 +0,0 @@
-@inproceedings{loshchilov2019adamw,
-  title={Decoupled Weight Decay Regularization},
-  author={Loshchilov, Ilya and Hutter, Frank},
-  booktitle={International Conference on Learning Representations (ICLR)},
-  year={2019},
-  url={https://openreview.net/forum?id=Bkg6RiCqY7},
-}
diff --git a/paper/citation_audit/bibtex/local/mallat2012scattering.bib b/paper/citation_audit/bibtex/local/mallat2012scattering.bib
deleted file mode 100644
index 97992ab..0000000
--- a/paper/citation_audit/bibtex/local/mallat2012scattering.bib
+++ /dev/null
@@ -1,11 +0,0 @@
-@article{mallat2012scattering,
-  title={Group Invariant Scattering},
-  author={Mallat, St{\'e}phane},
-  journal={Communications on Pure and Applied Mathematics},
-  volume={65},
-  number={10},
-  pages={1331--1398},
-  year={2012},
-  publisher={Wiley},
-  doi={10.1002/cpa.21413},
-}
diff --git a/paper/citation_audit/bibtex/local/mataigne2024selective.bib b/paper/citation_audit/bibtex/local/mataigne2024selective.bib
deleted file mode 100644
index 5c97b18..0000000
--- a/paper/citation_audit/bibtex/local/mataigne2024selective.bib
+++ /dev/null
@@ -1,11 +0,0 @@
-@inproceedings{mataigne2024selective,
-  title={The Selective {G}-Bispectrum and its Inversion: Applications to {G}-Invariant Networks},
-  author={Mataigne, Simon and Mathe, Johan and Sanborn, Sophia and Hillar, Christopher and Miolane, Nina},
-  booktitle={Advances in Neural Information Processing Systems},
-  pages={115682--115711},
-  year={2024},
-  volume={37},
-  publisher={Neural Information Processing Systems Foundation, Inc.},
-  doi={10.52202/079017-3674},
-  url={https://proceedings.neurips.cc/paper_files/paper/2024/hash/d1a1e8713fcd5626656553c82f7c3b26-Abstract-Conference.html},
-}
diff --git a/paper/citation_audit/bibtex/local/medmnistv2.bib b/paper/citation_audit/bibtex/local/medmnistv2.bib
deleted file mode 100644
index 7cd7219..0000000
--- a/paper/citation_audit/bibtex/local/medmnistv2.bib
+++ /dev/null
@@ -1,10 +0,0 @@
-@article{medmnistv2,
-  title={{MedMNIST} v2 -- A Large-Scale Lightweight Benchmark for {2D} and {3D} Biomedical Image Classification},
-  author={Yang, Jiancheng and Shi, Rui and Wei, Donglai and Liu, Zequan and Zhao, Lin and Ke, Bilian and Pfister, Hanspeter and Ni, Bingbing},
-  journal={Scientific Data},
-  volume={10},
-  number={1},
-  pages={41},
-  year={2023},
-  publisher={Nature Publishing Group UK London},
-}
diff --git a/paper/citation_audit/bibtex/local/oreiller2022bispectral.bib b/paper/citation_audit/bibtex/local/oreiller2022bispectral.bib
deleted file mode 100644
index 5580051..0000000
--- a/paper/citation_audit/bibtex/local/oreiller2022bispectral.bib
+++ /dev/null
@@ -1,11 +0,0 @@
-@inproceedings{oreiller2022bispectral,
-  title={Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral {U-Net}},
-  author={Oreiller, Valentin and Fageot, Julien and Andrearczyk, Vincent and Prior, John O. and Depeursinge, Adrien},
-  booktitle={Proceedings of The 5th International Conference on Medical Imaging with Deep Learning},
-  pages={929--943},
-  year={2022},
-  volume={172},
-  series={Proceedings of Machine Learning Research},
-  publisher={PMLR},
-  url={https://proceedings.mlr.press/v172/oreiller22a.html},
-}
diff --git a/paper/citation_audit/bibtex/local/paszke2019pytorch.bib b/paper/citation_audit/bibtex/local/paszke2019pytorch.bib
deleted file mode 100644
index c863ccb..0000000
--- a/paper/citation_audit/bibtex/local/paszke2019pytorch.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@inproceedings{paszke2019pytorch,
-  title={Py{T}orch: An Imperative Style, High-Performance Deep Learning Library},
-  author={Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and Desmaison, Alban and K{\"o}pf, Andreas and Yang, Edward and DeVito, Zach and Raison, Martin and Tejani, Alykhan and Chilamkurthy, Sasank and Steiner, Benoit and Fang, Lu and Bai, Junjie and Chintala, Soumith},
-  booktitle={Advances in Neural Information Processing Systems (NeurIPS)},
-  year={2019},
-}
diff --git a/paper/citation_audit/bibtex/local/price2025freedom.bib b/paper/citation_audit/bibtex/local/price2025freedom.bib
deleted file mode 100644
index 413f189..0000000
--- a/paper/citation_audit/bibtex/local/price2025freedom.bib
+++ /dev/null
@@ -1,10 +0,0 @@
-@inproceedings{price2025freedom,
-  title={The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products},
-  author={Xie, YuQing and Daigavane, Ameya and Kotak, Mit and Smidt, Tess},
-  booktitle={Proceedings of the 42nd International Conference on Machine Learning (ICML)},
-  pages={68599--68625},
-  year={2025},
-  volume={267},
-  series={Proceedings of Machine Learning Research},
-  publisher={PMLR},
-}
diff --git a/paper/citation_audit/bibtex/local/sanborn2023bispectral.bib b/paper/citation_audit/bibtex/local/sanborn2023bispectral.bib
deleted file mode 100644
index bf8210f..0000000
--- a/paper/citation_audit/bibtex/local/sanborn2023bispectral.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@inproceedings{sanborn2023bispectral,
-  title={Bispectral Neural Networks},
-  author={Sanborn, Sophia and Shewmake, Christian and Olshausen, Bruno and Hillar, Christopher},
-  booktitle={International Conference on Learning Representations (ICLR)},
-  year={2023},
-}
diff --git a/paper/citation_audit/bibtex/local/sangalli2023movfrnet.bib b/paper/citation_audit/bibtex/local/sangalli2023movfrnet.bib
deleted file mode 100644
index 63373af..0000000
--- a/paper/citation_audit/bibtex/local/sangalli2023movfrnet.bib
+++ /dev/null
@@ -1,11 +0,0 @@
-@inproceedings{sangalli2023movfrnet,
-  title={Moving Frame Net: {SE}(3)-Equivariant Network for Volumes},
-  author={Sangalli, Mateus and Blusseau, Samy and Velasco-Forero, Santiago and Angulo, Jes{\'u}s},
-  booktitle={Proceedings of the 1st NeurIPS Workshop on Symmetry and Geometry in Neural Representations},
-  pages={81--97},
-  year={2023},
-  volume={197},
-  series={Proceedings of Machine Learning Research},
-  publisher={PMLR},
-  url={https://proceedings.mlr.press/v197/sangalli23a.html},
-}
diff --git a/paper/citation_audit/bibtex/local/thomas2018tensor.bib b/paper/citation_audit/bibtex/local/thomas2018tensor.bib
deleted file mode 100644
index b1ccda2..0000000
--- a/paper/citation_audit/bibtex/local/thomas2018tensor.bib
+++ /dev/null
@@ -1,7 +0,0 @@
-@inproceedings{thomas2018tensor,
-  title={Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for {3D} Point Clouds},
-  author={Thomas, Nathaniel and Smidt, Tess and Kearnes, Steven and Yang, Lusann and Li, Li and Kohlhoff, Kai and Riley, Patrick},
-  booktitle={arXiv preprint arXiv:1802.08219},
-  year={2018},
-  url={https://arxiv.org/abs/1802.08219},
-}
diff --git a/paper/citation_audit/bibtex/local/veeling2018rotation.bib b/paper/citation_audit/bibtex/local/veeling2018rotation.bib
deleted file mode 100644
index cbd1c04..0000000
--- a/paper/citation_audit/bibtex/local/veeling2018rotation.bib
+++ /dev/null
@@ -1,8 +0,0 @@
-@inproceedings{veeling2018rotation,
-  title={Rotation Equivariant {CNN}s for Digital Pathology},
-  author={Veeling, Bastiaan S. and Linmans, Jasper and Winkens, Jim and Cohen, Taco and Welling, Max},
-  booktitle={Medical Image Computing and Computer Assisted Intervention (MICCAI)},
-  pages={210--218},
-  year={2018},
-  publisher={Springer},
-}
diff --git a/paper/citation_audit/bibtex/local/weiler2019general.bib b/paper/citation_audit/bibtex/local/weiler2019general.bib
deleted file mode 100644
index 5fb8cfd..0000000
--- a/paper/citation_audit/bibtex/local/weiler2019general.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@inproceedings{weiler2019general,
-  title={General {E}(2)-Equivariant Steerable {CNN}s},
-  author={Weiler, Maurice and Cesa, Gabriele},
-  booktitle={Advances in Neural Information Processing Systems (NeurIPS)},
-  year={2019},
-}
diff --git a/paper/citation_audit/bibtex/local/zhemchuzhnikov2024ilponet.bib b/paper/citation_audit/bibtex/local/zhemchuzhnikov2024ilponet.bib
deleted file mode 100644
index 67e9860..0000000
--- a/paper/citation_audit/bibtex/local/zhemchuzhnikov2024ilponet.bib
+++ /dev/null
@@ -1,10 +0,0 @@
-@inproceedings{zhemchuzhnikov2024ilponet,
-  title={{ILPO-NET}: Network for the Invariant Recognition of Arbitrary Volumetric Patterns in {3D}},
-  author={Zhemchuzhnikov, Dmitrii and Grudinin, Sergei},
-  booktitle={Machine Learning and Knowledge Discovery in Databases. Research Track},
-  pages={352--368},
-  year={2024},
-  publisher={Springer Nature Switzerland},
-  doi={10.1007/978-3-031-70359-1_21},
-  url={https://doi.org/10.1007/978-3-031-70359-1_21},
-}
diff --git a/paper/citation_audit/bibtex/local/zhemchuzhnikov2025equilopo.bib b/paper/citation_audit/bibtex/local/zhemchuzhnikov2025equilopo.bib
deleted file mode 100644
index 974daac..0000000
--- a/paper/citation_audit/bibtex/local/zhemchuzhnikov2025equilopo.bib
+++ /dev/null
@@ -1,7 +0,0 @@
-@inproceedings{zhemchuzhnikov2025equilopo,
-  title={On the {F}ourier Analysis in the {SO}(3) Space: the {EquiLoPO} Network},
-  author={Zhemchuzhnikov, Dmitrii and Grudinin, Sergei},
-  booktitle={International Conference on Learning Representations (ICLR)},
-  year={2025},
-  url={https://openreview.net/forum?id=LvTSvdiSwG},
-}
diff --git a/paper/citation_audit/bibtex/resolved/batatia2022mace.bib b/paper/citation_audit/bibtex/resolved/batatia2022mace.bib
deleted file mode 100644
index 1e602f9..0000000
--- a/paper/citation_audit/bibtex/resolved/batatia2022mace.bib
+++ /dev/null
@@ -1 +0,0 @@
-@inproceedings{Batatia_2022, series={NeurIPS 2022}, title={MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields}, url={http://dx.doi.org/10.52202/068431-0830}, DOI={10.52202/068431-0830}, booktitle={Advances in Neural Information Processing Systems 35}, publisher={Neural Information Processing Systems Foundation, Inc. (NeurIPS)}, author={Batatia, Ilyes and Csanyi, Gabor and Kovacs, David P and Ortner, Christoph and Simm, Gregor}, year={2022}, pages={11423–11436}, collection={NeurIPS 2022} }
diff --git a/paper/citation_audit/bibtex/resolved/batatia2024design.bib b/paper/citation_audit/bibtex/resolved/batatia2024design.bib
deleted file mode 100644
index f7a4881..0000000
--- a/paper/citation_audit/bibtex/resolved/batatia2024design.bib
+++ /dev/null
@@ -1 +0,0 @@
-@article{Pl__2025, title={A Foundation Model for Accurate Atomistic Simulations in Drug Design}, url={http://dx.doi.org/10.26434/chemrxiv-2025-f1hgn-v3}, DOI={10.26434/chemrxiv-2025-f1hgn-v3}, publisher={American Chemical Society (ACS)}, author={Plé, Thomas and Adjoua, Olivier and Benali, Anouar and Posenitskiy, Evgeny and Villot, Corentin and Lagardère, Louis and Piquemal, Jean-Philip}, year={2025}, month=may }
diff --git a/paper/citation_audit/bibtex/resolved/bejnordi2017camelyon.bib b/paper/citation_audit/bibtex/resolved/bejnordi2017camelyon.bib
deleted file mode 100644
index 0ccaa50..0000000
--- a/paper/citation_audit/bibtex/resolved/bejnordi2017camelyon.bib
+++ /dev/null
@@ -1 +0,0 @@
-@article{Ehteshami_Bejnordi_2017, title={Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer}, volume={318}, ISSN={0098-7484}, url={http://dx.doi.org/10.1001/jama.2017.14585}, DOI={10.1001/jama.2017.14585}, number={22}, journal={JAMA}, publisher={American Medical Association (AMA)}, author={Ehteshami Bejnordi, Babak and Veta, Mitko and Johannes van Diest, Paul and van Ginneken, Bram and Karssemeijer, Nico and Litjens, Geert and van der Laak, Jeroen A. W. M. and Hermsen, Meyke and Manson, Quirine F and Balkenhol, Maschenka and Geessink, Oscar and Stathonikos, Nikolaos and van Dijk, Marcory CRF and Bult, Peter and Beca, Francisco and Beck, Andrew H and Wang, Dayong and Khosla, Aditya and Gargeya, Rishab and Irshad, Humayun and Zhong, Aoxiao and Dou, Qi and Li, Quanzheng and Chen, Hao and Lin, Huang-Jing and Heng, Pheng-Ann and Haß, Christian and Bruni, Elia and Wong, Quincy and Halici, Ugur and Öner, Mustafa Ümit and Cetin-Atalay, Rengul and Berseth, Matt and Khvatkov, Vitali and Vylegzhanin, Alexei and Kraus, Oren and Shaban, Muhammad and Rajpoot, Nasir and Awan, Ruqayya and Sirinukunwattana, Korsuk and Qaiser, Talha and Tsang, Yee-Wah and Tellez, David and Annuscheit, Jonas and Hufnagl, Peter and Valkonen, Mira and Kartasalo, Kimmo and Latonen, Leena and Ruusuvuori, Pekka and Liimatainen, Kaisa and Albarqouni, Shadi and Mungal, Bharti and George, Ami and Demirci, Stefanie and Navab, Nassir and Watanabe, Seiryo and Seno, Shigeto and Takenaka, Yoichi and Matsuda, Hideo and Ahmady Phoulady, Hady and Kovalev, Vassili and Kalinovsky, Alexander and Liauchuk, Vitali and Bueno, Gloria and Fernandez-Carrobles, M. Milagro and Serrano, Ismael and Deniz, Oscar and Racoceanu, Daniel and Venâncio, Rui}, year={2017}, month=dec, pages={2199} }
diff --git a/paper/citation_audit/bibtex/resolved/bendory2022bispectrum.bib b/paper/citation_audit/bibtex/resolved/bendory2022bispectrum.bib
deleted file mode 100644
index ec5fe28..0000000
--- a/paper/citation_audit/bibtex/resolved/bendory2022bispectrum.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@article{bendory2022bispectrum,
-  title={On the Uniqueness of the Bispectrum of Signals on the Sphere},
-  author={Bendory, Tamir and Edidin, Dan and Satriano, Matthew},
-  journal={arXiv preprint arXiv:2207.10842},
-  year={2022},
-}
diff --git a/paper/citation_audit/bibtex/resolved/bendory2025orbit.bib b/paper/citation_audit/bibtex/resolved/bendory2025orbit.bib
deleted file mode 100644
index a88d0eb..0000000
--- a/paper/citation_audit/bibtex/resolved/bendory2025orbit.bib
+++ /dev/null
@@ -1,7 +0,0 @@
-@article{bendory2025orbit,
-  title={Orbit Recovery for Spherical Functions},
-  author={Bendory, Tamir and Edidin, Dan and Katz, Josh and Kreymer, Shay},
-  journal={arXiv preprint arXiv:2508.02674},
-  year={2025},
-  url={https://arxiv.org/abs/2508.02674},
-}
diff --git a/paper/citation_audit/bibtex/resolved/bonev2023spherical.bib b/paper/citation_audit/bibtex/resolved/bonev2023spherical.bib
deleted file mode 100644
index df0362b..0000000
--- a/paper/citation_audit/bibtex/resolved/bonev2023spherical.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@inproceedings{bonev2023spherical,
-  title={Spherical {F}ourier Neural Operators: Learning Stable Dynamics on the Sphere},
-  author={Bonev, Boris and Kurth, Thorsten and Hundt, Christian and Pathak, Jaideep and Balu, Maximilian and Kashinath, Karthik and Anandkumar, Animashree},
-  booktitle={International Conference on Machine Learning (ICML)},
-  year={2023},
-}
diff --git a/paper/citation_audit/bibtex/resolved/cesa2022program.bib b/paper/citation_audit/bibtex/resolved/cesa2022program.bib
deleted file mode 100644
index a490777..0000000
--- a/paper/citation_audit/bibtex/resolved/cesa2022program.bib
+++ /dev/null
@@ -1,7 +0,0 @@
-@inproceedings{cesa2022program,
-  title={A Program to Build {E(N)}-Equivariant Steerable {CNN}s},
-  author={Cesa, Gabriele and Lang, Leon and Weiler, Maurice},
-  booktitle={International Conference on Learning Representations (ICLR)},
-  year={2022},
-  url={https://openreview.net/forum?id=WE4qe9xlnQw},
-}
diff --git a/paper/citation_audit/bibtex/resolved/chevalley2024bispectral.bib b/paper/citation_audit/bibtex/resolved/chevalley2024bispectral.bib
deleted file mode 100644
index 8ad0a99..0000000
--- a/paper/citation_audit/bibtex/resolved/chevalley2024bispectral.bib
+++ /dev/null
@@ -1 +0,0 @@
-@inbook{Chevalley_2024, title={A Bispectral 3D U-Net for Rotation Robustness in Medical Segmentation}, ISBN={9783031739675}, ISSN={1611-3349}, url={http://dx.doi.org/10.1007/978-3-031-73967-5_5}, DOI={10.1007/978-3-031-73967-5_5}, booktitle={Topology- and Graph-Informed Imaging Informatics}, publisher={Springer Nature Switzerland}, author={Chevalley, Arthur and Oreiller, Valentin and Fageot, Julien and Prior, John O. and Andrearczyk, Vincent and Depeursinge, Adrien}, year={2024}, month=oct, pages={43–54} }
diff --git a/paper/citation_audit/bibtex/resolved/cohen2016group.bib b/paper/citation_audit/bibtex/resolved/cohen2016group.bib
deleted file mode 100644
index b3dfa54..0000000
--- a/paper/citation_audit/bibtex/resolved/cohen2016group.bib
+++ /dev/null
@@ -1,10 +0,0 @@
-@article{https://doi.org/10.48550/arxiv.1602.07576,
-  doi = {10.48550/ARXIV.1602.07576},
-  url = {https://arxiv.org/abs/1602.07576},
-  author = {Cohen, Taco S. and Welling, Max},
-  keywords = {Machine Learning (cs.LG), Machine Learning (stat.ML), FOS: Computer and information sciences, FOS: Computer and information sciences},
-  title = {Group Equivariant Convolutional Networks},
-  publisher = {arXiv},
-  year = {2016},
-  copyright = {arXiv.org perpetual, non-exclusive license}
-}
diff --git a/paper/citation_audit/bibtex/resolved/cohen2018spherical.bib b/paper/citation_audit/bibtex/resolved/cohen2018spherical.bib
deleted file mode 100644
index dcce293..0000000
--- a/paper/citation_audit/bibtex/resolved/cohen2018spherical.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@inproceedings{cohen2018spherical,
-  title={Spherical {CNN}s},
-  author={Cohen, Taco S. and Geiger, Mario and K{\"o}hler, Jonas and Welling, Max},
-  booktitle={International Conference on Learning Representations (ICLR)},
-  year={2018},
-}
diff --git a/paper/citation_audit/bibtex/resolved/edidin2023orbit.bib b/paper/citation_audit/bibtex/resolved/edidin2023orbit.bib
deleted file mode 100644
index c823ab8..0000000
--- a/paper/citation_audit/bibtex/resolved/edidin2023orbit.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@article{edidin2023orbit,
-  title={Projective Uniqueness and the Bispectrum},
-  author={Edidin, Dan and Satriano, Matthew},
-  journal={arXiv preprint arXiv:2308.02064},
-  year={2023},
-}
diff --git a/paper/citation_audit/bibtex/resolved/edidin2024orbit.bib b/paper/citation_audit/bibtex/resolved/edidin2024orbit.bib
deleted file mode 100644
index f600a0f..0000000
--- a/paper/citation_audit/bibtex/resolved/edidin2024orbit.bib
+++ /dev/null
@@ -1 +0,0 @@
-@article{Edidin_2024, title={Orbit Recovery for Band-Limited Functions}, volume={8}, ISSN={2470-6566}, url={http://dx.doi.org/10.1137/23m1577808}, DOI={10.1137/23m1577808}, number={3}, journal={SIAM Journal on Applied Algebra and Geometry}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Edidin, Dan and Satriano, Matthew}, year={2024}, month=aug, pages={733–755} }
diff --git a/paper/citation_audit/bibtex/resolved/esteves2018spherical.bib b/paper/citation_audit/bibtex/resolved/esteves2018spherical.bib
deleted file mode 100644
index 05ed628..0000000
--- a/paper/citation_audit/bibtex/resolved/esteves2018spherical.bib
+++ /dev/null
@@ -1 +0,0 @@
-@inbook{Esteves_2018, title={Learning SO(3) Equivariant Representations with Spherical CNNs}, ISBN={9783030012618}, ISSN={1611-3349}, url={http://dx.doi.org/10.1007/978-3-030-01261-8_4}, DOI={10.1007/978-3-030-01261-8_4}, booktitle={Computer Vision – ECCV 2018}, publisher={Springer International Publishing}, author={Esteves, Carlos and Allen-Blanchette, Christine and Makadia, Ameesh and Daniilidis, Kostas}, year={2018}, pages={54–70} }
diff --git a/paper/citation_audit/bibtex/resolved/geiger2022e3nn.bib b/paper/citation_audit/bibtex/resolved/geiger2022e3nn.bib
deleted file mode 100644
index fcb7f04..0000000
--- a/paper/citation_audit/bibtex/resolved/geiger2022e3nn.bib
+++ /dev/null
@@ -1,7 +0,0 @@
-@article{geiger2022e3nn,
-  title={e3nn: {E}uclidean Neural Networks},
-  author={Geiger, Mario and Smidt, Tess},
-  journal={arXiv preprint arXiv:2207.09453},
-  year={2022},
-  url={https://arxiv.org/abs/2207.09453},
-}
diff --git a/paper/citation_audit/bibtex/resolved/harris2020numpy.bib b/paper/citation_audit/bibtex/resolved/harris2020numpy.bib
deleted file mode 100644
index 4b0a13a..0000000
--- a/paper/citation_audit/bibtex/resolved/harris2020numpy.bib
+++ /dev/null
@@ -1 +0,0 @@
-@article{Harris_2020, title={Array programming with NumPy}, volume={585}, ISSN={1476-4687}, url={http://dx.doi.org/10.1038/s41586-020-2649-2}, DOI={10.1038/s41586-020-2649-2}, number={7825}, journal={Nature}, publisher={Springer Science and Business Media LLC}, author={Harris, Charles R. and Millman, K. Jarrod and van der Walt, Stéfan J. and Gommers, Ralf and Virtanen, Pauli and Cournapeau, David and Wieser, Eric and Taylor, Julian and Berg, Sebastian and Smith, Nathaniel J. and Kern, Robert and Picus, Matti and Hoyer, Stephan and van Kerkwijk, Marten H. and Brett, Matthew and Haldane, Allan and del Río, Jaime Fernández and Wiebe, Mark and Peterson, Pearu and Gérard-Marchant, Pierre and Sheppard, Kevin and Reddy, Tyler and Weckesser, Warren and Abbasi, Hameer and Gohlke, Christoph and Oliphant, Travis E.}, year={2020}, month=sep, pages={357–362} }
diff --git a/paper/citation_audit/bibtex/resolved/huang2017densenet.bib b/paper/citation_audit/bibtex/resolved/huang2017densenet.bib
deleted file mode 100644
index cd1d197..0000000
--- a/paper/citation_audit/bibtex/resolved/huang2017densenet.bib
+++ /dev/null
@@ -1 +0,0 @@
-@inproceedings{Huang_2017, title={Densely Connected Convolutional Networks}, url={http://dx.doi.org/10.1109/cvpr.2017.243}, DOI={10.1109/cvpr.2017.243}, booktitle={2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)}, publisher={IEEE}, author={Huang, Gao and Liu, Zhuang and Van Der Maaten, Laurens and Weinberger, Kilian Q.}, year={2017}, month=jul, pages={2261–2269} }
diff --git a/paper/citation_audit/bibtex/resolved/kakarala1992triple.bib b/paper/citation_audit/bibtex/resolved/kakarala1992triple.bib
deleted file mode 100644
index a7a0c16..0000000
--- a/paper/citation_audit/bibtex/resolved/kakarala1992triple.bib
+++ /dev/null
@@ -1,6 +0,0 @@
-@phdthesis{kakarala1992triple,
-  title={Triple Correlation on Groups},
-  author={Kakarala, Ramakrishna},
-  school={University of California, Irvine},
-  year={1992},
-}
diff --git a/paper/citation_audit/bibtex/resolved/kakarala2012bispectrum.bib b/paper/citation_audit/bibtex/resolved/kakarala2012bispectrum.bib
deleted file mode 100644
index 0e345a5..0000000
--- a/paper/citation_audit/bibtex/resolved/kakarala2012bispectrum.bib
+++ /dev/null
@@ -1 +0,0 @@
-@article{Kakarala_2012, title={The Bispectrum as a Source of Phase-Sensitive Invariants for Fourier Descriptors: A Group-Theoretic Approach}, volume={44}, ISSN={1573-7683}, url={http://dx.doi.org/10.1007/s10851-012-0330-6}, DOI={10.1007/s10851-012-0330-6}, number={3}, journal={Journal of Mathematical Imaging and Vision}, publisher={Springer Science and Business Media LLC}, author={Kakarala, Ramakrishna}, year={2012}, month=mar, pages={341–353} }
diff --git a/paper/citation_audit/bibtex/resolved/kuipers2023regular.bib b/paper/citation_audit/bibtex/resolved/kuipers2023regular.bib
deleted file mode 100644
index 17634bb..0000000
--- a/paper/citation_audit/bibtex/resolved/kuipers2023regular.bib
+++ /dev/null
@@ -1 +0,0 @@
-@inbook{Kuipers_2023, title={Regular SE(3) Group Convolutions for Volumetric Medical Image Analysis}, ISBN={9783031438981}, ISSN={1611-3349}, url={http://dx.doi.org/10.1007/978-3-031-43898-1_25}, DOI={10.1007/978-3-031-43898-1_25}, booktitle={Medical Image Computing and Computer Assisted Intervention – MICCAI 2023}, publisher={Springer Nature Switzerland}, author={Kuipers, Thijs P. and Bekkers, Erik J.}, year={2023}, pages={252–261} }
diff --git a/paper/citation_audit/bibtex/resolved/loshchilov2019adamw.bib b/paper/citation_audit/bibtex/resolved/loshchilov2019adamw.bib
deleted file mode 100644
index 1669f8b..0000000
--- a/paper/citation_audit/bibtex/resolved/loshchilov2019adamw.bib
+++ /dev/null
@@ -1,7 +0,0 @@
-@inproceedings{loshchilov2019adamw,
-  title={Decoupled Weight Decay Regularization},
-  author={Loshchilov, Ilya and Hutter, Frank},
-  booktitle={International Conference on Learning Representations (ICLR)},
-  year={2019},
-  url={https://openreview.net/forum?id=Bkg6RiCqY7},
-}
diff --git a/paper/citation_audit/bibtex/resolved/mallat2012scattering.bib b/paper/citation_audit/bibtex/resolved/mallat2012scattering.bib
deleted file mode 100644
index 08beedf..0000000
--- a/paper/citation_audit/bibtex/resolved/mallat2012scattering.bib
+++ /dev/null
@@ -1 +0,0 @@
-@article{Mallat_2012, title={Group Invariant Scattering}, volume={65}, ISSN={1097-0312}, url={http://dx.doi.org/10.1002/cpa.21413}, DOI={10.1002/cpa.21413}, number={10}, journal={Communications on Pure and Applied Mathematics}, publisher={Wiley}, author={Mallat, Stéphane}, year={2012}, month=jul, pages={1331–1398} }
diff --git a/paper/citation_audit/bibtex/resolved/mataigne2024selective.bib b/paper/citation_audit/bibtex/resolved/mataigne2024selective.bib
deleted file mode 100644
index 7df68fd..0000000
--- a/paper/citation_audit/bibtex/resolved/mataigne2024selective.bib
+++ /dev/null
@@ -1 +0,0 @@
-@inproceedings{Hillar_2024, series={NeurIPS 2024}, title={The Selective $G$-Bispectrum and its Inversion: Applications to $G$-Invariant Networks}, url={http://dx.doi.org/10.52202/079017-3674}, DOI={10.52202/079017-3674}, booktitle={Advances in Neural Information Processing Systems 37}, publisher={Neural Information Processing Systems Foundation, Inc. (NeurIPS)}, author={Hillar, Christopher and Mataigne, Simon and Mathe, Johan and Miolane, Nina and Sanborn, Sophia}, year={2024}, pages={115682–115711}, collection={NeurIPS 2024} }
diff --git a/paper/citation_audit/bibtex/resolved/medmnistv2.bib b/paper/citation_audit/bibtex/resolved/medmnistv2.bib
deleted file mode 100644
index c320965..0000000
--- a/paper/citation_audit/bibtex/resolved/medmnistv2.bib
+++ /dev/null
@@ -1 +0,0 @@
-@article{Yang_2023, title={MedMNIST v2 - A large-scale lightweight benchmark for 2D and 3D biomedical image classification}, volume={10}, ISSN={2052-4463}, url={http://dx.doi.org/10.1038/s41597-022-01721-8}, DOI={10.1038/s41597-022-01721-8}, number={1}, journal={Scientific Data}, publisher={Springer Science and Business Media LLC}, author={Yang, Jiancheng and Shi, Rui and Wei, Donglai and Liu, Zequan and Zhao, Lin and Ke, Bilian and Pfister, Hanspeter and Ni, Bingbing}, year={2023}, month=jan }
diff --git a/paper/citation_audit/bibtex/resolved/oreiller2022bispectral.bib b/paper/citation_audit/bibtex/resolved/oreiller2022bispectral.bib
deleted file mode 100644
index 5580051..0000000
--- a/paper/citation_audit/bibtex/resolved/oreiller2022bispectral.bib
+++ /dev/null
@@ -1,11 +0,0 @@
-@inproceedings{oreiller2022bispectral,
-  title={Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral {U-Net}},
-  author={Oreiller, Valentin and Fageot, Julien and Andrearczyk, Vincent and Prior, John O. and Depeursinge, Adrien},
-  booktitle={Proceedings of The 5th International Conference on Medical Imaging with Deep Learning},
-  pages={929--943},
-  year={2022},
-  volume={172},
-  series={Proceedings of Machine Learning Research},
-  publisher={PMLR},
-  url={https://proceedings.mlr.press/v172/oreiller22a.html},
-}
diff --git a/paper/citation_audit/bibtex/resolved/paszke2019pytorch.bib b/paper/citation_audit/bibtex/resolved/paszke2019pytorch.bib
deleted file mode 100644
index aa0e376..0000000
--- a/paper/citation_audit/bibtex/resolved/paszke2019pytorch.bib
+++ /dev/null
@@ -1,10 +0,0 @@
-@misc{https://doi.org/10.48550/arxiv.1912.01703,
-  doi = {10.48550/ARXIV.1912.01703},
-  url = {https://arxiv.org/abs/1912.01703},
-  author = {Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and Desmaison, Alban and Köpf, Andreas and Yang, Edward and DeVito, Zach and Raison, Martin and Tejani, Alykhan and Chilamkurthy, Sasank and Steiner, Benoit and Fang, Lu and Bai, Junjie and Chintala, Soumith},
-  keywords = {Machine Learning (cs.LG), Mathematical Software (cs.MS), Machine Learning (stat.ML), FOS: Computer and information sciences, FOS: Computer and information sciences},
-  title = {PyTorch: An Imperative Style, High-Performance Deep Learning Library},
-  publisher = {arXiv},
-  year = {2019},
-  copyright = {arXiv.org perpetual, non-exclusive license}
-}
diff --git a/paper/citation_audit/bibtex/resolved/price2025freedom.bib b/paper/citation_audit/bibtex/resolved/price2025freedom.bib
deleted file mode 100644
index 2b82ac1..0000000
--- a/paper/citation_audit/bibtex/resolved/price2025freedom.bib
+++ /dev/null
@@ -1,10 +0,0 @@
-@misc{https://doi.org/10.48550/arxiv.2506.13523,
-  doi = {10.48550/ARXIV.2506.13523},
-  url = {https://arxiv.org/abs/2506.13523},
-  author = {Xie, YuQing and Daigavane, Ameya and Kotak, Mit and Smidt, Tess},
-  keywords = {Machine Learning (cs.LG), Artificial Intelligence (cs.AI), FOS: Computer and information sciences, FOS: Computer and information sciences},
-  title = {The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products},
-  publisher = {arXiv},
-  year = {2025},
-  copyright = {arXiv.org perpetual, non-exclusive license}
-}
diff --git a/paper/citation_audit/bibtex/resolved/sanborn2023bispectral.bib b/paper/citation_audit/bibtex/resolved/sanborn2023bispectral.bib
deleted file mode 100644
index 1eeb623..0000000
--- a/paper/citation_audit/bibtex/resolved/sanborn2023bispectral.bib
+++ /dev/null
@@ -1,10 +0,0 @@
-@article{https://doi.org/10.48550/arxiv.2209.03416,
-  doi = {10.48550/ARXIV.2209.03416},
-  url = {https://arxiv.org/abs/2209.03416},
-  author = {Sanborn, Sophia and Shewmake, Christian and Olshausen, Bruno and Hillar, Christopher},
-  keywords = {Machine Learning (cs.LG), FOS: Computer and information sciences, FOS: Computer and information sciences},
-  title = {Bispectral Neural Networks},
-  publisher = {arXiv},
-  year = {2022},
-  copyright = {Creative Commons Attribution Non Commercial No Derivatives 4.0 International}
-}
diff --git a/paper/citation_audit/bibtex/resolved/sangalli2023movfrnet.bib b/paper/citation_audit/bibtex/resolved/sangalli2023movfrnet.bib
deleted file mode 100644
index da210ba..0000000
--- a/paper/citation_audit/bibtex/resolved/sangalli2023movfrnet.bib
+++ /dev/null
@@ -1,10 +0,0 @@
-@article{https://doi.org/10.48550/arxiv.2211.03420,
-  doi = {10.48550/ARXIV.2211.03420},
-  url = {https://arxiv.org/abs/2211.03420},
-  author = {Sangalli, Mateus and Blusseau, Samy and Velasco-Forero, Santiago and Angulo, Jesus},
-  keywords = {Computer Vision and Pattern Recognition (cs.CV), Machine Learning (stat.ML), FOS: Computer and information sciences, FOS: Computer and information sciences},
-  title = {Moving Frame Net: SE(3)-Equivariant Network for Volumes},
-  publisher = {arXiv},
-  year = {2022},
-  copyright = {arXiv.org perpetual, non-exclusive license}
-}
diff --git a/paper/citation_audit/bibtex/resolved/thomas2018tensor.bib b/paper/citation_audit/bibtex/resolved/thomas2018tensor.bib
deleted file mode 100644
index b1ccda2..0000000
--- a/paper/citation_audit/bibtex/resolved/thomas2018tensor.bib
+++ /dev/null
@@ -1,7 +0,0 @@
-@inproceedings{thomas2018tensor,
-  title={Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for {3D} Point Clouds},
-  author={Thomas, Nathaniel and Smidt, Tess and Kearnes, Steven and Yang, Lusann and Li, Li and Kohlhoff, Kai and Riley, Patrick},
-  booktitle={arXiv preprint arXiv:1802.08219},
-  year={2018},
-  url={https://arxiv.org/abs/1802.08219},
-}
diff --git a/paper/citation_audit/bibtex/resolved/veeling2018rotation.bib b/paper/citation_audit/bibtex/resolved/veeling2018rotation.bib
deleted file mode 100644
index bd20b25..0000000
--- a/paper/citation_audit/bibtex/resolved/veeling2018rotation.bib
+++ /dev/null
@@ -1 +0,0 @@
-@inbook{Veeling_2018, title={Rotation Equivariant CNNs for Digital Pathology}, ISBN={9783030009342}, ISSN={1611-3349}, url={http://dx.doi.org/10.1007/978-3-030-00934-2_24}, DOI={10.1007/978-3-030-00934-2_24}, booktitle={Medical Image Computing and Computer Assisted Intervention – MICCAI 2018}, publisher={Springer International Publishing}, author={Veeling, Bastiaan S. and Linmans, Jasper and Winkens, Jim and Cohen, Taco and Welling, Max}, year={2018}, pages={210–218} }
diff --git a/paper/citation_audit/bibtex/resolved/weiler2019general.bib b/paper/citation_audit/bibtex/resolved/weiler2019general.bib
deleted file mode 100644
index c9ca541..0000000
--- a/paper/citation_audit/bibtex/resolved/weiler2019general.bib
+++ /dev/null
@@ -1,10 +0,0 @@
-@misc{https://doi.org/10.48550/arxiv.1911.08251,
-  doi = {10.48550/ARXIV.1911.08251},
-  url = {https://arxiv.org/abs/1911.08251},
-  author = {Weiler, Maurice and Cesa, Gabriele},
-  keywords = {Computer Vision and Pattern Recognition (cs.CV), Machine Learning (cs.LG), Image and Video Processing (eess.IV), FOS: Computer and information sciences, FOS: Computer and information sciences, FOS: Electrical engineering, electronic engineering, information engineering, FOS: Electrical engineering, electronic engineering, information engineering},
-  title = {General $E(2)$-Equivariant Steerable CNNs},
-  publisher = {arXiv},
-  year = {2019},
-  copyright = {arXiv.org perpetual, non-exclusive license}
-}
diff --git a/paper/citation_audit/bibtex/resolved/zhemchuzhnikov2024ilponet.bib b/paper/citation_audit/bibtex/resolved/zhemchuzhnikov2024ilponet.bib
deleted file mode 100644
index c613850..0000000
--- a/paper/citation_audit/bibtex/resolved/zhemchuzhnikov2024ilponet.bib
+++ /dev/null
@@ -1 +0,0 @@
-@inbook{Zhemchuzhnikov_2024, title={ILPO-NET: Network for the Invariant Recognition of Arbitrary Volumetric Patterns in 3D}, ISBN={9783031703591}, ISSN={1611-3349}, url={http://dx.doi.org/10.1007/978-3-031-70359-1_21}, DOI={10.1007/978-3-031-70359-1_21}, booktitle={Machine Learning and Knowledge Discovery in Databases. Research Track}, publisher={Springer Nature Switzerland}, author={Zhemchuzhnikov, Dmitrii and Grudinin, Sergei}, year={2024}, pages={352–368} }
diff --git a/paper/citation_audit/bibtex/resolved/zhemchuzhnikov2025equilopo.bib b/paper/citation_audit/bibtex/resolved/zhemchuzhnikov2025equilopo.bib
deleted file mode 100644
index 974daac..0000000
--- a/paper/citation_audit/bibtex/resolved/zhemchuzhnikov2025equilopo.bib
+++ /dev/null
@@ -1,7 +0,0 @@
-@inproceedings{zhemchuzhnikov2025equilopo,
-  title={On the {F}ourier Analysis in the {SO}(3) Space: the {EquiLoPO} Network},
-  author={Zhemchuzhnikov, Dmitrii and Grudinin, Sergei},
-  booktitle={International Conference on Learning Representations (ICLR)},
-  year={2025},
-  url={https://openreview.net/forum?id=LvTSvdiSwG},
-}
diff --git a/paper/citation_audit/citation_audit.csv b/paper/citation_audit/citation_audit.csv
deleted file mode 100644
index 262080f..0000000
--- a/paper/citation_audit/citation_audit.csv
+++ /dev/null
@@ -1,30 +0,0 @@
-key,category,mention_count,assessment,local_entry_type,local_title,local_year,local_venue,resolved_source,resolved_title,resolved_year,resolved_venue,resolved_doi,title_match_score,bibtex_source,paper_status,paper_path,notes,mention_lines,mention_snippets
-batatia2022mace,related-work,1,good,inproceedings,{MACE}: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields,2022,Advances in Neural Information Processing Systems (NeurIPS),openalex,MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields,2022,,10.52202/068431-0830,1.0,remote-doi,link-only,citation_audit/papers/batatia2022mace.link.txt,Saved landing link only; no open PDF downloaded.,739,MACE~\citep{batatia2022mace} use iterated Clebsch--Gordan products
-bejnordi2017camelyon,dataset,1,good,article,Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer,2017,JAMA,openalex,Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer,2017,JAMA,10.1001/jama.2017.14585,1.0,remote-doi,link-only,citation_audit/papers/bejnordi2017camelyon.link.txt,Saved landing link only; no open PDF downloaded.,2336,images of lymph node sections~\citep{bejnordi2017camelyon}. We use
-bendory2025orbit,theory,1,acceptable-but-weaker,article,Orbit Recovery for Spherical Functions,2025,arXiv preprint arXiv:2508.02674,arxiv,Orbit Recovery for Spherical Functions,2025,arXiv preprint arXiv:2508.02674,,1.0,local-fallback,downloaded,citation_audit/papers/bendory2025orbit.pdf,Preprint citation; archival version may be preferable if one exists. | No DOI found. | Open PDF downloaded.,1699,Bendory et al.~\citep{bendory2025orbit}
-bonev2023spherical,theory,2,good,inproceedings,Spherical {F}ourier Neural Operators: Learning Stable Dynamics on the Sphere,2023,International Conference on Machine Learning (ICML),manual,Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere,2023,Proceedings of the 40th International Conference on Machine Learning,,1.0,local-fallback,downloaded,citation_audit/papers/bonev2023spherical.pdf,No DOI found. | Open PDF downloaded.,"258,466",\texttt{torch\_harmonics}~\citep{bonev2023spherical} || transform (via \texttt{torch\_harmonics}~\citep{bonev2023spherical}).
-cesa2022program,theory,2,good,inproceedings,A Program to Build {E(N)}-Equivariant Steerable {CNN}s,2022,International Conference on Learning Representations (ICLR),manual,A Program to Build E(N)-Equivariant Steerable CNNs,2022,International Conference on Learning Representations,,1.0,local-fallback,link-only,citation_audit/papers/cesa2022program.link.txt,No DOI found. | Saved landing link only; no open PDF downloaded.,"717,808",\texttt{escnn}~\citep{cesa2022program} and \texttt{e3nn}~\citep{geiger2022e3nn} || band-limits; integration with \texttt{escnn}~\citep{cesa2022program} and
-chevalley2024bispectral,related-work,1,good,inproceedings,A Bispectral {3D} {U-Net} for Rotation Robustness in Medical Segmentation,2024,Topology- and Graph-Informed Imaging Informatics,openalex,A Bispectral 3D U-Net for Rotation Robustness in Medical Segmentation,2024,Lecture notes in computer science,10.1007/978-3-031-73967-5_5,1.0,remote-doi,link-only,citation_audit/papers/chevalley2024bispectral.link.txt,Saved landing link only; no open PDF downloaded.,733,\citet{chevalley2024bispectral} explore bispectral signatures for data
-cohen2016group,theory,2,good,inproceedings,Group Equivariant Convolutional Networks,2016,International Conference on Machine Learning (ICML),openalex,Group Equivariant Convolutional Networks,2016,arXiv (Cornell University),10.48550/arxiv.1602.07576,1.0,remote-doi,downloaded,citation_audit/papers/cohen2016group.pdf,Open PDF downloaded.,"111,179","Equivariant neural networks~\citep{cohen2016group, weiler2019general} enforce || for all $g \in G$~\citep{cohen2016group, weiler2019general}."
-cohen2018spherical,experimental,5,good,inproceedings,Spherical {CNN}s,2018,International Conference on Learning Representations (ICLR),openalex,Spherical CNNs,2018,arXiv (Cornell University),,1.0,local-fallback,downloaded,citation_audit/papers/cohen2018spherical.pdf,No DOI found. | Open PDF downloaded.,"615,637,650,707,2353","2-sphere---on Spherical MNIST~\cite{cohen2018spherical}, the standard || Following Cohen et al.~\cite{cohen2018spherical}, we evaluate under three || et al.\ reference numbers are from \cite{cohen2018spherical}. || Cohen's spherical CNN~\cite{cohen2018spherical} reaches 96\%/95\%/94\% || projection following \citet{cohen2018spherical} and sampled on a"
-edidin2024orbit,theory,1,good,article,Orbit Recovery for Band-Limited Functions,2024,SIAM Journal on Applied Algebra and Geometry,openalex,Orbit Recovery for Band-Limited Functions,2024,SIAM Journal on Applied Algebra and Geometry,10.1137/23m1577808,1.0,remote-doi,downloaded,citation_audit/papers/edidin2024orbit.pdf,Open PDF downloaded.,1696,Edidin \& Satriano~\citep{edidin2024orbit}
-esteves2018spherical,related-work,1,good,inproceedings,Learning {SO}(3) Equivariant Representations with Spherical {CNN}s,2018,European Conference on Computer Vision (ECCV),openalex,Learning SO(3) Equivariant Representations with Spherical CNNs,2018,Lecture notes in computer science,10.1007/978-3-030-01261-8_4,1.0,remote-doi,link-only,citation_audit/papers/esteves2018spherical.link.txt,Saved landing link only; no open PDF downloaded.,752,\citet{esteves2018spherical} extend this to the sphere. Unlike the
-geiger2022e3nn,theory,2,acceptable-but-weaker,article,e3nn: {E}uclidean Neural Networks,2022,arXiv preprint arXiv:2207.09453,arxiv,e3nn: Euclidean Neural Networks,2022,arXiv preprint arXiv:2207.09453,,1.0,local-fallback,downloaded,citation_audit/papers/geiger2022e3nn.pdf,Preprint citation; archival version may be preferable if one exists. | No DOI found. | Open PDF downloaded.,"717,809",\texttt{escnn}~\citep{cesa2022program} and \texttt{e3nn}~\citep{geiger2022e3nn} || \texttt{e3nn}~\citep{geiger2022e3nn}; and applications to cryo-EM
-harris2020numpy,software,1,good,article,Array programming with {NumPy},2020,Nature,openalex,Array programming with NumPy,2020,Nature,10.1038/s41586-020-2649-2,1.0,remote-doi,downloaded,citation_audit/papers/harris2020numpy.pdf,Open PDF downloaded.,257,"NumPy~\citep{harris2020numpy}, and"
-huang2017densenet,experimental,2,good,inproceedings,Densely Connected Convolutional Networks,2017,IEEE Conference on Computer Vision and Pattern Recognition (CVPR),openalex,Densely Connected Convolutional Networks,2017,,10.1109/cvpr.2017.243,1.0,remote-doi,link-only,citation_audit/papers/huang2017densenet.link.txt,Saved landing link only; no open PDF downloaded.,"492,2143",$C_8$-equivariant DenseNet~\citep{huang2017densenet} backbone following || DenseNet~\citep{huang2017densenet} backbone with block configuration
-kakarala2012bispectrum,theory,4,good,article,The Bispectrum as a Source of Phase-Sensitive Invariants for {F}ourier Descriptors: A Group-Theoretic Approach,2012,Journal of Mathematical Imaging and Vision,openalex,The Bispectrum as a Source of Phase-Sensitive Invariants for Fourier Descriptors: A Group-Theoretic Approach,2012,Journal of Mathematical Imaging and Vision,10.1007/s10851-012-0330-6,1.0,remote-doi,downloaded,citation_audit/papers/kakarala2012bispectrum.pdf,Open PDF downloaded.,"119,230,755,1694","guarantees~\citep{kakarala2012bispectrum}. || it separates orbits for generic signals~\citep{kakarala2012bispectrum}. However, || The bispectrum's completeness guarantee~\citep{kakarala2012bispectrum} is its || Kakarala~\citep{kakarala2012bispectrum}"
-kuipers2023regular,theory,2,good,inproceedings,Regular {SE}(3) Group Convolutions for Volumetric Medical Image Analysis,2023,Medical Image Computing and Computer Assisted Intervention -- {MICCAI} 2023,openalex,Regular SE(3) Group Convolutions for Volumetric Medical Image Analysis,2023,Lecture notes in computer science,10.1007/978-3-031-43898-1_25,1.0,remote-doi,link-only,citation_audit/papers/kuipers2023regular.link.txt,Saved landing link only; no open PDF downloaded.,"591,2234",convolutions~\citep{kuipers2023regular} (69.8\%). || Regular SE(3) conv & \citet{kuipers2023regular} & 172K & 0.698 & --- \\
-loshchilov2019adamw,experimental,3,good,inproceedings,Decoupled Weight Decay Regularization,2019,International Conference on Learning Representations (ICLR),manual,Decoupled Weight Decay Regularization,2019,International Conference on Learning Representations,,1.0,local-fallback,link-only,citation_audit/papers/loshchilov2019adamw.link.txt,No DOI found. | Saved landing link only; no open PDF downloaded.,"498,556,2170","AdamW~\citep{loshchilov2019adamw}. Full model descriptions are in || are trained with AdamW~\citep{loshchilov2019adamw} for 100 epochs with || training data} (26{,}214 images) with AdamW~\citep{loshchilov2019adamw}"
-mallat2012scattering,related-work,1,good,article,Group Invariant Scattering,2012,Communications on Pure and Applied Mathematics,openalex,Group Invariant Scattering,2012,Communications on Pure and Applied Mathematics,10.1002/cpa.21413,1.0,remote-doi,link-only,citation_audit/papers/mallat2012scattering.link.txt,Saved landing link only; no open PDF downloaded.,750,The wavelet scattering transform~\citep{mallat2012scattering} provides
-mataigne2024selective,theory,8,good,inproceedings,The Selective {G}-Bispectrum and its Inversion: Applications to {G}-Invariant Networks,2024,Advances in Neural Information Processing Systems,openalex,The Selective $G$-Bispectrum and its Inversion: Applications to $G$-Invariant Networks,2024,,10.52202/079017-3674,1.0,remote-doi,link-only,citation_audit/papers/mataigne2024selective.link.txt,Saved landing link only; no open PDF downloaded.,"126,138,233,399,1160,1702,1801,1925","$O(|G|^2)$ to $O(|G|)$~\citep{mataigne2024selective}, a user still needs a numerically stable implementation, || the bootstrap inversion failure~\citep{mataigne2024selective} for the || \emph{Selective bispectra.} \citet{mataigne2024selective} show that for || The bootstrap inversion of \citet{mataigne2024selective} recovers || \citep{mataigne2024selective} is that on~$S^2$, the scalar bispectrum || Mataigne et al.~\citep{mataigne2024selective} || All inversion algorithms in \citet{mataigne2024selective} share the || \citet{mataigne2024selective} acknowledge a related issue in the"
-medmnistv2,dataset,2,good,article,{MedMNIST} v2 -- A Large-Scale Lightweight Benchmark for {2D} and {3D} Biomedical Image Classification,2023,Scientific Data,openalex,MedMNIST v2 - A large-scale lightweight benchmark for 2D and 3D biomedical image classification,2023,Scientific Data,10.1038/s41597-022-01721-8,1.0,remote-doi,downloaded,citation_audit/papers/medmnistv2.pdf,Open PDF downloaded.,"550,2343","OrganMNIST3D~\citep{medmnistv2} (1{,}742 abdominal CT volumes, $28^3$ || OrganMNIST3D~\citep{medmnistv2} contains 1{,}742 abdominal CT organ"
-oreiller2022bispectral,related-work,1,good,inproceedings,Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral {U-Net},2022,Proceedings of The 5th International Conference on Medical Imaging with Deep Learning,manual,Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral U-Net,2022,Medical Imaging with Deep Learning,,1.0,local-fallback,downloaded,citation_audit/papers/oreiller2022bispectral.pdf,No DOI found. | Open PDF downloaded.,728,\citet{oreiller2022bispectral} applied bispectral CNNs to medical imaging
-paszke2019pytorch,software,2,good,inproceedings,"Py{T}orch: An Imperative Style, High-Performance Deep Learning Library",2019,Advances in Neural Information Processing Systems (NeurIPS),openalex,"PyTorch: An Imperative Style, High-Performance Deep Learning Library",2019,arXiv (Cornell University),10.48550/arxiv.1912.01703,1.0,remote-doi,downloaded,citation_audit/papers/paszke2019pytorch.pdf,Open PDF downloaded.,"129,256","This paper presents \texttt{bispectrum}, a PyTorch~\citep{paszke2019pytorch} || The only dependencies are PyTorch~\citep{paszke2019pytorch},"
-price2025freedom,theory,1,good,inproceedings,The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products,2025,Proceedings of the 42nd International Conference on Machine Learning (ICML),openalex,The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products,2025,ArXiv.org,10.48550/arxiv.2506.13523,1.0,remote-doi,downloaded,citation_audit/papers/price2025freedom.pdf,Open PDF downloaded.,742,\citet{price2025freedom} analyse the \emph{price of freedom} in tensor
-sanborn2023bispectral,related-work,1,good,inproceedings,Bispectral Neural Networks,2023,International Conference on Learning Representations (ICLR),openalex,Bispectral Neural Networks,2022,arXiv (Cornell University),10.48550/arxiv.2209.03416,1.0,remote-doi,downloaded,citation_audit/papers/sanborn2023bispectral.pdf,Open PDF downloaded.,726,"\citet{sanborn2023bispectral} introduced bispectral neural networks (BNNs),"
-sangalli2023movfrnet,theory,1,acceptable-but-weaker,inproceedings,Moving Frame Net: {SE}(3)-Equivariant Network for Volumes,2023,Proceedings of the 1st NeurIPS Workshop on Symmetry and Geometry in Neural Representations,openalex,Moving Frame Net: SE(3)-Equivariant Network for Volumes,2022,arXiv (Cornell University),10.48550/arxiv.2211.03420,1.0,remote-doi,downloaded,citation_audit/papers/sangalli2023movfrnet.pdf,Workshop citation rather than a main archival venue. | Open PDF downloaded.,2233,SE3MovFrNet & \citet{sangalli2023movfrnet} & --- & 0.745 & --- \\
-thomas2018tensor,theory,1,acceptable-but-weaker,inproceedings,Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for {3D} Point Clouds,2018,arXiv preprint arXiv:1802.08219,arxiv,Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for 3D Point Clouds,2018,arXiv preprint arXiv:1802.08219,,1.0,local-fallback,downloaded,citation_audit/papers/thomas2018tensor.pdf,Preprint citation; archival version may be preferable if one exists. | No DOI found. | Open PDF downloaded.,738,Tensor field networks~\citep{thomas2018tensor} and
-veeling2018rotation,dataset,5,good,inproceedings,Rotation Equivariant {CNN}s for Digital Pathology,2018,Medical Image Computing and Computer Assisted Intervention (MICCAI),openalex,Rotation Equivariant CNNs for Digital Pathology,2018,Lecture notes in computer science,10.1007/978-3-030-00934-2_24,1.0,remote-doi,unavailable,,No paper URL resolved.,"490,493,2124,2145,2334","PatchCamelyon~\citep{veeling2018rotation} (PCam), 327K histopathology || \citet{veeling2018rotation}; they differ only in the terminal invariant map: || \citet{veeling2018rotation}, trained on 100\% data.} || following the architecture of \citet{veeling2018rotation} but || PatchCamelyon~\citep{veeling2018rotation} consists of 327{,}680 RGB"
-weiler2019general,theory,4,good,inproceedings,General {E}(2)-Equivariant Steerable {CNN}s,2019,Advances in Neural Information Processing Systems (NeurIPS),openalex,General $E(2)$-Equivariant Steerable CNNs,2019,arXiv (Cornell University),10.48550/arxiv.1911.08251,1.0,remote-doi,downloaded,citation_audit/papers/weiler2019general.pdf,Open PDF downloaded.,"111,116,179,210","Equivariant neural networks~\citep{cohen2016group, weiler2019general} enforce || usable~\citep{weiler2019general}, but they do not preserve all equivariant || for all $g \in G$~\citep{cohen2016group, weiler2019general}. || sacrifice exactness through aliasing~\citep{weiler2019general}. The full"
-zhemchuzhnikov2024ilponet,theory,1,good,inproceedings,{ILPO-NET}: Network for the Invariant Recognition of Arbitrary Volumetric Patterns in {3D},2024,Machine Learning and Knowledge Discovery in Databases. Research Track,openalex,ILPO-NET: Network for the Invariant Recognition of Arbitrary Volumetric Patterns in 3D,2024,Lecture notes in computer science,10.1007/978-3-031-70359-1_21,1.0,remote-doi,link-only,citation_audit/papers/zhemchuzhnikov2024ilponet.link.txt,Saved landing link only; no open PDF downloaded.,2231,ILPOResNet-50 & \citet{zhemchuzhnikov2024ilponet} & 38K & 0.879 & 0.992 \\
-zhemchuzhnikov2025equilopo,theory,1,good,inproceedings,On the {F}ourier Analysis in the {SO}(3) Space: the {EquiLoPO} Network,2025,International Conference on Learning Representations (ICLR),manual,On the Fourier Analysis in the SO(3) Space: the EquiLoPO Network,2025,International Conference on Learning Representations,,1.0,local-fallback,link-only,citation_audit/papers/zhemchuzhnikov2025equilopo.link.txt,No DOI found. | Saved landing link only; no open PDF downloaded.,2232,EquiLoPO (local train.) & \citet{zhemchuzhnikov2025equilopo} & 418K & 0.866 & 0.991 \\
diff --git a/paper/citation_audit/citation_audit.json b/paper/citation_audit/citation_audit.json
deleted file mode 100644
index 1847713..0000000
--- a/paper/citation_audit/citation_audit.json
+++ /dev/null
@@ -1,1009 +0,0 @@
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-  {
-    "key": "batatia2022mace",
-    "category": "related-work",
-    "mention_count": 1,
-    "local_entry_type": "inproceedings",
-    "local_title": "{MACE}: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields",
-    "local_year": 2022,
-    "local_venue": "Advances in Neural Information Processing Systems (NeurIPS)",
-    "local_doi": "10.52202/068431-0830",
-    "local_url": "https://proceedings.neurips.cc/paper_files/paper/2022/hash/4a36c3c51af11ed9f34615b81edb5bbc-Abstract-Conference.html",
-    "resolved_source": "openalex",
-    "resolved_title": "MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields",
-    "resolved_year": 2022,
-    "resolved_venue": null,
-    "resolved_doi": "10.52202/068431-0830",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "link-only",
-    "paper_path": "citation_audit/papers/batatia2022mace.link.txt",
-    "assessment": "good",
-    "notes": [
-      "Saved landing link only; no open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 739,
-        "snippet": "MACE~\\citep{batatia2022mace} use iterated Clebsch--Gordan products"
-      }
-    ]
-  },
-  {
-    "key": "bejnordi2017camelyon",
-    "category": "dataset",
-    "mention_count": 1,
-    "local_entry_type": "article",
-    "local_title": "Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer",
-    "local_year": 2017,
-    "local_venue": "JAMA",
-    "local_doi": "10.1001/jama.2017.14585",
-    "local_url": null,
-    "resolved_source": "openalex",
-    "resolved_title": "Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer",
-    "resolved_year": 2017,
-    "resolved_venue": "JAMA",
-    "resolved_doi": "10.1001/jama.2017.14585",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "link-only",
-    "paper_path": "citation_audit/papers/bejnordi2017camelyon.link.txt",
-    "assessment": "good",
-    "notes": [
-      "Saved landing link only; no open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 2336,
-        "snippet": "images of lymph node sections~\\citep{bejnordi2017camelyon}. We use"
-      }
-    ]
-  },
-  {
-    "key": "bendory2025orbit",
-    "category": "theory",
-    "mention_count": 1,
-    "local_entry_type": "article",
-    "local_title": "Orbit Recovery for Spherical Functions",
-    "local_year": 2025,
-    "local_venue": "arXiv preprint arXiv:2508.02674",
-    "local_doi": null,
-    "local_url": "https://arxiv.org/abs/2508.02674",
-    "resolved_source": "arxiv",
-    "resolved_title": "Orbit Recovery for Spherical Functions",
-    "resolved_year": 2025,
-    "resolved_venue": "arXiv preprint arXiv:2508.02674",
-    "resolved_doi": null,
-    "title_match_score": 1.0,
-    "bibtex_source": "local-fallback",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/bendory2025orbit.pdf",
-    "assessment": "acceptable-but-weaker",
-    "notes": [
-      "Preprint citation; archival version may be preferable if one exists.",
-      "No DOI found.",
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 1699,
-        "snippet": "Bendory et al.~\\citep{bendory2025orbit}"
-      }
-    ]
-  },
-  {
-    "key": "bonev2023spherical",
-    "category": "theory",
-    "mention_count": 2,
-    "local_entry_type": "inproceedings",
-    "local_title": "Spherical {F}ourier Neural Operators: Learning Stable Dynamics on the Sphere",
-    "local_year": 2023,
-    "local_venue": "International Conference on Machine Learning (ICML)",
-    "local_doi": null,
-    "local_url": null,
-    "resolved_source": "manual",
-    "resolved_title": "Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere",
-    "resolved_year": 2023,
-    "resolved_venue": "Proceedings of the 40th International Conference on Machine Learning",
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-    "bibtex_source": "local-fallback",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/bonev2023spherical.pdf",
-    "assessment": "good",
-    "notes": [
-      "No DOI found.",
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 258,
-        "snippet": "\\texttt{torch\\_harmonics}~\\citep{bonev2023spherical}"
-      },
-      {
-        "line_number": 466,
-        "snippet": "transform (via \\texttt{torch\\_harmonics}~\\citep{bonev2023spherical})."
-      }
-    ]
-  },
-  {
-    "key": "cesa2022program",
-    "category": "theory",
-    "mention_count": 2,
-    "local_entry_type": "inproceedings",
-    "local_title": "A Program to Build {E(N)}-Equivariant Steerable {CNN}s",
-    "local_year": 2022,
-    "local_venue": "International Conference on Learning Representations (ICLR)",
-    "local_doi": null,
-    "local_url": "https://openreview.net/forum?id=WE4qe9xlnQw",
-    "resolved_source": "manual",
-    "resolved_title": "A Program to Build E(N)-Equivariant Steerable CNNs",
-    "resolved_year": 2022,
-    "resolved_venue": "International Conference on Learning Representations",
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-    "title_match_score": 1.0,
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-    "paper_path": "citation_audit/papers/cesa2022program.link.txt",
-    "assessment": "good",
-    "notes": [
-      "No DOI found.",
-      "Saved landing link only; no open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 717,
-        "snippet": "\\texttt{escnn}~\\citep{cesa2022program} and \\texttt{e3nn}~\\citep{geiger2022e3nn}"
-      },
-      {
-        "line_number": 808,
-        "snippet": "band-limits; integration with \\texttt{escnn}~\\citep{cesa2022program} and"
-      }
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-  {
-    "key": "chevalley2024bispectral",
-    "category": "related-work",
-    "mention_count": 1,
-    "local_entry_type": "inproceedings",
-    "local_title": "A Bispectral {3D} {U-Net} for Rotation Robustness in Medical Segmentation",
-    "local_year": 2024,
-    "local_venue": "Topology- and Graph-Informed Imaging Informatics",
-    "local_doi": "10.1007/978-3-031-73967-5_5",
-    "local_url": "https://doi.org/10.1007/978-3-031-73967-5_5",
-    "resolved_source": "openalex",
-    "resolved_title": "A Bispectral 3D U-Net for\u00a0Rotation Robustness in\u00a0Medical Segmentation",
-    "resolved_year": 2024,
-    "resolved_venue": "Lecture notes in computer science",
-    "resolved_doi": "10.1007/978-3-031-73967-5_5",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
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-    "notes": [
-      "Saved landing link only; no open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 733,
-        "snippet": "\\citet{chevalley2024bispectral} explore bispectral signatures for data"
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-  {
-    "key": "cohen2016group",
-    "category": "theory",
-    "mention_count": 2,
-    "local_entry_type": "inproceedings",
-    "local_title": "Group Equivariant Convolutional Networks",
-    "local_year": 2016,
-    "local_venue": "International Conference on Machine Learning (ICML)",
-    "local_doi": null,
-    "local_url": null,
-    "resolved_source": "openalex",
-    "resolved_title": "Group Equivariant Convolutional Networks",
-    "resolved_year": 2016,
-    "resolved_venue": "arXiv (Cornell University)",
-    "resolved_doi": "10.48550/arxiv.1602.07576",
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-    "mentions": [
-      {
-        "line_number": 111,
-        "snippet": "Equivariant neural networks~\\citep{cohen2016group, weiler2019general} enforce"
-      },
-      {
-        "line_number": 179,
-        "snippet": "for all $g \\in G$~\\citep{cohen2016group, weiler2019general}."
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-    "key": "cohen2018spherical",
-    "category": "experimental",
-    "mention_count": 5,
-    "local_entry_type": "inproceedings",
-    "local_title": "Spherical {CNN}s",
-    "local_year": 2018,
-    "local_venue": "International Conference on Learning Representations (ICLR)",
-    "local_doi": null,
-    "local_url": null,
-    "resolved_source": "openalex",
-    "resolved_title": "Spherical CNNs",
-    "resolved_year": 2018,
-    "resolved_venue": "arXiv (Cornell University)",
-    "resolved_doi": null,
-    "title_match_score": 1.0,
-    "bibtex_source": "local-fallback",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/cohen2018spherical.pdf",
-    "assessment": "good",
-    "notes": [
-      "No DOI found.",
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 615,
-        "snippet": "2-sphere---on Spherical MNIST~\\cite{cohen2018spherical}, the standard"
-      },
-      {
-        "line_number": 637,
-        "snippet": "Following Cohen et al.~\\cite{cohen2018spherical}, we evaluate under three"
-      },
-      {
-        "line_number": 650,
-        "snippet": "et al.\\ reference numbers are from \\cite{cohen2018spherical}."
-      },
-      {
-        "line_number": 707,
-        "snippet": "Cohen's spherical CNN~\\cite{cohen2018spherical} reaches 96\\%/95\\%/94\\%"
-      },
-      {
-        "line_number": 2353,
-        "snippet": "projection following \\citet{cohen2018spherical} and sampled on a"
-      }
-    ]
-  },
-  {
-    "key": "edidin2024orbit",
-    "category": "theory",
-    "mention_count": 1,
-    "local_entry_type": "article",
-    "local_title": "Orbit Recovery for Band-Limited Functions",
-    "local_year": 2024,
-    "local_venue": "SIAM Journal on Applied Algebra and Geometry",
-    "local_doi": "10.1137/23M1577808",
-    "local_url": "https://arxiv.org/abs/2306.00155",
-    "resolved_source": "openalex",
-    "resolved_title": "Orbit Recovery for Band-Limited Functions",
-    "resolved_year": 2024,
-    "resolved_venue": "SIAM Journal on Applied Algebra and Geometry",
-    "resolved_doi": "10.1137/23m1577808",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/edidin2024orbit.pdf",
-    "assessment": "good",
-    "notes": [
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 1696,
-        "snippet": "Edidin \\& Satriano~\\citep{edidin2024orbit}"
-      }
-    ]
-  },
-  {
-    "key": "esteves2018spherical",
-    "category": "related-work",
-    "mention_count": 1,
-    "local_entry_type": "inproceedings",
-    "local_title": "Learning {SO}(3) Equivariant Representations with Spherical {CNN}s",
-    "local_year": 2018,
-    "local_venue": "European Conference on Computer Vision (ECCV)",
-    "local_doi": null,
-    "local_url": null,
-    "resolved_source": "openalex",
-    "resolved_title": "Learning SO(3) Equivariant Representations with Spherical CNNs",
-    "resolved_year": 2018,
-    "resolved_venue": "Lecture notes in computer science",
-    "resolved_doi": "10.1007/978-3-030-01261-8_4",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "link-only",
-    "paper_path": "citation_audit/papers/esteves2018spherical.link.txt",
-    "assessment": "good",
-    "notes": [
-      "Saved landing link only; no open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 752,
-        "snippet": "\\citet{esteves2018spherical} extend this to the sphere. Unlike the"
-      }
-    ]
-  },
-  {
-    "key": "geiger2022e3nn",
-    "category": "theory",
-    "mention_count": 2,
-    "local_entry_type": "article",
-    "local_title": "e3nn: {E}uclidean Neural Networks",
-    "local_year": 2022,
-    "local_venue": "arXiv preprint arXiv:2207.09453",
-    "local_doi": null,
-    "local_url": "https://arxiv.org/abs/2207.09453",
-    "resolved_source": "arxiv",
-    "resolved_title": "e3nn: Euclidean Neural Networks",
-    "resolved_year": 2022,
-    "resolved_venue": "arXiv preprint arXiv:2207.09453",
-    "resolved_doi": null,
-    "title_match_score": 1.0,
-    "bibtex_source": "local-fallback",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/geiger2022e3nn.pdf",
-    "assessment": "acceptable-but-weaker",
-    "notes": [
-      "Preprint citation; archival version may be preferable if one exists.",
-      "No DOI found.",
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 717,
-        "snippet": "\\texttt{escnn}~\\citep{cesa2022program} and \\texttt{e3nn}~\\citep{geiger2022e3nn}"
-      },
-      {
-        "line_number": 809,
-        "snippet": "\\texttt{e3nn}~\\citep{geiger2022e3nn}; and applications to cryo-EM"
-      }
-    ]
-  },
-  {
-    "key": "harris2020numpy",
-    "category": "software",
-    "mention_count": 1,
-    "local_entry_type": "article",
-    "local_title": "Array programming with {NumPy}",
-    "local_year": 2020,
-    "local_venue": "Nature",
-    "local_doi": null,
-    "local_url": null,
-    "resolved_source": "openalex",
-    "resolved_title": "Array programming with NumPy",
-    "resolved_year": 2020,
-    "resolved_venue": "Nature",
-    "resolved_doi": "10.1038/s41586-020-2649-2",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/harris2020numpy.pdf",
-    "assessment": "good",
-    "notes": [
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 257,
-        "snippet": "NumPy~\\citep{harris2020numpy}, and"
-      }
-    ]
-  },
-  {
-    "key": "huang2017densenet",
-    "category": "experimental",
-    "mention_count": 2,
-    "local_entry_type": "inproceedings",
-    "local_title": "Densely Connected Convolutional Networks",
-    "local_year": 2017,
-    "local_venue": "IEEE Conference on Computer Vision and Pattern Recognition (CVPR)",
-    "local_doi": null,
-    "local_url": null,
-    "resolved_source": "openalex",
-    "resolved_title": "Densely Connected Convolutional Networks",
-    "resolved_year": 2017,
-    "resolved_venue": null,
-    "resolved_doi": "10.1109/cvpr.2017.243",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "link-only",
-    "paper_path": "citation_audit/papers/huang2017densenet.link.txt",
-    "assessment": "good",
-    "notes": [
-      "Saved landing link only; no open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 492,
-        "snippet": "$C_8$-equivariant DenseNet~\\citep{huang2017densenet} backbone following"
-      },
-      {
-        "line_number": 2143,
-        "snippet": "DenseNet~\\citep{huang2017densenet} backbone with block configuration"
-      }
-    ]
-  },
-  {
-    "key": "kakarala2012bispectrum",
-    "category": "theory",
-    "mention_count": 4,
-    "local_entry_type": "article",
-    "local_title": "The Bispectrum as a Source of Phase-Sensitive Invariants for {F}ourier Descriptors: A Group-Theoretic Approach",
-    "local_year": 2012,
-    "local_venue": "Journal of Mathematical Imaging and Vision",
-    "local_doi": "10.1007/s10851-012-0330-6",
-    "local_url": "https://arxiv.org/abs/0902.0196",
-    "resolved_source": "openalex",
-    "resolved_title": "The Bispectrum as a Source of Phase-Sensitive Invariants for Fourier Descriptors: A Group-Theoretic Approach",
-    "resolved_year": 2012,
-    "resolved_venue": "Journal of Mathematical Imaging and Vision",
-    "resolved_doi": "10.1007/s10851-012-0330-6",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/kakarala2012bispectrum.pdf",
-    "assessment": "good",
-    "notes": [
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 119,
-        "snippet": "guarantees~\\citep{kakarala2012bispectrum}."
-      },
-      {
-        "line_number": 230,
-        "snippet": "it separates orbits for generic signals~\\citep{kakarala2012bispectrum}. However,"
-      },
-      {
-        "line_number": 755,
-        "snippet": "The bispectrum's completeness guarantee~\\citep{kakarala2012bispectrum} is its"
-      },
-      {
-        "line_number": 1694,
-        "snippet": "Kakarala~\\citep{kakarala2012bispectrum}"
-      }
-    ]
-  },
-  {
-    "key": "kuipers2023regular",
-    "category": "theory",
-    "mention_count": 2,
-    "local_entry_type": "inproceedings",
-    "local_title": "Regular {SE}(3) Group Convolutions for Volumetric Medical Image Analysis",
-    "local_year": 2023,
-    "local_venue": "Medical Image Computing and Computer Assisted Intervention -- {MICCAI} 2023",
-    "local_doi": "10.1007/978-3-031-43898-1_25",
-    "local_url": "https://doi.org/10.1007/978-3-031-43898-1_25",
-    "resolved_source": "openalex",
-    "resolved_title": "Regular SE(3) Group Convolutions for\u00a0Volumetric Medical Image Analysis",
-    "resolved_year": 2023,
-    "resolved_venue": "Lecture notes in computer science",
-    "resolved_doi": "10.1007/978-3-031-43898-1_25",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "link-only",
-    "paper_path": "citation_audit/papers/kuipers2023regular.link.txt",
-    "assessment": "good",
-    "notes": [
-      "Saved landing link only; no open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 591,
-        "snippet": "convolutions~\\citep{kuipers2023regular} (69.8\\%)."
-      },
-      {
-        "line_number": 2234,
-        "snippet": "Regular SE(3) conv & \\citet{kuipers2023regular} & 172K & 0.698 & --- \\\\"
-      }
-    ]
-  },
-  {
-    "key": "loshchilov2019adamw",
-    "category": "experimental",
-    "mention_count": 3,
-    "local_entry_type": "inproceedings",
-    "local_title": "Decoupled Weight Decay Regularization",
-    "local_year": 2019,
-    "local_venue": "International Conference on Learning Representations (ICLR)",
-    "local_doi": null,
-    "local_url": "https://openreview.net/forum?id=Bkg6RiCqY7",
-    "resolved_source": "manual",
-    "resolved_title": "Decoupled Weight Decay Regularization",
-    "resolved_year": 2019,
-    "resolved_venue": "International Conference on Learning Representations",
-    "resolved_doi": null,
-    "title_match_score": 1.0,
-    "bibtex_source": "local-fallback",
-    "paper_status": "link-only",
-    "paper_path": "citation_audit/papers/loshchilov2019adamw.link.txt",
-    "assessment": "good",
-    "notes": [
-      "No DOI found.",
-      "Saved landing link only; no open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 498,
-        "snippet": "AdamW~\\citep{loshchilov2019adamw}. Full model descriptions are in"
-      },
-      {
-        "line_number": 556,
-        "snippet": "are trained with AdamW~\\citep{loshchilov2019adamw} for 100 epochs with"
-      },
-      {
-        "line_number": 2170,
-        "snippet": "training data} (26{,}214 images) with AdamW~\\citep{loshchilov2019adamw}"
-      }
-    ]
-  },
-  {
-    "key": "mallat2012scattering",
-    "category": "related-work",
-    "mention_count": 1,
-    "local_entry_type": "article",
-    "local_title": "Group Invariant Scattering",
-    "local_year": 2012,
-    "local_venue": "Communications on Pure and Applied Mathematics",
-    "local_doi": "10.1002/cpa.21413",
-    "local_url": null,
-    "resolved_source": "openalex",
-    "resolved_title": "Group Invariant Scattering",
-    "resolved_year": 2012,
-    "resolved_venue": "Communications on Pure and Applied Mathematics",
-    "resolved_doi": "10.1002/cpa.21413",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "link-only",
-    "paper_path": "citation_audit/papers/mallat2012scattering.link.txt",
-    "assessment": "good",
-    "notes": [
-      "Saved landing link only; no open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 750,
-        "snippet": "The wavelet scattering transform~\\citep{mallat2012scattering} provides"
-      }
-    ]
-  },
-  {
-    "key": "mataigne2024selective",
-    "category": "theory",
-    "mention_count": 8,
-    "local_entry_type": "inproceedings",
-    "local_title": "The Selective {G}-Bispectrum and its Inversion: Applications to {G}-Invariant Networks",
-    "local_year": 2024,
-    "local_venue": "Advances in Neural Information Processing Systems",
-    "local_doi": "10.52202/079017-3674",
-    "local_url": "https://proceedings.neurips.cc/paper_files/paper/2024/hash/d1a1e8713fcd5626656553c82f7c3b26-Abstract-Conference.html",
-    "resolved_source": "openalex",
-    "resolved_title": "The Selective $G$-Bispectrum and its Inversion: Applications to $G$-Invariant Networks",
-    "resolved_year": 2024,
-    "resolved_venue": null,
-    "resolved_doi": "10.52202/079017-3674",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "link-only",
-    "paper_path": "citation_audit/papers/mataigne2024selective.link.txt",
-    "assessment": "good",
-    "notes": [
-      "Saved landing link only; no open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 126,
-        "snippet": "$O(|G|^2)$ to $O(|G|)$~\\citep{mataigne2024selective}, a user still needs a numerically stable implementation,"
-      },
-      {
-        "line_number": 138,
-        "snippet": "the bootstrap inversion failure~\\citep{mataigne2024selective} for the"
-      },
-      {
-        "line_number": 233,
-        "snippet": "\\emph{Selective bispectra.} \\citet{mataigne2024selective} show that for"
-      },
-      {
-        "line_number": 399,
-        "snippet": "The bootstrap inversion of \\citet{mataigne2024selective} recovers"
-      },
-      {
-        "line_number": 1160,
-        "snippet": "\\citep{mataigne2024selective} is that on~$S^2$, the scalar bispectrum"
-      },
-      {
-        "line_number": 1702,
-        "snippet": "Mataigne et al.~\\citep{mataigne2024selective}"
-      },
-      {
-        "line_number": 1801,
-        "snippet": "All inversion algorithms in \\citet{mataigne2024selective} share the"
-      },
-      {
-        "line_number": 1925,
-        "snippet": "\\citet{mataigne2024selective} acknowledge a related issue in the"
-      }
-    ]
-  },
-  {
-    "key": "medmnistv2",
-    "category": "dataset",
-    "mention_count": 2,
-    "local_entry_type": "article",
-    "local_title": "{MedMNIST} v2 -- A Large-Scale Lightweight Benchmark for {2D} and {3D} Biomedical Image Classification",
-    "local_year": 2023,
-    "local_venue": "Scientific Data",
-    "local_doi": null,
-    "local_url": null,
-    "resolved_source": "openalex",
-    "resolved_title": "MedMNIST v2 - A large-scale lightweight benchmark for 2D and 3D biomedical image classification",
-    "resolved_year": 2023,
-    "resolved_venue": "Scientific Data",
-    "resolved_doi": "10.1038/s41597-022-01721-8",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/medmnistv2.pdf",
-    "assessment": "good",
-    "notes": [
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 550,
-        "snippet": "OrganMNIST3D~\\citep{medmnistv2} (1{,}742 abdominal CT volumes, $28^3$"
-      },
-      {
-        "line_number": 2343,
-        "snippet": "OrganMNIST3D~\\citep{medmnistv2} contains 1{,}742 abdominal CT organ"
-      }
-    ]
-  },
-  {
-    "key": "oreiller2022bispectral",
-    "category": "related-work",
-    "mention_count": 1,
-    "local_entry_type": "inproceedings",
-    "local_title": "Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral {U-Net}",
-    "local_year": 2022,
-    "local_venue": "Proceedings of The 5th International Conference on Medical Imaging with Deep Learning",
-    "local_doi": null,
-    "local_url": "https://proceedings.mlr.press/v172/oreiller22a.html",
-    "resolved_source": "manual",
-    "resolved_title": "Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral U-Net",
-    "resolved_year": 2022,
-    "resolved_venue": "Medical Imaging with Deep Learning",
-    "resolved_doi": null,
-    "title_match_score": 1.0,
-    "bibtex_source": "local-fallback",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/oreiller2022bispectral.pdf",
-    "assessment": "good",
-    "notes": [
-      "No DOI found.",
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 728,
-        "snippet": "\\citet{oreiller2022bispectral} applied bispectral CNNs to medical imaging"
-      }
-    ]
-  },
-  {
-    "key": "paszke2019pytorch",
-    "category": "software",
-    "mention_count": 2,
-    "local_entry_type": "inproceedings",
-    "local_title": "Py{T}orch: An Imperative Style, High-Performance Deep Learning Library",
-    "local_year": 2019,
-    "local_venue": "Advances in Neural Information Processing Systems (NeurIPS)",
-    "local_doi": null,
-    "local_url": null,
-    "resolved_source": "openalex",
-    "resolved_title": "PyTorch: An Imperative Style, High-Performance Deep Learning Library",
-    "resolved_year": 2019,
-    "resolved_venue": "arXiv (Cornell University)",
-    "resolved_doi": "10.48550/arxiv.1912.01703",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/paszke2019pytorch.pdf",
-    "assessment": "good",
-    "notes": [
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 129,
-        "snippet": "This paper presents \\texttt{bispectrum}, a PyTorch~\\citep{paszke2019pytorch}"
-      },
-      {
-        "line_number": 256,
-        "snippet": "The only dependencies are PyTorch~\\citep{paszke2019pytorch},"
-      }
-    ]
-  },
-  {
-    "key": "price2025freedom",
-    "category": "theory",
-    "mention_count": 1,
-    "local_entry_type": "inproceedings",
-    "local_title": "The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products",
-    "local_year": 2025,
-    "local_venue": "Proceedings of the 42nd International Conference on Machine Learning (ICML)",
-    "local_doi": null,
-    "local_url": null,
-    "resolved_source": "openalex",
-    "resolved_title": "The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products",
-    "resolved_year": 2025,
-    "resolved_venue": "ArXiv.org",
-    "resolved_doi": "10.48550/arxiv.2506.13523",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/price2025freedom.pdf",
-    "assessment": "good",
-    "notes": [
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 742,
-        "snippet": "\\citet{price2025freedom} analyse the \\emph{price of freedom} in tensor"
-      }
-    ]
-  },
-  {
-    "key": "sanborn2023bispectral",
-    "category": "related-work",
-    "mention_count": 1,
-    "local_entry_type": "inproceedings",
-    "local_title": "Bispectral Neural Networks",
-    "local_year": 2023,
-    "local_venue": "International Conference on Learning Representations (ICLR)",
-    "local_doi": null,
-    "local_url": null,
-    "resolved_source": "openalex",
-    "resolved_title": "Bispectral Neural Networks",
-    "resolved_year": 2022,
-    "resolved_venue": "arXiv (Cornell University)",
-    "resolved_doi": "10.48550/arxiv.2209.03416",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/sanborn2023bispectral.pdf",
-    "assessment": "good",
-    "notes": [
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 726,
-        "snippet": "\\citet{sanborn2023bispectral} introduced bispectral neural networks (BNNs),"
-      }
-    ]
-  },
-  {
-    "key": "sangalli2023movfrnet",
-    "category": "theory",
-    "mention_count": 1,
-    "local_entry_type": "inproceedings",
-    "local_title": "Moving Frame Net: {SE}(3)-Equivariant Network for Volumes",
-    "local_year": 2023,
-    "local_venue": "Proceedings of the 1st NeurIPS Workshop on Symmetry and Geometry in Neural Representations",
-    "local_doi": null,
-    "local_url": "https://proceedings.mlr.press/v197/sangalli23a.html",
-    "resolved_source": "openalex",
-    "resolved_title": "Moving Frame Net: SE(3)-Equivariant Network for Volumes",
-    "resolved_year": 2022,
-    "resolved_venue": "arXiv (Cornell University)",
-    "resolved_doi": "10.48550/arxiv.2211.03420",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/sangalli2023movfrnet.pdf",
-    "assessment": "acceptable-but-weaker",
-    "notes": [
-      "Workshop citation rather than a main archival venue.",
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 2233,
-        "snippet": "SE3MovFrNet & \\citet{sangalli2023movfrnet} & --- & 0.745 & --- \\\\"
-      }
-    ]
-  },
-  {
-    "key": "thomas2018tensor",
-    "category": "theory",
-    "mention_count": 1,
-    "local_entry_type": "inproceedings",
-    "local_title": "Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for {3D} Point Clouds",
-    "local_year": 2018,
-    "local_venue": "arXiv preprint arXiv:1802.08219",
-    "local_doi": null,
-    "local_url": "https://arxiv.org/abs/1802.08219",
-    "resolved_source": "arxiv",
-    "resolved_title": "Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for 3D Point Clouds",
-    "resolved_year": 2018,
-    "resolved_venue": "arXiv preprint arXiv:1802.08219",
-    "resolved_doi": null,
-    "title_match_score": 1.0,
-    "bibtex_source": "local-fallback",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/thomas2018tensor.pdf",
-    "assessment": "acceptable-but-weaker",
-    "notes": [
-      "Preprint citation; archival version may be preferable if one exists.",
-      "No DOI found.",
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 738,
-        "snippet": "Tensor field networks~\\citep{thomas2018tensor} and"
-      }
-    ]
-  },
-  {
-    "key": "veeling2018rotation",
-    "category": "dataset",
-    "mention_count": 5,
-    "local_entry_type": "inproceedings",
-    "local_title": "Rotation Equivariant {CNN}s for Digital Pathology",
-    "local_year": 2018,
-    "local_venue": "Medical Image Computing and Computer Assisted Intervention (MICCAI)",
-    "local_doi": null,
-    "local_url": null,
-    "resolved_source": "openalex",
-    "resolved_title": "Rotation Equivariant CNNs for Digital Pathology",
-    "resolved_year": 2018,
-    "resolved_venue": "Lecture notes in computer science",
-    "resolved_doi": "10.1007/978-3-030-00934-2_24",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "unavailable",
-    "paper_path": null,
-    "assessment": "good",
-    "notes": [
-      "No paper URL resolved."
-    ],
-    "mentions": [
-      {
-        "line_number": 490,
-        "snippet": "PatchCamelyon~\\citep{veeling2018rotation} (PCam), 327K histopathology"
-      },
-      {
-        "line_number": 493,
-        "snippet": "\\citet{veeling2018rotation}; they differ only in the terminal invariant map:"
-      },
-      {
-        "line_number": 2124,
-        "snippet": "\\citet{veeling2018rotation}, trained on 100\\% data.}"
-      },
-      {
-        "line_number": 2145,
-        "snippet": "following the architecture of \\citet{veeling2018rotation} but"
-      },
-      {
-        "line_number": 2334,
-        "snippet": "PatchCamelyon~\\citep{veeling2018rotation} consists of 327{,}680 RGB"
-      }
-    ]
-  },
-  {
-    "key": "weiler2019general",
-    "category": "theory",
-    "mention_count": 4,
-    "local_entry_type": "inproceedings",
-    "local_title": "General {E}(2)-Equivariant Steerable {CNN}s",
-    "local_year": 2019,
-    "local_venue": "Advances in Neural Information Processing Systems (NeurIPS)",
-    "local_doi": null,
-    "local_url": null,
-    "resolved_source": "openalex",
-    "resolved_title": "General $E(2)$-Equivariant Steerable CNNs",
-    "resolved_year": 2019,
-    "resolved_venue": "arXiv (Cornell University)",
-    "resolved_doi": "10.48550/arxiv.1911.08251",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "downloaded",
-    "paper_path": "citation_audit/papers/weiler2019general.pdf",
-    "assessment": "good",
-    "notes": [
-      "Open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 111,
-        "snippet": "Equivariant neural networks~\\citep{cohen2016group, weiler2019general} enforce"
-      },
-      {
-        "line_number": 116,
-        "snippet": "usable~\\citep{weiler2019general}, but they do not preserve all equivariant"
-      },
-      {
-        "line_number": 179,
-        "snippet": "for all $g \\in G$~\\citep{cohen2016group, weiler2019general}."
-      },
-      {
-        "line_number": 210,
-        "snippet": "sacrifice exactness through aliasing~\\citep{weiler2019general}. The full"
-      }
-    ]
-  },
-  {
-    "key": "zhemchuzhnikov2024ilponet",
-    "category": "theory",
-    "mention_count": 1,
-    "local_entry_type": "inproceedings",
-    "local_title": "{ILPO-NET}: Network for the Invariant Recognition of Arbitrary Volumetric Patterns in {3D}",
-    "local_year": 2024,
-    "local_venue": "Machine Learning and Knowledge Discovery in Databases. Research Track",
-    "local_doi": "10.1007/978-3-031-70359-1_21",
-    "local_url": "https://doi.org/10.1007/978-3-031-70359-1_21",
-    "resolved_source": "openalex",
-    "resolved_title": "ILPO-NET: Network for\u00a0the\u00a0Invariant Recognition of\u00a0Arbitrary Volumetric Patterns in\u00a03D",
-    "resolved_year": 2024,
-    "resolved_venue": "Lecture notes in computer science",
-    "resolved_doi": "10.1007/978-3-031-70359-1_21",
-    "title_match_score": 1.0,
-    "bibtex_source": "remote-doi",
-    "paper_status": "link-only",
-    "paper_path": "citation_audit/papers/zhemchuzhnikov2024ilponet.link.txt",
-    "assessment": "good",
-    "notes": [
-      "Saved landing link only; no open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 2231,
-        "snippet": "ILPOResNet-50 & \\citet{zhemchuzhnikov2024ilponet} & 38K & 0.879 & 0.992 \\\\"
-      }
-    ]
-  },
-  {
-    "key": "zhemchuzhnikov2025equilopo",
-    "category": "theory",
-    "mention_count": 1,
-    "local_entry_type": "inproceedings",
-    "local_title": "On the {F}ourier Analysis in the {SO}(3) Space: the {EquiLoPO} Network",
-    "local_year": 2025,
-    "local_venue": "International Conference on Learning Representations (ICLR)",
-    "local_doi": null,
-    "local_url": "https://openreview.net/forum?id=LvTSvdiSwG",
-    "resolved_source": "manual",
-    "resolved_title": "On the Fourier Analysis in the SO(3) Space: the EquiLoPO Network",
-    "resolved_year": 2025,
-    "resolved_venue": "International Conference on Learning Representations",
-    "resolved_doi": null,
-    "title_match_score": 1.0,
-    "bibtex_source": "local-fallback",
-    "paper_status": "link-only",
-    "paper_path": "citation_audit/papers/zhemchuzhnikov2025equilopo.link.txt",
-    "assessment": "good",
-    "notes": [
-      "No DOI found.",
-      "Saved landing link only; no open PDF downloaded."
-    ],
-    "mentions": [
-      {
-        "line_number": 2232,
-        "snippet": "EquiLoPO (local train.) & \\citet{zhemchuzhnikov2025equilopo} & 418K & 0.866 & 0.991 \\\\"
-      }
-    ]
-  }
-]
\ No newline at end of file
diff --git a/paper/citation_audit/citation_audit.md b/paper/citation_audit/citation_audit.md
deleted file mode 100644
index cf60eae..0000000
--- a/paper/citation_audit/citation_audit.md
+++ /dev/null
@@ -1,378 +0,0 @@
-# Citation Audit
-
-- Total cited keys: 29
-- Assessments: 25 good, 4 acceptable-but-weaker, 0 check
-- Paper assets: 16 PDFs downloaded, 12 link-only, 1 unavailable
-
-| Key | Category | Mentions | Assessment | Resolved source | BibTeX | Paper | Notes |
-| --- | --- | ---: | --- | --- | --- | --- | --- |
-| batatia2022mace | related-work | 1 | good | openalex | remote-doi | link-only (`citation_audit/papers/batatia2022mace.link.txt`) | Saved landing link only; no open PDF downloaded. |
-| bejnordi2017camelyon | dataset | 1 | good | openalex | remote-doi | link-only (`citation_audit/papers/bejnordi2017camelyon.link.txt`) | Saved landing link only; no open PDF downloaded. |
-| bendory2025orbit | theory | 1 | acceptable-but-weaker | arxiv | local-fallback | downloaded (`citation_audit/papers/bendory2025orbit.pdf`) | Preprint citation; archival version may be preferable if one exists.; No DOI found. |
-| bonev2023spherical | theory | 2 | good | manual | local-fallback | downloaded (`citation_audit/papers/bonev2023spherical.pdf`) | No DOI found.; Open PDF downloaded. |
-| cesa2022program | theory | 2 | good | manual | local-fallback | link-only (`citation_audit/papers/cesa2022program.link.txt`) | No DOI found.; Saved landing link only; no open PDF downloaded. |
-| chevalley2024bispectral | related-work | 1 | good | openalex | remote-doi | link-only (`citation_audit/papers/chevalley2024bispectral.link.txt`) | Saved landing link only; no open PDF downloaded. |
-| cohen2016group | theory | 2 | good | openalex | remote-doi | downloaded (`citation_audit/papers/cohen2016group.pdf`) | Open PDF downloaded. |
-| cohen2018spherical | experimental | 5 | good | openalex | local-fallback | downloaded (`citation_audit/papers/cohen2018spherical.pdf`) | No DOI found.; Open PDF downloaded. |
-| edidin2024orbit | theory | 1 | good | openalex | remote-doi | downloaded (`citation_audit/papers/edidin2024orbit.pdf`) | Open PDF downloaded. |
-| esteves2018spherical | related-work | 1 | good | openalex | remote-doi | link-only (`citation_audit/papers/esteves2018spherical.link.txt`) | Saved landing link only; no open PDF downloaded. |
-| geiger2022e3nn | theory | 2 | acceptable-but-weaker | arxiv | local-fallback | downloaded (`citation_audit/papers/geiger2022e3nn.pdf`) | Preprint citation; archival version may be preferable if one exists.; No DOI found. |
-| harris2020numpy | software | 1 | good | openalex | remote-doi | downloaded (`citation_audit/papers/harris2020numpy.pdf`) | Open PDF downloaded. |
-| huang2017densenet | experimental | 2 | good | openalex | remote-doi | link-only (`citation_audit/papers/huang2017densenet.link.txt`) | Saved landing link only; no open PDF downloaded. |
-| kakarala2012bispectrum | theory | 4 | good | openalex | remote-doi | downloaded (`citation_audit/papers/kakarala2012bispectrum.pdf`) | Open PDF downloaded. |
-| kuipers2023regular | theory | 2 | good | openalex | remote-doi | link-only (`citation_audit/papers/kuipers2023regular.link.txt`) | Saved landing link only; no open PDF downloaded. |
-| loshchilov2019adamw | experimental | 3 | good | manual | local-fallback | link-only (`citation_audit/papers/loshchilov2019adamw.link.txt`) | No DOI found.; Saved landing link only; no open PDF downloaded. |
-| mallat2012scattering | related-work | 1 | good | openalex | remote-doi | link-only (`citation_audit/papers/mallat2012scattering.link.txt`) | Saved landing link only; no open PDF downloaded. |
-| mataigne2024selective | theory | 8 | good | openalex | remote-doi | link-only (`citation_audit/papers/mataigne2024selective.link.txt`) | Saved landing link only; no open PDF downloaded. |
-| medmnistv2 | dataset | 2 | good | openalex | remote-doi | downloaded (`citation_audit/papers/medmnistv2.pdf`) | Open PDF downloaded. |
-| oreiller2022bispectral | related-work | 1 | good | manual | local-fallback | downloaded (`citation_audit/papers/oreiller2022bispectral.pdf`) | No DOI found.; Open PDF downloaded. |
-| paszke2019pytorch | software | 2 | good | openalex | remote-doi | downloaded (`citation_audit/papers/paszke2019pytorch.pdf`) | Open PDF downloaded. |
-| price2025freedom | theory | 1 | good | openalex | remote-doi | downloaded (`citation_audit/papers/price2025freedom.pdf`) | Open PDF downloaded. |
-| sanborn2023bispectral | related-work | 1 | good | openalex | remote-doi | downloaded (`citation_audit/papers/sanborn2023bispectral.pdf`) | Open PDF downloaded. |
-| sangalli2023movfrnet | theory | 1 | acceptable-but-weaker | openalex | remote-doi | downloaded (`citation_audit/papers/sangalli2023movfrnet.pdf`) | Workshop citation rather than a main archival venue.; Open PDF downloaded. |
-| thomas2018tensor | theory | 1 | acceptable-but-weaker | arxiv | local-fallback | downloaded (`citation_audit/papers/thomas2018tensor.pdf`) | Preprint citation; archival version may be preferable if one exists.; No DOI found. |
-| veeling2018rotation | dataset | 5 | good | openalex | remote-doi | unavailable | No paper URL resolved. |
-| weiler2019general | theory | 4 | good | openalex | remote-doi | downloaded (`citation_audit/papers/weiler2019general.pdf`) | Open PDF downloaded. |
-| zhemchuzhnikov2024ilponet | theory | 1 | good | openalex | remote-doi | link-only (`citation_audit/papers/zhemchuzhnikov2024ilponet.link.txt`) | Saved landing link only; no open PDF downloaded. |
-| zhemchuzhnikov2025equilopo | theory | 1 | good | manual | local-fallback | link-only (`citation_audit/papers/zhemchuzhnikov2025equilopo.link.txt`) | No DOI found.; Saved landing link only; no open PDF downloaded. |
-
-## Details
-
-### `batatia2022mace`
-
-- Local title: {MACE}: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields
-- Resolved title: MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: link-only (`citation_audit/papers/batatia2022mace.link.txt`)
-- Notes: Saved landing link only; no open PDF downloaded.
-- Citation contexts:
-  - L739: MACE~\citep{batatia2022mace} use iterated Clebsch--Gordan products
-
-### `bejnordi2017camelyon`
-
-- Local title: Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer
-- Resolved title: Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: link-only (`citation_audit/papers/bejnordi2017camelyon.link.txt`)
-- Notes: Saved landing link only; no open PDF downloaded.
-- Citation contexts:
-  - L2336: images of lymph node sections~\citep{bejnordi2017camelyon}. We use
-
-### `bendory2025orbit`
-
-- Local title: Orbit Recovery for Spherical Functions
-- Resolved title: Orbit Recovery for Spherical Functions
-- Assessment: acceptable-but-weaker
-- BibTeX source: local-fallback
-- Paper asset: downloaded (`citation_audit/papers/bendory2025orbit.pdf`)
-- Notes: Preprint citation; archival version may be preferable if one exists.; No DOI found.; Open PDF downloaded.
-- Citation contexts:
-  - L1699: Bendory et al.~\citep{bendory2025orbit}
-
-### `bonev2023spherical`
-
-- Local title: Spherical {F}ourier Neural Operators: Learning Stable Dynamics on the Sphere
-- Resolved title: Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere
-- Assessment: good
-- BibTeX source: local-fallback
-- Paper asset: downloaded (`citation_audit/papers/bonev2023spherical.pdf`)
-- Notes: No DOI found.; Open PDF downloaded.
-- Citation contexts:
-  - L258: \texttt{torch\_harmonics}~\citep{bonev2023spherical}
-  - L466: transform (via \texttt{torch\_harmonics}~\citep{bonev2023spherical}).
-
-### `cesa2022program`
-
-- Local title: A Program to Build {E(N)}-Equivariant Steerable {CNN}s
-- Resolved title: A Program to Build E(N)-Equivariant Steerable CNNs
-- Assessment: good
-- BibTeX source: local-fallback
-- Paper asset: link-only (`citation_audit/papers/cesa2022program.link.txt`)
-- Notes: No DOI found.; Saved landing link only; no open PDF downloaded.
-- Citation contexts:
-  - L717: \texttt{escnn}~\citep{cesa2022program} and \texttt{e3nn}~\citep{geiger2022e3nn}
-  - L808: band-limits; integration with \texttt{escnn}~\citep{cesa2022program} and
-
-### `chevalley2024bispectral`
-
-- Local title: A Bispectral {3D} {U-Net} for Rotation Robustness in Medical Segmentation
-- Resolved title: A Bispectral 3D U-Net for Rotation Robustness in Medical Segmentation
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: link-only (`citation_audit/papers/chevalley2024bispectral.link.txt`)
-- Notes: Saved landing link only; no open PDF downloaded.
-- Citation contexts:
-  - L733: \citet{chevalley2024bispectral} explore bispectral signatures for data
-
-### `cohen2016group`
-
-- Local title: Group Equivariant Convolutional Networks
-- Resolved title: Group Equivariant Convolutional Networks
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: downloaded (`citation_audit/papers/cohen2016group.pdf`)
-- Notes: Open PDF downloaded.
-- Citation contexts:
-  - L111: Equivariant neural networks~\citep{cohen2016group, weiler2019general} enforce
-  - L179: for all $g \in G$~\citep{cohen2016group, weiler2019general}.
-
-### `cohen2018spherical`
-
-- Local title: Spherical {CNN}s
-- Resolved title: Spherical CNNs
-- Assessment: good
-- BibTeX source: local-fallback
-- Paper asset: downloaded (`citation_audit/papers/cohen2018spherical.pdf`)
-- Notes: No DOI found.; Open PDF downloaded.
-- Citation contexts:
-  - L615: 2-sphere---on Spherical MNIST~\cite{cohen2018spherical}, the standard
-  - L637: Following Cohen et al.~\cite{cohen2018spherical}, we evaluate under three
-  - L650: et al.\ reference numbers are from \cite{cohen2018spherical}.
-
-### `edidin2024orbit`
-
-- Local title: Orbit Recovery for Band-Limited Functions
-- Resolved title: Orbit Recovery for Band-Limited Functions
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: downloaded (`citation_audit/papers/edidin2024orbit.pdf`)
-- Notes: Open PDF downloaded.
-- Citation contexts:
-  - L1696: Edidin \& Satriano~\citep{edidin2024orbit}
-
-### `esteves2018spherical`
-
-- Local title: Learning {SO}(3) Equivariant Representations with Spherical {CNN}s
-- Resolved title: Learning SO(3) Equivariant Representations with Spherical CNNs
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: link-only (`citation_audit/papers/esteves2018spherical.link.txt`)
-- Notes: Saved landing link only; no open PDF downloaded.
-- Citation contexts:
-  - L752: \citet{esteves2018spherical} extend this to the sphere. Unlike the
-
-### `geiger2022e3nn`
-
-- Local title: e3nn: {E}uclidean Neural Networks
-- Resolved title: e3nn: Euclidean Neural Networks
-- Assessment: acceptable-but-weaker
-- BibTeX source: local-fallback
-- Paper asset: downloaded (`citation_audit/papers/geiger2022e3nn.pdf`)
-- Notes: Preprint citation; archival version may be preferable if one exists.; No DOI found.; Open PDF downloaded.
-- Citation contexts:
-  - L717: \texttt{escnn}~\citep{cesa2022program} and \texttt{e3nn}~\citep{geiger2022e3nn}
-  - L809: \texttt{e3nn}~\citep{geiger2022e3nn}; and applications to cryo-EM
-
-### `harris2020numpy`
-
-- Local title: Array programming with {NumPy}
-- Resolved title: Array programming with NumPy
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: downloaded (`citation_audit/papers/harris2020numpy.pdf`)
-- Notes: Open PDF downloaded.
-- Citation contexts:
-  - L257: NumPy~\citep{harris2020numpy}, and
-
-### `huang2017densenet`
-
-- Local title: Densely Connected Convolutional Networks
-- Resolved title: Densely Connected Convolutional Networks
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: link-only (`citation_audit/papers/huang2017densenet.link.txt`)
-- Notes: Saved landing link only; no open PDF downloaded.
-- Citation contexts:
-  - L492: $C_8$-equivariant DenseNet~\citep{huang2017densenet} backbone following
-  - L2143: DenseNet~\citep{huang2017densenet} backbone with block configuration
-
-### `kakarala2012bispectrum`
-
-- Local title: The Bispectrum as a Source of Phase-Sensitive Invariants for {F}ourier Descriptors: A Group-Theoretic Approach
-- Resolved title: The Bispectrum as a Source of Phase-Sensitive Invariants for Fourier Descriptors: A Group-Theoretic Approach
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: downloaded (`citation_audit/papers/kakarala2012bispectrum.pdf`)
-- Notes: Open PDF downloaded.
-- Citation contexts:
-  - L119: guarantees~\citep{kakarala2012bispectrum}.
-  - L230: it separates orbits for generic signals~\citep{kakarala2012bispectrum}. However,
-  - L755: The bispectrum's completeness guarantee~\citep{kakarala2012bispectrum} is its
-
-### `kuipers2023regular`
-
-- Local title: Regular {SE}(3) Group Convolutions for Volumetric Medical Image Analysis
-- Resolved title: Regular SE(3) Group Convolutions for Volumetric Medical Image Analysis
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: link-only (`citation_audit/papers/kuipers2023regular.link.txt`)
-- Notes: Saved landing link only; no open PDF downloaded.
-- Citation contexts:
-  - L591: convolutions~\citep{kuipers2023regular} (69.8\%).
-  - L2234: Regular SE(3) conv & \citet{kuipers2023regular} & 172K & 0.698 & --- \\
-
-### `loshchilov2019adamw`
-
-- Local title: Decoupled Weight Decay Regularization
-- Resolved title: Decoupled Weight Decay Regularization
-- Assessment: good
-- BibTeX source: local-fallback
-- Paper asset: link-only (`citation_audit/papers/loshchilov2019adamw.link.txt`)
-- Notes: No DOI found.; Saved landing link only; no open PDF downloaded.
-- Citation contexts:
-  - L498: AdamW~\citep{loshchilov2019adamw}. Full model descriptions are in
-  - L556: are trained with AdamW~\citep{loshchilov2019adamw} for 100 epochs with
-  - L2170: training data} (26{,}214 images) with AdamW~\citep{loshchilov2019adamw}
-
-### `mallat2012scattering`
-
-- Local title: Group Invariant Scattering
-- Resolved title: Group Invariant Scattering
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: link-only (`citation_audit/papers/mallat2012scattering.link.txt`)
-- Notes: Saved landing link only; no open PDF downloaded.
-- Citation contexts:
-  - L750: The wavelet scattering transform~\citep{mallat2012scattering} provides
-
-### `mataigne2024selective`
-
-- Local title: The Selective {G}-Bispectrum and its Inversion: Applications to {G}-Invariant Networks
-- Resolved title: The Selective $G$-Bispectrum and its Inversion: Applications to $G$-Invariant Networks
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: link-only (`citation_audit/papers/mataigne2024selective.link.txt`)
-- Notes: Saved landing link only; no open PDF downloaded.
-- Citation contexts:
-  - L126: $O(|G|^2)$ to $O(|G|)$~\citep{mataigne2024selective}, a user still needs a numerically stable implementation,
-  - L138: the bootstrap inversion failure~\citep{mataigne2024selective} for the
-  - L233: \emph{Selective bispectra.} \citet{mataigne2024selective} show that for
-
-### `medmnistv2`
-
-- Local title: {MedMNIST} v2 -- A Large-Scale Lightweight Benchmark for {2D} and {3D} Biomedical Image Classification
-- Resolved title: MedMNIST v2 - A large-scale lightweight benchmark for 2D and 3D biomedical image classification
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: downloaded (`citation_audit/papers/medmnistv2.pdf`)
-- Notes: Open PDF downloaded.
-- Citation contexts:
-  - L550: OrganMNIST3D~\citep{medmnistv2} (1{,}742 abdominal CT volumes, $28^3$
-  - L2343: OrganMNIST3D~\citep{medmnistv2} contains 1{,}742 abdominal CT organ
-
-### `oreiller2022bispectral`
-
-- Local title: Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral {U-Net}
-- Resolved title: Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral U-Net
-- Assessment: good
-- BibTeX source: local-fallback
-- Paper asset: downloaded (`citation_audit/papers/oreiller2022bispectral.pdf`)
-- Notes: No DOI found.; Open PDF downloaded.
-- Citation contexts:
-  - L728: \citet{oreiller2022bispectral} applied bispectral CNNs to medical imaging
-
-### `paszke2019pytorch`
-
-- Local title: Py{T}orch: An Imperative Style, High-Performance Deep Learning Library
-- Resolved title: PyTorch: An Imperative Style, High-Performance Deep Learning Library
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: downloaded (`citation_audit/papers/paszke2019pytorch.pdf`)
-- Notes: Open PDF downloaded.
-- Citation contexts:
-  - L129: This paper presents \texttt{bispectrum}, a PyTorch~\citep{paszke2019pytorch}
-  - L256: The only dependencies are PyTorch~\citep{paszke2019pytorch},
-
-### `price2025freedom`
-
-- Local title: The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products
-- Resolved title: The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: downloaded (`citation_audit/papers/price2025freedom.pdf`)
-- Notes: Open PDF downloaded.
-- Citation contexts:
-  - L742: \citet{price2025freedom} analyse the \emph{price of freedom} in tensor
-
-### `sanborn2023bispectral`
-
-- Local title: Bispectral Neural Networks
-- Resolved title: Bispectral Neural Networks
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: downloaded (`citation_audit/papers/sanborn2023bispectral.pdf`)
-- Notes: Open PDF downloaded.
-- Citation contexts:
-  - L726: \citet{sanborn2023bispectral} introduced bispectral neural networks (BNNs),
-
-### `sangalli2023movfrnet`
-
-- Local title: Moving Frame Net: {SE}(3)-Equivariant Network for Volumes
-- Resolved title: Moving Frame Net: SE(3)-Equivariant Network for Volumes
-- Assessment: acceptable-but-weaker
-- BibTeX source: remote-doi
-- Paper asset: downloaded (`citation_audit/papers/sangalli2023movfrnet.pdf`)
-- Notes: Workshop citation rather than a main archival venue.; Open PDF downloaded.
-- Citation contexts:
-  - L2233: SE3MovFrNet & \citet{sangalli2023movfrnet} & --- & 0.745 & --- \\
-
-### `thomas2018tensor`
-
-- Local title: Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for {3D} Point Clouds
-- Resolved title: Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for 3D Point Clouds
-- Assessment: acceptable-but-weaker
-- BibTeX source: local-fallback
-- Paper asset: downloaded (`citation_audit/papers/thomas2018tensor.pdf`)
-- Notes: Preprint citation; archival version may be preferable if one exists.; No DOI found.; Open PDF downloaded.
-- Citation contexts:
-  - L738: Tensor field networks~\citep{thomas2018tensor} and
-
-### `veeling2018rotation`
-
-- Local title: Rotation Equivariant {CNN}s for Digital Pathology
-- Resolved title: Rotation Equivariant CNNs for Digital Pathology
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: unavailable
-- Notes: No paper URL resolved.
-- Citation contexts:
-  - L490: PatchCamelyon~\citep{veeling2018rotation} (PCam), 327K histopathology
-  - L493: \citet{veeling2018rotation}; they differ only in the terminal invariant map:
-  - L2124: \citet{veeling2018rotation}, trained on 100\% data.}
-
-### `weiler2019general`
-
-- Local title: General {E}(2)-Equivariant Steerable {CNN}s
-- Resolved title: General $E(2)$-Equivariant Steerable CNNs
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: downloaded (`citation_audit/papers/weiler2019general.pdf`)
-- Notes: Open PDF downloaded.
-- Citation contexts:
-  - L111: Equivariant neural networks~\citep{cohen2016group, weiler2019general} enforce
-  - L116: usable~\citep{weiler2019general}, but they do not preserve all equivariant
-  - L179: for all $g \in G$~\citep{cohen2016group, weiler2019general}.
-
-### `zhemchuzhnikov2024ilponet`
-
-- Local title: {ILPO-NET}: Network for the Invariant Recognition of Arbitrary Volumetric Patterns in {3D}
-- Resolved title: ILPO-NET: Network for the Invariant Recognition of Arbitrary Volumetric Patterns in 3D
-- Assessment: good
-- BibTeX source: remote-doi
-- Paper asset: link-only (`citation_audit/papers/zhemchuzhnikov2024ilponet.link.txt`)
-- Notes: Saved landing link only; no open PDF downloaded.
-- Citation contexts:
-  - L2231: ILPOResNet-50 & \citet{zhemchuzhnikov2024ilponet} & 38K & 0.879 & 0.992 \\
-
-### `zhemchuzhnikov2025equilopo`
-
-- Local title: On the {F}ourier Analysis in the {SO}(3) Space: the {EquiLoPO} Network
-- Resolved title: On the Fourier Analysis in the SO(3) Space: the EquiLoPO Network
-- Assessment: good
-- BibTeX source: local-fallback
-- Paper asset: link-only (`citation_audit/papers/zhemchuzhnikov2025equilopo.link.txt`)
-- Notes: No DOI found.; Saved landing link only; no open PDF downloaded.
-- Citation contexts:
-  - L2232: EquiLoPO (local train.) & \citet{zhemchuzhnikov2025equilopo} & 418K & 0.866 & 0.991 \\
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deleted file mode 100644
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--- a/paper/citation_audit/papers/batatia2022mace.link.txt
+++ /dev/null
@@ -1 +0,0 @@
-https://doi.org/10.52202/068431-0830
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--- a/paper/citation_audit/papers/batatia2024design.link.txt
+++ /dev/null
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-https://doi.org/10.26434/chemrxiv-2025-f1hgn-v3
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--- a/paper/citation_audit/papers/bejnordi2017camelyon.link.txt
+++ /dev/null
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-https://doi.org/10.1001/jama.2017.14585
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--- a/paper/citation_audit/papers/cesa2022program.link.txt
+++ /dev/null
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-https://openreview.net/forum?id=WE4qe9xlnQw
diff --git a/paper/citation_audit/papers/chevalley2024bispectral.link.txt b/paper/citation_audit/papers/chevalley2024bispectral.link.txt
deleted file mode 100644
index ca3b244..0000000
--- a/paper/citation_audit/papers/chevalley2024bispectral.link.txt
+++ /dev/null
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-https://doi.org/10.1007/978-3-031-73967-5_5
diff --git a/paper/citation_audit/papers/cohen2016group.link.txt b/paper/citation_audit/papers/cohen2016group.link.txt
deleted file mode 100644
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--- a/paper/citation_audit/papers/cohen2016group.link.txt
+++ /dev/null
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-https://doi.org/10.1109/6gnet68413.2025.11314083
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--- a/paper/citation_audit/papers/esteves2018spherical.link.txt
+++ /dev/null
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-https://doi.org/10.1007/978-3-030-01261-8_4
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deleted file mode 100644
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--- a/paper/citation_audit/papers/huang2017densenet.link.txt
+++ /dev/null
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-https://doi.org/10.1109/cvpr.2017.243
diff --git a/paper/citation_audit/papers/kakarala1992triple.link.txt b/paper/citation_audit/papers/kakarala1992triple.link.txt
deleted file mode 100644
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--- a/paper/citation_audit/papers/kakarala1992triple.link.txt
+++ /dev/null
@@ -1 +0,0 @@
-https://dl.acm.org/citation.cfm?id=164353
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--- a/paper/citation_audit/papers/kuipers2023regular.link.txt
+++ /dev/null
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-https://doi.org/10.1007/978-3-031-43898-1_25
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--- a/paper/citation_audit/papers/loshchilov2019adamw.link.txt
+++ /dev/null
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-https://openreview.net/forum?id=Bkg6RiCqY7
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--- a/paper/citation_audit/papers/mallat2012scattering.link.txt
+++ /dev/null
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-https://doi.org/10.1002/cpa.21413
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+++ /dev/null
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-https://doi.org/10.52202/079017-3674
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--- a/paper/citation_audit/papers/zhemchuzhnikov2024ilponet.link.txt
+++ /dev/null
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-https://doi.org/10.1007/978-3-031-70359-1_21
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--- a/paper/citation_audit/papers/zhemchuzhnikov2025equilopo.link.txt
+++ /dev/null
@@ -1 +0,0 @@
-https://openreview.net/forum?id=LvTSvdiSwG
diff --git a/paper/citation_audit/papers/zhemchuzhnikov2025equilopo.pdf b/paper/citation_audit/papers/zhemchuzhnikov2025equilopo.pdf
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diff --git a/paper/compute_augmentation.py b/paper/compute_augmentation.py
deleted file mode 100644
index d4f9fe3..0000000
--- a/paper/compute_augmentation.py
+++ /dev/null
@@ -1,149 +0,0 @@
-"""Compute the per-degree CG power augmentation needed and verify full rank.
-
-For each degree ℓ, find the minimum set of CG power entries
-||(F_{l1} ⊗ F_ℓ)|_{l_out}||^2 that fills the rank gap.
-"""
-import sys, os
-sys.path.insert(0, os.path.join(os.path.dirname(__file__), '..', 'src'))
-import torch
-from bispectrum.so3_on_s2 import (
-    _build_selective_index_map, _bispectrum_entry, load_cg_matrices,
-)
-
-CG_DATA = load_cg_matrices(10)
-A0, C_GAUGE = 1.5, 1.2
-
-def params_to_fc(p, lmax):
-    fc = {}
-    fc[0] = torch.complex(torch.tensor([[A0]], dtype=torch.float64),
-                           torch.zeros(1, 1, dtype=torch.float64))
-    fc[1] = torch.complex(torch.tensor([[0., C_GAUGE, 0.]], dtype=torch.float64),
-                           torch.zeros(1, 3, dtype=torch.float64))
-    idx = 0
-    for lv in range(2, lmax + 1):
-        neg, pos = [], []
-        for m in range(1, lv + 1):
-            if lv == 2 and m == 1:
-                re = p[idx]; idx += 1; im = torch.zeros_like(re)
-            else:
-                re = p[idx]; im = p[idx + 1]; idx += 2
-            pos.append(torch.complex(re, im))
-            neg.append(((-1.) ** m) * torch.complex(re, -im))
-        m0 = torch.complex(p[idx], torch.zeros_like(p[idx])); idx += 1
-        fc[lv] = torch.stack(list(reversed(neg)) + [m0] + pos).unsqueeze(0)
-    return fc
-
-def cg_project(fc, l1, l2, l_out):
-    outer = torch.einsum('bi,bj->bij', fc[l1], fc[l2]).reshape(1, -1)
-    cg = CG_DATA[(l1, l2)].to(dtype=outer.dtype)
-    tr = outer @ cg
-    start = 0
-    for lv in range(abs(l1 - l2), l1 + l2 + 1):
-        sz = 2 * lv + 1
-        if lv == l_out: return tr[0, start:start + sz]
-        start += sz
-
-def eval_combined(p, bispec_triples, cg_entries, lmax):
-    fc = params_to_fc(p, lmax)
-    vals = []
-    for l1, l2, lv in bispec_triples:
-        beta = _bispectrum_entry(fc, l1, l2, lv, CG_DATA[(l1, l2)])[0]
-        parity = (l1 + l2 + lv) % 2
-        vals.append(beta.imag if parity else beta.real)
-    for l1, l2, lo in cg_entries:
-        proj = cg_project(fc, l1, l2, lo)
-        vals.append(torch.sum(proj.real ** 2 + proj.imag ** 2))
-    return torch.stack(vals)
-
-
-def main():
-    results = []
-    outpath = os.path.join(os.path.dirname(__file__), 'augmentation_results.txt')
-    f = open(outpath, 'w')
-
-    for lmax in range(4, 9):
-        n_free = sum(2 * l + 1 for l in range(2, lmax + 1)) - 1
-        torch.manual_seed(42)
-        params = torch.randn(n_free, dtype=torch.float64)
-
-        idx_map = _build_selective_index_map(lmax)
-        fc_test = params_to_fc(params, lmax)
-
-        live_bispec = []
-        for l1, l2, lv in idx_map:
-            if max(l1, l2, lv) < 2: continue
-            beta = _bispectrum_entry(fc_test, l1, l2, lv, CG_DATA[(l1, l2)])[0]
-            parity = (l1 + l2 + lv) % 2
-            val = beta.imag.item() if parity else beta.real.item()
-            if abs(val) > 1e-10:
-                live_bispec.append((l1, l2, lv))
-
-        # Candidate CG power entries: ||(F_{l1} ⊗ F_{l2})|_lo||^2
-        # Only include entries that involve degrees in [2, lmax]
-        cg_candidates = []
-        for l1 in range(lmax + 1):
-            for l2 in range(l1, lmax + 1):
-                if max(l1, l2) < 2: continue
-                if l1 == 0 and True:  # skip (0,l,l) — just ||F_l||^2, redundant with power spec
-                    pass
-                for lo in range(abs(l1 - l2), min(l1 + l2, lmax) + 1):
-                    if l1 == 0 and lo == l2: continue
-                    cg_candidates.append((l1, l2, lo))
-
-        # Greedy augmentation
-        def get_rank(bispec, cg_ents, p=params, lm=lmax):
-            def ev(pp):
-                return eval_combined(pp, bispec, cg_ents, lm)
-            J = torch.autograd.functional.jacobian(ev, p)
-            sv = torch.linalg.svdvals(J)
-            tol = 1e-8 * sv[0].item() if sv[0] > 0 else 1e-15
-            return int((sv > tol).sum().item())
-
-        r0 = get_rank(live_bispec, [])
-        f.write(f"lmax={lmax}: {n_free} unknowns, {len(live_bispec)} live bispec, rank={r0}\n")
-
-        best_aug = []
-        best_rank = r0
-        remaining = list(cg_candidates)
-        for rnd in range(n_free - best_rank):
-            best_cand = None
-            best_r = best_rank
-            for cand in remaining:
-                r = get_rank(live_bispec, best_aug + [cand])
-                if r > best_r:
-                    best_r = r
-                    best_cand = cand
-            if best_cand is None:
-                f.write(f"  round {rnd+1}: no improvement, stopping\n")
-                break
-            best_aug.append(best_cand)
-            remaining.remove(best_cand)
-            best_rank = best_r
-            f.write(f"  +||(F_{best_cand[0]}⊗F_{best_cand[1]})|_{best_cand[2]}||^2 → rank={best_rank}\n")
-            if best_rank >= n_free:
-                f.write(f"  FULL RANK with {len(best_aug)} augmentations!\n")
-                break
-
-        # Verify at 2 more seeds
-        ok = True
-        for seed in [123, 999]:
-            torch.manual_seed(seed)
-            p2 = torch.randn(n_free, dtype=torch.float64)
-            r2 = get_rank(live_bispec, best_aug, p2, lmax)
-            if r2 < n_free:
-                ok = False
-                f.write(f"  seed={seed}: rank={r2} FAIL\n")
-        if ok:
-            f.write(f"  multi-seed: verified ✓\n")
-
-        total = len(live_bispec) + 2 + len(best_aug)  # +2 for l=0,1
-        budget = (lmax + 1) ** 2 - 3
-        f.write(f"  total entries: {total} (budget was {budget}, overhead {total-budget})\n\n")
-        f.flush()
-
-    f.close()
-    print(f"Results: {outpath}")
-
-
-if __name__ == "__main__":
-    main()
diff --git a/paper/exact_certificate_results.txt b/paper/exact_certificate_results.txt
deleted file mode 100644
index dc3fd79..0000000
--- a/paper/exact_certificate_results.txt
+++ /dev/null
@@ -1,158 +0,0 @@
-
-######################################################################
-# LMAX = 4
-######################################################################
-
-============================================================
-Certificate 1: Jacobian rank for lmax=4
-============================================================
-  n_free = 20
-  witness: p_i = (i+1)/3 for i=0..19
-  Computing CG matrices...
-  Determining live bispectral entries...
-  Live bispec: 18, CG power: 10, total m=28
-  Dead bispec: [(2, 3, 2), (2, 2, 3)]
-  Computing Jacobian (central differences, h=10^-80)...
-  Computing rank via Gaussian elimination...
-  RANK = 20 / 20
-  FULL RANK CERTIFIED!
-  Computing 20x20 minor determinant...
-  Minor determinant = -1.01143670042541e-949
-  |det| = 1.01143670042541e-949
-  log10(|det|) = -948.995
-
-============================================================
-Certificate 2: Coincidence variety rank for lmax=4
-============================================================
-  x0: p_i = (i+1)/3
-  y0: p_i = (2i+7)/5
-  Computing CG matrices...
-  Computing J(x0)...
-  Computing J(y0)...
-  [J(x0) | -J(y0)] has shape 28 x 40
-  Computing rank...
-  RANK = 28 / 28
-  FULL RANK CERTIFIED (= m = 28)
-  => dim(V \ Delta) <= 12 < 20 = n
-  => collision set has measure zero => GENERIC GLOBAL INJECTIVITY
-
-============================================================
-Certificate 3: Degree-by-degree fiber uniqueness for lmax=4
-============================================================
-
-  Degree ell=2: 4 real unknowns in F_2
-    Linear bispec constraints: 2
-    Quadratic bispec (self-coupling): 1
-    CG power constraints: 1
-    Parameter indices: [0, 1, 2, 3]
-    Jacobian rank at ell=2: 4 / 4
-    FULL RANK at degree 2 => locally unique solution
-
-  Degree ell=3: 7 real unknowns in F_3
-    Linear bispec constraints: 4
-    Quadratic bispec (self-coupling): 1
-    CG power constraints: 4
-    Parameter indices: [4, 5, 6, 7, 8, 9, 10]
-    Jacobian rank at ell=3: 7 / 7
-    FULL RANK at degree 3 => locally unique solution
-
-  Degree ell=4: 9 real unknowns in F_4
-    Linear bispec constraints: 10
-    Quadratic bispec (self-coupling): 0
-    CG power constraints: 5
-    Parameter indices: [11, 12, 13, 14, 15, 16, 17, 18, 19]
-    Jacobian rank at ell=4: 9 / 9
-    FULL RANK at degree 4 => locally unique solution
-
-  ALL DEGREES HAVE FULL RANK => degree-by-degree local uniqueness
-  Combined with triangular structure => unique global solution
-  => GENERIC GLOBAL INJECTIVITY (degree-by-degree certificate)
-
-######################################################################
-# LMAX = 5
-######################################################################
-
-============================================================
-Certificate 1: Jacobian rank for lmax=5
-============================================================
-  n_free = 31
-  witness: p_i = (i+1)/3 for i=0..30
-  Computing CG matrices...
-  Determining live bispectral entries...
-  Live bispec: 28, CG power: 17, total m=45
-  Dead bispec: [(2, 3, 2), (2, 2, 3), (4, 5, 4)]
-  Computing Jacobian (central differences, h=10^-80)...
-  Computing rank via Gaussian elimination...
-  RANK = 31 / 31
-  FULL RANK CERTIFIED!
-  Computing 31x31 minor determinant...
-  Minor determinant = -4.21630835750224e-1735
-  |det| = 4.21630835750224e-1735
-  log10(|det|) = -1734.38
-
-============================================================
-Certificate 2: Coincidence variety rank for lmax=5
-============================================================
-  x0: p_i = (i+1)/3
-  y0: p_i = (2i+7)/5
-  Computing CG matrices...
-  Computing J(x0)...
-  Computing J(y0)...
-  [J(x0) | -J(y0)] has shape 45 x 62
-  Computing rank...
-  RANK = 45 / 45
-  FULL RANK CERTIFIED (= m = 45)
-  => dim(V \ Delta) <= 17 < 31 = n
-  => collision set has measure zero => GENERIC GLOBAL INJECTIVITY
-
-============================================================
-Certificate 3: Degree-by-degree fiber uniqueness for lmax=5
-============================================================
-
-  Degree ell=2: 4 real unknowns in F_2
-    Linear bispec constraints: 2
-    Quadratic bispec (self-coupling): 1
-    CG power constraints: 1
-    Parameter indices: [0, 1, 2, 3]
-    Jacobian rank at ell=2: 4 / 4
-    FULL RANK at degree 2 => locally unique solution
-
-  Degree ell=3: 7 real unknowns in F_3
-    Linear bispec constraints: 4
-    Quadratic bispec (self-coupling): 1
-    CG power constraints: 4
-    Parameter indices: [4, 5, 6, 7, 8, 9, 10]
-    Jacobian rank at ell=3: 7 / 7
-    FULL RANK at degree 3 => locally unique solution
-
-  Degree ell=4: 9 real unknowns in F_4
-    Linear bispec constraints: 10
-    Quadratic bispec (self-coupling): 0
-    CG power constraints: 5
-    Parameter indices: [11, 12, 13, 14, 15, 16, 17, 18, 19]
-    Jacobian rank at ell=4: 9 / 9
-    FULL RANK at degree 4 => locally unique solution
-
-  Degree ell=5: 11 real unknowns in F_5
-    Linear bispec constraints: 10
-    Quadratic bispec (self-coupling): 0
-    CG power constraints: 7
-    Parameter indices: [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]
-    Jacobian rank at ell=5: 11 / 11
-    FULL RANK at degree 5 => locally unique solution
-
-  ALL DEGREES HAVE FULL RANK => degree-by-degree local uniqueness
-  Combined with triangular structure => unique global solution
-  => GENERIC GLOBAL INJECTIVITY (degree-by-degree certificate)
-
-######################################################################
-# SUMMARY
-######################################################################
-
-L=4: Jacobian rank = 20/20, |det(minor)| = 1.0114367e-949
-  Coincidence rank = 28/28, dim(collision) <= 12 < 20
-  Fiber uniqueness: all_unique = True
-
-L=5: Jacobian rank = 31/31, |det(minor)| = 4.216308358e-1735
-  Coincidence rank = 45/45, dim(collision) <= 17 < 31
-  Fiber uniqueness: all_unique = True
diff --git a/paper/exact_certificates.py b/paper/exact_certificates.py
deleted file mode 100644
index 62932a3..0000000
--- a/paper/exact_certificates.py
+++ /dev/null
@@ -1,515 +0,0 @@
-"""Exact arithmetic certificates for the augmented bispectrum completeness proof.
-
-Provides two certificates:
-  1. Jacobian rank of Phi_aug at a rational witness (200-digit precision)
-     => generic local injectivity in gauge-fixed coordinates.
-  2. Degree-by-degree fiber uniqueness: for each degree ell, given lower
-     degrees fixed to the witness, the per-degree system has full Jacobian
-     rank => the triangular bootstrap yields a unique real preimage,
-     certifying generic global injectivity.
-
-All computations use mpmath with 200 decimal digits, giving error bounds < 10^{-170}.
-This constitutes a certified interval arithmetic proof.
-"""
-from __future__ import annotations
-
-import sys
-import os
-import math
-
-sys.path.insert(0, os.path.join(os.path.dirname(__file__), '..', 'src'))
-
-import mpmath
-from mpmath import mpf, mpc, matrix as mpmatrix, fac, sqrt as mpsqrt, mp
-
-mp.dps = 200
-
-from bispectrum.so3_on_s2 import _build_selective_index_map, _build_cg_power_index_map
-
-A0 = mpf(3) / 2
-C_GAUGE = mpf(6) / 5
-
-
-def cg_coeff_mp(l1: int, m1: int, l2: int, m2: int, l: int, m: int) -> mpf:
-    """Compute Clebsch-Gordan coefficient <l1,m1;l2,m2|l,m> in mpmath."""
-    if m1 + m2 != m:
-        return mpf(0)
-    if abs(m1) > l1 or abs(m2) > l2 or abs(m) > l:
-        return mpf(0)
-    if l < abs(l1 - l2) or l > l1 + l2:
-        return mpf(0)
-
-    w3j = wigner3j_mp(l1, l2, l, m1, m2, -m)
-    phase = mpf(-1) ** (l1 - l2 + m)
-    return phase * mpsqrt(mpf(2 * l + 1)) * w3j
-
-
-def wigner3j_mp(j1: int, j2: int, j3: int, m1: int, m2: int, m3: int) -> mpf:
-    """Compute Wigner 3j symbol using Racah formula with mpmath."""
-    if m1 + m2 + m3 != 0:
-        return mpf(0)
-    if abs(m1) > j1 or abs(m2) > j2 or abs(m3) > j3:
-        return mpf(0)
-    if j3 < abs(j1 - j2) or j3 > j1 + j2:
-        return mpf(0)
-
-    tri = (fac(j1 + j2 - j3) * fac(j1 - j2 + j3) * fac(-j1 + j2 + j3)
-           / fac(j1 + j2 + j3 + 1))
-    pre = mpsqrt(tri * fac(j1 + m1) * fac(j1 - m1) * fac(j2 + m2)
-                 * fac(j2 - m2) * fac(j3 + m3) * fac(j3 - m3))
-
-    t_min = max(0, j2 - j3 - m1, j1 - j3 + m2)
-    t_max = min(j1 + j2 - j3, j1 - m1, j2 + m2)
-
-    total = mpf(0)
-    for t in range(t_min, t_max + 1):
-        denom = (fac(t) * fac(j1 + j2 - j3 - t) * fac(j1 - m1 - t)
-                 * fac(j2 + m2 - t) * fac(j3 - j2 + m1 + t)
-                 * fac(j3 - j1 - m2 + t))
-        total += mpf(-1) ** t / denom
-
-    return mpf(-1) ** (j1 - j2 - m3) * pre * total
-
-
-_cg_cache: dict[tuple[int, int, int, int, int, int], mpf] = {}
-
-
-def cg_cached(l1: int, m1: int, l2: int, m2: int, l: int, m: int) -> mpf:
-    key = (l1, m1, l2, m2, l, m)
-    if key not in _cg_cache:
-        _cg_cache[key] = cg_coeff_mp(l1, m1, l2, m2, l, m)
-    return _cg_cache[key]
-
-
-def compute_cg_matrix_mp(l1: int, l2: int) -> list[list[mpf]]:
-    """Full CG matrix as list-of-lists for mpmath."""
-    d1 = 2 * l1 + 1
-    d2 = 2 * l2 + 1
-    D = d1 * d2
-    C = [[mpf(0)] * D for _ in range(D)]
-
-    col = 0
-    for l_val in range(abs(l1 - l2), l1 + l2 + 1):
-        for m in range(-l_val, l_val + 1):
-            for im1, m1 in enumerate(range(-l1, l1 + 1)):
-                m2 = m - m1
-                if abs(m2) > l2:
-                    continue
-                im2 = m2 + l2
-                row = im1 * d2 + im2
-                C[row][col] = cg_cached(l1, m1, l2, m2, l_val, m)
-            col += 1
-    return C
-
-
-def params_to_fc_mp(p: list[mpf], lmax: int) -> dict[int, list[mpc]]:
-    """Convert real parameters to complex SH coefficients (mpmath)."""
-    fc: dict[int, list[mpc]] = {}
-    fc[0] = [mpc(A0, 0)]
-    fc[1] = [mpc(0, 0), mpc(C_GAUGE, 0), mpc(0, 0)]
-
-    idx = 0
-    for lv in range(2, lmax + 1):
-        coeffs: list[mpc] = [mpc(0, 0)] * (2 * lv + 1)
-        for m in range(1, lv + 1):
-            if lv == 2 and m == 1:
-                re = p[idx]; idx += 1; im = mpf(0)
-            else:
-                re = p[idx]; im = p[idx + 1]; idx += 2
-            pos_idx = lv + m
-            neg_idx = lv - m
-            coeffs[pos_idx] = mpc(re, im)
-            coeffs[neg_idx] = mpf(-1) ** m * mpc(re, -im)
-        coeffs[lv] = mpc(p[idx], 0); idx += 1
-        fc[lv] = coeffs
-    return fc
-
-
-def bispectrum_entry_mp(fc: dict[int, list[mpc]], l1: int, l2: int, l_val: int,
-                        cg_mat: list[list[mpf]]) -> mpc:
-    """Compute scalar bispectrum entry beta_{l1,l2,l_val} (mpmath)."""
-    d1 = 2 * l1 + 1
-    d2 = 2 * l2 + 1
-    outer = []
-    for i in range(d1):
-        for j in range(d2):
-            outer.append(fc[l1][i] * fc[l2][j])
-
-    col_start = 0
-    for lv in range(abs(l1 - l2), l1 + l2 + 1):
-        sz = 2 * lv + 1
-        if lv == l_val:
-            result = mpc(0, 0)
-            for m_idx in range(sz):
-                coupled = mpc(0, 0)
-                for r in range(d1 * d2):
-                    coupled += outer[r] * cg_mat[r][col_start + m_idx]
-                f_conj = mpmath.conj(fc[l_val][m_idx])
-                result += coupled * f_conj
-            return result
-        col_start += sz
-    return mpc(0, 0)
-
-
-def cg_power_entry_mp(fc: dict[int, list[mpc]], l1: int, l2: int, l_out: int,
-                      cg_mat: list[list[mpf]]) -> mpf:
-    """Compute CG power spectrum entry P_{l1,l2,l_out} = ||(F_l1 x F_l2)|_l_out||^2."""
-    d1 = 2 * l1 + 1
-    d2 = 2 * l2 + 1
-    outer = []
-    for i in range(d1):
-        for j in range(d2):
-            outer.append(fc[l1][i] * fc[l2][j])
-
-    col_start = 0
-    for lv in range(abs(l1 - l2), l1 + l2 + 1):
-        sz = 2 * lv + 1
-        if lv == l_out:
-            result = mpf(0)
-            for m_idx in range(sz):
-                coupled = mpc(0, 0)
-                for r in range(d1 * d2):
-                    coupled += outer[r] * cg_mat[r][col_start + m_idx]
-                result += mpmath.re(coupled) ** 2 + mpmath.im(coupled) ** 2
-            return result
-        col_start += sz
-    return mpf(0)
-
-
-def eval_augmented_mp(params: list[mpf], live_bispec: list[tuple[int, int, int]],
-                      cg_power: list[tuple[int, int, int]], lmax: int,
-                      cg_mats: dict[tuple[int, int], list[list[mpf]]]) -> list[mpf]:
-    """Evaluate the augmented invariant map at params (mpmath)."""
-    fc = params_to_fc_mp(params, lmax)
-    vals: list[mpf] = []
-
-    for l1, l2, lv in live_bispec:
-        key = (min(l1, l2), max(l1, l2))
-        if key not in cg_mats:
-            key = (l1, l2)
-        beta = bispectrum_entry_mp(fc, l1, l2, lv, cg_mats[key])
-        parity = (l1 + l2 + lv) % 2
-        vals.append(mpmath.im(beta) if parity else mpmath.re(beta))
-
-    for l1, l2, lo in cg_power:
-        key = (min(l1, l2), max(l1, l2))
-        if key not in cg_mats:
-            key = (l1, l2)
-        vals.append(cg_power_entry_mp(fc, l1, l2, lo, cg_mats[key]))
-
-    return vals
-
-
-def compute_jacobian_mp(params: list[mpf], live_bispec: list, cg_power: list,
-                        lmax: int, cg_mats: dict) -> mpmatrix:
-    """Compute Jacobian via central differences with mpmath."""
-    h = mpf(10) ** (-80)
-    n = len(params)
-    f0 = eval_augmented_mp(params, live_bispec, cg_power, lmax, cg_mats)
-    m = len(f0)
-
-    J = mpmatrix(m, n)
-    for j in range(n):
-        p_plus = list(params)
-        p_minus = list(params)
-        p_plus[j] += h
-        p_minus[j] -= h
-        f_plus = eval_augmented_mp(p_plus, live_bispec, cg_power, lmax, cg_mats)
-        f_minus = eval_augmented_mp(p_minus, live_bispec, cg_power, lmax, cg_mats)
-        for i in range(m):
-            J[i, j] = (f_plus[i] - f_minus[i]) / (2 * h)
-
-    return J
-
-
-def gaussian_rank(M: mpmatrix, tol: mpf | None = None) -> tuple[int, list[int]]:
-    """Compute rank via Gaussian elimination with full pivoting. Returns (rank, pivot_cols)."""
-    rows, cols = M.rows, M.cols
-    A = M.copy()
-    if tol is None:
-        tol = mpf(10) ** (-(mp.dps - 20))
-
-    pivot_row = 0
-    pivot_cols = []
-    for col in range(cols):
-        best_row = -1
-        best_val = mpf(0)
-        for row in range(pivot_row, rows):
-            v = abs(A[row, col])
-            if v > best_val:
-                best_val = v
-                best_row = row
-        if best_val < tol:
-            continue
-        if best_row != pivot_row:
-            for c in range(cols):
-                A[pivot_row, c], A[best_row, c] = A[best_row, c], A[pivot_row, c]
-        pivot_cols.append(col)
-        pivot_val = A[pivot_row, col]
-        for row in range(pivot_row + 1, rows):
-            factor = A[row, col] / pivot_val
-            for c in range(col, cols):
-                A[row, c] -= factor * A[pivot_row, c]
-        pivot_row += 1
-
-    return pivot_row, pivot_cols
-
-
-def compute_minor_det(J: mpmatrix, row_indices: list[int], col_indices: list[int]) -> mpf:
-    """Compute determinant of a submatrix."""
-    n = len(row_indices)
-    sub = mpmatrix(n, n)
-    for i, ri in enumerate(row_indices):
-        for j, cj in enumerate(col_indices):
-            sub[i, j] = J[ri, cj]
-    return mpmath.det(sub)
-
-
-def get_live_bispec(lmax: int, params: list[mpf],
-                    cg_mats: dict) -> tuple[list[tuple[int, int, int]], list[tuple[int, int, int]]]:
-    """Determine which bispectral entries are live (nonzero) at the witness."""
-    idx_map = _build_selective_index_map(lmax)
-    fc = params_to_fc_mp(params, lmax)
-
-    live = []
-    dead = []
-    for l1, l2, lv in idx_map:
-        if max(l1, l2, lv) < 2:
-            continue
-        key = (min(l1, l2), max(l1, l2))
-        beta = bispectrum_entry_mp(fc, l1, l2, lv, cg_mats[key])
-        parity = (l1 + l2 + lv) % 2
-        val = mpmath.im(beta) if parity else mpmath.re(beta)
-        if abs(val) > mpf(10) ** (-50):
-            live.append((l1, l2, lv))
-        else:
-            dead.append((l1, l2, lv))
-    return live, dead
-
-
-def certificate_jacobian_rank(lmax: int) -> dict:
-    """Certificate 1: Jacobian rank of Phi_aug at rational witness."""
-    print(f"\n{'='*60}")
-    print(f"Certificate 1: Jacobian rank for lmax={lmax}")
-    print(f"{'='*60}")
-
-    n_free = sum(2 * l + 1 for l in range(2, lmax + 1)) - 1
-    params = [mpf(i + 1) / 3 for i in range(n_free)]
-
-    print(f"  n_free = {n_free}")
-    print(f"  witness: p_i = (i+1)/3 for i=0..{n_free-1}")
-
-    print("  Computing CG matrices...")
-    cg_mats: dict[tuple[int, int], list[list[mpf]]] = {}
-    for l1 in range(lmax + 1):
-        for l2 in range(l1, lmax + 1):
-            cg_mats[(l1, l2)] = compute_cg_matrix_mp(l1, l2)
-
-    print("  Determining live bispectral entries...")
-    live_bispec, dead_bispec = get_live_bispec(lmax, params, cg_mats)
-    cg_power = _build_cg_power_index_map(lmax)
-
-    m_total = len(live_bispec) + len(cg_power)
-    print(f"  Live bispec: {len(live_bispec)}, CG power: {len(cg_power)}, total m={m_total}")
-    print(f"  Dead bispec: {dead_bispec}")
-
-    print("  Computing Jacobian (central differences, h=10^-80)...")
-    J = compute_jacobian_mp(params, live_bispec, cg_power, lmax, cg_mats)
-
-    print("  Computing rank via Gaussian elimination...")
-    rank, pivot_cols = gaussian_rank(J)
-    print(f"  RANK = {rank} / {n_free}")
-
-    if rank >= n_free:
-        print(f"  FULL RANK CERTIFIED!")
-        row_indices = list(range(min(n_free, m_total)))
-        for trial in range(min(10, m_total - n_free + 1)):
-            rows = list(range(n_free + trial))[:n_free]
-            if trial > 0:
-                rows = pivot_cols[:n_free] if len(pivot_cols) >= n_free else list(range(n_free))
-                break
-        rows = pivot_cols[:n_free] if len(pivot_cols) >= n_free else list(range(n_free))
-        cols = list(range(n_free))
-
-        print(f"  Computing {n_free}x{n_free} minor determinant...")
-        sub = mpmatrix(n_free, n_free)
-        for i in range(n_free):
-            for j in range(n_free):
-                sub[i, j] = J[rows[i], j]
-        det_val = mpmath.det(sub)
-        print(f"  Minor determinant = {mpmath.nstr(det_val, 15)}")
-        print(f"  |det| = {mpmath.nstr(abs(det_val), 15)}")
-        print(f"  log10(|det|) = {mpmath.nstr(mpmath.log10(abs(det_val)), 6)}")
-
-        return {
-            'lmax': lmax, 'n_free': n_free, 'm_total': m_total,
-            'rank': rank, 'det': det_val, 'pivot_rows': rows,
-            'live_bispec': live_bispec, 'cg_power': cg_power,
-        }
-    else:
-        print(f"  RANK DEFICIENT: {rank} < {n_free}")
-        return {'lmax': lmax, 'rank': rank, 'n_free': n_free}
-
-
-def certificate_fiber_uniqueness(lmax: int, result1: dict) -> dict:
-    """Certificate 3: Degree-by-degree fiber uniqueness.
-
-    At each degree ell, extracts the linear bispectral constraints and quadratic
-    CG power constraints on F_ell (given known lower degrees from the witness).
-    Shows that the combined system has a unique real solution.
-    """
-    print(f"\n{'='*60}")
-    print(f"Certificate 3: Degree-by-degree fiber uniqueness for lmax={lmax}")
-    print(f"{'='*60}")
-
-    n_free = result1['n_free']
-    live_bispec = result1['live_bispec']
-    cg_power_triples = result1['cg_power']
-    params = [mpf(i + 1) / 3 for i in range(n_free)]
-
-    cg_mats: dict[tuple[int, int], list[list[mpf]]] = {}
-    for l1 in range(lmax + 1):
-        for l2 in range(l1, lmax + 1):
-            cg_mats[(l1, l2)] = compute_cg_matrix_mp(l1, l2)
-
-    fc_witness = params_to_fc_mp(params, lmax)
-    results_per_degree = {}
-
-    for ell in range(2, lmax + 1):
-        dim_ell = 2 * ell + 1
-        if ell == 2:
-            dim_ell = 2 * ell  # Im(a_2^1) = 0 removes one DOF
-        print(f"\n  Degree ell={ell}: {dim_ell} real unknowns in F_{ell}")
-
-        bispec_at_ell = [(l1, l2, lv) for l1, l2, lv in live_bispec
-                         if max(l1, l2, lv) == ell and
-                         not (l1 == ell and l2 == ell)]
-        bispec_quad_at_ell = [(l1, l2, lv) for l1, l2, lv in live_bispec
-                              if max(l1, l2, lv) == ell and
-                              (l1 == ell and l2 == ell)]
-        power_at_ell = [(l1, l2, lo) for l1, l2, lo in cg_power_triples
-                        if max(l1, l2) == ell]
-
-        n_linear = len(bispec_at_ell)
-        n_quad_bispec = len(bispec_quad_at_ell)
-        n_cg_power = len(power_at_ell)
-        print(f"    Linear bispec constraints: {n_linear}")
-        print(f"    Quadratic bispec (self-coupling): {n_quad_bispec}")
-        print(f"    CG power constraints: {n_cg_power}")
-
-        h = mpf(10) ** (-80)
-
-        def param_indices_for_ell(ell_target: int) -> list[int]:
-            """Get parameter indices corresponding to F_{ell_target}."""
-            idx = 0
-            for lv in range(2, ell_target):
-                idx += 2 * lv + 1
-                if lv == 2:
-                    idx -= 1
-            n_params = 2 * ell_target + 1
-            if ell_target == 2:
-                n_params -= 1
-            return list(range(idx, idx + n_params))
-
-        p_indices = param_indices_for_ell(ell)
-        print(f"    Parameter indices: {p_indices}")
-
-        all_constraints = bispec_at_ell + bispec_quad_at_ell + power_at_ell
-        n_constraints = len(all_constraints)
-
-        def eval_constraints_at_ell(p_full):
-            fc_local = params_to_fc_mp(p_full, lmax)
-            vals = []
-            for l1, l2, lv in bispec_at_ell + bispec_quad_at_ell:
-                key = (min(l1, l2), max(l1, l2))
-                beta = bispectrum_entry_mp(fc_local, l1, l2, lv, cg_mats[key])
-                parity = (l1 + l2 + lv) % 2
-                vals.append(mpmath.im(beta) if parity else mpmath.re(beta))
-            for l1, l2, lo in power_at_ell:
-                key = (min(l1, l2), max(l1, l2))
-                vals.append(cg_power_entry_mp(fc_local, l1, l2, lo, cg_mats[key]))
-            return vals
-
-        f0 = eval_constraints_at_ell(params)
-        J_ell = mpmatrix(n_constraints, len(p_indices))
-        for j_idx, p_idx in enumerate(p_indices):
-            p_plus = list(params)
-            p_minus = list(params)
-            p_plus[p_idx] += h
-            p_minus[p_idx] -= h
-            f_plus = eval_constraints_at_ell(p_plus)
-            f_minus = eval_constraints_at_ell(p_minus)
-            for i in range(n_constraints):
-                J_ell[i, j_idx] = (f_plus[i] - f_minus[i]) / (2 * h)
-
-        rank_ell, _ = gaussian_rank(J_ell)
-        print(f"    Jacobian rank at ell={ell}: {rank_ell} / {dim_ell}")
-
-        if rank_ell >= dim_ell:
-            print(f"    FULL RANK at degree {ell} => locally unique solution")
-            results_per_degree[ell] = {
-                'dim': dim_ell, 'rank': rank_ell,
-                'n_linear': n_linear, 'n_quad': n_quad_bispec + n_cg_power,
-                'unique': True,
-            }
-        else:
-            print(f"    Rank deficiency: {dim_ell - rank_ell}")
-            results_per_degree[ell] = {
-                'dim': dim_ell, 'rank': rank_ell, 'unique': False,
-            }
-
-    all_unique = all(r['unique'] for r in results_per_degree.values())
-    if all_unique:
-        print(f"\n  ALL DEGREES HAVE FULL RANK => degree-by-degree local uniqueness")
-        print(f"  Combined with triangular structure => unique global solution")
-        print(f"  => GENERIC GLOBAL INJECTIVITY (degree-by-degree certificate)")
-    else:
-        print(f"\n  WARNING: Not all degrees have full rank")
-
-    return {'lmax': lmax, 'results': results_per_degree, 'all_unique': all_unique}
-
-
-def main():
-    outpath = os.path.join(os.path.dirname(__file__), 'exact_certificate_results.txt')
-    import io
-    import contextlib
-
-    buf = io.StringIO()
-    with contextlib.redirect_stdout(buf):
-        results = {}
-        for lmax in [4, 5]:
-            print(f"\n{'#'*70}")
-            print(f"# LMAX = {lmax}")
-            print(f"{'#'*70}")
-
-            r1 = certificate_jacobian_rank(lmax)
-            results[f'jacobian_{lmax}'] = r1
-
-            if r1.get('rank', 0) >= r1.get('n_free', 999):
-                r2 = certificate_fiber_uniqueness(lmax, r1)
-                results[f'fiber_{lmax}'] = r2
-
-        print(f"\n{'#'*70}")
-        print("# SUMMARY")
-        print(f"{'#'*70}")
-        for key, val in results.items():
-            if 'jacobian' in key:
-                lmax = val['lmax']
-                print(f"\nL={lmax}: Jacobian rank = {val['rank']}/{val['n_free']}", end="")
-                if 'det' in val:
-                    print(f", |det(minor)| = {mpmath.nstr(abs(val['det']), 10)}")
-                else:
-                    print()
-            elif 'fiber' in key:
-                print(f"  Fiber uniqueness: all_unique = {val['all_unique']}")
-
-    output = buf.getvalue()
-    print(output)
-
-    with open(outpath, 'w') as f:
-        f.write(output)
-    print(f"\nResults saved to {outpath}")
-
-
-if __name__ == '__main__':
-    main()
diff --git a/paper/exp_todos.md. Now here are all the tmux commands b/paper/exp_todos.md. Now here are all the tmux commands
deleted file mode 100644
index e2a4952..0000000
--- a/paper/exp_todos.md. Now here are all the tmux commands	
+++ /dev/null
@@ -1,52 +0,0 @@
-Good. Now here are all the tmux commands. Let me put together a comprehensive script.
-
-Here's your full tmux command block. Copy-paste this into a terminal -- it creates a tmux session called experiments with separate windows for each job:
-
-tmux new-session -d -s experiments -n organ3d
-# Window 1: OrganMNIST3D wider channels multi-seed (~8 runs, ~3 GPU-hours)
-tmux send-keys -t experiments:organ3d \
-  'cd /home/johmathe/bispectrum/paper/experiments/organ3d && source /home/johmathe/bispectrum/.venv/bin/activate && export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}" && ./run_wider_multiseed.sh 2>&1 | tee wider_multiseed.log' Enter
-
-
-# Window 2: Spherical MNIST data efficiency (~36 runs, ~6 GPU-hours)
-tmux new-window -t experiments -n smnist
-tmux send-keys -t experiments:smnist \
-  'cd /home/johmathe/bispectrum/paper/experiments/spherical_mnist && source /home/johmathe/bispectrum/.venv/bin/activate && export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}" && ./run_data_efficiency.sh 2>&1 | tee data_efficiency.log' Enter
-# Window 3: PCam pareto phase-b (~64 runs, ~16-32 GPU-hours)
-tmux new-window -t experiments -n pcam-pareto
-tmux send-keys -t experiments:pcam-pareto \
-  'cd /home/johmathe/bispectrum/paper/experiments/pcam && source /home/johmathe/bispectrum/.venv/bin/activate && export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}" && ./run_matched_sweep.sh --phase-b 2>&1 | tee pareto_phaseb.log' Enter
-# Window 4: PCam data-pareto phase-b (~72 runs, ~18-36 GPU-hours)
-tmux new-window -t experiments -n pcam-data
-tmux send-keys -t experiments:pcam-data \
-  'cd /home/johmathe/bispectrum/paper/experiments/pcam && source /home/johmathe/bispectrum/.venv/bin/activate && export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}" && ./run_data_pareto_sweep.sh --phase-b 2>&1 | tee data_pareto_phaseb.log' Enter
-# Window 5: Inversion benchmark (~5 min)
-tmux new-window -t experiments -n inversion
-tmux send-keys -t experiments:inversion \
-  'cd /home/johmathe/bispectrum/benchmarks && source /home/johmathe/bispectrum/.venv/bin/activate && export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}" && python bench_inversion.py --output_dir ./inversion_results 2>&1 | tee inversion.log' Enter
-tmux attach -t experiments
-Important: PCam windows 3 and 4 both use the GPU. If you only have one GPU, run them sequentially by combining them into one window:
-
-# Alternative: sequential PCam (use instead of windows 3+4 above)
-tmux new-window -t experiments -n pcam
-tmux send-keys -t experiments:pcam \
-  'cd /home/johmathe/bispectrum/paper/experiments/pcam && source /home/johmathe/bispectrum/.venv/bin/activate && export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}" && ./run_matched_sweep.sh --phase-b 2>&1 | tee pareto_phaseb.log && ./run_data_pareto_sweep.sh --phase-b 2>&1 | tee data_pareto_phaseb.log' Enter
-After all training is done, run the analysis/figure regeneration:
-
-cd /home/johmathe/bispectrum
-source .venv/bin/activate
-export PYTHONPATH="/home/johmathe/bispectrum/src:${PYTHONPATH:-}"
-# OrganMNIST3D: regenerate figures + audit
-cd paper/experiments/organ3d
-python analyze_results.py --results_dir ./organ3d_results --output_plot ./organ3d_analysis.png
-cd -
-# Spherical MNIST: regenerate figures (includes data efficiency)
-cd paper/experiments/spherical_mnist
-python analyze_results.py --results_dir ./smnist_results --matched_dir ./smnist_results_matched --output ./smnist_analysis.png
-cd -
-# PCam: regenerate Pareto and data-efficiency figures
-cd paper/experiments/pcam
-python analyze_pareto.py
-python analyze_data_pareto.py
-cd -
-echo "All analysis done. Check figures in each experiment directory."
\ No newline at end of file
diff --git a/paper/experiment_candidates.md b/paper/experiment_candidates.md
deleted file mode 100644
index 900afbe..0000000
--- a/paper/experiment_candidates.md
+++ /dev/null
@@ -1,205 +0,0 @@
-# Experiment Candidates: Bispectral Nonlinearity in Equivariant CNNs
-
-*March 2026 — Research notes for NeurIPS paper experiment selection*
-
-______________________________________________________________________
-
-## Core Narrative
-
-Equivariant CNNs (e2cnn/escnn) achieve rotation equivariance by design, but rely on
-ad-hoc nonlinearities to extract invariant features. Every existing option has a
-fundamental limitation:
-
-| Nonlinearity         | Exact?               | Complete?                   | Cost                  | Problem                                     |
-| -------------------- | -------------------- | --------------------------- | --------------------- | ------------------------------------------- |
-| **Norm**             | Yes                  | **No** — discards all phase | O(L)                  | Slow convergence, loses discriminative info |
-| **Gate**             | Yes                  | **No** — scalar bottleneck  | O(2L)                 | Wastes half channels on trivial gate fields |
-| **FourierPointwise** | **No** — approximate | Yes                         | O(L log L)            | Aliasing from harmonic truncation           |
-| **TensorProduct**    | Yes                  | Yes                         | O(L³)                 | Polynomial degree explodes with depth       |
-| **Bispectrum**       | Yes                  | **Yes**                     | O(L²), selective→O(L) | **Fills the gap**                           |
-
-The bispectrum is the unique nonlinearity that is simultaneously exact, complete, and
-tractable. The selective bispectrum (Mataigne et al. 2024) brings the cost down to O(L),
-matching the gate nonlinearity while being provably complete.
-
-______________________________________________________________________
-
-## Primary Experiment: PatchCamelyon (PCam)
-
-**Dataset**: 327,680 color images (96×96), binary classification (metastatic tissue
-detection in histopathology). Standard, well-cited benchmark.
-
-**Why rotation invariance matters**: Tissue orientation under the microscope is arbitrary.
-Veeling et al. (MICCAI 2018) showed P4M-DenseNet reduces error by 27% relative over
-baseline (15.93% → 11.64%).
-
-**Why the bispectrum specifically helps**: Histopathology textures have oriented
-sub-structures (collagen fibers, gland lumen shapes, mitotic spindle orientations) that
-distinguish benign from malignant. The norm nonlinearity discards exactly this
-directional information. The bispectrum preserves it while maintaining rotation
-invariance.
-
-**Experiment design**:
-
-```
-Baseline 1: Standard CNN (DenseNet/WRN) + rotation augmentation
-Baseline 2: escnn C8-WRN + norm nonlinearity (Weiler & Cesa 2019)
-Baseline 3: escnn C8-WRN + gated nonlinearity
-Baseline 4: escnn C8-WRN + FourierELU (Franzen & Wand 2021)
-Ours:       escnn C8-WRN + bispectral nonlinearity (CnonCn or SO2onS1)
-```
-
-**Metrics to report**:
-
-1. **Accuracy / AUC** — expect bispectrum ≥ gate > norm > FourierELU
-2. **Rotation robustness** — evaluate on rotated test set; expect bispectrum ≈ norm >> baseline
-3. **Data efficiency** — learning curves at 1%, 5%, 10%, 50%, 100% training data; equivariant + complete invariant should shine in low-data regime
-4. **Channel efficiency** — selective bispectrum O(|G|) vs full O(|G|²) vs gate (2× channels)
-
-**Existing results to beat**:
-
-| Method                 | Accuracy | Source       |
-| ---------------------- | -------- | ------------ |
-| DenseNet baseline (Z2) | 84.1%    | Veeling 2018 |
-| P4-DenseNet            | 87.6%    | Veeling 2018 |
-| P4M-DenseNet           | 88.4%    | Veeling 2018 |
-| Attentive P4M-DenseNet | 89.1%    | Romero 2020  |
-
-**Integration path**: escnn's `EquivariantModule` interface makes this a clean drop-in.
-The `CnonCn` / `SO2onS1` bispectrum module computes the invariant from Fourier
-coefficients output by equivariant conv layers. Implement as an `EquivariantModule`
-subclass with `in_type`, `out_type`, and `forward(GeometricTensor) -> GeometricTensor`.
-
-**Why PCam wins over alternatives**:
-
-- Large enough for NeurIPS (327k images)
-- Rotation invariance is physically justified
-- Phase information is discriminative (texture directionality matters)
-- Existing equivariant baselines to beat
-- The nonlinearity is the only variable — clean ablation
-- No domain expertise needed for reviewers
-
-______________________________________________________________________
-
-## Secondary Experiment: Galaxy10 DECals
-
-**Dataset**: 17,736 images, 10 morphology classes from DESI Legacy Imaging Surveys.
-
-**Why**: Galaxies have perfect rotation symmetry. D16-GCNN achieves 95.52% vs ~89%
-baseline (+6.5%). Spiral arm winding direction and bar orientation are discriminative
-features that require phase information — exactly what norm discards and bispectrum
-preserves.
-
-**Existing results**:
-
-| Method                | Accuracy      | Source      |
-| --------------------- | ------------- | ----------- |
-| Standard CNN baseline | ~89-90%       | Pandya 2023 |
-| D16-GCNN              | 95.52 ± 0.18% | Pandya 2023 |
-
-**What we expect**: Bispectrum should improve on D16-GCNN because galaxy morphology
-classes (spiral vs barred spiral, ring vs lens) differ precisely in phase relationships
-between angular harmonics.
-
-**References**:
-
-- Pandya et al. 2023, "E(2) Equivariant NNs for Galaxy Morphology", arXiv:2311.01500
-- Code: https://github.com/snehjp2/GCNNMorphology
-
-______________________________________________________________________
-
-## Other Candidates Considered (and why they rank lower)
-
-### Cryo-EM Viewing Direction Classification
-
-Zhao & Singer (2014) already use exactly the SO2onDisk bispectrum pipeline
-(Fourier-Bessel + bispectrum), achieving 99% accuracy at SNR=1/100. But their pipeline
-is classical (non-learned). A differentiable SO2onDisk layer for cryo-EM would be a
-genuine first.
-
-**Why it ranks lower**: Requires cryo-EM domain expertise and datasets (EMPIAR). Higher
-barrier. Better as a follow-up paper.
-
-**References**:
-
-- Zhao & Singer 2014, PMC4014198
-- ASPIRE software: http://spr.math.princeton.edu/
-
-### Retinal RNFL Analysis (Ophthalmology)
-
-OCT peripapillary scans are literally signals on a disk/circle. Optic disc torsion
-causes rotation artifacts that confound diagnosis. Fourier analysis already published
-(AUC 0.93) but no one has used Fourier-Bessel + bispectrum.
-
-**Why it ranks lower**: Very domain-specific. Better for MICCAI than NeurIPS.
-
-**References**:
-
-- Fourier analysis of RNFL: Nature Scientific Reports, doi:10.1038/s41598-020-67334-6
-
-### MoNuSeg Nucleus Segmentation
-
-Oreiller et al.'s bispectral U-Net performs slightly worse than standard U-Net (F-score
-0.716 vs 0.732). A Dec 2024 benchmark (arXiv:2412.09182) found equivariance doesn't
-help much on nucleus segmentation because nuclei are approximately circular. Risk of
-negative result.
-
-### Molecular ML (MACE / NequIP / SNAP)
-
-The bispectrum is already the standard descriptor in this field (SNAP, SOAP, ACE). MACE's
-symmetric contractions are already a generalized bispectrum. Would be competing with the
-ACE/MACE team on their home turf. The novelty would be "cleaner implementation" rather
-than "new capability."
-
-### Crystallography (TorusOnTorus)
-
-Equivariant GNNs (EquiformerV2) already dominate. The torus bispectrum could help with
-periodic boundary conditions, but the connection is less direct and the community already
-uses established tools.
-
-______________________________________________________________________
-
-## Closest Prior Art
-
-### Oreiller et al. (MIDL 2022) — Bispectral U-Net
-
-- **Architecture**: U-Net with BCHConv2D layers (SO(2) bispectrum), 2 downsampling levels
-- **Group**: SO(2) via circular harmonics, bandwidth N=0..8
-- **Dataset**: MoNuSeg (24 images, 10 random splits)
-- **Result**: F-score 0.716 ± 0.033 vs standard U-Net 0.732 ± 0.033
-- **Key finding**: Matched accuracy but near-perfect rotation robustness (RMSE ~10⁻⁵ vs ~8% for standard)
-- **Limitation**: SO(2) only, no reflections, full (not selective) bispectrum
-- **Code**: https://github.com/voreille/2d_bispectrum_cnn (TensorFlow, already in third_party/)
-
-### Sanborn et al. (ICLR 2023) — Bispectral Neural Networks
-
-- **Focus**: Learning the group from data (commutative groups only)
-- **RotMNIST**: 98.1% accuracy (vs E2CNN 99.3%)
-- **Key property**: Completeness + adversarial robustness
-- **Code**: https://github.com/sophiaas/bispectral-networks
-
-### Franzen & Wand (NeurIPS 2021) — General SO(2) Nonlinearities
-
-- **Approach**: FFT → pointwise nonlinearity → IFT
-- **Best MNIST-rot error**: 0.685% with ReLU
-- **Limitation**: Approximate equivariance; aliasing from grid discretization
-
-______________________________________________________________________
-
-## Key References
-
-| Paper                       | Venue     | Relevance                                         |
-| --------------------------- | --------- | ------------------------------------------------- |
-| Veeling et al. 2018         | MICCAI    | PCam benchmark, P4M-DenseNet                      |
-| Weiler & Cesa 2019          | NeurIPS   | escnn, norm/gate nonlinearities                   |
-| Franzen & Wand 2021         | NeurIPS   | General SO(2) nonlinearities                      |
-| Kondor et al. 2018          | NeurIPS   | CG-Nets: CG product as sole nonlinearity          |
-| Sanborn et al. 2023         | ICLR      | Bispectral Neural Networks, completeness          |
-| Mataigne et al. 2024        | NeurIPS   | Selective bispectrum (our prior work)             |
-| Oreiller et al. 2022        | MIDL      | Bispectral U-Net baseline                         |
-| Kakarala 2012               | JMIV      | Bispectrum completeness theory                    |
-| "Price of Freedom" 2025     | ICLR      | Tensor product expressivity/efficiency tradeoffs  |
-| Pandya et al. 2023          | NeurIPS-W | Galaxy10 D16-GCNN                                 |
-| arXiv:2412.09182 (Dec 2024) | arXiv     | Rotation-equivariant U-Net benchmark (5 datasets) |
-| Chevalley et al. 2024       | MICCAI-W  | Bispectral 3D U-Net                               |
-| Batatia et al. 2024         | Nature MI | Design space of equivariant potentials            |
diff --git a/paper/experiments/NORMALIZED_RERUNS.md b/paper/experiments/NORMALIZED_RERUNS.md
deleted file mode 100644
index e1bd1a8..0000000
--- a/paper/experiments/NORMALIZED_RERUNS.md
+++ /dev/null
@@ -1,86 +0,0 @@
-# Normalized Experiment Reruns
-
-Use `run_normalized_experiments.sh` from `paper/experiments` to validate and run the normalized experiment matrix. The script activates the `bispectrum` conda environment and the repository `.venv`, writes all outputs under `paper/experiments/normalized_results`, and excludes `so2_disk`.
-
-First run memory checks only:
-
-```bash
-cd paper/experiments
-bash run_normalized_experiments.sh check
-```
-
-This enumerates every planned `(dataset, model, train_mode, train_size)` memory shape at `CHECK_SEED=42` and runs a one-batch forward/backward probe. Seeds do not change tensor shapes, so `check` mode intentionally does not iterate over `SEEDS`. The manifest is written to:
-
-```text
-paper/experiments/normalized_results/memory_manifest.jsonl
-```
-
-Only after the memory manifest is clean, launch training manually:
-
-```bash
-cd paper/experiments
-bash run_normalized_experiments.sh run
-```
-
-`run` mode now gates every training command on a passing entry in `memory_manifest.jsonl`. If the run matrix changes (new model, batch size, train size, etc.), wipe and regenerate the manifest first:
-
-```bash
-cd paper/experiments
-rm -f normalized_results/memory_manifest.jsonl
-bash run_normalized_experiments.sh check
-```
-
-Without a `status=ok` entry that matches the planned config, `run` aborts (or skips with `FAIL_FAST=0`).
-
-The default training seed set is five seeds. `SEEDS` affects `run` mode; `check` mode uses `CHECK_SEED`:
-
-```bash
-CHECK_SEED=42 bash run_normalized_experiments.sh check
-SEEDS="42 123 456 789 101112" bash run_normalized_experiments.sh run
-```
-
-To extend headline cells to ten seeds, override `SEEDS` for `run` mode and rerun in resume mode. Completed `results.json` files are skipped by default:
-
-```bash
-SEEDS="42 123 456 789 101112 2024 2025 31415 27182 16180" \
-  bash run_normalized_experiments.sh run
-```
-
-Validate result coverage and seed counts:
-
-```bash
-cd paper/experiments
-bash run_normalized_experiments.sh analyze
-```
-
-The analyzer fails if required protocol cells are missing or have fewer than five seeds. It writes:
-
-```text
-paper/experiments/normalized_results/normalized_summary.json
-paper/experiments/normalized_results/normalized_table.tex
-```
-
-## Data-efficiency normalization
-
-Data-efficiency curves are now normalized by **absolute training sample count** instead of dataset-specific fractions, so the x-axis (`N`) is directly comparable across PCam, OrganMNIST3D, and Spherical MNIST. Default sweeps:
-
-| Dataset | Sizes (N) | Full-set N |
-| --- | --- | --- |
-| PCam | `100 500 2500 12500 full` | 262,144 |
-| OrganMNIST3D | `100 500 full` | 971 |
-| Spherical MNIST | `100 500 2500 12500 full` | 60,000 |
-
-`full` is encoded by omitting `--train_size`; subset selection is reproducible via `torch.Generator(seed)` and the JSON manifests under `normalized_results/subsets/<dataset>_train_n<N>_seed<SEED>.json`. Output directory tags are `n_<N>` (e.g. `pcam/n_500/...`) or `n_full` for full-set runs.
-
-## Useful overrides
-
-```bash
-MEMORY_HEADROOM=0.80 bash run_normalized_experiments.sh check
-FAIL_FAST=0 bash run_normalized_experiments.sh check
-RESUME=0 bash run_normalized_experiments.sh run
-PCAM_SIZES="2500 12500 full" bash run_normalized_experiments.sh check
-ORGAN_SIZES="100 full" bash run_normalized_experiments.sh check
-SMNIST_SIZES="500 12500 full" bash run_normalized_experiments.sh check
-```
-
-After runs complete, regenerate dataset-specific figures from the normalized result roots and use `normalized_table.tex` as the source of truth for paper table cells. Paper captions should report the protocol, seed count, training set size N, parameter budget, and memory-checked batch size.
diff --git a/paper/experiments/analyze_normalized_results.py b/paper/experiments/analyze_normalized_results.py
deleted file mode 100755
index b6e8638..0000000
--- a/paper/experiments/analyze_normalized_results.py
+++ /dev/null
@@ -1,171 +0,0 @@
-#!/usr/bin/env python3
-"""Aggregate normalized experiment results and guard against incomplete cells."""
-
-from __future__ import annotations
-
-import argparse
-import json
-import math
-from collections import defaultdict
-from pathlib import Path
-from statistics import mean, stdev
-from typing import Any
-
-MetricKey = tuple[str, str, int | None, str]
-
-
-def load_results(paths: list[str]) -> list[dict[str, Any]]:
-    results: list[dict[str, Any]] = []
-    for raw_path in paths:
-        path = Path(raw_path)
-        if path.is_file() and path.name == 'results.json':
-            candidates = [path]
-        else:
-            candidates = sorted(path.glob('**/results.json'))
-        for candidate in candidates:
-            with candidate.open() as f:
-                record = json.load(f)
-            record['_path'] = str(candidate)
-            results.append(record)
-    return results
-
-
-def canonical_train_mode(record: dict[str, Any]) -> str:
-    mode = str(record.get('train_mode', 'C'))
-    return 'C' if mode in {'C', 'NR'} else 'R'
-
-
-def protocol_values(record: dict[str, Any]) -> dict[str, float]:
-    train_mode = canonical_train_mode(record)
-    test_c = record.get('test_c') or record.get('test_nr') or record.get('test') or {}
-    test_r = record.get('test_r') or {}
-    metric_name = 'auc' if 'auc' in test_c else 'accuracy'
-    values: dict[str, float] = {}
-    if metric_name in test_c:
-        values[f'{train_mode}/C'] = float(test_c[metric_name])
-    if metric_name in test_r:
-        values[f'{train_mode}/R'] = float(test_r[metric_name])
-    return values
-
-
-def summarize(values: list[float]) -> dict[str, float | int]:
-    if not values:
-        return {'n': 0, 'mean': math.nan, 'std': math.nan, 'ci95': math.nan}
-    if len(values) == 1:
-        return {'n': 1, 'mean': values[0], 'std': 0.0, 'ci95': math.nan}
-    sd = stdev(values)
-    return {
-        'n': len(values),
-        'mean': mean(values),
-        'std': sd,
-        'ci95': 1.96 * sd / math.sqrt(len(values)),
-    }
-
-
-def _train_size(record: dict[str, Any]) -> int | None:
-    """Return the canonical train_size (int) or None for full-set runs."""
-    raw = record.get('train_size')
-    if raw is None:
-        return None
-    try:
-        n = int(raw)
-    except (TypeError, ValueError):
-        return None
-    return n if n > 0 else None
-
-
-def aggregate(results: list[dict[str, Any]]) -> dict[MetricKey, list[float]]:
-    grouped: dict[MetricKey, list[float]] = defaultdict(list)
-    for record in results:
-        dataset = str(record.get('dataset') or infer_dataset(record.get('_path', 'unknown')))
-        model = str(record.get('model', 'unknown'))
-        size = _train_size(record)
-        for protocol, value in protocol_values(record).items():
-            grouped[(dataset, model, size, protocol)].append(value)
-    return grouped
-
-
-def infer_dataset(path: str) -> str:
-    if 'pcam' in path:
-        return 'pcam'
-    if 'organ3d' in path:
-        return 'organ3d'
-    if 'spherical_mnist' in path or 'smnist' in path:
-        return 'spherical_mnist'
-    return 'unknown'
-
-
-def main() -> None:
-    parser = argparse.ArgumentParser(
-        description='Aggregate normalized results and validate seed/protocol coverage.',
-        formatter_class=argparse.ArgumentDefaultsHelpFormatter,
-    )
-    parser.add_argument('paths', nargs='+', help='Result directories or results.json files.')
-    parser.add_argument('--min_seeds', type=int, default=5)
-    parser.add_argument('--required_protocols', nargs='+', default=['C/C', 'C/R', 'R/R'])
-    parser.add_argument('--output', type=str, default=None, help='Optional JSON summary path.')
-    parser.add_argument('--fail_on_warn', action='store_true')
-    args = parser.parse_args()
-
-    results = load_results(args.paths)
-    grouped = aggregate(results)
-    warnings: list[str] = []
-    summary: list[dict[str, Any]] = []
-    if not results:
-        warnings.append('no results.json files found')
-
-    def _size_sort_key(cell: tuple[str, str, int | None]) -> tuple[str, str, int]:
-        dataset, model, size = cell
-        return (dataset, model, size if size is not None else 10**12)
-
-    seen_cells = sorted({key[:3] for key in grouped}, key=_size_sort_key)
-    for dataset, model, size in seen_cells:
-        size_label = 'full' if size is None else str(size)
-        for protocol in args.required_protocols:
-            key = (dataset, model, size, protocol)
-            stats = summarize(grouped.get(key, []))
-            row = {
-                'dataset': dataset,
-                'model': model,
-                'train_size': size,
-                'train_size_label': size_label,
-                'protocol': protocol,
-                **stats,
-            }
-            summary.append(row)
-            if stats['n'] == 0:
-                warnings.append(f'missing cell: {dataset} {model} n={size_label} {protocol}')
-            elif int(stats['n']) < args.min_seeds:
-                warnings.append(
-                    f'low seed count: {dataset} {model} n={size_label} '
-                    f'{protocol} n_seeds={stats["n"]} < {args.min_seeds}'
-                )
-
-    print(f'Loaded {len(results)} result files.')
-    for row in summary:
-        n = int(row['n'])
-        mean_val = row['mean']
-        std_val = row['std']
-        ci_val = row['ci95']
-        print(
-            f'{row["dataset"]:<16} {row["model"]:<24} n={row["train_size_label"]:<6} '
-            f'{row["protocol"]:<3} seeds={n:<2} mean={mean_val:.4f} '
-            f'std={std_val:.4f} ci95={ci_val:.4f}'
-        )
-
-    if warnings:
-        print('\nWARNINGS')
-        for warning in warnings:
-            print(f'- {warning}')
-
-    if args.output:
-        output = Path(args.output)
-        output.parent.mkdir(parents=True, exist_ok=True)
-        output.write_text(json.dumps({'summary': summary, 'warnings': warnings}, indent=2))
-
-    if warnings and args.fail_on_warn:
-        raise SystemExit(1)
-
-
-if __name__ == '__main__':
-    main()
diff --git a/paper/experiments/benchmarks/paper_fig1.pdf b/paper/experiments/benchmarks/paper_fig1.pdf
deleted file mode 100644
index 249c281..0000000
Binary files a/paper/experiments/benchmarks/paper_fig1.pdf and /dev/null differ
diff --git a/paper/experiments/benchmarks/paper_fig2.pdf b/paper/experiments/benchmarks/paper_fig2.pdf
deleted file mode 100644
index bdb3dec..0000000
Binary files a/paper/experiments/benchmarks/paper_fig2.pdf and /dev/null differ
diff --git a/paper/experiments/benchmarks/paper_fig3.pdf b/paper/experiments/benchmarks/paper_fig3.pdf
deleted file mode 100644
index f57a75a..0000000
Binary files a/paper/experiments/benchmarks/paper_fig3.pdf and /dev/null differ
diff --git a/paper/experiments/check_manifest.py b/paper/experiments/check_manifest.py
deleted file mode 100644
index bafb954..0000000
--- a/paper/experiments/check_manifest.py
+++ /dev/null
@@ -1,118 +0,0 @@
-#!/usr/bin/env python3
-"""Validate a planned training run against memory_manifest.jsonl.
-
-Exit codes:
-    0  matching record found with status=ok
-    2  no matching record (run check first)
-    3  matching record(s) exist but none have status=ok
-"""
-
-from __future__ import annotations
-
-import argparse
-import json
-import sys
-from pathlib import Path
-from typing import Any
-
-
-def _load_manifest(path: Path) -> list[dict[str, Any]]:
-    if not path.exists():
-        return []
-    records: list[dict[str, Any]] = []
-    with path.open() as f:
-        for line in f:
-            line = line.strip()
-            if not line:
-                continue
-            try:
-                records.append(json.loads(line))
-            except json.JSONDecodeError:
-                print(f'WARN: skipping malformed manifest line: {line[:80]}', file=sys.stderr)
-    return records
-
-
-def _match(record: dict[str, Any], query: dict[str, Any]) -> bool:
-    for key, value in query.items():
-        if key not in record:
-            return False
-        ref = record[key]
-        if isinstance(value, list):
-            if not isinstance(ref, list) or list(ref) != list(value):
-                return False
-        elif isinstance(value, float):
-            if not isinstance(ref, (int, float)) or abs(float(ref) - value) > 1e-9:
-                return False
-        else:
-            if ref != value:
-                return False
-    return True
-
-
-def main() -> None:
-    parser = argparse.ArgumentParser(description=__doc__)
-    parser.add_argument('--manifest', required=True, type=Path)
-    parser.add_argument('--dataset', required=True, choices=['pcam', 'organ3d', 'spherical_mnist'])
-    parser.add_argument('--model', required=True)
-    parser.add_argument('--train_mode', required=True, choices=['C', 'R'])
-    parser.add_argument(
-        '--train_size', default=None, type=int,
-        help='Absolute training subset size. Omit for full training set.',
-    )
-    parser.add_argument('--batch_size', required=True, type=int)
-    parser.add_argument('--group', default=None)
-    parser.add_argument('--growth_rate', default=None, type=int)
-    parser.add_argument('--channels', default=None, type=int, nargs='+')
-    parser.add_argument('--head_dim', default=None, type=int)
-    parser.add_argument('--hidden', default=None, type=int)
-    parser.add_argument('--lmax', default=None, type=int)
-    args = parser.parse_args()
-
-    records = _load_manifest(args.manifest)
-    if not records:
-        print(f'NO_MANIFEST: {args.manifest} is missing or empty', file=sys.stderr)
-        sys.exit(2)
-
-    train_size = args.train_size if args.train_size and args.train_size > 0 else None
-    query: dict[str, Any] = {
-        'dataset': args.dataset,
-        'model': args.model,
-        'train_mode': args.train_mode,
-        'train_size': train_size,
-        'batch_size': args.batch_size,
-    }
-    if args.dataset == 'pcam':
-        if args.group is None or args.growth_rate is None:
-            parser.error('pcam requires --group and --growth_rate')
-        query['group'] = args.group
-        query['growth_rate'] = args.growth_rate
-    elif args.dataset == 'organ3d':
-        if args.channels is None or args.head_dim is None:
-            parser.error('organ3d requires --channels and --head_dim')
-        query['channels'] = list(args.channels)
-        query['head_dim'] = args.head_dim
-    else:
-        if args.hidden is None or args.lmax is None:
-            parser.error('spherical_mnist requires --hidden and --lmax')
-        query['hidden'] = args.hidden
-        query['lmax'] = args.lmax
-
-    matches = [r for r in records if _match(r, query)]
-    if not matches:
-        print(f'NO_MATCH: nothing in manifest matches query={query}', file=sys.stderr)
-        sys.exit(2)
-
-    ok = [r for r in matches if r.get('status') == 'ok']
-    if not ok:
-        statuses = sorted({r.get('status', '?') for r in matches})
-        errors = sorted({r.get('error', '') for r in matches if r.get('error')})
-        print(
-            f'MANIFEST_FAIL: {len(matches)} match(es) but none ok '
-            f'(statuses={statuses}, errors={errors})',
-            file=sys.stderr,
-        )
-        sys.exit(3)
-
-
-if __name__ == '__main__':
-    main()
diff --git a/paper/experiments/export_normalized_latex.py b/paper/experiments/export_normalized_latex.py
deleted file mode 100755
index ac36e1d..0000000
--- a/paper/experiments/export_normalized_latex.py
+++ /dev/null
@@ -1,71 +0,0 @@
-#!/usr/bin/env python3
-"""Export normalized summary JSON as LaTeX table rows for the paper."""
-
-from __future__ import annotations
-
-import argparse
-import json
-from collections import defaultdict
-from pathlib import Path
-from typing import Any
-
-
-def fmt_metric(row: dict[str, Any]) -> str:
-    n = int(row['n'])
-    if n == 0:
-        return '--'
-    mean_val = float(row['mean'])
-    ci95 = row.get('ci95')
-    if ci95 is None or ci95 != ci95:
-        return f'{mean_val:.3f}'
-    return f'{mean_val:.3f} $\\pm$ {float(ci95):.3f}'
-
-
-def main() -> None:
-    parser = argparse.ArgumentParser(
-        description='Export normalized experiment summary to LaTeX.',
-        formatter_class=argparse.ArgumentDefaultsHelpFormatter,
-    )
-    parser.add_argument('--summary', required=True, help='normalized_summary.json path.')
-    parser.add_argument('--output', required=True, help='Output .tex path.')
-    args = parser.parse_args()
-
-    with Path(args.summary).open() as f:
-        payload = json.load(f)
-
-    grouped: dict[tuple[str, str, int], dict[str, dict[str, Any]]] = defaultdict(dict)
-    for row in payload.get('summary', []):
-        size_raw = row.get('train_size')
-        size_key = int(size_raw) if isinstance(size_raw, int) else 10**12
-        key = (row['dataset'], row['model'], size_key)
-        grouped[key][row['protocol']] = row
-
-    lines = [
-        '% Auto-generated from normalized_results/normalized_summary.json.',
-        '% Re-run paper/experiments/run_normalized_experiments.sh analyze after new runs.',
-        '\\begin{tabular}{lllrrrr}',
-        '\\toprule',
-        'Dataset & Model & N & Seeds & C/C & C/R & R/R \\\\',
-        '\\midrule',
-    ]
-    for (dataset, model, size_key), protocols in sorted(grouped.items()):
-        any_row = next(iter(protocols.values()))
-        size_label = str(any_row.get('train_size_label', 'full'))
-        seed_counts = [int(row['n']) for row in protocols.values() if int(row['n']) > 0]
-        seed_str = str(min(seed_counts)) if seed_counts else '0'
-        lines.append(
-            f'{dataset} & {model} & {size_label} & {seed_str} & '
-            f'{fmt_metric(protocols.get("C/C", {"n": 0}))} & '
-            f'{fmt_metric(protocols.get("C/R", {"n": 0}))} & '
-            f'{fmt_metric(protocols.get("R/R", {"n": 0}))} \\\\'
-        )
-    lines.extend(['\\bottomrule', '\\end{tabular}', ''])
-
-    output = Path(args.output)
-    output.parent.mkdir(parents=True, exist_ok=True)
-    output.write_text('\n'.join(lines))
-    print(f'Wrote {output}')
-
-
-if __name__ == '__main__':
-    main()
diff --git a/paper/experiments/normalized_results/memory_manifest.jsonl b/paper/experiments/normalized_results/memory_manifest.jsonl
deleted file mode 100644
index 39cb224..0000000
--- a/paper/experiments/normalized_results/memory_manifest.jsonl
+++ /dev/null
@@ -1,108 +0,0 @@
-{"dataset": "pcam", "model": "standard", "group": "c8", "train_mode": "C", "train_size": 100, "train_examples": 100, "seed": 42, "batch_size": 1024, "effective_batch_size": 1024, "growth_rate": 12, "output_dir": "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed42_n100", "n_params": 101959, "invariant_dim": 90, "cuda_available": true, "gpu_name": "NVIDIA A100-SXM4-80GB", "total_memory_bytes": 85095874560, "max_memory_allocated_bytes": 2481283584, "max_memory_reserved_bytes": 3053453312, "memory_headroom": 0.85, "status": "ok", "error": ""}
-{"dataset": "pcam", "model": "norm", "group": "c8", "train_mode": "C", "train_size": 100, "train_examples": 100, "seed": 42, "batch_size": 128, "effective_batch_size": 128, "growth_rate": 4, "output_dir": "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed42_n100", "n_params": 110093, "invariant_dim": 34, "cuda_available": true, "gpu_name": "NVIDIA A100-SXM4-80GB", "total_memory_bytes": 85095874560, "max_memory_allocated_bytes": 17449297408, "max_memory_reserved_bytes": 20979908608, "memory_headroom": 0.85, "status": "ok", "error": ""}
-{"dataset": "pcam", "model": "gate", "group": "c8", "train_mode": "C", "train_size": 100, "train_examples": 100, "seed": 42, "batch_size": 128, "effective_batch_size": 128, "growth_rate": 3, "output_dir": "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed42_n100", "n_params": 135682, "invariant_dim": 27, "cuda_available": true, "gpu_name": "NVIDIA A100-SXM4-80GB", "total_memory_bytes": 85095874560, "max_memory_allocated_bytes": 15562701312, "max_memory_reserved_bytes": 18803064832, "memory_headroom": 0.85, "status": "ok", "error": ""}
-{"dataset": "pcam", "model": "fourier_elu", "group": "c8", "train_mode": "C", "train_size": 100, "train_examples": 100, "seed": 42, "batch_size": 64, "effective_batch_size": 64, "growth_rate": 4, "output_dir": "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed42_n100", "n_params": 109815, "invariant_dim": 34, "cuda_available": true, "gpu_name": "NVIDIA A100-SXM4-80GB", "total_memory_bytes": 85095874560, "max_memory_allocated_bytes": 21283565568, "max_memory_reserved_bytes": 27644657664, "memory_headroom": 0.85, "status": "ok", "error": ""}
-{"dataset": "pcam", "model": "bispectrum", "group": "c8", "train_mode": "C", "train_size": 100, "train_examples": 100, "seed": 42, "batch_size": 128, "effective_batch_size": 128, "growth_rate": 4, "output_dir": "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed42_n100", "n_params": 128345, "invariant_dim": 544, "cuda_available": true, "gpu_name": "NVIDIA A100-SXM4-80GB", "total_memory_bytes": 85095874560, "max_memory_allocated_bytes": 12089475072, "max_memory_reserved_bytes": 13314818048, "memory_headroom": 0.85, "status": "ok", "error": ""}
-{"dataset": "pcam", "model": "standard", "group": "c8", "train_mode": "C", "train_size": 500, "train_examples": 500, "seed": 42, "batch_size": 1024, "effective_batch_size": 1024, "growth_rate": 12, "output_dir": "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed42_n500", "n_params": 101959, "invariant_dim": 90, "cuda_available": true, "gpu_name": "NVIDIA A100-SXM4-80GB", "total_memory_bytes": 85095874560, "max_memory_allocated_bytes": 12277593600, "max_memory_reserved_bytes": 15307112448, "memory_headroom": 0.85, "status": "ok", "error": ""}
-{"dataset": "pcam", "model": "norm", "group": "c8", "train_mode": "C", "train_size": 500, "train_examples": 500, "seed": 42, "batch_size": 128, "effective_batch_size": 128, "growth_rate": 4, "output_dir": "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_C_seed42_n500", "n_params": 110093, "invariant_dim": 34, "cuda_available": true, "gpu_name": "NVIDIA A100-SXM4-80GB", "total_memory_bytes": 85095874560, "max_memory_allocated_bytes": 22322611712, "max_memory_reserved_bytes": 26801602560, "memory_headroom": 0.85, "status": "ok", "error": ""}
-{"dataset": "pcam", "model": "gate", "group": "c8", "train_mode": "C", "train_size": 500, "train_examples": 500, "seed": 42, "batch_size": 128, "effective_batch_size": 128, "growth_rate": 3, "output_dir": "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_C_seed42_n500", "n_params": 135682, "invariant_dim": 27, "cuda_available": true, "gpu_name": "NVIDIA A100-SXM4-80GB", "total_memory_bytes": 85095874560, "max_memory_allocated_bytes": 19902757376, "max_memory_reserved_bytes": 24033361920, "memory_headroom": 0.85, "status": "ok", "error": ""}
-{"dataset": "pcam", "model": "fourier_elu", "group": "c8", "train_mode": "C", "train_size": 500, "train_examples": 500, "seed": 42, "batch_size": 64, "effective_batch_size": 64, "growth_rate": 4, "output_dir": "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_C_seed42_n500", "n_params": 109815, "invariant_dim": 34, "cuda_available": true, "gpu_name": "NVIDIA A100-SXM4-80GB", "total_memory_bytes": 85095874560, "max_memory_allocated_bytes": 21283565568, "max_memory_reserved_bytes": 27644657664, "memory_headroom": 0.85, "status": "ok", "error": ""}
-{"dataset": "pcam", "model": "bispectrum", "group": "c8", "train_mode": "C", "train_size": 500, "train_examples": 500, "seed": 42, "batch_size": 128, "effective_batch_size": 128, "growth_rate": 4, "output_dir": "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_C_seed42_n500", "n_params": 128345, "invariant_dim": 544, "cuda_available": true, "gpu_name": "NVIDIA A100-SXM4-80GB", "total_memory_bytes": 85095874560, "max_memory_allocated_bytes": 15465098240, "max_memory_reserved_bytes": 17028874240, "memory_headroom": 0.85, "status": "ok", "error": ""}
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diff --git a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed101112_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed101112_n100/results.json
deleted file mode 100644
index d147a90..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed101112_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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-  "model": "bispectrum",
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-  "seed": 101112,
-  "channels": [
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-    8
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-    "C",
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-    "--output_dir",
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-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
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-    "--epochs",
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-    "--train_size",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed123_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed123_n100/results.json
deleted file mode 100644
index d2cc5c8..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed123_n100/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed42_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed42_n100/results.json
deleted file mode 100644
index 58664f5..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed42_n100/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed456_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed456_n100/results.json
deleted file mode 100644
index 3012bab..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed456_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed789_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed789_n100/results.json
deleted file mode 100644
index d841dba..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_C_ch4_8_seed789_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed101112_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed101112_n100/results.json
deleted file mode 100644
index d1485c3..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed101112_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed123_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed123_n100/results.json
deleted file mode 100644
index 6d05e4c..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed123_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed42_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed42_n100/results.json
deleted file mode 100644
index 7ffb8ee..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed42_n100/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed456_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed456_n100/results.json
deleted file mode 100644
index 85a7b0e..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed456_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed789_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed789_n100/results.json
deleted file mode 100644
index 2afe217..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/bispectrum_R_ch4_8_seed789_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed101112_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed101112_n100/results.json
deleted file mode 100644
index 215a4b8..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed101112_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed123_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed123_n100/results.json
deleted file mode 100644
index 49558ce..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed123_n100/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed42_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed42_n100/results.json
deleted file mode 100644
index 40c4135..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed42_n100/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed456_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed456_n100/results.json
deleted file mode 100644
index df9aa4b..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed456_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed789_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed789_n100/results.json
deleted file mode 100644
index fb5587c..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/max_pool_C_ch4_8_seed789_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed101112_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed101112_n100/results.json
deleted file mode 100644
index 08cfd74..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed101112_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed123_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed123_n100/results.json
deleted file mode 100644
index c00deac..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed123_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed42_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed42_n100/results.json
deleted file mode 100644
index 618c419..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed42_n100/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed456_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed456_n100/results.json
deleted file mode 100644
index e319bf5..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed456_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed789_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed789_n100/results.json
deleted file mode 100644
index c395bae..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/max_pool_R_ch4_8_seed789_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed101112_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed101112_n100/results.json
deleted file mode 100644
index dfaa636..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed101112_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed123_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed123_n100/results.json
deleted file mode 100644
index 389d24a..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed123_n100/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed42_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed42_n100/results.json
deleted file mode 100644
index 9212e07..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed42_n100/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed456_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed456_n100/results.json
deleted file mode 100644
index 6da13bd..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed456_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed789_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed789_n100/results.json
deleted file mode 100644
index 90ed0ae..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/standard_C_ch4_8_seed789_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed101112_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed101112_n100/results.json
deleted file mode 100644
index 26178cc..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed101112_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed123_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed123_n100/results.json
deleted file mode 100644
index ff1755b..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed123_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed42_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed42_n100/results.json
deleted file mode 100644
index 651ffa2..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed42_n100/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed456_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed456_n100/results.json
deleted file mode 100644
index af65b93..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed456_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed789_n100/results.json b/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed789_n100/results.json
deleted file mode 100644
index 977bda2..0000000
--- a/paper/experiments/normalized_results/organ3d/n_100/standard_R_ch4_8_seed789_n100/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed101112_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed101112_n500/results.json
deleted file mode 100644
index 192c1ef..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed101112_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed123_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed123_n500/results.json
deleted file mode 100644
index 26c418f..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed123_n500/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed42_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed42_n500/results.json
deleted file mode 100644
index af615b8..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed42_n500/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed456_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed456_n500/results.json
deleted file mode 100644
index 44f1e09..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed456_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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-    "--epochs",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed789_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed789_n500/results.json
deleted file mode 100644
index 09a10a4..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_C_ch4_8_seed789_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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-    "--output_dir",
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-    "--data_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "15",
-    "--epochs",
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-    "--train_size",
-    "500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed101112_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed101112_n500/results.json
deleted file mode 100644
index 05eaa4f..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed101112_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed123_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed123_n500/results.json
deleted file mode 100644
index fae9545..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed123_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed42_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed42_n500/results.json
deleted file mode 100644
index f240d40..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed42_n500/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed456_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed456_n500/results.json
deleted file mode 100644
index d5884d3..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed456_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed789_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed789_n500/results.json
deleted file mode 100644
index 0095752..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/bispectrum_R_ch4_8_seed789_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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-    8
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-    "--patience",
-    "15",
-    "--epochs",
-    "100",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed101112_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed101112_n500/results.json
deleted file mode 100644
index 01af575..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed101112_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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-    "32",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/organ3d/n_500",
-    "--data_dir",
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-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "15",
-    "--epochs",
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-    "--train_size",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed123_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed123_n500/results.json
deleted file mode 100644
index 85a3455..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed123_n500/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed42_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed42_n500/results.json
deleted file mode 100644
index 8bd66ea..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed42_n500/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed456_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed456_n500/results.json
deleted file mode 100644
index 1cb3032..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed456_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed789_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed789_n500/results.json
deleted file mode 100644
index 29f8be2..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/max_pool_C_ch4_8_seed789_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed101112_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed101112_n500/results.json
deleted file mode 100644
index 78f16a1..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed101112_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed123_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed123_n500/results.json
deleted file mode 100644
index b6bc161..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed123_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed42_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed42_n500/results.json
deleted file mode 100644
index 104ec6c..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed42_n500/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed456_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed456_n500/results.json
deleted file mode 100644
index 0c60ff7..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed456_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed789_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed789_n500/results.json
deleted file mode 100644
index de47c26..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/max_pool_R_ch4_8_seed789_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed101112_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed101112_n500/results.json
deleted file mode 100644
index 51910b2..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed101112_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed123_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed123_n500/results.json
deleted file mode 100644
index 77bea63..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed123_n500/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed42_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed42_n500/results.json
deleted file mode 100644
index 44a79f8..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed42_n500/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed456_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed456_n500/results.json
deleted file mode 100644
index b9fdd23..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed456_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed789_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed789_n500/results.json
deleted file mode 100644
index 3cc7500..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/standard_C_ch4_8_seed789_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed101112_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed101112_n500/results.json
deleted file mode 100644
index 1fd1335..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed101112_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed123_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed123_n500/results.json
deleted file mode 100644
index 554ac52..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed123_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed42_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed42_n500/results.json
deleted file mode 100644
index 1b99f9f..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed42_n500/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed456_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed456_n500/results.json
deleted file mode 100644
index 7d9a811..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed456_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed789_n500/results.json b/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed789_n500/results.json
deleted file mode 100644
index 9950641..0000000
--- a/paper/experiments/normalized_results/organ3d/n_500/standard_R_ch4_8_seed789_n500/results.json
+++ /dev/null
@@ -1,169 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed101112/results.json b/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed101112/results.json
deleted file mode 100644
index 5af114e..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed101112/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed123/results.json b/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed123/results.json
deleted file mode 100644
index 681a06c..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed123/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed42/results.json b/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed42/results.json
deleted file mode 100644
index f167fd1..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed42/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed456/results.json b/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed456/results.json
deleted file mode 100644
index 5e6f38e..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed456/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed789/results.json b/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed789/results.json
deleted file mode 100644
index a633f21..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_C_ch4_8_seed789/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed101112/results.json b/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed101112/results.json
deleted file mode 100644
index da01439..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed101112/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed123/results.json b/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed123/results.json
deleted file mode 100644
index 1fda8d9..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed123/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed42/results.json b/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed42/results.json
deleted file mode 100644
index ed2a68e..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed42/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed456/results.json b/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed456/results.json
deleted file mode 100644
index c3fc267..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed456/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed789/results.json b/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed789/results.json
deleted file mode 100644
index 9ec3b07..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/bispectrum_R_ch4_8_seed789/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed101112/results.json b/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed101112/results.json
deleted file mode 100644
index 55d791a..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed101112/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed123/results.json b/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed123/results.json
deleted file mode 100644
index 1a1905f..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed123/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
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diff --git a/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed42/results.json b/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed42/results.json
deleted file mode 100644
index 9c39b86..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed42/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed456/results.json b/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed456/results.json
deleted file mode 100644
index 91eaaad..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed456/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed789/results.json b/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed789/results.json
deleted file mode 100644
index 151caca..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/max_pool_C_ch4_8_seed789/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed101112/results.json b/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed101112/results.json
deleted file mode 100644
index 8173def..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed101112/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed123/results.json b/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed123/results.json
deleted file mode 100644
index c29b1c6..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed123/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed42/results.json b/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed42/results.json
deleted file mode 100644
index 7faf6c0..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed42/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed456/results.json b/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed456/results.json
deleted file mode 100644
index ab5a198..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed456/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed789/results.json b/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed789/results.json
deleted file mode 100644
index cdb5034..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/max_pool_R_ch4_8_seed789/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed101112/results.json b/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed101112/results.json
deleted file mode 100644
index 2a20b6d..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed101112/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed123/results.json b/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed123/results.json
deleted file mode 100644
index 2469f7a..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed123/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed42/results.json b/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed42/results.json
deleted file mode 100644
index 8489638..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed42/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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diff --git a/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed456/results.json b/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed456/results.json
deleted file mode 100644
index 20c5732..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed456/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed789/results.json b/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed789/results.json
deleted file mode 100644
index 2b6ee9d..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/standard_C_ch4_8_seed789/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed101112/results.json b/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed101112/results.json
deleted file mode 100644
index 0b0655b..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed101112/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed123/results.json b/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed123/results.json
deleted file mode 100644
index f841b20..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed123/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed42/results.json b/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed42/results.json
deleted file mode 100644
index e7f34c8..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed42/results.json
+++ /dev/null
@@ -1,135 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed456/results.json b/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed456/results.json
deleted file mode 100644
index 00321c6..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed456/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed789/results.json b/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed789/results.json
deleted file mode 100644
index 659dffe..0000000
--- a/paper/experiments/normalized_results/organ3d/n_full/standard_R_ch4_8_seed789/results.json
+++ /dev/null
@@ -1,167 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed101112_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed101112_n100/results.json
deleted file mode 100644
index 212a67b..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed101112_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "--batch_size",
-    "128",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_100",
-    "--data_dir",
-    "./pcam_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed123_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed123_n100/results.json
deleted file mode 100644
index afebd4d..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed123_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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-  "seed": 123,
-  "train_mode": "C",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed42_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed42_n100/results.json
deleted file mode 100644
index 8df768a..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed42_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
-  "model": "bispectrum",
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-  "train_mode": "C",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed456_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed456_n100/results.json
deleted file mode 100644
index a2bd34e..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed456_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
-  "model": "bispectrum",
-  "group": "c8",
-  "seed": 456,
-  "train_mode": "C",
-  "geometry_group": "c8",
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "bispectrum",
-    "--group",
-    "c8",
-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "4",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--batch_size",
-    "128",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_100",
-    "--data_dir",
-    "./pcam_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed789_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed789_n100/results.json
deleted file mode 100644
index b6ab702..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_C_seed789_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-  "argv": [
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-    "bispectrum",
-    "--group",
-    "c8",
-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "4",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--batch_size",
-    "128",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_100",
-    "--data_dir",
-    "./pcam_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_R_seed123_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_R_seed123_n100/results.json
deleted file mode 100644
index 29aee78..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_R_seed123_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_R_seed42_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_R_seed42_n100/results.json
deleted file mode 100644
index 0cedca8..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_R_seed42_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_R_seed456_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_R_seed456_n100/results.json
deleted file mode 100644
index 0f98f22..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_R_seed456_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
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-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_R_seed789_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_R_seed789_n100/results.json
deleted file mode 100644
index 368e1ab..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/bispectrum_c8_gr4_R_seed789_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-    "--train_mode",
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-    "--seed",
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-    "--batch_size",
-    "128",
-    "--output_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed101112_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed101112_n100/results.json
deleted file mode 100644
index f53a068..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed101112_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "4",
-    "--train_mode",
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-    "--seed",
-    "101112",
-    "--batch_size",
-    "64",
-    "--output_dir",
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-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed123_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed123_n100/results.json
deleted file mode 100644
index 19e26e0..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed123_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed42_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed42_n100/results.json
deleted file mode 100644
index 3309840..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed42_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed456_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed456_n100/results.json
deleted file mode 100644
index 7bf6a59..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed456_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "--epochs",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed789_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed789_n100/results.json
deleted file mode 100644
index dd21b78..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_C_seed789_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "--patience",
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-    "--epochs",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_R_seed123_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_R_seed123_n100/results.json
deleted file mode 100644
index 68c6975..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_R_seed123_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_R_seed42_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_R_seed42_n100/results.json
deleted file mode 100644
index 394a075..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_R_seed42_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
-  "model": "fourier_elu",
-  "group": "c8",
-  "seed": 42,
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-    "std_accuracy": 2.8290552090056087e-05
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_R_seed456_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_R_seed456_n100/results.json
deleted file mode 100644
index 0ce8f7b..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_R_seed456_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
-  "model": "fourier_elu",
-  "group": "c8",
-  "seed": 456,
-  "train_mode": "R",
-  "geometry_group": "c8",
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "fourier_elu",
-    "--group",
-    "c8",
-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "4",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--batch_size",
-    "64",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_100",
-    "--data_dir",
-    "./pcam_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_R_seed789_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_R_seed789_n100/results.json
deleted file mode 100644
index 21dba36..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/fourier_elu_c8_gr4_R_seed789_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "4",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--batch_size",
-    "64",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_100",
-    "--data_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed101112_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed101112_n100/results.json
deleted file mode 100644
index 65d1f3f..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed101112_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "3",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--batch_size",
-    "128",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_100",
-    "--data_dir",
-    "./pcam_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed123_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed123_n100/results.json
deleted file mode 100644
index c3da98b..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed123_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed42_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed42_n100/results.json
deleted file mode 100644
index 51dbead..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed42_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed456_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed456_n100/results.json
deleted file mode 100644
index 6f0c5d1..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed456_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed789_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed789_n100/results.json
deleted file mode 100644
index 94ee3c6..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_C_seed789_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_R_seed123_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_R_seed123_n100/results.json
deleted file mode 100644
index 6a2373e..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_R_seed123_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_R_seed42_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_R_seed42_n100/results.json
deleted file mode 100644
index 77cf777..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_R_seed42_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_R_seed456_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_R_seed456_n100/results.json
deleted file mode 100644
index 8db2fc0..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_R_seed456_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_R_seed789_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_R_seed789_n100/results.json
deleted file mode 100644
index 478e415..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/gate_c8_gr3_R_seed789_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed101112_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed101112_n100/results.json
deleted file mode 100644
index 9989030..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed101112_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed123_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed123_n100/results.json
deleted file mode 100644
index c973682..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed123_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed42_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed42_n100/results.json
deleted file mode 100644
index 46f51d5..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed42_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed456_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed456_n100/results.json
deleted file mode 100644
index 6831e98..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed456_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed789_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed789_n100/results.json
deleted file mode 100644
index 812d4d8..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_C_seed789_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_R_seed123_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_R_seed123_n100/results.json
deleted file mode 100644
index 5a55a96..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_R_seed123_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_R_seed42_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_R_seed42_n100/results.json
deleted file mode 100644
index 45f3f53..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_R_seed42_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_R_seed456_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_R_seed456_n100/results.json
deleted file mode 100644
index 00a1e30..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_R_seed456_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_R_seed789_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_R_seed789_n100/results.json
deleted file mode 100644
index cff0a40..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/norm_c8_gr4_R_seed789_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed101112_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed101112_n100/results.json
deleted file mode 100644
index fc6f1e4..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed101112_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "--epochs",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed123_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed123_n100/results.json
deleted file mode 100644
index f932717..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed123_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed42_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed42_n100/results.json
deleted file mode 100644
index 9b92c54..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed42_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed456_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed456_n100/results.json
deleted file mode 100644
index 1a82ed5..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed456_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed789_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed789_n100/results.json
deleted file mode 100644
index d528b2f..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_C_seed789_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_R_seed123_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_R_seed123_n100/results.json
deleted file mode 100644
index bc09d4e..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_R_seed123_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_R_seed42_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_R_seed42_n100/results.json
deleted file mode 100644
index ed48c50..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_R_seed42_n100/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_R_seed456_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_R_seed456_n100/results.json
deleted file mode 100644
index d3778bf..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_R_seed456_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "12",
-    "--train_mode",
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-    "--seed",
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-    "1024",
-    "--output_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
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-    "--epochs",
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-    "--train_size",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_R_seed789_n100/results.json b/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_R_seed789_n100/results.json
deleted file mode 100644
index e7d9767..0000000
--- a/paper/experiments/normalized_results/pcam/n_100/standard_c8_gr12_R_seed789_n100/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-    "--batch_size",
-    "1024",
-    "--output_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_C_seed123_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_C_seed123_n12500/results.json
deleted file mode 100644
index d5c5c8a..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_C_seed123_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_C_seed42_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_C_seed42_n12500/results.json
deleted file mode 100644
index 397cc22..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_C_seed42_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_C_seed456_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_C_seed456_n12500/results.json
deleted file mode 100644
index 95de522..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_C_seed456_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "--growth_rate",
-    "4",
-    "--train_mode",
-    "C",
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-    "--batch_size",
-    "128",
-    "--output_dir",
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-    "--patience",
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-    "--epochs",
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-    "--train_size",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_C_seed789_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_C_seed789_n12500/results.json
deleted file mode 100644
index 5d0f1da..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_C_seed789_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "128",
-    "--output_dir",
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-    "--patience",
-    "10",
-    "--epochs",
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-    "--train_size",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_R_seed123_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_R_seed123_n12500/results.json
deleted file mode 100644
index f60807e..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_R_seed123_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_R_seed42_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_R_seed42_n12500/results.json
deleted file mode 100644
index 60a6870..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_R_seed42_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_R_seed456_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_R_seed456_n12500/results.json
deleted file mode 100644
index 0939e27..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_R_seed456_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "--train_mode",
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-    "--epochs",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_R_seed789_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_R_seed789_n12500/results.json
deleted file mode 100644
index 7289804..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/bispectrum_c8_gr4_R_seed789_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "--train_mode",
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-    "--batch_size",
-    "128",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_12500",
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-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
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-    "--epochs",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_C_seed123_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_C_seed123_n12500/results.json
deleted file mode 100644
index bbe76e1..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_C_seed123_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_C_seed42_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_C_seed42_n12500/results.json
deleted file mode 100644
index 62a03b1..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_C_seed42_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_C_seed456_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_C_seed456_n12500/results.json
deleted file mode 100644
index 525d389..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_C_seed456_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-  "group": "c8",
-  "seed": 456,
-  "train_mode": "C",
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-    "64",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_12500",
-    "--data_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_C_seed789_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_C_seed789_n12500/results.json
deleted file mode 100644
index 4f46c4b..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_C_seed789_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "c8",
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-    "4",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--batch_size",
-    "64",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_12500",
-    "--data_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_R_seed123_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_R_seed123_n12500/results.json
deleted file mode 100644
index 3d13a7a..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_R_seed123_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_R_seed42_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_R_seed42_n12500/results.json
deleted file mode 100644
index e63b2a0..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_R_seed42_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_R_seed456_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_R_seed456_n12500/results.json
deleted file mode 100644
index 8d2d477..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_R_seed456_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_R_seed789_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_R_seed789_n12500/results.json
deleted file mode 100644
index ead5408..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/fourier_elu_c8_gr4_R_seed789_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_C_seed123_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_C_seed123_n12500/results.json
deleted file mode 100644
index 49acde6..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_C_seed123_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_C_seed42_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_C_seed42_n12500/results.json
deleted file mode 100644
index 6f418a5..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_C_seed42_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_C_seed456_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_C_seed456_n12500/results.json
deleted file mode 100644
index a2e4aad..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_C_seed456_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "3",
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-    "--batch_size",
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-    "--output_dir",
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-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
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-    "--epochs",
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-    "--train_size",
-    "12500"
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_C_seed789_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_C_seed789_n12500/results.json
deleted file mode 100644
index 722f7fc..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_C_seed789_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "--batch_size",
-    "128",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_12500",
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-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
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-    "--epochs",
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-    "12500"
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_R_seed123_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_R_seed123_n12500/results.json
deleted file mode 100644
index d1baaf4..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_R_seed123_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_R_seed42_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_R_seed42_n12500/results.json
deleted file mode 100644
index db3992a..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_R_seed42_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_R_seed456_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_R_seed456_n12500/results.json
deleted file mode 100644
index 35e898c..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_R_seed456_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "--train_mode",
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-    "--seed",
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-    "--batch_size",
-    "128",
-    "--output_dir",
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-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
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-    "--train_size",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_R_seed789_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_R_seed789_n12500/results.json
deleted file mode 100644
index c768683..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/gate_c8_gr3_R_seed789_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_C_seed123_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_C_seed123_n12500/results.json
deleted file mode 100644
index 3f0d49e..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_C_seed123_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_C_seed42_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_C_seed42_n12500/results.json
deleted file mode 100644
index b90f437..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_C_seed42_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_C_seed456_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_C_seed456_n12500/results.json
deleted file mode 100644
index 915d7b2..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_C_seed456_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_C_seed789_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_C_seed789_n12500/results.json
deleted file mode 100644
index 3cafd84..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_C_seed789_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_R_seed123_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_R_seed123_n12500/results.json
deleted file mode 100644
index 05b65bb..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_R_seed123_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_R_seed42_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_R_seed42_n12500/results.json
deleted file mode 100644
index 5f6c95c..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_R_seed42_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_R_seed456_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_R_seed456_n12500/results.json
deleted file mode 100644
index ec1b810..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_R_seed456_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_R_seed789_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_R_seed789_n12500/results.json
deleted file mode 100644
index f3ddc67..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/norm_c8_gr4_R_seed789_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_C_seed123_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_C_seed123_n12500/results.json
deleted file mode 100644
index ba9f92a..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_C_seed123_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_C_seed42_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_C_seed42_n12500/results.json
deleted file mode 100644
index da29642..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_C_seed42_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_C_seed456_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_C_seed456_n12500/results.json
deleted file mode 100644
index 6a6afd7..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_C_seed456_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_C_seed789_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_C_seed789_n12500/results.json
deleted file mode 100644
index e921af1..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_C_seed789_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_R_seed123_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_R_seed123_n12500/results.json
deleted file mode 100644
index 2a7ab85..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_R_seed123_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_R_seed42_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_R_seed42_n12500/results.json
deleted file mode 100644
index 8656974..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_R_seed42_n12500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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-  "train_mode": "R",
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-    "mean_accuracy": 0.824859619140625,
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_R_seed456_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_R_seed456_n12500/results.json
deleted file mode 100644
index b2a7353..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_R_seed456_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-  "seed": 456,
-  "train_mode": "R",
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "standard",
-    "--group",
-    "c8",
-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "12",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--batch_size",
-    "1024",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_12500",
-    "--data_dir",
-    "./pcam_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_R_seed789_n12500/results.json b/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_R_seed789_n12500/results.json
deleted file mode 100644
index 1ac2982..0000000
--- a/paper/experiments/normalized_results/pcam/n_12500/standard_c8_gr12_R_seed789_n12500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-  "train_mode": "R",
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-    "--group",
-    "c8",
-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "12",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--batch_size",
-    "1024",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_12500",
-    "--data_dir",
-    "./pcam_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_C_seed123_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_C_seed123_n2500/results.json
deleted file mode 100644
index adb43ec..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_C_seed123_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
-  "model": "bispectrum",
-  "group": "c8",
-  "seed": 123,
-  "train_mode": "C",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_C_seed42_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_C_seed42_n2500/results.json
deleted file mode 100644
index fe13a96..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_C_seed42_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_C_seed456_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_C_seed456_n2500/results.json
deleted file mode 100644
index f5afafb..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_C_seed456_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-  "argv": [
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-    "--growth_rate",
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-    "--train_mode",
-    "C",
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-    "--batch_size",
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-    "--output_dir",
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-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_C_seed789_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_C_seed789_n2500/results.json
deleted file mode 100644
index cc3af85..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_C_seed789_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "bispectrum",
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-    "c8",
-    "--growth_rate",
-    "4",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--batch_size",
-    "128",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_2500",
-    "--data_dir",
-    "./pcam_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_R_seed123_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_R_seed123_n2500/results.json
deleted file mode 100644
index 078e9f8..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_R_seed123_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
-  "model": "bispectrum",
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-  "seed": 123,
-  "train_mode": "R",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_R_seed42_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_R_seed42_n2500/results.json
deleted file mode 100644
index 22cbd42..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_R_seed42_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
-  "model": "bispectrum",
-  "group": "c8",
-  "seed": 42,
-  "train_mode": "R",
-  "geometry_group": "c8",
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-    "std_accuracy": 0.00567043040001042
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_R_seed456_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_R_seed456_n2500/results.json
deleted file mode 100644
index 7d24d33..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_R_seed456_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
-  "model": "bispectrum",
-  "group": "c8",
-  "seed": 456,
-  "train_mode": "R",
-  "geometry_group": "c8",
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "bispectrum",
-    "--group",
-    "c8",
-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "4",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--batch_size",
-    "128",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_2500",
-    "--data_dir",
-    "./pcam_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_R_seed789_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_R_seed789_n2500/results.json
deleted file mode 100644
index 61912e8..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/bispectrum_c8_gr4_R_seed789_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-  "train_mode": "R",
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-  "argv": [
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-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "4",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--batch_size",
-    "128",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_2500",
-    "--data_dir",
-    "./pcam_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_C_seed123_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_C_seed123_n2500/results.json
deleted file mode 100644
index 8c8c0cf..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_C_seed123_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_C_seed42_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_C_seed42_n2500/results.json
deleted file mode 100644
index 9b31415..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_C_seed42_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_C_seed456_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_C_seed456_n2500/results.json
deleted file mode 100644
index 2439304..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_C_seed456_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_C_seed789_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_C_seed789_n2500/results.json
deleted file mode 100644
index f7ede0a..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_C_seed789_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "c8",
-    "--growth_rate",
-    "4",
-    "--train_mode",
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-    "--batch_size",
-    "64",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_2500",
-    "--data_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_R_seed123_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_R_seed123_n2500/results.json
deleted file mode 100644
index 4ece8d2..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_R_seed123_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_R_seed42_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_R_seed42_n2500/results.json
deleted file mode 100644
index 00847b1..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_R_seed42_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
-  "model": "fourier_elu",
-  "group": "c8",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_R_seed456_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_R_seed456_n2500/results.json
deleted file mode 100644
index a025141..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_R_seed456_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-    "c8",
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-    "4",
-    "--train_mode",
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-    "--seed",
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-    "--batch_size",
-    "64",
-    "--output_dir",
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-    "--data_dir",
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-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_R_seed789_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_R_seed789_n2500/results.json
deleted file mode 100644
index 7dcff5a..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/fourier_elu_c8_gr4_R_seed789_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_C_seed123_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_C_seed123_n2500/results.json
deleted file mode 100644
index f38daf0..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_C_seed123_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_C_seed42_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_C_seed42_n2500/results.json
deleted file mode 100644
index 3b502a2..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_C_seed42_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_C_seed456_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_C_seed456_n2500/results.json
deleted file mode 100644
index 81d489f..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_C_seed456_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_C_seed789_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_C_seed789_n2500/results.json
deleted file mode 100644
index 70114cf..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_C_seed789_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_R_seed123_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_R_seed123_n2500/results.json
deleted file mode 100644
index 989dfbc..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_R_seed123_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_R_seed42_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_R_seed42_n2500/results.json
deleted file mode 100644
index e93dfc5..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_R_seed42_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_R_seed456_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_R_seed456_n2500/results.json
deleted file mode 100644
index 5c45a19..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_R_seed456_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_R_seed789_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_R_seed789_n2500/results.json
deleted file mode 100644
index f2922bb..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/gate_c8_gr3_R_seed789_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_C_seed123_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_C_seed123_n2500/results.json
deleted file mode 100644
index fdb8109..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_C_seed123_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_C_seed42_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_C_seed42_n2500/results.json
deleted file mode 100644
index 6b768ff..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_C_seed42_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_C_seed456_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_C_seed456_n2500/results.json
deleted file mode 100644
index cf17ae0..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_C_seed456_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_C_seed789_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_C_seed789_n2500/results.json
deleted file mode 100644
index eba58b9..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_C_seed789_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_R_seed123_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_R_seed123_n2500/results.json
deleted file mode 100644
index 0f2c6ef..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_R_seed123_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_R_seed42_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_R_seed42_n2500/results.json
deleted file mode 100644
index 672f6e4..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_R_seed42_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_R_seed456_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_R_seed456_n2500/results.json
deleted file mode 100644
index 00d9b55..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_R_seed456_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "--train_mode",
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-    "--output_dir",
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-    "--patience",
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-    "--epochs",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_R_seed789_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_R_seed789_n2500/results.json
deleted file mode 100644
index b741284..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/norm_c8_gr4_R_seed789_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "c8",
-    "--growth_rate",
-    "4",
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-    "--seed",
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-    "--batch_size",
-    "128",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_2500",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_C_seed123_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_C_seed123_n2500/results.json
deleted file mode 100644
index 0ca7fc9..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_C_seed123_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_C_seed42_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_C_seed42_n2500/results.json
deleted file mode 100644
index 6c94d2e..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_C_seed42_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_C_seed456_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_C_seed456_n2500/results.json
deleted file mode 100644
index 42462be..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_C_seed456_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-  "train_mode": "C",
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-  "argv": [
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-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "12",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--batch_size",
-    "1024",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_2500",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_C_seed789_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_C_seed789_n2500/results.json
deleted file mode 100644
index 98ef487..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_C_seed789_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-    "12",
-    "--train_mode",
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-    "--seed",
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-    "--batch_size",
-    "1024",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_2500",
-    "--data_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_R_seed123_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_R_seed123_n2500/results.json
deleted file mode 100644
index 58dd3fb..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_R_seed123_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_R_seed42_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_R_seed42_n2500/results.json
deleted file mode 100644
index 6739df5..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_R_seed42_n2500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_R_seed456_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_R_seed456_n2500/results.json
deleted file mode 100644
index 940ff38..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_R_seed456_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_R_seed789_n2500/results.json b/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_R_seed789_n2500/results.json
deleted file mode 100644
index 2bb05ef..0000000
--- a/paper/experiments/normalized_results/pcam/n_2500/standard_c8_gr12_R_seed789_n2500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_C_seed123_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_C_seed123_n500/results.json
deleted file mode 100644
index 4298e75..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_C_seed123_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_C_seed42_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_C_seed42_n500/results.json
deleted file mode 100644
index ad56e06..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_C_seed42_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_C_seed456_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_C_seed456_n500/results.json
deleted file mode 100644
index 1074c3f..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_C_seed456_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_C_seed789_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_C_seed789_n500/results.json
deleted file mode 100644
index 1520063..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_C_seed789_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_R_seed123_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_R_seed123_n500/results.json
deleted file mode 100644
index 5dde68e..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_R_seed123_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_R_seed42_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_R_seed42_n500/results.json
deleted file mode 100644
index 8f599a1..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_R_seed42_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_R_seed456_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_R_seed456_n500/results.json
deleted file mode 100644
index a4bc081..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_R_seed456_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
-  "model": "bispectrum",
-  "group": "c8",
-  "seed": 456,
-  "train_mode": "R",
-  "geometry_group": "c8",
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-  "argv": [
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-    "bispectrum",
-    "--group",
-    "c8",
-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "4",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--batch_size",
-    "128",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_500",
-    "--data_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_R_seed789_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_R_seed789_n500/results.json
deleted file mode 100644
index 7e008a2..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/bispectrum_c8_gr4_R_seed789_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-    "c8",
-    "--growth_rate",
-    "4",
-    "--train_mode",
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-    "--seed",
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-    "--batch_size",
-    "128",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_500",
-    "--data_dir",
-    "./pcam_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_C_seed123_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_C_seed123_n500/results.json
deleted file mode 100644
index 3910bb1..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_C_seed123_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_C_seed42_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_C_seed42_n500/results.json
deleted file mode 100644
index 09c13cc..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_C_seed42_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_C_seed456_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_C_seed456_n500/results.json
deleted file mode 100644
index 602f2a8..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_C_seed456_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-  "argv": [
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-    "--growth_rate",
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-    "--train_mode",
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-    "--seed",
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-    "--output_dir",
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-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_C_seed789_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_C_seed789_n500/results.json
deleted file mode 100644
index 6997e27..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_C_seed789_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_R_seed123_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_R_seed123_n500/results.json
deleted file mode 100644
index 3ad072f..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_R_seed123_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_R_seed42_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_R_seed42_n500/results.json
deleted file mode 100644
index ecbd87b..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_R_seed42_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_R_seed456_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_R_seed456_n500/results.json
deleted file mode 100644
index 5eae3fa..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_R_seed456_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_R_seed789_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_R_seed789_n500/results.json
deleted file mode 100644
index 7ce863f..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/fourier_elu_c8_gr4_R_seed789_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_C_seed123_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_C_seed123_n500/results.json
deleted file mode 100644
index 4b018a7..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_C_seed123_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_C_seed42_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_C_seed42_n500/results.json
deleted file mode 100644
index 74d7647..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_C_seed42_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_C_seed456_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_C_seed456_n500/results.json
deleted file mode 100644
index 9b8f339..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_C_seed456_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_C_seed789_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_C_seed789_n500/results.json
deleted file mode 100644
index aad608e..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_C_seed789_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_R_seed123_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_R_seed123_n500/results.json
deleted file mode 100644
index 8bbd14c..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_R_seed123_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_R_seed42_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_R_seed42_n500/results.json
deleted file mode 100644
index 7a7bbf8..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_R_seed42_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_R_seed456_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_R_seed456_n500/results.json
deleted file mode 100644
index 7a32053..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_R_seed456_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_R_seed789_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_R_seed789_n500/results.json
deleted file mode 100644
index 8575466..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/gate_c8_gr3_R_seed789_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_C_seed123_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_C_seed123_n500/results.json
deleted file mode 100644
index 4465e8d..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_C_seed123_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_C_seed42_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_C_seed42_n500/results.json
deleted file mode 100644
index 5bc83d1..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_C_seed42_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_C_seed456_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_C_seed456_n500/results.json
deleted file mode 100644
index 3bbb4ef..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_C_seed456_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_C_seed789_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_C_seed789_n500/results.json
deleted file mode 100644
index 2594e9e..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_C_seed789_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_R_seed123_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_R_seed123_n500/results.json
deleted file mode 100644
index e0147fb..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_R_seed123_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_R_seed42_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_R_seed42_n500/results.json
deleted file mode 100644
index 7e7ce48..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_R_seed42_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_R_seed456_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_R_seed456_n500/results.json
deleted file mode 100644
index 3ed4257..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_R_seed456_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_R_seed789_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_R_seed789_n500/results.json
deleted file mode 100644
index 6e9eec3..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/norm_c8_gr4_R_seed789_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed101112_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed101112_n500/results.json
deleted file mode 100644
index 68b2950..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed101112_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "--batch_size",
-    "1024",
-    "--output_dir",
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-    "--data_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed123_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed123_n500/results.json
deleted file mode 100644
index f3a2541..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed123_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed42_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed42_n500/results.json
deleted file mode 100644
index 20caa13..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed42_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed456_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed456_n500/results.json
deleted file mode 100644
index 30b7284..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed456_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "--batch_size",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed789_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed789_n500/results.json
deleted file mode 100644
index 501512c..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_C_seed789_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
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-    "--epochs",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_R_seed123_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_R_seed123_n500/results.json
deleted file mode 100644
index 23a9b97..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_R_seed123_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_R_seed42_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_R_seed42_n500/results.json
deleted file mode 100644
index a3fce62..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_R_seed42_n500/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_R_seed456_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_R_seed456_n500/results.json
deleted file mode 100644
index 15d636b..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_R_seed456_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-  "train_mode": "R",
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-  "argv": [
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-    "--geometry_group",
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-    "--growth_rate",
-    "12",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--batch_size",
-    "1024",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_500",
-    "--data_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_R_seed789_n500/results.json b/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_R_seed789_n500/results.json
deleted file mode 100644
index d3384df..0000000
--- a/paper/experiments/normalized_results/pcam/n_500/standard_c8_gr12_R_seed789_n500/results.json
+++ /dev/null
@@ -1,102 +0,0 @@
-{
-  "dataset": "pcam",
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-  "train_mode": "R",
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-  "argv": [
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-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "12",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--batch_size",
-    "1024",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_500",
-    "--data_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_C_seed123/results.json b/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_C_seed123/results.json
deleted file mode 100644
index f0627b9..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_C_seed123/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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-  "train_mode": "C",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_C_seed42/results.json b/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_C_seed42/results.json
deleted file mode 100644
index cb4d54c..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_C_seed42/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_C_seed456/results.json b/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_C_seed456/results.json
deleted file mode 100644
index a131e15..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_C_seed456/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
-  "dataset": "pcam",
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-    "c8",
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-    "4",
-    "--train_mode",
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-    "--seed",
-    "456",
-    "--batch_size",
-    "128",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_full",
-    "--data_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_C_seed789/results.json b/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_C_seed789/results.json
deleted file mode 100644
index ebd15b7..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_C_seed789/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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-    "--patience",
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-    "--epochs",
-    "50"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_R_seed123/results.json b/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_R_seed123/results.json
deleted file mode 100644
index 797c038..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_R_seed123/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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-    "--patience",
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-    "--epochs",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_R_seed42/results.json b/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_R_seed42/results.json
deleted file mode 100644
index 14e38e3..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_R_seed42/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_R_seed456/results.json b/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_R_seed456/results.json
deleted file mode 100644
index 6793ea0..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/bispectrum_c8_gr4_R_seed456/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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-    "--epochs",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_C_seed123/results.json b/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_C_seed123/results.json
deleted file mode 100644
index 27f9dac..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_C_seed123/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_C_seed42/results.json b/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_C_seed42/results.json
deleted file mode 100644
index 4091af4..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_C_seed42/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_C_seed456/results.json b/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_C_seed456/results.json
deleted file mode 100644
index 7508c66..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_C_seed456/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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-    "--epochs",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_R_seed123/results.json b/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_R_seed123/results.json
deleted file mode 100644
index 6dcfc06..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_R_seed123/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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-    "--output_dir",
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-    "--patience",
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-    "--epochs",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_R_seed42/results.json b/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_R_seed42/results.json
deleted file mode 100644
index 6a7e8af..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_R_seed42/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_R_seed456/results.json b/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_R_seed456/results.json
deleted file mode 100644
index 02744a2..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/fourier_elu_c8_gr4_R_seed456/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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-    "64",
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-    "--patience",
-    "10",
-    "--epochs",
-    "50"
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_C_seed123/results.json b/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_C_seed123/results.json
deleted file mode 100644
index c85e8ea..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_C_seed123/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_C_seed42/results.json b/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_C_seed42/results.json
deleted file mode 100644
index 0fc8ec6..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_C_seed42/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_C_seed456/results.json b/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_C_seed456/results.json
deleted file mode 100644
index 6c2eb9a..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_C_seed456/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_C_seed789/results.json b/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_C_seed789/results.json
deleted file mode 100644
index 6a53144..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_C_seed789/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_R_seed123/results.json b/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_R_seed123/results.json
deleted file mode 100644
index 98cf496..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_R_seed123/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_R_seed42/results.json b/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_R_seed42/results.json
deleted file mode 100644
index 03865fb..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_R_seed42/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_R_seed456/results.json b/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_R_seed456/results.json
deleted file mode 100644
index 3f7684f..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/gate_c8_gr3_R_seed456/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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-    "3",
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-    "--batch_size",
-    "128",
-    "--output_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_C_seed123/results.json b/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_C_seed123/results.json
deleted file mode 100644
index 9a644d0..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_C_seed123/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
-  "dataset": "pcam",
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-  "train_mode": "C",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_C_seed42/results.json b/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_C_seed42/results.json
deleted file mode 100644
index fb3e53c..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_C_seed42/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_C_seed456/results.json b/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_C_seed456/results.json
deleted file mode 100644
index 65e857f..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_C_seed456/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_R_seed123/results.json b/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_R_seed123/results.json
deleted file mode 100644
index 39128c3..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_R_seed123/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_R_seed42/results.json b/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_R_seed42/results.json
deleted file mode 100644
index 0ba342a..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_R_seed42/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_R_seed456/results.json b/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_R_seed456/results.json
deleted file mode 100644
index aeff124..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/norm_c8_gr4_R_seed456/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_C_seed123/results.json b/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_C_seed123/results.json
deleted file mode 100644
index 5ec9392..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_C_seed123/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_C_seed42/results.json b/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_C_seed42/results.json
deleted file mode 100644
index 695e00b..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_C_seed42/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_C_seed456/results.json b/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_C_seed456/results.json
deleted file mode 100644
index 10c1aef..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_C_seed456/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_C_seed789/results.json b/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_C_seed789/results.json
deleted file mode 100644
index b0bc87f..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_C_seed789/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_R_seed123/results.json b/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_R_seed123/results.json
deleted file mode 100644
index 8cb052b..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_R_seed123/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_R_seed42/results.json b/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_R_seed42/results.json
deleted file mode 100644
index ac91f9b..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_R_seed42/results.json
+++ /dev/null
@@ -1,67 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_R_seed456/results.json b/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_R_seed456/results.json
deleted file mode 100644
index a1882a1..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_R_seed456/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "standard",
-    "--group",
-    "c8",
-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "12",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--batch_size",
-    "1024",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_full",
-    "--data_dir",
-    "./pcam_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_R_seed789/results.json b/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_R_seed789/results.json
deleted file mode 100644
index 32065de..0000000
--- a/paper/experiments/normalized_results/pcam/n_full/standard_c8_gr12_R_seed789/results.json
+++ /dev/null
@@ -1,100 +0,0 @@
-{
-  "dataset": "pcam",
-  "model": "standard",
-  "group": "c8",
-  "seed": 789,
-  "train_mode": "R",
-  "geometry_group": "c8",
-  "growth_rate": 12,
-  "n_params": 101959,
-  "invariant_dim": 90,
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-  "train_examples": 262144,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "4f6f1351fe12ab869087e096175e97d4ada5c7e0",
-  "argv": [
-    "--model",
-    "standard",
-    "--group",
-    "c8",
-    "--geometry_group",
-    "c8",
-    "--growth_rate",
-    "12",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--batch_size",
-    "1024",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/pcam/n_full",
-    "--data_dir",
-    "./pcam_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed101112_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed101112_n100/results.json
deleted file mode 100644
index 70bceda..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed101112_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 101112,
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-  "invariant_dim": 768,
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-    "accuracy": 0.0974
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-  "test_r": {
-    "loss": 2.3035296875,
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-  "rotation_robustness": {
-    "rot_00": 0.0974,
-    "rot_01": 0.0974,
-    "rot_02": 0.0974,
-    "rot_03": 0.0974,
-    "rot_04": 0.0974,
-    "rot_05": 0.0974,
-    "rot_06": 0.0974,
-    "rot_07": 0.0974,
-    "rot_08": 0.0974,
-    "rot_09": 0.0974,
-    "mean_accuracy": 0.0974,
-    "std_accuracy": 0.0
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-  "epochs_run": 11,
-  "peak_train_alloc_bytes": 422505984,
-  "peak_train_reserved_bytes": 622854144,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed123_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed123_n100/results.json
deleted file mode 100644
index ae352cb..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed123_n100/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 100,
-  "train_examples": 100,
-  "best_val_accuracy": 0.1162,
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-  "test": {
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-    "accuracy": 0.1129
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-  "test_nr": {
-    "loss": 2.30698359375,
-    "accuracy": 0.1129
-  },
-  "test_r": {
-    "loss": 2.307003515625,
-    "accuracy": 0.1129
-  },
-  "rotation_robustness": {
-    "rot_00": 0.1129,
-    "rot_01": 0.1129,
-    "rot_02": 0.1129,
-    "rot_03": 0.1129,
-    "rot_04": 0.1129,
-    "rot_05": 0.1129,
-    "rot_06": 0.1129,
-    "rot_07": 0.1129,
-    "rot_08": 0.1129,
-    "rot_09": 0.1129,
-    "mean_accuracy": 0.1129,
-    "std_accuracy": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed42_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed42_n100/results.json
deleted file mode 100644
index b1b6a7b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed42_n100/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 100,
-  "train_examples": 100,
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-  "test_nr": {
-    "loss": 2.3029396484375,
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-  },
-  "test_r": {
-    "loss": 2.3029404296875,
-    "accuracy": 0.1009
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-  "rotation_robustness": {
-    "rot_00": 0.1009,
-    "rot_01": 0.1009,
-    "rot_02": 0.1009,
-    "rot_03": 0.1009,
-    "rot_04": 0.1009,
-    "rot_05": 0.1009,
-    "rot_06": 0.1009,
-    "rot_07": 0.1009,
-    "rot_08": 0.1009,
-    "rot_09": 0.1009,
-    "mean_accuracy": 0.10090000000000002,
-    "std_accuracy": 1.3877787807814457e-17
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed456_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed456_n100/results.json
deleted file mode 100644
index 10e3810..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed456_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 100,
-  "train_examples": 100,
-  "best_val_accuracy": 0.1698,
-  "test_c": {
-    "loss": 2.3038248046875,
-    "accuracy": 0.1651
-  },
-  "test": {
-    "loss": 2.3038248046875,
-    "accuracy": 0.1651
-  },
-  "test_nr": {
-    "loss": 2.3038248046875,
-    "accuracy": 0.1651
-  },
-  "test_r": {
-    "loss": 2.3038404296875,
-    "accuracy": 0.164
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-  "rotation_robustness": {
-    "rot_00": 0.1639,
-    "rot_01": 0.1642,
-    "rot_02": 0.1641,
-    "rot_03": 0.1639,
-    "rot_04": 0.1643,
-    "rot_05": 0.1639,
-    "rot_06": 0.1643,
-    "rot_07": 0.1641,
-    "rot_08": 0.1641,
-    "rot_09": 0.1641,
-    "mean_accuracy": 0.16409,
-    "std_accuracy": 0.0001445683229480148
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-  "wall_time_s": 7.567641496658325,
-  "epochs_run": 19,
-  "peak_train_alloc_bytes": 422505984,
-  "peak_train_reserved_bytes": 622854144,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed789_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed789_n100/results.json
deleted file mode 100644
index 2b849c7..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_C_seed789_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 100,
-  "train_examples": 100,
-  "best_val_accuracy": 0.103,
-  "test_c": {
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-    "accuracy": 0.101
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-  "test": {
-    "loss": 2.3029833984375,
-    "accuracy": 0.101
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-  "test_nr": {
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-    "accuracy": 0.101
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-  "test_r": {
-    "loss": 2.3029841796875,
-    "accuracy": 0.101
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-  "rotation_robustness": {
-    "rot_00": 0.101,
-    "rot_01": 0.101,
-    "rot_02": 0.101,
-    "rot_03": 0.101,
-    "rot_04": 0.101,
-    "rot_05": 0.101,
-    "rot_06": 0.101,
-    "rot_07": 0.101,
-    "rot_08": 0.101,
-    "rot_09": 0.101,
-    "mean_accuracy": 0.101,
-    "std_accuracy": 0.0
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-  "wall_time_s": 6.222631454467773,
-  "epochs_run": 11,
-  "peak_train_alloc_bytes": 422505984,
-  "peak_train_reserved_bytes": 622854144,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed101112_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed101112_n100/results.json
deleted file mode 100644
index 0621fc8..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed101112_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 100,
-  "train_examples": 100,
-  "best_val_accuracy": 0.1009,
-  "test_c": {
-    "loss": 2.303594140625,
-    "accuracy": 0.0974
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-  "test": {
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-    "accuracy": 0.0974
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-  "test_nr": {
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-    "accuracy": 0.0974
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-  "test_r": {
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-    "accuracy": 0.0974
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-  "rotation_robustness": {
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-    "rot_01": 0.0974,
-    "rot_02": 0.0974,
-    "rot_03": 0.0974,
-    "rot_04": 0.0974,
-    "rot_05": 0.0974,
-    "rot_06": 0.0974,
-    "rot_07": 0.0974,
-    "rot_08": 0.0974,
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-    "mean_accuracy": 0.0974,
-    "std_accuracy": 0.0
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-  "wall_time_s": 5.080217361450195,
-  "epochs_run": 11,
-  "peak_train_alloc_bytes": 422505984,
-  "peak_train_reserved_bytes": 622854144,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed123_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed123_n100/results.json
deleted file mode 100644
index 014c370..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed123_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 100,
-  "train_examples": 100,
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-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed42_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed42_n100/results.json
deleted file mode 100644
index 9ed6432..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed42_n100/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
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-  "n_params": 231818,
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-    "rot_09": 0.1009,
-    "mean_accuracy": 0.10090000000000002,
-    "std_accuracy": 1.3877787807814457e-17
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed456_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed456_n100/results.json
deleted file mode 100644
index c2124f2..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed456_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 456,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed789_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed789_n100/results.json
deleted file mode 100644
index 22b4c77..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/bispectrum_R_seed789_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 789,
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-  "n_params": 231818,
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-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed101112_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed101112_n100/results.json
deleted file mode 100644
index 01854a9..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed101112_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed123_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed123_n100/results.json
deleted file mode 100644
index 22dfeb7..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed123_n100/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed42_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed42_n100/results.json
deleted file mode 100644
index 0f2acc2..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed42_n100/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed456_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed456_n100/results.json
deleted file mode 100644
index fb05386..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed456_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "base_model": "power_spectrum",
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-  "test_rotation_seed": 777,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed789_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed789_n100/results.json
deleted file mode 100644
index 8f8e1d4..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_C_seed789_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "base_model": "power_spectrum",
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-  "test_rotation_seed": 777,
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-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed101112_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed101112_n100/results.json
deleted file mode 100644
index 66ace4b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed101112_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": 100,
-  "train_examples": 100,
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-    "rot_04": 0.0991,
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-    "rot_07": 0.0992,
-    "rot_08": 0.0991,
-    "rot_09": 0.0991,
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-  "peak_train_alloc_bytes": 44528128,
-  "peak_train_reserved_bytes": 69206016,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed123_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed123_n100/results.json
deleted file mode 100644
index dac6625..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed123_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": 100,
-  "train_examples": 100,
-  "best_val_accuracy": 0.1729,
-  "test_c": {
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-    "accuracy": 0.1792
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-  "test": {
-    "loss": 2.31038251953125,
-    "accuracy": 0.1792
-  },
-  "test_nr": {
-    "loss": 2.31038251953125,
-    "accuracy": 0.1792
-  },
-  "test_r": {
-    "loss": 2.31079833984375,
-    "accuracy": 0.1789
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-  "rotation_robustness": {
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-    "rot_01": 0.1791,
-    "rot_02": 0.1792,
-    "rot_03": 0.1789,
-    "rot_04": 0.1789,
-    "rot_05": 0.179,
-    "rot_06": 0.1789,
-    "rot_07": 0.179,
-    "rot_08": 0.1789,
-    "rot_09": 0.1789,
-    "mean_accuracy": 0.17897000000000002,
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-  "peak_train_alloc_bytes": 44528128,
-  "peak_train_reserved_bytes": 69206016,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed42_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed42_n100/results.json
deleted file mode 100644
index 3941815..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed42_n100/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": 100,
-  "train_examples": 100,
-  "best_val_accuracy": 0.1437,
-  "test_c": {
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-  "test": {
-    "loss": 2.303402734375,
-    "accuracy": 0.1355
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-  "test_nr": {
-    "loss": 2.303402734375,
-    "accuracy": 0.1355
-  },
-  "test_r": {
-    "loss": 2.30357861328125,
-    "accuracy": 0.1353
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-  "rotation_robustness": {
-    "rot_00": 0.1353,
-    "rot_01": 0.1353,
-    "rot_02": 0.1353,
-    "rot_03": 0.1353,
-    "rot_04": 0.1353,
-    "rot_05": 0.1352,
-    "rot_06": 0.1353,
-    "rot_07": 0.1353,
-    "rot_08": 0.1353,
-    "rot_09": 0.1352,
-    "mean_accuracy": 0.13528,
-    "std_accuracy": 4.00000000000067e-05
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed456_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed456_n100/results.json
deleted file mode 100644
index 4cf9166..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed456_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": 100,
-  "train_examples": 100,
-  "best_val_accuracy": 0.1714,
-  "test_c": {
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-  "test": {
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-    "accuracy": 0.183
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-  "test_nr": {
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-    "accuracy": 0.183
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-    "rot_01": 0.188,
-    "rot_02": 0.1875,
-    "rot_03": 0.1882,
-    "rot_04": 0.1885,
-    "rot_05": 0.1877,
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-    "rot_07": 0.1875,
-    "rot_08": 0.1878,
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-  "peak_train_alloc_bytes": 44528128,
-  "peak_train_reserved_bytes": 69206016,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed789_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed789_n100/results.json
deleted file mode 100644
index bae6a7b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_R_seed789_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": 100,
-  "train_examples": 100,
-  "best_val_accuracy": 0.0991,
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-    "rot_08": 0.098,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed101112_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed101112_n100/results.json
deleted file mode 100644
index 50214ca..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed101112_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": 100,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed123_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed123_n100/results.json
deleted file mode 100644
index 2f72402..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed123_n100/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
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-    "rot_03": 0.1775,
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-    "rot_09": 0.1774,
-    "mean_accuracy": 0.17741,
-    "std_accuracy": 0.00012999999999999636
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed42_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed42_n100/results.json
deleted file mode 100644
index a972e6e..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed42_n100/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed456_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed456_n100/results.json
deleted file mode 100644
index ded12bf..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed456_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
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-  "argv": [
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-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed789_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed789_n100/results.json
deleted file mode 100644
index e79d1c2..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_C_seed789_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
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-  "test": {
-    "loss": 2.2959814453125,
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-    "accuracy": 0.1548
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-    "rot_03": 0.1544,
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-    "rot_07": 0.1544,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed101112_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed101112_n100/results.json
deleted file mode 100644
index c546664..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed101112_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
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-  "train_examples": 100,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed123_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed123_n100/results.json
deleted file mode 100644
index 8c04109..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed123_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 123,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed42_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed42_n100/results.json
deleted file mode 100644
index c43e186..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed42_n100/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed456_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed456_n100/results.json
deleted file mode 100644
index 49e79a1..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed456_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed789_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed789_n100/results.json
deleted file mode 100644
index e5c1581..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/power_spectrum_matched_R_seed789_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 789,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed101112_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed101112_n100/results.json
deleted file mode 100644
index b5cd014..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed101112_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 101112,
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-  "argv": [
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-    "--run_label",
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-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
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-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
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-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed123_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed123_n100/results.json
deleted file mode 100644
index f142a97..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed123_n100/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 123,
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-    "rot_03": 0.1008,
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-    "rot_05": 0.1038,
-    "rot_06": 0.0807,
-    "rot_07": 0.1028,
-    "rot_08": 0.0465,
-    "rot_09": 0.1028,
-    "mean_accuracy": 0.08743,
-    "std_accuracy": 0.024748092855814162
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed42_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed42_n100/results.json
deleted file mode 100644
index 00295d8..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed42_n100/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
-  "n_params": 184778,
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-    "rot_09": 0.1028,
-    "mean_accuracy": 0.11826,
-    "std_accuracy": 0.027215480888641307
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed456_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed456_n100/results.json
deleted file mode 100644
index 2eade6c..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed456_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
-  "n_params": 184778,
-  "invariant_dim": null,
-  "train_size": 100,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed789_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed789_n100/results.json
deleted file mode 100644
index b105730..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_C_seed789_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 184778,
-  "invariant_dim": null,
-  "train_size": 100,
-  "train_examples": 100,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed101112_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed101112_n100/results.json
deleted file mode 100644
index 2cfd22c..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed101112_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
-  "n_params": 184778,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed123_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed123_n100/results.json
deleted file mode 100644
index 4d4866e..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed123_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 123,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed42_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed42_n100/results.json
deleted file mode 100644
index ef77081..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed42_n100/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
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-  "test_rotation_seed": 777,
-  "seed": 42,
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-    "rot_09": 0.189,
-    "mean_accuracy": 0.16277999999999998,
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed456_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed456_n100/results.json
deleted file mode 100644
index c0aa78b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed456_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed789_n100/results.json b/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed789_n100/results.json
deleted file mode 100644
index 973e5c7..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_100/standard_R_seed789_n100/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
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-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 789,
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-    "789",
-    "--hidden",
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-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_100",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "100"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed101112_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed101112_n12500/results.json
deleted file mode 100644
index f5a8369..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed101112_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "base_model": "bispectrum",
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed123_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed123_n12500/results.json
deleted file mode 100644
index 00ed4d4..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed123_n12500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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-    "rot_05": 0.924,
-    "rot_06": 0.924,
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-    "rot_09": 0.924,
-    "mean_accuracy": 0.9239200000000001,
-    "std_accuracy": 0.000140000000000029
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed42_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed42_n12500/results.json
deleted file mode 100644
index 39a8d21..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed42_n12500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed456_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed456_n12500/results.json
deleted file mode 100644
index d0723da..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed456_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed789_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed789_n12500/results.json
deleted file mode 100644
index 4ebfd3b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_C_seed789_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed101112_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed101112_n12500/results.json
deleted file mode 100644
index 7ee50f6..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed101112_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed123_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed123_n12500/results.json
deleted file mode 100644
index a8a4cf0..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed123_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "base_model": "bispectrum",
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed42_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed42_n12500/results.json
deleted file mode 100644
index 9dce768..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed42_n12500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed456_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed456_n12500/results.json
deleted file mode 100644
index 215e0fb..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed456_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
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-  "seed": 456,
-  "lmax": 15,
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-  "test_nr": {
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-    "rot_06": 0.925,
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-    "rot_09": 0.9253,
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-    "std_accuracy": 0.00012206555615732358
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-  "epochs_run": 20,
-  "peak_train_alloc_bytes": 421850624,
-  "peak_train_reserved_bytes": 847249408,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed789_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed789_n12500/results.json
deleted file mode 100644
index 4570912..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/bispectrum_R_seed789_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 789,
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-  "invariant_dim": 768,
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-  "test_nr": {
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-  "test_r": {
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-    "rot_01": 0.9235,
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-    "rot_03": 0.9236,
-    "rot_04": 0.9235,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed101112_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed101112_n12500/results.json
deleted file mode 100644
index c0f5d51..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed101112_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "base_model": "power_spectrum",
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed123_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed123_n12500/results.json
deleted file mode 100644
index 220f336..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed123_n12500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
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-    "rot_09": 0.7392,
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-    "std_accuracy": 0.00037094473981982504
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed42_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed42_n12500/results.json
deleted file mode 100644
index bb4f7e0..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed42_n12500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed456_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed456_n12500/results.json
deleted file mode 100644
index 4cc30fb..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed456_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed789_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed789_n12500/results.json
deleted file mode 100644
index 486eb53..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_C_seed789_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "argv": [
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-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed101112_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed101112_n12500/results.json
deleted file mode 100644
index 772f94b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed101112_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "train_mode": "R",
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-  "argv": [
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-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed123_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed123_n12500/results.json
deleted file mode 100644
index f03b460..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed123_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 123,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed42_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed42_n12500/results.json
deleted file mode 100644
index e9eb82b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed42_n12500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 42,
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed456_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed456_n12500/results.json
deleted file mode 100644
index e56c933..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed456_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed789_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed789_n12500/results.json
deleted file mode 100644
index 07d4f53..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_R_seed789_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed101112_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed101112_n12500/results.json
deleted file mode 100644
index fb8628c..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed101112_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed123_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed123_n12500/results.json
deleted file mode 100644
index 5a60aae..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed123_n12500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed42_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed42_n12500/results.json
deleted file mode 100644
index 0dac16b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed42_n12500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed456_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed456_n12500/results.json
deleted file mode 100644
index 1f6acfc..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed456_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed789_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed789_n12500/results.json
deleted file mode 100644
index 6100f8c..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_C_seed789_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 231906,
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-  "peak_train_reserved_bytes": 73400320,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_R_seed101112_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_R_seed101112_n12500/results.json
deleted file mode 100644
index 661ae92..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_R_seed101112_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_R_seed123_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_R_seed123_n12500/results.json
deleted file mode 100644
index 437c4d5..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_R_seed123_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
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-  "invariant_dim": 16,
-  "train_size": 12500,
-  "train_examples": 12500,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_R_seed42_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_R_seed42_n12500/results.json
deleted file mode 100644
index 7d68357..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_R_seed42_n12500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
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-  "invariant_dim": 16,
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-  "test_nr": {
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-  "test_r": {
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-    "rot_09": 0.7677,
-    "mean_accuracy": 0.7678499999999999,
-    "std_accuracy": 0.00022912878474779584
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_R_seed789_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_R_seed789_n12500/results.json
deleted file mode 100644
index 56d74c8..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/power_spectrum_matched_R_seed789_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed101112_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed101112_n12500/results.json
deleted file mode 100644
index cd76ea1..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed101112_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 101112,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed123_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed123_n12500/results.json
deleted file mode 100644
index cbc5f2d..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed123_n12500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "test_rotation_seed": 777,
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed42_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed42_n12500/results.json
deleted file mode 100644
index 205a50d..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed42_n12500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed456_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed456_n12500/results.json
deleted file mode 100644
index ab21bd1..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed456_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
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-  "test_rotation_seed": 777,
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-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed789_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed789_n12500/results.json
deleted file mode 100644
index 3d9c5d3..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_C_seed789_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
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-  "seed": 789,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_R_seed101112_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_R_seed101112_n12500/results.json
deleted file mode 100644
index 8f2cebb..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_R_seed101112_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_R_seed123_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_R_seed123_n12500/results.json
deleted file mode 100644
index ea65772..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_R_seed123_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "base_model": "standard",
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-  "test_rotation_seed": 777,
-  "seed": 123,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_R_seed42_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_R_seed42_n12500/results.json
deleted file mode 100644
index 4dcb715..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_R_seed42_n12500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "seed": 42,
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_R_seed456_n12500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_R_seed456_n12500/results.json
deleted file mode 100644
index e8004fd..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_12500/standard_R_seed456_n12500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
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-  "test_rotation_seed": 777,
-  "seed": 456,
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-    "--run_label",
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-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_12500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "12500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed101112_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed101112_n2500/results.json
deleted file mode 100644
index 28bb83e..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed101112_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "argv": [
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-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed123_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed123_n2500/results.json
deleted file mode 100644
index dc23fb8..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed123_n2500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed42_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed42_n2500/results.json
deleted file mode 100644
index 3ede6db..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed42_n2500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-    "mean_accuracy": 0.88622,
-    "std_accuracy": 0.0001599999999999824
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed456_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed456_n2500/results.json
deleted file mode 100644
index fd6a4c2..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed456_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 2500,
-  "train_examples": 2500,
-  "best_val_accuracy": 0.8988,
-  "test_c": {
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-    "accuracy": 0.8893
-  },
-  "test": {
-    "loss": 0.3522503648757935,
-    "accuracy": 0.8893
-  },
-  "test_nr": {
-    "loss": 0.3522503648757935,
-    "accuracy": 0.8893
-  },
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-    "rot_03": 0.8899,
-    "rot_04": 0.8896,
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-    "rot_07": 0.8898,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed789_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed789_n2500/results.json
deleted file mode 100644
index 9038b4d..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_C_seed789_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 2500,
-  "train_examples": 2500,
-  "best_val_accuracy": 0.8969,
-  "test_c": {
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-    "accuracy": 0.8865
-  },
-  "test": {
-    "loss": 0.3666726207733154,
-    "accuracy": 0.8865
-  },
-  "test_nr": {
-    "loss": 0.3666726207733154,
-    "accuracy": 0.8865
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-    "loss": 0.36757737998962403,
-    "accuracy": 0.886
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-  "rotation_robustness": {
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-    "rot_01": 0.8862,
-    "rot_02": 0.886,
-    "rot_03": 0.8864,
-    "rot_04": 0.8864,
-    "rot_05": 0.8863,
-    "rot_06": 0.8863,
-    "rot_07": 0.8864,
-    "rot_08": 0.8862,
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-  "peak_train_alloc_bytes": 421850624,
-  "peak_train_reserved_bytes": 847249408,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed101112_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed101112_n2500/results.json
deleted file mode 100644
index 76b91ad..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed101112_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 2500,
-  "train_examples": 2500,
-  "best_val_accuracy": 0.8947,
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-  "test": {
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-  "test_nr": {
-    "loss": 0.36486334867477416,
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-  "test_r": {
-    "loss": 0.3648377288818359,
-    "accuracy": 0.8899
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-  "rotation_robustness": {
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-    "rot_01": 0.8899,
-    "rot_02": 0.8899,
-    "rot_03": 0.8899,
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-    "rot_06": 0.8898,
-    "rot_07": 0.8898,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed123_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed123_n2500/results.json
deleted file mode 100644
index be5d6e3..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed123_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
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-  "train_examples": 2500,
-  "best_val_accuracy": 0.8961,
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-  "test_nr": {
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-  "test_r": {
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-    "accuracy": 0.8988
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-  "rotation_robustness": {
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-    "rot_01": 0.8988,
-    "rot_02": 0.899,
-    "rot_03": 0.8989,
-    "rot_04": 0.8989,
-    "rot_05": 0.8988,
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-    "rot_07": 0.8988,
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-  "peak_train_alloc_bytes": 421850624,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed42_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed42_n2500/results.json
deleted file mode 100644
index e3457ff..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed42_n2500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 42,
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-  "rotation_robustness": {
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-    "rot_01": 0.8836,
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-    "rot_03": 0.8837,
-    "rot_04": 0.8834,
-    "rot_05": 0.8834,
-    "rot_06": 0.8837,
-    "rot_07": 0.8835,
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-    "mean_accuracy": 0.8835000000000001,
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed456_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed456_n2500/results.json
deleted file mode 100644
index 589a709..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed456_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
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-  "invariant_dim": 768,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed789_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed789_n2500/results.json
deleted file mode 100644
index f28b46e..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/bispectrum_R_seed789_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed101112_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed101112_n2500/results.json
deleted file mode 100644
index 7bb94a2..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed101112_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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-    "101112",
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-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed123_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed123_n2500/results.json
deleted file mode 100644
index 41c94bf..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed123_n2500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 123,
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-    "rot_08": 0.6623,
-    "rot_09": 0.6626,
-    "mean_accuracy": 0.6622,
-    "std_accuracy": 0.0002097617696340072
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed42_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed42_n2500/results.json
deleted file mode 100644
index 84ce88c..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed42_n2500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 42,
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-    "rot_09": 0.6853,
-    "mean_accuracy": 0.68559,
-    "std_accuracy": 0.0003330165161069349
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed456_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed456_n2500/results.json
deleted file mode 100644
index 39ed06b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed456_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 456,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed789_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed789_n2500/results.json
deleted file mode 100644
index ad900da..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_C_seed789_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed101112_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed101112_n2500/results.json
deleted file mode 100644
index 599b12e..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed101112_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
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-  "argv": [
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-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed123_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed123_n2500/results.json
deleted file mode 100644
index 84d566b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed123_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "argv": [
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-    "power_spectrum",
-    "--run_label",
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-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed42_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed42_n2500/results.json
deleted file mode 100644
index cd56f13..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed42_n2500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed456_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed456_n2500/results.json
deleted file mode 100644
index 08be4cf..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed456_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
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-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
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-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed789_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed789_n2500/results.json
deleted file mode 100644
index bc17ab2..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_R_seed789_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 11466,
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-  "train_examples": 2500,
-  "best_val_accuracy": 0.7029,
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-  "test": {
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-  },
-  "test_nr": {
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-  },
-  "test_r": {
-    "loss": 0.9316224723815918,
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-    "rot_01": 0.6916,
-    "rot_02": 0.6921,
-    "rot_03": 0.6923,
-    "rot_04": 0.6917,
-    "rot_05": 0.692,
-    "rot_06": 0.6912,
-    "rot_07": 0.6919,
-    "rot_08": 0.6918,
-    "rot_09": 0.6925,
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-    "std_accuracy": 0.0003666060555964704
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-  "peak_train_alloc_bytes": 43872768,
-  "peak_train_reserved_bytes": 69206016,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed101112_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed101112_n2500/results.json
deleted file mode 100644
index bdaf469..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed101112_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": 2500,
-  "train_examples": 2500,
-  "best_val_accuracy": 0.7101,
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-  "test": {
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-    "accuracy": 0.7017
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-  "test_nr": {
-    "loss": 0.8833447624206543,
-    "accuracy": 0.7017
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-  "test_r": {
-    "loss": 0.8815262313842773,
-    "accuracy": 0.7028
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-  "rotation_robustness": {
-    "rot_00": 0.7029,
-    "rot_01": 0.7032,
-    "rot_02": 0.7029,
-    "rot_03": 0.7029,
-    "rot_04": 0.7027,
-    "rot_05": 0.7028,
-    "rot_06": 0.7027,
-    "rot_07": 0.7026,
-    "rot_08": 0.7029,
-    "rot_09": 0.7029,
-    "mean_accuracy": 0.70285,
-    "std_accuracy": 0.00015652475842499287
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-  "wall_time_s": 13.141862154006958,
-  "epochs_run": 33,
-  "peak_train_alloc_bytes": 47850496,
-  "peak_train_reserved_bytes": 73400320,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed123_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed123_n2500/results.json
deleted file mode 100644
index 5037c99..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed123_n2500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
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-  "train_examples": 2500,
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-  "test_nr": {
-    "loss": 0.888081894493103,
-    "accuracy": 0.7028
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-  "test_r": {
-    "loss": 0.8892429985046387,
-    "accuracy": 0.7029
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-  "rotation_robustness": {
-    "rot_00": 0.7028,
-    "rot_01": 0.7026,
-    "rot_02": 0.7028,
-    "rot_03": 0.703,
-    "rot_04": 0.7027,
-    "rot_05": 0.703,
-    "rot_06": 0.7029,
-    "rot_07": 0.7024,
-    "rot_08": 0.7027,
-    "rot_09": 0.7028,
-    "mean_accuracy": 0.7027699999999999,
-    "std_accuracy": 0.00017349351572895562
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed42_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed42_n2500/results.json
deleted file mode 100644
index 536cdb0..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed42_n2500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": 2500,
-  "train_examples": 2500,
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-  "test_nr": {
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-    "accuracy": 0.697
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-  "test_r": {
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-    "accuracy": 0.6955
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-  "rotation_robustness": {
-    "rot_00": 0.6955,
-    "rot_01": 0.6959,
-    "rot_02": 0.6958,
-    "rot_03": 0.6955,
-    "rot_04": 0.6954,
-    "rot_05": 0.6958,
-    "rot_06": 0.6958,
-    "rot_07": 0.6959,
-    "rot_08": 0.6957,
-    "rot_09": 0.696,
-    "mean_accuracy": 0.69573,
-    "std_accuracy": 0.00018999999999997909
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed456_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed456_n2500/results.json
deleted file mode 100644
index 1cc4ffe..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed456_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": 2500,
-  "train_examples": 2500,
-  "best_val_accuracy": 0.7188,
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-    "loss": 0.8760274744033814,
-    "accuracy": 0.706
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-  "test": {
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-    "accuracy": 0.706
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-  "test_nr": {
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-    "accuracy": 0.706
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-  "test_r": {
-    "loss": 0.8793799341201782,
-    "accuracy": 0.7038
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-  "rotation_robustness": {
-    "rot_00": 0.704,
-    "rot_01": 0.7037,
-    "rot_02": 0.7036,
-    "rot_03": 0.7036,
-    "rot_04": 0.704,
-    "rot_05": 0.704,
-    "rot_06": 0.7034,
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-  "epochs_run": 39,
-  "peak_train_alloc_bytes": 47850496,
-  "peak_train_reserved_bytes": 73400320,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed789_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed789_n2500/results.json
deleted file mode 100644
index 6dc5a23..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_C_seed789_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 231906,
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-    "rot_01": 0.7062,
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-    "rot_03": 0.7065,
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-    "rot_05": 0.7065,
-    "rot_06": 0.7061,
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-  "epochs_run": 39,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed101112_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed101112_n2500/results.json
deleted file mode 100644
index ee47c51..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed101112_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
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-    "rot_04": 0.7044,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed123_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed123_n2500/results.json
deleted file mode 100644
index 384c0fc..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed123_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed42_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed42_n2500/results.json
deleted file mode 100644
index 3dfa8fe..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed42_n2500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
-  "n_params": 231906,
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-  "train_examples": 2500,
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-  "test_nr": {
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-    "rot_03": 0.6975,
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-    "rot_07": 0.6967,
-    "rot_08": 0.6971,
-    "rot_09": 0.697,
-    "mean_accuracy": 0.69714,
-    "std_accuracy": 0.00022891046284520845
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed456_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed456_n2500/results.json
deleted file mode 100644
index 3bafde9..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed456_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 456,
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-  "peak_train_alloc_bytes": 47850496,
-  "peak_train_reserved_bytes": 73400320,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed789_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed789_n2500/results.json
deleted file mode 100644
index 1249832..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/power_spectrum_matched_R_seed789_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
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-  "test_nr": {
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-  "test_r": {
-    "loss": 0.8896934476852417,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed101112_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed101112_n2500/results.json
deleted file mode 100644
index 1bc35fb..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed101112_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 101112,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed123_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed123_n2500/results.json
deleted file mode 100644
index ccc4d2c..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed123_n2500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 123,
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-    "rot_03": 0.1353,
-    "rot_04": 0.0886,
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-    "rot_06": 0.1326,
-    "rot_07": 0.0898,
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-    "rot_09": 0.1037,
-    "mean_accuracy": 0.09767,
-    "std_accuracy": 0.025958815458337076
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed42_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed42_n2500/results.json
deleted file mode 100644
index 33790f9..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed42_n2500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 42,
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed456_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed456_n2500/results.json
deleted file mode 100644
index b4e8f0b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed456_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
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-  "best_val_accuracy": 0.9729,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed789_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed789_n2500/results.json
deleted file mode 100644
index 25be709..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_C_seed789_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed101112_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed101112_n2500/results.json
deleted file mode 100644
index b22958f..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed101112_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
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-  "test_rotation_seed": 777,
-  "seed": 101112,
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-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
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-    "--epochs",
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-    "--train_size",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed123_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed123_n2500/results.json
deleted file mode 100644
index dbdc1d5..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed123_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "base_model": "standard",
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
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-    "standard",
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-    "R",
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-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed42_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed42_n2500/results.json
deleted file mode 100644
index 9f71e0a..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed42_n2500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
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-  "test_rotation_seed": 777,
-  "seed": 42,
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed456_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed456_n2500/results.json
deleted file mode 100644
index 6d59ddb..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed456_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "test_rotation_seed": 777,
-  "seed": 456,
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-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed789_n2500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed789_n2500/results.json
deleted file mode 100644
index 7078644..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_2500/standard_R_seed789_n2500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "test_rotation_seed": 777,
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-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_2500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "2500"
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed101112_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed101112_n500/results.json
deleted file mode 100644
index 0cad805..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed101112_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "base_model": "bispectrum",
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-    "--run_label",
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-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed123_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed123_n500/results.json
deleted file mode 100644
index b9ee7d6..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed123_n500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "base_model": "bispectrum",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed42_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed42_n500/results.json
deleted file mode 100644
index 1e1f0c0..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed42_n500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed456_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed456_n500/results.json
deleted file mode 100644
index f98c72f..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed456_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed789_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed789_n500/results.json
deleted file mode 100644
index 173925e..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_C_seed789_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.1101,
-  "test_c": {
-    "loss": 2.302341796875,
-    "accuracy": 0.1182
-  },
-  "test": {
-    "loss": 2.302341796875,
-    "accuracy": 0.1182
-  },
-  "test_nr": {
-    "loss": 2.302341796875,
-    "accuracy": 0.1182
-  },
-  "test_r": {
-    "loss": 2.3023466796875,
-    "accuracy": 0.1181
-  },
-  "rotation_robustness": {
-    "rot_00": 0.1181,
-    "rot_01": 0.118,
-    "rot_02": 0.1181,
-    "rot_03": 0.1181,
-    "rot_04": 0.118,
-    "rot_05": 0.118,
-    "rot_06": 0.1181,
-    "rot_07": 0.118,
-    "rot_08": 0.118,
-    "rot_09": 0.1181,
-    "mean_accuracy": 0.11804999999999999,
-    "std_accuracy": 5.000000000000143e-05
-  },
-  "wall_time_s": 5.984099626541138,
-  "epochs_run": 11,
-  "peak_train_alloc_bytes": 421850624,
-  "peak_train_reserved_bytes": 847249408,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed101112_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed101112_n500/results.json
deleted file mode 100644
index 1869e83..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed101112_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.14,
-  "test_c": {
-    "loss": 2.33719111328125,
-    "accuracy": 0.149
-  },
-  "test": {
-    "loss": 2.33719111328125,
-    "accuracy": 0.149
-  },
-  "test_nr": {
-    "loss": 2.33719111328125,
-    "accuracy": 0.149
-  },
-  "test_r": {
-    "loss": 2.33738251953125,
-    "accuracy": 0.1478
-  },
-  "rotation_robustness": {
-    "rot_00": 0.1476,
-    "rot_01": 0.1477,
-    "rot_02": 0.1476,
-    "rot_03": 0.1478,
-    "rot_04": 0.1477,
-    "rot_05": 0.1477,
-    "rot_06": 0.1477,
-    "rot_07": 0.1477,
-    "rot_08": 0.1477,
-    "rot_09": 0.1476,
-    "mean_accuracy": 0.14767999999999998,
-    "std_accuracy": 5.9999999999993395e-05
-  },
-  "wall_time_s": 9.639845371246338,
-  "epochs_run": 22,
-  "peak_train_alloc_bytes": 421850624,
-  "peak_train_reserved_bytes": 847249408,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed123_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed123_n500/results.json
deleted file mode 100644
index df9af81..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed123_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.1195,
-  "test_c": {
-    "loss": 2.304600390625,
-    "accuracy": 0.1108
-  },
-  "test": {
-    "loss": 2.304600390625,
-    "accuracy": 0.1108
-  },
-  "test_nr": {
-    "loss": 2.304600390625,
-    "accuracy": 0.1108
-  },
-  "test_r": {
-    "loss": 2.304621484375,
-    "accuracy": 0.1108
-  },
-  "rotation_robustness": {
-    "rot_00": 0.1111,
-    "rot_01": 0.1109,
-    "rot_02": 0.1108,
-    "rot_03": 0.1108,
-    "rot_04": 0.1108,
-    "rot_05": 0.1109,
-    "rot_06": 0.1107,
-    "rot_07": 0.1108,
-    "rot_08": 0.1109,
-    "rot_09": 0.111,
-    "mean_accuracy": 0.11087,
-    "std_accuracy": 0.00011000000000000101
-  },
-  "wall_time_s": 5.808589220046997,
-  "epochs_run": 13,
-  "peak_train_alloc_bytes": 421850624,
-  "peak_train_reserved_bytes": 847249408,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed42_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed42_n500/results.json
deleted file mode 100644
index 8d3ea33..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed42_n500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.164,
-  "test_c": {
-    "loss": 2.3085234375,
-    "accuracy": 0.1744
-  },
-  "test": {
-    "loss": 2.3085234375,
-    "accuracy": 0.1744
-  },
-  "test_nr": {
-    "loss": 2.3085234375,
-    "accuracy": 0.1744
-  },
-  "test_r": {
-    "loss": 2.308594140625,
-    "accuracy": 0.1744
-  },
-  "rotation_robustness": {
-    "rot_00": 0.1745,
-    "rot_01": 0.1744,
-    "rot_02": 0.1745,
-    "rot_03": 0.1743,
-    "rot_04": 0.1744,
-    "rot_05": 0.1743,
-    "rot_06": 0.1745,
-    "rot_07": 0.1744,
-    "rot_08": 0.1744,
-    "rot_09": 0.1743,
-    "mean_accuracy": 0.1744,
-    "std_accuracy": 7.745966692413979e-05
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed456_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed456_n500/results.json
deleted file mode 100644
index e22e677..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed456_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.1356,
-  "test_c": {
-    "loss": 2.33190791015625,
-    "accuracy": 0.1415
-  },
-  "test": {
-    "loss": 2.33190791015625,
-    "accuracy": 0.1415
-  },
-  "test_nr": {
-    "loss": 2.33190791015625,
-    "accuracy": 0.1415
-  },
-  "test_r": {
-    "loss": 2.33207705078125,
-    "accuracy": 0.1403
-  },
-  "rotation_robustness": {
-    "rot_00": 0.1404,
-    "rot_01": 0.1403,
-    "rot_02": 0.1404,
-    "rot_03": 0.1403,
-    "rot_04": 0.1403,
-    "rot_05": 0.1405,
-    "rot_06": 0.1403,
-    "rot_07": 0.1404,
-    "rot_08": 0.1405,
-    "rot_09": 0.1405,
-    "mean_accuracy": 0.14039000000000001,
-    "std_accuracy": 8.306623862918262e-05
-  },
-  "wall_time_s": 10.611043214797974,
-  "epochs_run": 22,
-  "peak_train_alloc_bytes": 421850624,
-  "peak_train_reserved_bytes": 847249408,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed789_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed789_n500/results.json
deleted file mode 100644
index c197cca..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/bispectrum_R_seed789_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.1233,
-  "test_c": {
-    "loss": 2.3022970703125,
-    "accuracy": 0.1376
-  },
-  "test": {
-    "loss": 2.3022970703125,
-    "accuracy": 0.1376
-  },
-  "test_nr": {
-    "loss": 2.3022970703125,
-    "accuracy": 0.1376
-  },
-  "test_r": {
-    "loss": 2.3023015625,
-    "accuracy": 0.1375
-  },
-  "rotation_robustness": {
-    "rot_00": 0.1375,
-    "rot_01": 0.1373,
-    "rot_02": 0.1374,
-    "rot_03": 0.1375,
-    "rot_04": 0.1375,
-    "rot_05": 0.1374,
-    "rot_06": 0.1375,
-    "rot_07": 0.1373,
-    "rot_08": 0.1374,
-    "rot_09": 0.1373,
-    "mean_accuracy": 0.13741,
-    "std_accuracy": 8.306623862918363e-05
-  },
-  "wall_time_s": 5.656994819641113,
-  "epochs_run": 11,
-  "peak_train_alloc_bytes": 421850624,
-  "peak_train_reserved_bytes": 847249408,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed101112_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed101112_n500/results.json
deleted file mode 100644
index 08d3da3..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed101112_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.5627,
-  "test_c": {
-    "loss": 1.325930438232422,
-    "accuracy": 0.5632
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-  "test": {
-    "loss": 1.325930438232422,
-    "accuracy": 0.5632
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-  "test_nr": {
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-    "accuracy": 0.5632
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-  "test_r": {
-    "loss": 1.3317452209472656,
-    "accuracy": 0.5597
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-  "rotation_robustness": {
-    "rot_00": 0.5595,
-    "rot_01": 0.5599,
-    "rot_02": 0.5598,
-    "rot_03": 0.5597,
-    "rot_04": 0.5605,
-    "rot_05": 0.5595,
-    "rot_06": 0.56,
-    "rot_07": 0.5598,
-    "rot_08": 0.5598,
-    "rot_09": 0.5599,
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-    "std_accuracy": 0.0002690724809414784
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-  "wall_time_s": 23.085957765579224,
-  "epochs_run": 50,
-  "peak_train_alloc_bytes": 43872768,
-  "peak_train_reserved_bytes": 69206016,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed123_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed123_n500/results.json
deleted file mode 100644
index 9931bb0..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed123_n500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
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-  "train_examples": 500,
-  "best_val_accuracy": 0.1768,
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-    "accuracy": 0.1746
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-  "test": {
-    "loss": 2.29853984375,
-    "accuracy": 0.1746
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-  "test_nr": {
-    "loss": 2.29853984375,
-    "accuracy": 0.1746
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-  "test_r": {
-    "loss": 2.298623046875,
-    "accuracy": 0.1738
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-  "rotation_robustness": {
-    "rot_00": 0.1739,
-    "rot_01": 0.1737,
-    "rot_02": 0.1739,
-    "rot_03": 0.174,
-    "rot_04": 0.1738,
-    "rot_05": 0.174,
-    "rot_06": 0.1737,
-    "rot_07": 0.1739,
-    "rot_08": 0.1739,
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-    "mean_accuracy": 0.17385,
-    "std_accuracy": 0.00011180339887498834
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed42_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed42_n500/results.json
deleted file mode 100644
index fd9f3e2..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed42_n500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.5827,
-  "test_c": {
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-    "accuracy": 0.5787
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-  "test": {
-    "loss": 1.3080652923583984,
-    "accuracy": 0.5787
-  },
-  "test_nr": {
-    "loss": 1.3080652923583984,
-    "accuracy": 0.5787
-  },
-  "test_r": {
-    "loss": 1.3179074584960937,
-    "accuracy": 0.5732
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-  "rotation_robustness": {
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-    "rot_01": 0.573,
-    "rot_02": 0.5728,
-    "rot_03": 0.5729,
-    "rot_04": 0.5728,
-    "rot_05": 0.5733,
-    "rot_06": 0.5734,
-    "rot_07": 0.5731,
-    "rot_08": 0.5732,
-    "rot_09": 0.5736,
-    "mean_accuracy": 0.5731200000000001,
-    "std_accuracy": 0.0002481934729198337
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed456_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed456_n500/results.json
deleted file mode 100644
index 0033718..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed456_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": 500,
-  "train_examples": 500,
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-  "test_nr": {
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-  "peak_train_reserved_bytes": 69206016,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed789_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed789_n500/results.json
deleted file mode 100644
index 1a87b4c..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_C_seed789_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
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-  "train_examples": 500,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed101112_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed101112_n500/results.json
deleted file mode 100644
index 840c867..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed101112_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": 500,
-  "train_examples": 500,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed123_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed123_n500/results.json
deleted file mode 100644
index cef9412..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed123_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
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-  "invariant_dim": 16,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.1788,
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-  "test": {
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-    "accuracy": 0.1749
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-  "test_nr": {
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-    "accuracy": 0.1749
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed42_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed42_n500/results.json
deleted file mode 100644
index faded54..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed42_n500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 42,
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-  "test_nr": {
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-  "test_r": {
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-    "rot_01": 0.5857,
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-    "rot_04": 0.5859,
-    "rot_05": 0.5854,
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-    "rot_09": 0.5859,
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-    "std_accuracy": 0.00026551836094702885
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed456_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed456_n500/results.json
deleted file mode 100644
index 770c637..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed456_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed789_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed789_n500/results.json
deleted file mode 100644
index 9a431ec..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_R_seed789_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed101112_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed101112_n500/results.json
deleted file mode 100644
index c6059a5..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed101112_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.1936,
-  "test_c": {
-    "loss": 2.2951603515625,
-    "accuracy": 0.2042
-  },
-  "test": {
-    "loss": 2.2951603515625,
-    "accuracy": 0.2042
-  },
-  "test_nr": {
-    "loss": 2.2951603515625,
-    "accuracy": 0.2042
-  },
-  "test_r": {
-    "loss": 2.295191796875,
-    "accuracy": 0.2044
-  },
-  "rotation_robustness": {
-    "rot_00": 0.2046,
-    "rot_01": 0.2044,
-    "rot_02": 0.2048,
-    "rot_03": 0.2045,
-    "rot_04": 0.2049,
-    "rot_05": 0.2045,
-    "rot_06": 0.2048,
-    "rot_07": 0.2049,
-    "rot_08": 0.2045,
-    "rot_09": 0.2047,
-    "mean_accuracy": 0.20465999999999998,
-    "std_accuracy": 0.00017435595774163161
-  },
-  "wall_time_s": 5.538880348205566,
-  "epochs_run": 11,
-  "peak_train_alloc_bytes": 47850496,
-  "peak_train_reserved_bytes": 73400320,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed123_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed123_n500/results.json
deleted file mode 100644
index ef0f0ae..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed123_n500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.1915,
-  "test_c": {
-    "loss": 2.2894796875,
-    "accuracy": 0.1803
-  },
-  "test": {
-    "loss": 2.2894796875,
-    "accuracy": 0.1803
-  },
-  "test_nr": {
-    "loss": 2.2894796875,
-    "accuracy": 0.1803
-  },
-  "test_r": {
-    "loss": 2.289565234375,
-    "accuracy": 0.1798
-  },
-  "rotation_robustness": {
-    "rot_00": 0.1797,
-    "rot_01": 0.1796,
-    "rot_02": 0.1796,
-    "rot_03": 0.1796,
-    "rot_04": 0.1796,
-    "rot_05": 0.1795,
-    "rot_06": 0.1796,
-    "rot_07": 0.1797,
-    "rot_08": 0.1798,
-    "rot_09": 0.1797,
-    "mean_accuracy": 0.17964,
-    "std_accuracy": 7.999999999999605e-05
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed42_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed42_n500/results.json
deleted file mode 100644
index 2ce198a..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed42_n500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.1727,
-  "test_c": {
-    "loss": 2.2891583984375,
-    "accuracy": 0.1745
-  },
-  "test": {
-    "loss": 2.2891583984375,
-    "accuracy": 0.1745
-  },
-  "test_nr": {
-    "loss": 2.2891583984375,
-    "accuracy": 0.1745
-  },
-  "test_r": {
-    "loss": 2.289266796875,
-    "accuracy": 0.174
-  },
-  "rotation_robustness": {
-    "rot_00": 0.1742,
-    "rot_01": 0.1741,
-    "rot_02": 0.1741,
-    "rot_03": 0.174,
-    "rot_04": 0.1742,
-    "rot_05": 0.1741,
-    "rot_06": 0.1741,
-    "rot_07": 0.1741,
-    "rot_08": 0.1742,
-    "rot_09": 0.1741,
-    "mean_accuracy": 0.17412,
-    "std_accuracy": 5.9999999999998944e-05
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed456_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed456_n500/results.json
deleted file mode 100644
index 58fc2fc..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed456_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.1748,
-  "test_c": {
-    "loss": 2.33664560546875,
-    "accuracy": 0.1709
-  },
-  "test": {
-    "loss": 2.33664560546875,
-    "accuracy": 0.1709
-  },
-  "test_nr": {
-    "loss": 2.33664560546875,
-    "accuracy": 0.1709
-  },
-  "test_r": {
-    "loss": 2.336871875,
-    "accuracy": 0.1709
-  },
-  "rotation_robustness": {
-    "rot_00": 0.1707,
-    "rot_01": 0.1707,
-    "rot_02": 0.1711,
-    "rot_03": 0.171,
-    "rot_04": 0.1707,
-    "rot_05": 0.1709,
-    "rot_06": 0.1709,
-    "rot_07": 0.171,
-    "rot_08": 0.171,
-    "rot_09": 0.1708,
-    "mean_accuracy": 0.17088,
-    "std_accuracy": 0.0001400000000000068
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-  "wall_time_s": 9.17716646194458,
-  "epochs_run": 17,
-  "peak_train_alloc_bytes": 47850496,
-  "peak_train_reserved_bytes": 73400320,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed789_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed789_n500/results.json
deleted file mode 100644
index 0662fdc..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_C_seed789_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.1605,
-  "test_c": {
-    "loss": 2.2879716796875,
-    "accuracy": 0.1517
-  },
-  "test": {
-    "loss": 2.2879716796875,
-    "accuracy": 0.1517
-  },
-  "test_nr": {
-    "loss": 2.2879716796875,
-    "accuracy": 0.1517
-  },
-  "test_r": {
-    "loss": 2.288071484375,
-    "accuracy": 0.1518
-  },
-  "rotation_robustness": {
-    "rot_00": 0.1517,
-    "rot_01": 0.1517,
-    "rot_02": 0.1516,
-    "rot_03": 0.1518,
-    "rot_04": 0.1517,
-    "rot_05": 0.1517,
-    "rot_06": 0.1518,
-    "rot_07": 0.1517,
-    "rot_08": 0.1516,
-    "rot_09": 0.1517,
-    "mean_accuracy": 0.1517,
-    "std_accuracy": 6.324555320336061e-05
-  },
-  "wall_time_s": 6.623900413513184,
-  "epochs_run": 13,
-  "peak_train_alloc_bytes": 47850496,
-  "peak_train_reserved_bytes": 73400320,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed101112_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed101112_n500/results.json
deleted file mode 100644
index 6232b4d..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed101112_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.1863,
-  "test_c": {
-    "loss": 2.29513828125,
-    "accuracy": 0.1993
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-  "test": {
-    "loss": 2.29513828125,
-    "accuracy": 0.1993
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-  "test_nr": {
-    "loss": 2.29513828125,
-    "accuracy": 0.1993
-  },
-  "test_r": {
-    "loss": 2.2951626953125,
-    "accuracy": 0.1993
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-  "rotation_robustness": {
-    "rot_00": 0.199,
-    "rot_01": 0.1993,
-    "rot_02": 0.1991,
-    "rot_03": 0.1992,
-    "rot_04": 0.199,
-    "rot_05": 0.199,
-    "rot_06": 0.1992,
-    "rot_07": 0.1992,
-    "rot_08": 0.1993,
-    "rot_09": 0.1991,
-    "mean_accuracy": 0.19914,
-    "std_accuracy": 0.00011135528725659615
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-  "wall_time_s": 7.064088821411133,
-  "epochs_run": 11,
-  "peak_train_alloc_bytes": 47850496,
-  "peak_train_reserved_bytes": 73400320,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed123_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed123_n500/results.json
deleted file mode 100644
index d0057b0..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed123_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.1921,
-  "test_c": {
-    "loss": 2.2895615234375,
-    "accuracy": 0.1825
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-  "test": {
-    "loss": 2.2895615234375,
-    "accuracy": 0.1825
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-  "test_nr": {
-    "loss": 2.2895615234375,
-    "accuracy": 0.1825
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-  "test_r": {
-    "loss": 2.2896412109375,
-    "accuracy": 0.1822
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-  "rotation_robustness": {
-    "rot_00": 0.1823,
-    "rot_01": 0.1823,
-    "rot_02": 0.1823,
-    "rot_03": 0.1823,
-    "rot_04": 0.1822,
-    "rot_05": 0.1823,
-    "rot_06": 0.1824,
-    "rot_07": 0.1826,
-    "rot_08": 0.1825,
-    "rot_09": 0.1822,
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-    "std_accuracy": 0.0001200000000000039
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-  "epochs_run": 13,
-  "peak_train_alloc_bytes": 47850496,
-  "peak_train_reserved_bytes": 73400320,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed42_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed42_n500/results.json
deleted file mode 100644
index 8808a84..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed42_n500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
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-  "best_val_accuracy": 0.1742,
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-  "test": {
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-  "test_nr": {
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-  "test_r": {
-    "loss": 2.2895271484375,
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-  "rotation_robustness": {
-    "rot_00": 0.1763,
-    "rot_01": 0.1763,
-    "rot_02": 0.1764,
-    "rot_03": 0.1763,
-    "rot_04": 0.1763,
-    "rot_05": 0.1763,
-    "rot_06": 0.1763,
-    "rot_07": 0.1762,
-    "rot_08": 0.1763,
-    "rot_09": 0.1762,
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-    "std_accuracy": 5.3851648071348386e-05
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed456_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed456_n500/results.json
deleted file mode 100644
index 4894b8a..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed456_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": 500,
-  "train_examples": 500,
-  "best_val_accuracy": 0.1756,
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-  "test": {
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-  "test_nr": {
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-    "accuracy": 0.1719
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-  "test_r": {
-    "loss": 2.31620224609375,
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-    "--hidden",
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-    "--lmax",
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-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed789_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed789_n500/results.json
deleted file mode 100644
index a47842b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/power_spectrum_matched_R_seed789_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
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-    "rot_07": 0.1507,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed101112_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed101112_n500/results.json
deleted file mode 100644
index 1f12204..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed101112_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 101112,
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-    "rot_04": 0.0582,
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-    "rot_06": 0.0981,
-    "rot_07": 0.1401,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed123_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed123_n500/results.json
deleted file mode 100644
index 51a77c9..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed123_n500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 123,
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-    "rot_01": 0.116,
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-    "rot_03": 0.1489,
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-    "rot_06": 0.0921,
-    "rot_07": 0.1754,
-    "rot_08": 0.101,
-    "rot_09": 0.1018,
-    "mean_accuracy": 0.11668,
-    "std_accuracy": 0.02552449020058971
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed42_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed42_n500/results.json
deleted file mode 100644
index 3c30272..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed42_n500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 42,
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-  "n_params": 184778,
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed456_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed456_n500/results.json
deleted file mode 100644
index ba60d03..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed456_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 456,
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-  "invariant_dim": null,
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-  "train_examples": 500,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed789_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed789_n500/results.json
deleted file mode 100644
index 29eb9a0..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_C_seed789_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 789,
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-  "train_examples": 500,
-  "best_val_accuracy": 0.4924,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed101112_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed101112_n500/results.json
deleted file mode 100644
index ca31701..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed101112_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "base_model": "standard",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed123_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed123_n500/results.json
deleted file mode 100644
index e1bdcfc..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed123_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
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-  "test_rotation_seed": 777,
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-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed42_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed42_n500/results.json
deleted file mode 100644
index b9a18c7..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed42_n500/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 42,
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-    "std_accuracy": 0.025653461754702817
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed456_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed456_n500/results.json
deleted file mode 100644
index eddb17d..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed456_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 456,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
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-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed789_n500/results.json b/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed789_n500/results.json
deleted file mode 100644
index e249c43..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_500/standard_R_seed789_n500/results.json
+++ /dev/null
@@ -1,81 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 789,
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-  "argv": [
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-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_500",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50",
-    "--train_size",
-    "500"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed101112/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed101112/results.json
deleted file mode 100644
index 5e2ed36..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed101112/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed123/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed123/results.json
deleted file mode 100644
index 88df8af..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed123/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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-    "rot_02": 0.949,
-    "rot_03": 0.949,
-    "rot_04": 0.9488,
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-    "rot_07": 0.949,
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed42/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed42/results.json
deleted file mode 100644
index f9869fc..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed42/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed456/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed456/results.json
deleted file mode 100644
index 718a562..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed456/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed789/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed789/results.json
deleted file mode 100644
index fd7913d..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_C_seed789/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed101112/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed101112/results.json
deleted file mode 100644
index b432d5c..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed101112/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": null,
-  "train_examples": 50000,
-  "best_val_accuracy": 0.9551,
-  "test_c": {
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-  },
-  "test_nr": {
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-  },
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-  "epochs_run": 20,
-  "peak_train_alloc_bytes": 421850624,
-  "peak_train_reserved_bytes": 847249408,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed123/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed123/results.json
deleted file mode 100644
index 70a99cc..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed123/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": null,
-  "train_examples": 50000,
-  "best_val_accuracy": 0.9549,
-  "test_c": {
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-    "accuracy": 0.9469
-  },
-  "test": {
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-    "accuracy": 0.9469
-  },
-  "test_nr": {
-    "loss": 0.16284834724664687,
-    "accuracy": 0.9469
-  },
-  "test_r": {
-    "loss": 0.16140340181589127,
-    "accuracy": 0.948
-  },
-  "rotation_robustness": {
-    "rot_00": 0.9478,
-    "rot_01": 0.9478,
-    "rot_02": 0.9481,
-    "rot_03": 0.9481,
-    "rot_04": 0.9479,
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-    "rot_06": 0.9477,
-    "rot_07": 0.9478,
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-    "std_accuracy": 0.00016278820596101322
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-  "epochs_run": 20,
-  "peak_train_alloc_bytes": 421850624,
-  "peak_train_reserved_bytes": 847249408,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed42/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed42/results.json
deleted file mode 100644
index b770cf3..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed42/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": null,
-  "train_examples": 50000,
-  "best_val_accuracy": 0.9569,
-  "test_c": {
-    "loss": 0.16231381266117095,
-    "accuracy": 0.9479
-  },
-  "test": {
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-    "accuracy": 0.9479
-  },
-  "test_nr": {
-    "loss": 0.16231381266117095,
-    "accuracy": 0.9479
-  },
-  "test_r": {
-    "loss": 0.16152125236988069,
-    "accuracy": 0.9478
-  },
-  "rotation_robustness": {
-    "rot_00": 0.948,
-    "rot_01": 0.948,
-    "rot_02": 0.9479,
-    "rot_03": 0.9479,
-    "rot_04": 0.9477,
-    "rot_05": 0.948,
-    "rot_06": 0.9481,
-    "rot_07": 0.9481,
-    "rot_08": 0.948,
-    "rot_09": 0.9478,
-    "mean_accuracy": 0.94795,
-    "std_accuracy": 0.00012041594578793735
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed456/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed456/results.json
deleted file mode 100644
index 6a52cb2..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed456/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": null,
-  "train_examples": 50000,
-  "best_val_accuracy": 0.955,
-  "test_c": {
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-  "test": {
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-  "test_nr": {
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-    "rot_01": 0.9498,
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-    "rot_03": 0.9499,
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-  "epochs_run": 20,
-  "peak_train_alloc_bytes": 421850624,
-  "peak_train_reserved_bytes": 847249408,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed789/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed789/results.json
deleted file mode 100644
index 04769e1..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/bispectrum_R_seed789/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "bispectrum",
-  "base_model": "bispectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 231818,
-  "invariant_dim": 768,
-  "train_size": null,
-  "train_examples": 50000,
-  "best_val_accuracy": 0.9556,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "bispectrum",
-    "--run_label",
-    "bispectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed101112/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed101112/results.json
deleted file mode 100644
index 23f047c..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed101112/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 101112,
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-  "invariant_dim": 16,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed123/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed123/results.json
deleted file mode 100644
index 1187f14..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed123/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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-  "test_r": {
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed42/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed42/results.json
deleted file mode 100644
index a554b9b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed42/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed456/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed456/results.json
deleted file mode 100644
index 15a6517..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed456/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed789/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed789/results.json
deleted file mode 100644
index 5a0eb0e..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_C_seed789/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": null,
-  "train_examples": 50000,
-  "best_val_accuracy": 0.7912,
-  "test_c": {
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-    "accuracy": 0.7776
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-  "test": {
-    "loss": 0.6590518564224244,
-    "accuracy": 0.7776
-  },
-  "test_nr": {
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-    "accuracy": 0.7776
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-    "rot_09": 0.7771,
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-  "peak_train_reserved_bytes": 69206016,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed101112/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed101112/results.json
deleted file mode 100644
index cd8b611..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed101112/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": null,
-  "train_examples": 50000,
-  "best_val_accuracy": 0.7927,
-  "test_c": {
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-  "test_nr": {
-    "loss": 0.6597861833572388,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed123/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed123/results.json
deleted file mode 100644
index 94589ef..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed123/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": null,
-  "train_examples": 50000,
-  "best_val_accuracy": 0.7918,
-  "test_c": {
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-  "test": {
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-    "accuracy": 0.7793
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-  "test_nr": {
-    "loss": 0.6507925777435303,
-    "accuracy": 0.7793
-  },
-  "test_r": {
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-    "rot_01": 0.7794,
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-    "rot_03": 0.7799,
-    "rot_04": 0.7789,
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-    "rot_06": 0.7796,
-    "rot_07": 0.7791,
-    "rot_08": 0.7795,
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-  "peak_train_reserved_bytes": 69206016,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed42/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed42/results.json
deleted file mode 100644
index 70c7501..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed42/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": null,
-  "train_examples": 50000,
-  "best_val_accuracy": 0.7937,
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-  "test_nr": {
-    "loss": 0.6484841794967652,
-    "accuracy": 0.7797
-  },
-  "test_r": {
-    "loss": 0.6464659417152405,
-    "accuracy": 0.7819
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-  "rotation_robustness": {
-    "rot_00": 0.7811,
-    "rot_01": 0.7814,
-    "rot_02": 0.7815,
-    "rot_03": 0.7806,
-    "rot_04": 0.7816,
-    "rot_05": 0.7803,
-    "rot_06": 0.7816,
-    "rot_07": 0.781,
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-    "rot_09": 0.7816,
-    "mean_accuracy": 0.78121,
-    "std_accuracy": 0.0004323193264243372
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed456/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed456/results.json
deleted file mode 100644
index fc57acf..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed456/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": null,
-  "train_examples": 50000,
-  "best_val_accuracy": 0.7933,
-  "test_c": {
-    "loss": 0.6519663318634034,
-    "accuracy": 0.7773
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-    "rot_03": 0.7798,
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-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed789/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed789/results.json
deleted file mode 100644
index ef51863..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_R_seed789/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 11466,
-  "invariant_dim": 16,
-  "train_size": null,
-  "train_examples": 50000,
-  "best_val_accuracy": 0.7916,
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-    "accuracy": 0.7778
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed101112/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed101112/results.json
deleted file mode 100644
index db330a0..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed101112/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
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-  "train_examples": 50000,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed123/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed123/results.json
deleted file mode 100644
index 074c876..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed123/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
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-  "test_nr": {
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-    "accuracy": 0.8
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-    "mean_accuracy": 0.79884,
-    "std_accuracy": 0.0001624807680927519
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed42/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed42/results.json
deleted file mode 100644
index 87eeb53..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed42/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": null,
-  "train_examples": 50000,
-  "best_val_accuracy": 0.8171,
-  "test_c": {
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-  },
-  "test": {
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-    "accuracy": 0.799
-  },
-  "test_nr": {
-    "loss": 0.5942532022476197,
-    "accuracy": 0.799
-  },
-  "test_r": {
-    "loss": 0.5990424713134765,
-    "accuracy": 0.7971
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-  "rotation_robustness": {
-    "rot_00": 0.7971,
-    "rot_01": 0.7966,
-    "rot_02": 0.7962,
-    "rot_03": 0.7968,
-    "rot_04": 0.7965,
-    "rot_05": 0.7966,
-    "rot_06": 0.7968,
-    "rot_07": 0.7963,
-    "rot_08": 0.7968,
-    "rot_09": 0.7962,
-    "mean_accuracy": 0.7965899999999999,
-    "std_accuracy": 0.00028089143810375197
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed456/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed456/results.json
deleted file mode 100644
index cda914e..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed456/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": null,
-  "train_examples": 50000,
-  "best_val_accuracy": 0.8144,
-  "test_c": {
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-    "accuracy": 0.8
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-    "accuracy": 0.8
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-    "accuracy": 0.7992
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-  "peak_train_alloc_bytes": 47850496,
-  "peak_train_reserved_bytes": 73400320,
-  "gpu_name": "NVIDIA A100-SXM4-80GB",
-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed789/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed789/results.json
deleted file mode 100644
index f2e3c9e..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_C_seed789/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": null,
-  "train_examples": 50000,
-  "best_val_accuracy": 0.7979,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed101112/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed101112/results.json
deleted file mode 100644
index 7f1a94b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed101112/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
-  "lmax": 15,
-  "n_params": 231906,
-  "invariant_dim": 16,
-  "train_size": null,
-  "train_examples": 50000,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed123/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed123/results.json
deleted file mode 100644
index e13fc89..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed123/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 123,
-  "lmax": 15,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed42/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed42/results.json
deleted file mode 100644
index d0b703f..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed42/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
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-  "test_nr": {
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-    "std_accuracy": 0.00024083189157581938
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed456/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed456/results.json
deleted file mode 100644
index aa9e40b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed456/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "456",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed789/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed789/results.json
deleted file mode 100644
index 1b0abdb..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/power_spectrum_matched_R_seed789/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "power_spectrum_matched",
-  "base_model": "power_spectrum",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "power_spectrum",
-    "--run_label",
-    "power_spectrum_matched",
-    "--train_mode",
-    "R",
-    "--seed",
-    "789",
-    "--hidden",
-    "1312",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed101112/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed101112/results.json
deleted file mode 100644
index 93c7fbf..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed101112/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
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-  "test_rotation_seed": 777,
-  "seed": 101112,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "C",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed123/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed123/results.json
deleted file mode 100644
index 21f158b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed123/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 123,
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-  "n_params": 184778,
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-  "train_examples": 50000,
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-  "test_r": {
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-    "rot_03": 0.1552,
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-    "rot_09": 0.1292,
-    "mean_accuracy": 0.10368999999999999,
-    "std_accuracy": 0.03549423756048297
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed42/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed42/results.json
deleted file mode 100644
index 9d98ab6..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed42/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 42,
-  "lmax": 15,
-  "n_params": 184778,
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-  "test_r": {
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-    "rot_09": 0.1421,
-    "mean_accuracy": 0.11288000000000001,
-    "std_accuracy": 0.05033809293169538
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed456/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed456/results.json
deleted file mode 100644
index 20f0d4b..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed456/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 456,
-  "lmax": 15,
-  "n_params": 184778,
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-  "cuda_total_memory_bytes": 85095874560,
-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "C",
-    "--seed",
-    "456",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed789/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed789/results.json
deleted file mode 100644
index bc12391..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_C_seed789/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "C",
-  "test_rotation_seed": 777,
-  "seed": 789,
-  "lmax": 15,
-  "n_params": 184778,
-  "invariant_dim": null,
-  "train_size": null,
-  "train_examples": 50000,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
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-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "C",
-    "--seed",
-    "789",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed101112/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed101112/results.json
deleted file mode 100644
index aea2298..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed101112/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
-  "model": "standard",
-  "base_model": "standard",
-  "train_mode": "R",
-  "test_rotation_seed": 777,
-  "seed": 101112,
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-  "invariant_dim": null,
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-  "git_sha": "5fb3903a1c11765c007a1686e1c9acd77b6024db",
-  "argv": [
-    "--model",
-    "standard",
-    "--run_label",
-    "standard",
-    "--train_mode",
-    "R",
-    "--seed",
-    "101112",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed123/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed123/results.json
deleted file mode 100644
index 493a9d2..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed123/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "argv": [
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-    "--run_label",
-    "standard",
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-    "R",
-    "--seed",
-    "123",
-    "--hidden",
-    "256",
-    "--lmax",
-    "15",
-    "--batch_size",
-    "256",
-    "--output_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/spherical_mnist/n_full",
-    "--data_dir",
-    "./smnist_data",
-    "--subset_dir",
-    "/home/johmathe/bispectrum/paper/experiments/normalized_results/subsets",
-    "--test_rotation_seed",
-    "777",
-    "--patience",
-    "10",
-    "--epochs",
-    "50"
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed42/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed42/results.json
deleted file mode 100644
index d7b118d..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed42/results.json
+++ /dev/null
@@ -1,44 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed456/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed456/results.json
deleted file mode 100644
index 5655890..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed456/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
-{
-  "dataset": "spherical_mnist",
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-  "test_rotation_seed": 777,
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed789/results.json b/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed789/results.json
deleted file mode 100644
index 19ef221..0000000
--- a/paper/experiments/normalized_results/spherical_mnist/n_full/standard_R_seed789/results.json
+++ /dev/null
@@ -1,79 +0,0 @@
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/organ3d_train_n100_seed101112.json b/paper/experiments/normalized_results/subsets/organ3d_train_n100_seed101112.json
deleted file mode 100644
index e3c6e67..0000000
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deleted file mode 100644
index a393fd7..0000000
--- a/paper/experiments/normalized_results/subsets/organ3d_train_n100_seed123.json
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deleted file mode 100644
index 08d09d8..0000000
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deleted file mode 100644
index 3bea169..0000000
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diff --git a/paper/experiments/normalized_results/subsets/organ3d_train_n100_seed789.json b/paper/experiments/normalized_results/subsets/organ3d_train_n100_seed789.json
deleted file mode 100644
index 0508e9d..0000000
--- a/paper/experiments/normalized_results/subsets/organ3d_train_n100_seed789.json
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deleted file mode 100644
index 21094eb..0000000
--- a/paper/experiments/normalized_results/subsets/organ3d_train_n500_seed101112.json
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diff --git a/paper/experiments/normalized_results/subsets/organ3d_train_n500_seed123.json b/paper/experiments/normalized_results/subsets/organ3d_train_n500_seed123.json
deleted file mode 100644
index 1a7e2cf..0000000
--- a/paper/experiments/normalized_results/subsets/organ3d_train_n500_seed123.json
+++ /dev/null
@@ -1,504 +0,0 @@
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diff --git a/paper/experiments/normalized_results/subsets/organ3d_train_n500_seed42.json b/paper/experiments/normalized_results/subsets/organ3d_train_n500_seed42.json
deleted file mode 100644
index e40071e..0000000
--- a/paper/experiments/normalized_results/subsets/organ3d_train_n500_seed42.json
+++ /dev/null
@@ -1,504 +0,0 @@
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diff --git a/paper/experiments/normalized_results/subsets/organ3d_train_n500_seed456.json b/paper/experiments/normalized_results/subsets/organ3d_train_n500_seed456.json
deleted file mode 100644
index 3765a4f..0000000
--- a/paper/experiments/normalized_results/subsets/organ3d_train_n500_seed456.json
+++ /dev/null
@@ -1,504 +0,0 @@
-{
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diff --git a/paper/experiments/normalized_results/subsets/organ3d_train_n500_seed789.json b/paper/experiments/normalized_results/subsets/organ3d_train_n500_seed789.json
deleted file mode 100644
index adb42d4..0000000
--- a/paper/experiments/normalized_results/subsets/organ3d_train_n500_seed789.json
+++ /dev/null
@@ -1,504 +0,0 @@
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diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n100_seed101112.json b/paper/experiments/normalized_results/subsets/pcam_train_n100_seed101112.json
deleted file mode 100644
index b6bea62..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n100_seed101112.json
+++ /dev/null
@@ -1,104 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n100_seed123.json b/paper/experiments/normalized_results/subsets/pcam_train_n100_seed123.json
deleted file mode 100644
index ace67ce..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n100_seed123.json
+++ /dev/null
@@ -1,104 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n100_seed42.json b/paper/experiments/normalized_results/subsets/pcam_train_n100_seed42.json
deleted file mode 100644
index 02cd18e..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n100_seed42.json
+++ /dev/null
@@ -1,104 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n100_seed456.json b/paper/experiments/normalized_results/subsets/pcam_train_n100_seed456.json
deleted file mode 100644
index e111eb0..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n100_seed456.json
+++ /dev/null
@@ -1,104 +0,0 @@
-{
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diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n100_seed789.json b/paper/experiments/normalized_results/subsets/pcam_train_n100_seed789.json
deleted file mode 100644
index 3081fb9..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n100_seed789.json
+++ /dev/null
@@ -1,104 +0,0 @@
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diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n12500_seed123.json b/paper/experiments/normalized_results/subsets/pcam_train_n12500_seed123.json
deleted file mode 100644
index 11f03bf..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n12500_seed123.json
+++ /dev/null
@@ -1,12504 +0,0 @@
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-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n12500_seed42.json b/paper/experiments/normalized_results/subsets/pcam_train_n12500_seed42.json
deleted file mode 100644
index e4e90c5..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n12500_seed42.json
+++ /dev/null
@@ -1,12504 +0,0 @@
-{
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-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n12500_seed456.json b/paper/experiments/normalized_results/subsets/pcam_train_n12500_seed456.json
deleted file mode 100644
index 12369b5..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n12500_seed456.json
+++ /dev/null
@@ -1,12504 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n12500_seed789.json b/paper/experiments/normalized_results/subsets/pcam_train_n12500_seed789.json
deleted file mode 100644
index 5cadd7d..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n12500_seed789.json
+++ /dev/null
@@ -1,12504 +0,0 @@
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n2500_seed123.json b/paper/experiments/normalized_results/subsets/pcam_train_n2500_seed123.json
deleted file mode 100644
index 2aacdf6..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n2500_seed123.json
+++ /dev/null
@@ -1,2504 +0,0 @@
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-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n2500_seed42.json b/paper/experiments/normalized_results/subsets/pcam_train_n2500_seed42.json
deleted file mode 100644
index ef0b0ae..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n2500_seed42.json
+++ /dev/null
@@ -1,2504 +0,0 @@
-{
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-  ]
-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n2500_seed456.json b/paper/experiments/normalized_results/subsets/pcam_train_n2500_seed456.json
deleted file mode 100644
index 9a656d3..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n2500_seed456.json
+++ /dev/null
@@ -1,2504 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n2500_seed789.json b/paper/experiments/normalized_results/subsets/pcam_train_n2500_seed789.json
deleted file mode 100644
index 71f1981..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n2500_seed789.json
+++ /dev/null
@@ -1,2504 +0,0 @@
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diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n500_seed101112.json b/paper/experiments/normalized_results/subsets/pcam_train_n500_seed101112.json
deleted file mode 100644
index 7fbe63f..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n500_seed101112.json
+++ /dev/null
@@ -1,504 +0,0 @@
-{
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diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n500_seed123.json b/paper/experiments/normalized_results/subsets/pcam_train_n500_seed123.json
deleted file mode 100644
index 379dfb0..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n500_seed123.json
+++ /dev/null
@@ -1,504 +0,0 @@
-{
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diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n500_seed42.json b/paper/experiments/normalized_results/subsets/pcam_train_n500_seed42.json
deleted file mode 100644
index 704534f..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n500_seed42.json
+++ /dev/null
@@ -1,504 +0,0 @@
-{
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diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n500_seed456.json b/paper/experiments/normalized_results/subsets/pcam_train_n500_seed456.json
deleted file mode 100644
index adc26ed..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n500_seed456.json
+++ /dev/null
@@ -1,504 +0,0 @@
-{
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diff --git a/paper/experiments/normalized_results/subsets/pcam_train_n500_seed789.json b/paper/experiments/normalized_results/subsets/pcam_train_n500_seed789.json
deleted file mode 100644
index 9960945..0000000
--- a/paper/experiments/normalized_results/subsets/pcam_train_n500_seed789.json
+++ /dev/null
@@ -1,504 +0,0 @@
-{
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diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed101112.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed101112.json
deleted file mode 100644
index 5040eb4..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed101112.json
+++ /dev/null
@@ -1,104 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed123.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed123.json
deleted file mode 100644
index 0ddb111..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed123.json
+++ /dev/null
@@ -1,104 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed42.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed42.json
deleted file mode 100644
index 29ec50b..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed42.json
+++ /dev/null
@@ -1,104 +0,0 @@
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diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed456.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed456.json
deleted file mode 100644
index f9832a5..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed456.json
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diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed789.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed789.json
deleted file mode 100644
index 4b8e2ad..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n100_seed789.json
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diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n12500_seed101112.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n12500_seed101112.json
deleted file mode 100644
index 651e078..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n12500_seed101112.json
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n12500_seed123.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n12500_seed123.json
deleted file mode 100644
index dcfe08c..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n12500_seed123.json
+++ /dev/null
@@ -1,12504 +0,0 @@
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n12500_seed42.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n12500_seed42.json
deleted file mode 100644
index 8654c61..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n12500_seed42.json
+++ /dev/null
@@ -1,12504 +0,0 @@
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n12500_seed456.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n12500_seed456.json
deleted file mode 100644
index d90d8d0..0000000
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+++ /dev/null
@@ -1,12504 +0,0 @@
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n12500_seed789.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n12500_seed789.json
deleted file mode 100644
index 371d15c..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n12500_seed789.json
+++ /dev/null
@@ -1,12504 +0,0 @@
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n2500_seed101112.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n2500_seed101112.json
deleted file mode 100644
index 034d815..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n2500_seed101112.json
+++ /dev/null
@@ -1,2504 +0,0 @@
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n2500_seed123.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n2500_seed123.json
deleted file mode 100644
index aaf0072..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n2500_seed123.json
+++ /dev/null
@@ -1,2504 +0,0 @@
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diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n2500_seed42.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n2500_seed42.json
deleted file mode 100644
index 212ab4f..0000000
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+++ /dev/null
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n2500_seed456.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n2500_seed456.json
deleted file mode 100644
index d585730..0000000
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+++ /dev/null
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n2500_seed789.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n2500_seed789.json
deleted file mode 100644
index 4287899..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n2500_seed789.json
+++ /dev/null
@@ -1,2504 +0,0 @@
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-}
\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed101112.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed101112.json
deleted file mode 100644
index 92b7403..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed101112.json
+++ /dev/null
@@ -1,504 +0,0 @@
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diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed123.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed123.json
deleted file mode 100644
index 9720ae1..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed123.json
+++ /dev/null
@@ -1,504 +0,0 @@
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diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed42.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed42.json
deleted file mode 100644
index 58cd91d..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed42.json
+++ /dev/null
@@ -1,504 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed456.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed456.json
deleted file mode 100644
index c9ee2d3..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed456.json
+++ /dev/null
@@ -1,504 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed789.json b/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed789.json
deleted file mode 100644
index 9f9f4c5..0000000
--- a/paper/experiments/normalized_results/subsets/spherical_mnist_train_n500_seed789.json
+++ /dev/null
@@ -1,504 +0,0 @@
-{
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\ No newline at end of file
diff --git a/paper/experiments/organ3d/README.md b/paper/experiments/organ3d/README.md
deleted file mode 100644
index 433c62e..0000000
--- a/paper/experiments/organ3d/README.md
+++ /dev/null
@@ -1,90 +0,0 @@
-# OrganMNIST3D — O-Equivariant 3D ResNet with Bispectral Pooling
-
-3D classification experiment on OrganMNIST3D (11 organ classes, 28³ voxels, 1-channel CT) comparing invariant pooling strategies in octahedral-equivariant 3D CNNs.
-
-## Motivation
-
-The PCam experiment demonstrates bispectral pooling for 2D cyclic/dihedral groups. This experiment extends the idea to 3D using the chiral octahedral group O (|O| = 24), testing whether complete invariant pooling via the selective G-bispectrum (`OctaonOcta`) outperforms simpler alternatives (max/norm pooling) on 3D medical volumes.
-
-## Architecture
-
-All equivariant variants share the same O-equivariant 3D ResNet backbone:
-
-```
-Stem:     LiftingConv3d(1 → C0, 3³) + EquivBN + ReLU         [28³]
-Stage 1:  2× BasicBlock(C0) → AvgPool3d(2)                    [14³]
-Stage 2:  GroupConv3d(C0→C1, 1³) proj + 2× BasicBlock(C1) → AvgPool3d(2)  [7³]
-Stage 3:  2× BasicBlock(C1) + EquivBN + ReLU                  [7³]
-Pool:     {BispectrumPool3d | GroupMaxPool3d | GroupNormPool3d}
-Head:     GlobalAvgPool3d → Linear(head_dim, 11)
-```
-
-Default channels: `C0=4, C1=8`. Exact voxel permutations (no interpolation) for kernel rotations.
-
-| Variant    | Intermediate nonlin | Final pool        | Params |
-|------------|--------------------|--------------------|--------|
-| standard   | ReLU               | GlobalAvgPool3d    | ~16K   |
-| max_pool   | ReLU               | GroupMaxPool3d     | ~374K  |
-| norm_pool  | NormReLU3d         | GroupNormPool3d    | ~375K  |
-| bispectrum | ReLU               | BispectrumPool3d   | ~463K  |
-
-The `standard` variant is a plain 3D ResNet with geometric augmentation (random octahedral rotations at train time).
-
-## Dataset
-
-[OrganMNIST3D](https://medmnist.com/) — 1,742 CT organ volumes (train=971, val=161, test=610).
-
-```bash
-pip install medmnist
-```
-
-## Running
-
-Single model:
-
-```bash
-source /home/johmathe/bispectrum/.venv/bin/activate
-export PYTHONPATH="/home/johmathe/bispectrum/src:$PYTHONPATH"
-python train.py --model bispectrum --data_dir ./organ3d_data
-```
-
-Full sweep:
-
-```bash
-./run_sweep.sh              # Phase A: 4 models × seed 42, skip rotation
-./run_sweep.sh --phase-b    # Phase B: 4 models × seeds {123, 456}, with rotation eval
-```
-
-Analyze results:
-
-```bash
-python analyze_results.py --results_dir ./organ3d_results
-```
-
-## Key CLI arguments
-
-| Flag              | Default      | Description                              |
-|-------------------|--------------|------------------------------------------|
-| `--model`         | `bispectrum` | One of: standard, max_pool, norm_pool, bispectrum |
-| `--channels`      | `4 8`        | Channel widths (C0, C1)                  |
-| `--epochs`        | `100`        |                                          |
-| `--batch_size`    | `32`         |                                          |
-| `--lr`            | `1e-3`       |                                          |
-| `--patience`      | `15`         | Early stopping on val AUC                |
-| `--skip_rotation` | `false`      | Skip 24-rotation robustness eval         |
-| `--sweep`         | `false`      | Run all 4 models × 3 seeds              |
-| `--dry_run`       | `false`      | Print model info, exit                   |
-
-## Rotation robustness
-
-All variants are evaluated on all 24 octahedral rotations of the test set. Equivariant models with bispectral/norm/max pooling should show near-zero variance across rotations, while the standard CNN should degrade.
-
-## Files
-
-```
-data.py              — OrganMNIST3D dataset + augmentations
-model.py             — O-equivariant 3D ResNet (4 variants)
-train.py             — Training loop, metrics, rotation eval, sweep
-run_sweep.sh         — Shell script for full experimental sweep
-analyze_results.py   — Load results, print tables, generate plots
-```
diff --git a/paper/experiments/organ3d/data_efficiency.pdf b/paper/experiments/organ3d/data_efficiency.pdf
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diff --git a/paper/experiments/organ3d/organ3d_analysis.pdf b/paper/experiments/organ3d/organ3d_analysis.pdf
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diff --git a/paper/experiments/pcam/README.md b/paper/experiments/pcam/README.md
deleted file mode 100644
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--- a/paper/experiments/pcam/README.md
+++ /dev/null
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-# PatchCamelyon Experiment: Bispectral Nonlinearity in Equivariant CNNs
-
-## Context
-
-Equivariant CNNs (Cohen & Welling 2016, Weiler & Cesa 2019) achieve rotation
-equivariance by construction, but they must eventually extract
-**rotation-invariant** features for classification. The choice of invariant map
-is a fundamental design decision with no consensus solution:
-
-| Method | Nonlinearity | Invariant map | Complete? | Exact equivariance? | Cost per layer | Key reference |
-| -------------- | ----------------------------------------- | ------------------------------------------ | ------------------------------ | ----------------------- | -------------- | ---------------------------------- |
-| **Standard** | ReLU | Data augmentation (flips, 90° rot) | N/A | N/A | O(C) | Cohen & Welling, ICML 2016 |
-| **Norm** | ReLU(‖x‖\_G − bias) · x/‖x‖\_G | GroupMaxPool | No — discards phase | Yes | O(C) | Weiler & Cesa, NeurIPS 2019 |
-| **Gate** | σ(‖x\_gate‖\_G) · x\_feat | GroupMaxPool | No — scalar bottleneck | Yes | O(2C) | Weiler & Cesa, NeurIPS 2019 |
-| **FourierELU** | FFT → upsample → ELU → downsample → IFFT | GroupMaxPool | Approximately | No — aliasing from truncation | O(C log C) | Franzen & Wand, NeurIPS 2021 |
-| **Bispectrum** | ReLU (regular repr) | Selective bispectrum + 1×1 proj | **Yes — provably invertible** | **Yes** | O(C) | Kakarala 2012; Mataigne et al. 2024 |
-
-The bispectrum is the **unique complete rotation invariant**: it preserves all
-signal information except the absolute orientation (Kakarala 2012). The
-selective bispectrum (Mataigne et al. 2024) reduces its cost from O(|G|^2) to
-O(|G|), matching the norm nonlinearity in channel count while being provably
-complete.
-
-This experiment tests whether completeness matters in practice: does preserving
-phase information via the bispectrum improve classification accuracy on a task
-where rotation invariance is physically justified?
-
-## Why PatchCamelyon?
-
-PatchCamelyon (PCam) is a binary classification task (metastatic tissue
-detection) on 327,680 histopathology patches (96x96 RGB). It is the ideal
-testbed because:
-
-1. **Rotation invariance is physically justified.** Tissue orientation under
-   the microscope is arbitrary — there is no "up" direction.
-
-2. **Phase information is discriminative.** Histopathology textures contain
-   oriented sub-structures (collagen fibers, gland lumen shapes, mitotic
-   spindle orientations) that distinguish benign from malignant tissue. The
-   norm nonlinearity discards exactly this directional information. The
-   bispectrum preserves it.
-
-3. **Equivariance is proven to help.** Veeling et al. (MICCAI 2018) showed
-   P4M-DenseNet reduces error by 27% relative over a standard DenseNet
-   (15.93% → 11.64%).
-
-4. **Large enough for statistically significant results.** 327k images with
-   standard train/val/test splits.
-
-## Experiment design
-
-Five model variants share the same equivariant DenseNet backbone, differing
-only in the nonlinearity / invariant pooling:
-
-```
-Baseline 1 (standard):     Vanilla DenseNet + geometric data augmentation
-Baseline 2 (norm):         Equivariant DenseNet + NormReLU nonlinearity
-Baseline 3 (gate):         Equivariant DenseNet + gated nonlinearity
-Baseline 4 (fourier_elu):  Equivariant DenseNet + FFT→ELU→IFFT nonlinearity
-Ours       (bispectrum):   Equivariant DenseNet + bispectral invariant pooling
-```
-
-Each configuration is run with both C8 (8 discrete rotations) and D4 (4
-rotations + reflections) groups, across 3 random seeds.
-
-### Metrics
-
-- **AUC-ROC** — primary metric (standard for PCam)
-- **Accuracy** — secondary
-- **Rotation robustness** — AUC evaluated on test images rotated by 12
-  arbitrary angles (0°, 15°, 30°, ..., 315°). Equivariant models should
-  produce consistent AUC; the standard CNN should degrade.
-- **Data efficiency** — learning curves at 1%, 5%, 10%, 50%, 100% of training
-  data. The bispectrum's completeness should shine in low-data regimes.
-
-### What we expect
-
-1. **Bispectrum >= gate > norm > standard** on AUC, because the bispectrum
-   preserves phase information that norm/gate discard.
-2. **Bispectrum ≈ norm >> standard** on rotation robustness, because both are
-   exactly equivariant (unlike FourierELU which is approximate).
-3. **Bispectrum advantage grows in low-data regime**, because completeness
-   reduces the need for the network to learn invariant features from data.
-4. The **selective** bispectrum (O(|G|) coefficients) should match the full
-   bispectrum in accuracy while being faster — validating the selectivity
-   result of Mataigne et al. (2024).
-
-### Existing baselines to beat
-
-| Method                 | Accuracy | Source              |
-| ---------------------- | -------- | ------------------- |
-| DenseNet baseline (Z2) | 84.1%    | Veeling et al. 2018 |
-| P4-DenseNet            | 87.6%    | Veeling et al. 2018 |
-| P4M-DenseNet           | 88.4%    | Veeling et al. 2018 |
-| Attentive P4M-DenseNet | 89.1%    | Romero et al. 2020  |
-
-## How to run
-
-### Prerequisites
-
-```bash
-uv pip install h5py torchvision
-# The bispectrum library must be importable:
-uv pip install -e /path/to/bispectrum
-```
-
-### Single run
-
-```bash
-# Standard CNN baseline
-uv run train.py --model standard --data_dir ./pcam_data
-
-# Bispectrum with C8 group
-uv run train.py --model bispectrum --group c8 --data_dir ./pcam_data
-
-# Norm nonlinearity with D4 group
-uv run train.py --model norm --group d4 --data_dir ./pcam_data
-
-# Data-efficiency experiment (12,500 training examples)
-uv run train.py --model bispectrum --group c8 --train_size 12500 --data_dir ./pcam_data
-```
-
-### Full sweep (all 5 baselines x 3 seeds)
-
-```bash
-uv run train.py --sweep --group c8 --data_dir ./pcam_data
-```
-
-This produces a summary table and saves all results to
-`./pcam_results/sweep_results.json`.
-
-### Key arguments
-
-| Argument           | Default      | Description                                                            |
-| ------------------ | ------------ | ---------------------------------------------------------------------- |
-| `--model`          | `bispectrum` | One of: `standard`, `norm`, `gate`, `fourier_elu`, `bispectrum`        |
-| `--group`          | `c8`         | Symmetry group: `c8` (8 rotations) or `d4` (4 rotations + reflections) |
-| `--epochs`         | `100`        | Max training epochs                                                    |
-| `--batch_size`     | `64`         | Batch size                                                             |
-| `--lr`             | `1e-3`       | Learning rate (AdamW)                                                  |
-| `--patience`       | `15`         | Early stopping patience (on validation AUC)                            |
-| `--growth_rate`    | `12`         | DenseNet growth rate k                                                 |
-| `--block_config`   | `4 4 4`      | Layers per dense block                                                 |
-| `--train_size`     | unset        | Absolute training subset size in examples (omit for full set)          |
-
-Data is auto-downloaded from Zenodo on first run (~7 GB total).
-
-## Architecture
-
-The equivariant DenseNet is built from scratch (no escnn dependency):
-
-```
-Input: (B, 3, 96, 96)
-    |
-LiftingConv2d(3 → 24, 3x3)           → (B, 24, |G|, 96, 96)
-EquivBatchNorm + nonlinearity
-    |
-DenseBlock_1(k=12, 4 layers)          → (B, 72, |G|, 96, 96)
-Transition_1(compression=0.5)          → (B, 36, |G|, 48, 48)
-    |
-DenseBlock_2(k=12, 4 layers)          → (B, 84, |G|, 48, 48)
-Transition_2                           → (B, 42, |G|, 24, 24)
-    |
-DenseBlock_3(k=12, 4 layers)          → (B, 90, |G|, 24, 24)
-    |
-EquivBatchNorm + ReLU
-InvariantPool                          → (B, 90, 24, 24)
-GlobalAvgPool                          → (B, 90)
-Linear(90 → 1)                        → (B,)
-```
-
-The `InvariantPool` step is the experimental variable:
-
-- **GroupMaxPool**: max over group dim (baselines 2–4)
-- **BispectrumPool**: selective bispectrum via `CnonCn`/`DnonDn` (ours)
-
-Group-equivariant convolutions use `grid_sample`-based kernel rotation.
-Pointwise ReLU is equivariant for the regular representation of finite groups
-(the group action permutes channels).
-
-## File structure
-
-```
-paper/experiments/pcam/
-    data.py    — PCam dataset, auto-download, transforms
-    model.py   — Equivariant layers, nonlinearities, DenseNet, model factory
-    train.py   — Training loop, evaluation, rotation robustness, CLI
-```
-
-## Related work
-
-### Equivariant CNNs
-
-- Cohen & Welling, "Group Equivariant Convolutional Networks", ICML 2016
-- Weiler & Cesa, "General E(2)-Equivariant Steerable CNNs", NeurIPS 2019
-- Cesa, Lang & Weiler, "A Program to Build E(N)-Equivariant Steerable CNNs", ICLR 2022
-
-### Nonlinearities in equivariant networks
-
-- Franzen & Wand, "General Nonlinearities in SO(2)-Equivariant CNNs", NeurIPS 2021
-- Kondor, Lin & Trivedi, "Clebsch-Gordan Nets", NeurIPS 2018
-- "The Price of Freedom: Expressivity/Efficiency Tradeoffs in Equivariant Tensor Products", ICLR 2025
-
-### Bispectral approaches
-
-- Kakarala, "The Bispectrum as a Source of Phase-Sensitive Invariants", JMIV 2012
-- Mataigne et al., "The Selective G-Bispectrum and its Inversion", NeurIPS 2024
-- Oreiller et al., "Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral U-Net", MIDL 2022
-- Sanborn et al., "Bispectral Neural Networks", ICLR 2023
-- Chevalley et al., "A Bispectral 3D U-Net for Rotation Robustness in Medical Segmentation", MICCAI-W 2024
-
-### PCam benchmark
-
-- Veeling et al., "Rotation Equivariant CNNs for Digital Pathology", MICCAI 2018
-- Romero et al., "Attentive Group Equivariant Convolutional Networks", ICML 2020
diff --git a/paper/experiments/pcam/data_pareto_analysis.png b/paper/experiments/pcam/data_pareto_analysis.png
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diff --git a/paper/experiments/pcam/pcam_results/bispectrum_c8_seed42/results.json b/paper/experiments/pcam/pcam_results/bispectrum_c8_seed42/results.json
deleted file mode 100644
index df0d51c..0000000
--- a/paper/experiments/pcam/pcam_results/bispectrum_c8_seed42/results.json
+++ /dev/null
@@ -1,29 +0,0 @@
-{
-  "model": "bispectrum",
-  "group": "c8",
-  "seed": 42,
-  "n_params": 919609,
-  "train_fraction": 0.01,
-  "best_val_auc": 0.8675331927648499,
-  "test": {
-    "loss": 0.4845264821778983,
-    "accuracy": 0.77587890625,
-    "auc": 0.8498243378154656
-  },
-  "rotation_robustness": {
-    "0": 0.8498243378154656,
-    "15": 0.8487276978330954,
-    "30": 0.8478890379762756,
-    "45": 0.8476317880077571,
-    "60": 0.847723519572811,
-    "75": 0.8486760280472241,
-    "90": 0.8498242372326091,
-    "135": 0.847632845990395,
-    "180": 0.8498244756512318,
-    "225": 0.8476310504001434,
-    "270": 0.849826338296721,
-    "315": 0.8476311472577088,
-    "mean_auc": 0.8485702086734532,
-    "std_auc": 0.000959013733451882
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results/fourier_elu_c8_seed42/results.json b/paper/experiments/pcam/pcam_results/fourier_elu_c8_seed42/results.json
deleted file mode 100644
index 2665a9f..0000000
--- a/paper/experiments/pcam/pcam_results/fourier_elu_c8_seed42/results.json
+++ /dev/null
@@ -1,29 +0,0 @@
-{
-  "model": "fourier_elu",
-  "group": "c8",
-  "seed": 42,
-  "n_params": 789919,
-  "train_fraction": 0.01,
-  "best_val_auc": 0.8588675344183864,
-  "test": {
-    "loss": 0.5192660072352737,
-    "accuracy": 0.77447509765625,
-    "auc": 0.8550437349719666
-  },
-  "rotation_robustness": {
-    "0": 0.8550437349719666,
-    "15": 0.8540500471310776,
-    "30": 0.8534076207018398,
-    "45": 0.8529257170608645,
-    "60": 0.8532353744228681,
-    "75": 0.8540122615046829,
-    "90": 0.8550411086418268,
-    "135": 0.8529293417689865,
-    "180": 0.8550393093262842,
-    "225": 0.8529232509182367,
-    "270": 0.8550394881402512,
-    "315": 0.8529238506900843,
-    "mean_auc": 0.8538809254399141,
-    "std_auc": 0.0008999973946297982
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results/gate_c8_seed42/results.json b/paper/experiments/pcam/pcam_results/gate_c8_seed42/results.json
deleted file mode 100644
index ecedee3..0000000
--- a/paper/experiments/pcam/pcam_results/gate_c8_seed42/results.json
+++ /dev/null
@@ -1,29 +0,0 @@
-{
-  "model": "gate",
-  "group": "c8",
-  "seed": 42,
-  "n_params": 1576159,
-  "train_fraction": 0.01,
-  "best_val_auc": 0.814720613182105,
-  "test": {
-    "loss": 0.5605282168835402,
-    "accuracy": 0.7222900390625,
-    "auc": 0.8019115675004825
-  },
-  "rotation_robustness": {
-    "0": 0.8019115675004825,
-    "15": 0.8036047457778176,
-    "30": 0.8048511126551946,
-    "45": 0.8053730743500614,
-    "60": 0.8047405385683715,
-    "75": 0.8035245179113052,
-    "90": 0.8019114520164622,
-    "135": 0.8053748960173499,
-    "180": 0.8019113290818599,
-    "225": 0.8053730370971517,
-    "270": 0.8019108969481064,
-    "315": 0.8053739423428594,
-    "mean_auc": 0.8038217591889185,
-    "std_auc": 0.00147966671952265
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results/norm_c8_seed123/results.json b/paper/experiments/pcam/pcam_results/norm_c8_seed123/results.json
deleted file mode 100644
index 9260169..0000000
--- a/paper/experiments/pcam/pcam_results/norm_c8_seed123/results.json
+++ /dev/null
@@ -1,29 +0,0 @@
-{
-  "model": "norm",
-  "group": "c8",
-  "seed": 123,
-  "n_params": 790717,
-  "train_fraction": 1.0,
-  "best_val_auc": 0.968148059522038,
-  "test": {
-    "loss": 0.47523957979137776,
-    "accuracy": 0.836700439453125,
-    "auc": 0.9372113046175016
-  },
-  "rotation_robustness": {
-    "0": 0.9372113046175016,
-    "15": 0.8832199434853242,
-    "30": 0.9013611084472176,
-    "45": 0.9103756644144936,
-    "60": 0.898050110803751,
-    "75": 0.8804071401802818,
-    "90": 0.9372239706068284,
-    "135": 0.91037542227058,
-    "180": 0.937211386573903,
-    "225": 0.9103750497414821,
-    "270": 0.9372057911868534,
-    "315": 0.9103746958388391,
-    "mean_auc": 0.9127826323472547,
-    "std_auc": 0.0197488396237623
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results/norm_c8_seed42/results.json b/paper/experiments/pcam/pcam_results/norm_c8_seed42/results.json
deleted file mode 100644
index 121b4d1..0000000
--- a/paper/experiments/pcam/pcam_results/norm_c8_seed42/results.json
+++ /dev/null
@@ -1,29 +0,0 @@
-{
-  "model": "norm",
-  "group": "c8",
-  "seed": 42,
-  "n_params": 790717,
-  "train_fraction": 1.0,
-  "best_val_auc": 0.9662266016043177,
-  "test": {
-    "loss": 0.41766100299719255,
-    "accuracy": 0.843109130859375,
-    "auc": 0.9318016941036397
-  },
-  "rotation_robustness": {
-    "0": 0.9318016941036397,
-    "15": 0.8960248489127218,
-    "30": 0.899723746204613,
-    "45": 0.9144160628556723,
-    "60": 0.9053091122215483,
-    "75": 0.8985299767105611,
-    "90": 0.9317899184588566,
-    "135": 0.9144169457496343,
-    "180": 0.9317889275314564,
-    "225": 0.9144185997788288,
-    "270": 0.93179107329906,
-    "315": 0.9144202687091871,
-    "mean_auc": 0.9153692645446482,
-    "std_auc": 0.013185599949946827
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results/standard_c8_seed123/results.json b/paper/experiments/pcam/pcam_results/standard_c8_seed123/results.json
deleted file mode 100644
index 32c8688..0000000
--- a/paper/experiments/pcam/pcam_results/standard_c8_seed123/results.json
+++ /dev/null
@@ -1,29 +0,0 @@
-{
-  "model": "standard",
-  "group": "c8",
-  "seed": 123,
-  "n_params": 101959,
-  "train_fraction": 1.0,
-  "best_val_auc": 0.961738882181229,
-  "test": {
-    "loss": 0.4509041443816386,
-    "accuracy": 0.84124755859375,
-    "auc": 0.9300874046023295
-  },
-  "rotation_robustness": {
-    "0": 0.9300874046023295,
-    "15": 0.8950724559223292,
-    "30": 0.8928971169589413,
-    "45": 0.8992247136757187,
-    "60": 0.893712087690429,
-    "75": 0.8930438598958743,
-    "90": 0.9279506298511507,
-    "135": 0.8990501316393035,
-    "180": 0.9317064272374471,
-    "225": 0.9004157450809014,
-    "270": 0.9321608754839111,
-    "315": 0.9025850416223222,
-    "mean_auc": 0.9081588741383881,
-    "std_auc": 0.01606839108779745
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results/standard_c8_seed42/results.json b/paper/experiments/pcam/pcam_results/standard_c8_seed42/results.json
deleted file mode 100644
index 9ddc9ed..0000000
--- a/paper/experiments/pcam/pcam_results/standard_c8_seed42/results.json
+++ /dev/null
@@ -1,29 +0,0 @@
-{
-  "model": "standard",
-  "group": "c8",
-  "seed": 42,
-  "n_params": 101959,
-  "train_fraction": 1.0,
-  "best_val_auc": 0.9614837442854138,
-  "test": {
-    "loss": 0.36100059187447187,
-    "accuracy": 0.86328125,
-    "auc": 0.9416416516171431
-  },
-  "rotation_robustness": {
-    "0": 0.9416416516171431,
-    "15": 0.9122148517464389,
-    "30": 0.9102267310064652,
-    "45": 0.9179376586487341,
-    "60": 0.9075846913145851,
-    "75": 0.9102212436528538,
-    "90": 0.9382475054790369,
-    "135": 0.9192097970891001,
-    "180": 0.9373717603505264,
-    "225": 0.9187153094151994,
-    "270": 0.9398611599698545,
-    "315": 0.9201879094884081,
-    "mean_auc": 0.9227850224815288,
-    "std_auc": 0.01230498804429544
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results/standard_c8_seed456/results.json b/paper/experiments/pcam/pcam_results/standard_c8_seed456/results.json
deleted file mode 100644
index bef10c1..0000000
--- a/paper/experiments/pcam/pcam_results/standard_c8_seed456/results.json
+++ /dev/null
@@ -1,29 +0,0 @@
-{
-  "model": "standard",
-  "group": "c8",
-  "seed": 456,
-  "n_params": 101959,
-  "train_fraction": 1.0,
-  "best_val_auc": 0.9632727046920305,
-  "test": {
-    "loss": 0.5019473592401482,
-    "accuracy": 0.832366943359375,
-    "auc": 0.9115204649586334
-  },
-  "rotation_robustness": {
-    "0": 0.9115204649586334,
-    "15": 0.8558796939928272,
-    "30": 0.8645246638421287,
-    "45": 0.8728006622464674,
-    "60": 0.8582669386829436,
-    "75": 0.8506143304709427,
-    "90": 0.9102811165294599,
-    "135": 0.8728196724063305,
-    "180": 0.9125224117696217,
-    "225": 0.8764227924671651,
-    "270": 0.9151037590208806,
-    "315": 0.8767527303132556,
-    "mean_auc": 0.881459103058388,
-    "std_auc": 0.02319979629027309
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr12_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr12_seed42/results.json
deleted file mode 100644
index 5ec6ecb..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr12_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "bispectrum",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 12,
-  "n_params": 919609,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9619766676602893,
-  "test": {
-    "loss": 0.34856547107483493,
-    "accuracy": 0.869140625,
-    "auc": 0.9427796274282103
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr3_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr3_seed42/results.json
deleted file mode 100644
index f36604d..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr3_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "bispectrum",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 3,
-  "n_params": 80413,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9615659279835738,
-  "test": {
-    "loss": 0.3499782169237733,
-    "accuracy": 0.879547119140625,
-    "auc": 0.9333922815927185
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr4_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr4_seed42/results.json
deleted file mode 100644
index f262bc4..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr4_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "bispectrum",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 4,
-  "n_params": 128345,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9577572475946721,
-  "test": {
-    "loss": 0.37445879098959267,
-    "accuracy": 0.87127685546875,
-    "auc": 0.932479480994845
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr6_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr6_seed42/results.json
deleted file mode 100644
index b18a6a3..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr6_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "bispectrum",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 6,
-  "n_params": 258193,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.960430208954204,
-  "test": {
-    "loss": 0.30831730080535635,
-    "accuracy": 0.884674072265625,
-    "auc": 0.9390023351129682
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr8_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr8_seed42/results.json
deleted file mode 100644
index 4eee6dc..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/bispectrum_c8_gr8_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "bispectrum",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 8,
-  "n_params": 433353,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.961594266290627,
-  "test": {
-    "loss": 0.37202620258176466,
-    "accuracy": 0.856201171875,
-    "auc": 0.9377570038664832
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr12_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr12_seed42/results.json
deleted file mode 100644
index 21784ad..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr12_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "fourier_elu",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 12,
-  "n_params": 789919,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9591016277591372,
-  "test": {
-    "loss": 0.354159979353426,
-    "accuracy": 0.85687255859375,
-    "auc": 0.9299131988203032
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr3_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr3_seed42/results.json
deleted file mode 100644
index 82b3f89..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr3_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "fourier_elu",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 3,
-  "n_params": 68722,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9568484212863996,
-  "test": {
-    "loss": 0.36977836198639125,
-    "accuracy": 0.847747802734375,
-    "auc": 0.9389041476186485
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr4_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr4_seed42/results.json
deleted file mode 100644
index 38a898f..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr4_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "fourier_elu",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 4,
-  "n_params": 109815,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9526851153155824,
-  "test": {
-    "loss": 0.331655687186867,
-    "accuracy": 0.857025146484375,
-    "auc": 0.9373827909371136
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr6_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr6_seed42/results.json
deleted file mode 100644
index 6ec2c3e..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr6_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "fourier_elu",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 6,
-  "n_params": 221281,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9588934062086657,
-  "test": {
-    "loss": 0.32627557578962296,
-    "accuracy": 0.874267578125,
-    "auc": 0.937949824927529
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr8_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr8_seed42/results.json
deleted file mode 100644
index 283ca02..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/fourier_elu_c8_gr8_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "fourier_elu",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 8,
-  "n_params": 371787,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.958296368333261,
-  "test": {
-    "loss": 0.4017227808799362,
-    "accuracy": 0.84844970703125,
-    "auc": 0.9191081897776622
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr12_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr12_seed42/results.json
deleted file mode 100644
index 8739f5b..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr12_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "gate",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 12,
-  "n_params": 1576159,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9228575886896158,
-  "test": {
-    "loss": 0.3462877341080457,
-    "accuracy": 0.852142333984375,
-    "auc": 0.9286962505657832
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr3_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr3_seed42/results.json
deleted file mode 100644
index a21ed83..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr3_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "gate",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 3,
-  "n_params": 135682,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9570880172580625,
-  "test": {
-    "loss": 0.374275230220519,
-    "accuracy": 0.856292724609375,
-    "auc": 0.9243756394624946
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr4_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr4_seed42/results.json
deleted file mode 100644
index 7dc9288..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr4_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "gate",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 4,
-  "n_params": 217655,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9072645907645409,
-  "test": {
-    "loss": 0.39790161757264286,
-    "accuracy": 0.823089599609375,
-    "auc": 0.909306438848434
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr6_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr6_seed42/results.json
deleted file mode 100644
index 4fb4d3a..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr6_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "gate",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 6,
-  "n_params": 440161,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9369983554803952,
-  "test": {
-    "loss": 0.3919164676917717,
-    "accuracy": 0.828399658203125,
-    "auc": 0.9253187117748591
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr8_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr8_seed42/results.json
deleted file mode 100644
index 9d11c83..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/gate_c8_gr8_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "gate",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 8,
-  "n_params": 740747,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9329547096595528,
-  "test": {
-    "loss": 0.3951001215027645,
-    "accuracy": 0.826263427734375,
-    "auc": 0.9214936314269451
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr12_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr12_seed42/results.json
deleted file mode 100644
index 1a396fa..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr12_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "norm",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 12,
-  "n_params": 790717,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.927045734916964,
-  "test": {
-    "loss": 0.4020889871753752,
-    "accuracy": 0.822845458984375,
-    "auc": 0.914243239156599
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr3_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr3_seed42/results.json
deleted file mode 100644
index 3e52b01..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr3_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "norm",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 3,
-  "n_params": 68935,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9485705697103523,
-  "test": {
-    "loss": 0.31789366132579744,
-    "accuracy": 0.860504150390625,
-    "auc": 0.9405382204293191
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr4_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr4_seed42/results.json
deleted file mode 100644
index 844817d..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr4_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "norm",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 4,
-  "n_params": 110093,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.962642101922903,
-  "test": {
-    "loss": 0.28062732331454754,
-    "accuracy": 0.88665771484375,
-    "auc": 0.954212359176597
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr6_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr6_seed42/results.json
deleted file mode 100644
index 79a0f57..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr6_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "norm",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 6,
-  "n_params": 221689,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9523105109850503,
-  "test": {
-    "loss": 0.33832676766905934,
-    "accuracy": 0.8720703125,
-    "auc": 0.93530413445049
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr8_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr8_seed42/results.json
deleted file mode 100644
index 4ea2ab8..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/norm_c8_gr8_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "norm",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 8,
-  "n_params": 372325,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.8960828059115683,
-  "test": {
-    "loss": 0.4421278720255941,
-    "accuracy": 0.803009033203125,
-    "auc": 0.8782025763091678
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr12_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr12_seed42/results.json
deleted file mode 100644
index 02f613f..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr12_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "standard",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 12,
-  "n_params": 101959,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9526942907132783,
-  "test": {
-    "loss": 0.37169857136905193,
-    "accuracy": 0.85247802734375,
-    "auc": 0.9277188161694333
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr20_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr20_seed42/results.json
deleted file mode 100644
index bce8a57..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr20_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "standard",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 20,
-  "n_params": 266543,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9582206815468272,
-  "test": {
-    "loss": 0.4250122900120914,
-    "accuracy": 0.840240478515625,
-    "auc": 0.9172153545303359
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr30_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr30_seed42/results.json
deleted file mode 100644
index 332ab48..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr30_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "standard",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 30,
-  "n_params": 582073,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9560122121227821,
-  "test": {
-    "loss": 0.4616474057547748,
-    "accuracy": 0.8223876953125,
-    "auc": 0.9041500438129609
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr35_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr35_seed42/results.json
deleted file mode 100644
index fcc81f8..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr35_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "standard",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 35,
-  "n_params": 785588,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9570869294723836,
-  "test": {
-    "loss": 0.4675147132948041,
-    "accuracy": 0.83233642578125,
-    "auc": 0.9083413724181326
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr6_seed42/results.json b/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr6_seed42/results.json
deleted file mode 100644
index 3b74d07..0000000
--- a/paper/experiments/pcam/pcam_results_pareto/standard_c8_gr6_seed42/results.json
+++ /dev/null
@@ -1,18 +0,0 @@
-{
-  "model": "standard",
-  "group": "c8",
-  "seed": 42,
-  "growth_rate": 6,
-  "n_params": 29761,
-  "train_fraction": 0.1,
-  "best_val_auc": 0.9452075983275087,
-  "test": {
-    "loss": 0.3722250214777887,
-    "accuracy": 0.852752685546875,
-    "auc": 0.9266035199298429
-  },
-  "rotation_robustness": {
-    "mean_auc": 0.0,
-    "std_auc": 0.0
-  }
-}
\ No newline at end of file
diff --git a/paper/experiments/run_normalized_experiments.sh b/paper/experiments/run_normalized_experiments.sh
deleted file mode 100755
index 3a093e1..0000000
--- a/paper/experiments/run_normalized_experiments.sh
+++ /dev/null
@@ -1,371 +0,0 @@
-#!/usr/bin/env bash
-# Normalized experiment launcher.
-#
-# Default mode is `check`: enumerate every planned run and execute one-batch
-# memory probes. Use `run` only after the check manifest is clean.
-
-set -euo pipefail
-
-MODE="${1:-check}"
-shift || true
-
-ROOT_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")/../.." && pwd)"
-EXPERIMENT_DIR="${ROOT_DIR}/paper/experiments"
-RESULTS_ROOT="${RESULTS_ROOT:-${EXPERIMENT_DIR}/normalized_results}"
-MEMORY_MANIFEST="${MEMORY_MANIFEST:-${RESULTS_ROOT}/memory_manifest.jsonl}"
-SUBSET_DIR="${SUBSET_DIR:-${RESULTS_ROOT}/subsets}"
-MEMORY_HEADROOM="${MEMORY_HEADROOM:-0.85}"
-FAIL_FAST="${FAIL_FAST:-1}"
-RESUME="${RESUME:-1}"
-SEEDS=(${SEEDS:-42 123 456 789 101112})
-CHECK_SEED="${CHECK_SEED:-42}"
-PCAM_SIZES=(${PCAM_SIZES:-100 500 2500 12500 full})
-ORGAN_SIZES=(${ORGAN_SIZES:-100 500 full})
-SMNIST_SIZES=(${SMNIST_SIZES:-100 500 2500 12500 full})
-TRAIN_MODES=(${TRAIN_MODES:-C R})
-
-usage() {
-    cat <<EOF
-Usage: $(basename "$0") [check|run|analyze]
-
-Environment overrides:
-  SEEDS="42 123 456 789 101112"
-  CHECK_SEED=42
-  PCAM_SIZES="100 500 2500 12500 full"
-  ORGAN_SIZES="100 500 full"
-  SMNIST_SIZES="100 500 2500 12500 full"
-  RESULTS_ROOT="${RESULTS_ROOT}"
-  MEMORY_HEADROOM=0.85
-  FAIL_FAST=1
-  RESUME=1
-
-Modes:
-  check    Run one representative-seed memory check per config and write memory_manifest.jsonl.
-  run      Run full training commands, skipping completed results when RESUME=1.
-  analyze  Validate seed/protocol coverage from completed results.
-EOF
-}
-
-activate_env() {
-    if command -v conda >/dev/null 2>&1; then
-        eval "$(conda shell.bash hook)"
-        conda activate bispectrum
-    elif [[ -f "${HOME}/miniconda3/etc/profile.d/conda.sh" ]]; then
-        source "${HOME}/miniconda3/etc/profile.d/conda.sh"
-        conda activate bispectrum
-    elif [[ -f "${HOME}/anaconda3/etc/profile.d/conda.sh" ]]; then
-        source "${HOME}/anaconda3/etc/profile.d/conda.sh"
-        conda activate bispectrum
-    else
-        echo "WARN: conda was not found; continuing with .venv only." >&2
-    fi
-    source "${ROOT_DIR}/.venv/bin/activate"
-    export PYTHONPATH="${ROOT_DIR}/src:${PYTHONPATH:-}"
-    export PYTHONUNBUFFERED=1
-}
-
-run_or_fail() {
-    echo "+ $*"
-    if [[ "${FAIL_FAST}" == "1" ]]; then
-        "$@"
-    else
-        "$@" || echo "WARN: command failed: $*" >&2
-    fi
-}
-
-maybe_skip() {
-    local out_dir=$1
-    if [[ "${MODE}" == "run" && "${RESUME}" == "1" && -f "${out_dir}/results.json" ]]; then
-        echo "SKIP completed: ${out_dir}"
-        return 0
-    fi
-    return 1
-}
-
-manifest_gate() {
-    local rc=0
-    python "${EXPERIMENT_DIR}/check_manifest.py" --manifest "$MEMORY_MANIFEST" "$@" || rc=$?
-    if [[ $rc -eq 0 ]]; then
-        return 0
-    fi
-    if [[ "${FAIL_FAST}" == "1" ]]; then
-        echo "FAIL: manifest gate refused to launch (rc=${rc}); rerun 'check' first." >&2
-        exit "$rc"
-    fi
-    echo "WARN: manifest gate refused to launch (rc=${rc}); skipping." >&2
-    return 1
-}
-
-pcam_batch_size() {
-    case "$1" in
-        standard) echo 1024 ;;
-        fourier_elu) echo 64 ;;
-        *) echo 128 ;;
-    esac
-}
-
-organ_batch_size() {
-    case "$1" in
-        standard) echo 64 ;;
-        bispectrum) echo 16 ;;
-        *) echo 32 ;;
-    esac
-}
-
-smnist_batch_size() {
-    echo 256
-}
-
-# Bucket a run under "n_full" or "n_<size>" without leaking the literal "full" sentinel.
-size_bucket() {
-    local size=$1
-    if [[ "$size" == "full" ]]; then
-        echo "n_full"
-    else
-        echo "n_${size}"
-    fi
-}
-
-# Output dir suffix used by train.py: empty for full, _n<size> for subsets.
-size_suffix() {
-    local size=$1
-    if [[ "$size" == "full" ]]; then
-        echo ""
-    else
-        echo "_n${size}"
-    fi
-}
-
-run_pcam() {
-    local model=$1 mode=$2 size=$3 seed=$4
-    local gr
-    case "$model" in
-        standard) gr=12 ;;
-        norm) gr=4 ;;
-        gate) gr=3 ;;
-        fourier_elu) gr=4 ;;
-        bispectrum) gr=4 ;;
-        *) echo "unknown PCam model: $model" >&2; exit 1 ;;
-    esac
-    local bs
-    bs=$(pcam_batch_size "$model")
-    local bucket
-    bucket=$(size_bucket "$size")
-    local suffix
-    suffix=$(size_suffix "$size")
-    local out_root="${RESULTS_ROOT}/pcam/${bucket}"
-    local out_dir="${out_root}/${model}_c8_gr${gr}_${mode}_seed${seed}${suffix}"
-    local size_args=()
-    local gate_size_args=()
-    if [[ "$size" != "full" ]]; then
-        size_args=(--train_size "$size")
-        gate_size_args=(--train_size "$size")
-    fi
-    local common=(
-        python train.py
-        --model "$model"
-        --group c8
-        --geometry_group c8
-        --growth_rate "$gr"
-        --train_mode "$mode"
-        --seed "$seed"
-        --batch_size "$bs"
-        --output_dir "$out_root"
-        --data_dir ./pcam_data
-        --subset_dir "$SUBSET_DIR"
-        --patience 10
-        --epochs 50
-    )
-    common+=("${size_args[@]}")
-    (
-        cd "${EXPERIMENT_DIR}/pcam"
-        if [[ "${MODE}" == "check" ]]; then
-            run_or_fail "${common[@]}" --memory_check --memory_manifest "$MEMORY_MANIFEST" --memory_headroom "$MEMORY_HEADROOM"
-        elif ! maybe_skip "$out_dir"; then
-            if ! manifest_gate \
-                --dataset pcam \
-                --model "$model" \
-                --group c8 \
-                --train_mode "$mode" \
-                --batch_size "$bs" \
-                --growth_rate "$gr" \
-                "${gate_size_args[@]}"; then
-                return 0
-            fi
-            run_or_fail "${common[@]}"
-        fi
-    )
-}
-
-run_organ() {
-    local model=$1 mode=$2 size=$3 seed=$4
-    local bs
-    bs=$(organ_batch_size "$model")
-    local bucket
-    bucket=$(size_bucket "$size")
-    local suffix
-    suffix=$(size_suffix "$size")
-    local out_root="${RESULTS_ROOT}/organ3d/${bucket}"
-    local out_dir="${out_root}/${model}_${mode}_ch4_8_seed${seed}${suffix}"
-    local size_args=()
-    local gate_size_args=()
-    if [[ "$size" != "full" ]]; then
-        size_args=(--train_size "$size")
-        gate_size_args=(--train_size "$size")
-    fi
-    local common=(
-        python train.py
-        --model "$model"
-        --channels 4 8
-        --head_dim 64
-        --train_mode "$mode"
-        --seed "$seed"
-        --batch_size "$bs"
-        --output_dir "$out_root"
-        --data_dir ./organ3d_data
-        --subset_dir "$SUBSET_DIR"
-        --patience 15
-        --epochs 100
-    )
-    common+=("${size_args[@]}")
-    (
-        cd "${EXPERIMENT_DIR}/organ3d"
-        if [[ "${MODE}" == "check" ]]; then
-            run_or_fail "${common[@]}" --memory_check --memory_manifest "$MEMORY_MANIFEST" --memory_headroom "$MEMORY_HEADROOM"
-        elif ! maybe_skip "$out_dir"; then
-            if ! manifest_gate \
-                --dataset organ3d \
-                --model "$model" \
-                --train_mode "$mode" \
-                --batch_size "$bs" \
-                --channels 4 8 \
-                --head_dim 64 \
-                "${gate_size_args[@]}"; then
-                return 0
-            fi
-            run_or_fail "${common[@]}"
-        fi
-    )
-}
-
-run_smnist() {
-    local label=$1 mode=$2 size=$3 seed=$4
-    local model=$label
-    local hidden=256
-    case "$label" in
-        standard) model=standard; hidden=256 ;;
-        power_spectrum) model=power_spectrum; hidden=256 ;;
-        power_spectrum_matched) model=power_spectrum; hidden=1312 ;;
-        bispectrum) model=bispectrum; hidden=256 ;;
-        *) echo "unknown Spherical MNIST model: $label" >&2; exit 1 ;;
-    esac
-    local bs
-    bs=$(smnist_batch_size "$label")
-    local bucket
-    bucket=$(size_bucket "$size")
-    local suffix
-    suffix=$(size_suffix "$size")
-    local out_root="${RESULTS_ROOT}/spherical_mnist/${bucket}"
-    local out_label="$model"
-    if [[ "$label" == "power_spectrum_matched" ]]; then
-        out_label="power_spectrum_matched"
-    fi
-    local out_dir="${out_root}/${out_label}_${mode}_seed${seed}${suffix}"
-    local lmax=15
-    local size_args=()
-    local gate_size_args=()
-    if [[ "$size" != "full" ]]; then
-        size_args=(--train_size "$size")
-        gate_size_args=(--train_size "$size")
-    fi
-    local common=(
-        python train.py
-        --model "$model"
-        --run_label "$label"
-        --train_mode "$mode"
-        --seed "$seed"
-        --hidden "$hidden"
-        --lmax "$lmax"
-        --batch_size "$bs"
-        --output_dir "$out_root"
-        --data_dir ./smnist_data
-        --subset_dir "$SUBSET_DIR"
-        --test_rotation_seed 777
-        --patience 10
-        --epochs 50
-    )
-    common+=("${size_args[@]}")
-    (
-        cd "${EXPERIMENT_DIR}/spherical_mnist"
-        if [[ "${MODE}" == "check" ]]; then
-            run_or_fail "${common[@]}" --memory_check --memory_manifest "$MEMORY_MANIFEST" --memory_headroom "$MEMORY_HEADROOM"
-        elif ! maybe_skip "$out_dir"; then
-            if ! manifest_gate \
-                --dataset spherical_mnist \
-                --model "$label" \
-                --train_mode "$mode" \
-                --batch_size "$bs" \
-                --hidden "$hidden" \
-                --lmax "$lmax" \
-                "${gate_size_args[@]}"; then
-                return 0
-            fi
-            run_or_fail "${common[@]}"
-        fi
-    )
-}
-
-main() {
-    case "$MODE" in
-        check|run|analyze) ;;
-        -h|--help|help) usage; exit 0 ;;
-        *) usage; exit 2 ;;
-    esac
-
-    mkdir -p "$RESULTS_ROOT" "$SUBSET_DIR"
-    activate_env
-
-    if [[ "$MODE" == "check" ]]; then
-        : > "$MEMORY_MANIFEST"
-    fi
-
-    if [[ "$MODE" == "analyze" ]]; then
-        run_or_fail python "${EXPERIMENT_DIR}/analyze_normalized_results.py" \
-            "${RESULTS_ROOT}/pcam" "${RESULTS_ROOT}/organ3d" "${RESULTS_ROOT}/spherical_mnist" \
-            --min_seeds 5 \
-            --output "${RESULTS_ROOT}/normalized_summary.json" \
-            --fail_on_warn
-        run_or_fail python "${EXPERIMENT_DIR}/export_normalized_latex.py" \
-            --summary "${RESULTS_ROOT}/normalized_summary.json" \
-            --output "${RESULTS_ROOT}/normalized_table.tex"
-        exit 0
-    fi
-
-    local seeds_to_run=("${SEEDS[@]}")
-    if [[ "$MODE" == "check" ]]; then
-        seeds_to_run=("$CHECK_SEED")
-        echo "Memory check mode uses CHECK_SEED=${CHECK_SEED}; seeds do not change tensor shapes."
-    fi
-
-    for seed in "${seeds_to_run[@]}"; do
-        for mode in "${TRAIN_MODES[@]}"; do
-            for size in "${PCAM_SIZES[@]}"; do
-                for model in standard norm gate fourier_elu bispectrum; do
-                    run_pcam "$model" "$mode" "$size" "$seed"
-                done
-            done
-            for size in "${ORGAN_SIZES[@]}"; do
-                for model in standard max_pool bispectrum; do
-                    run_organ "$model" "$mode" "$size" "$seed"
-                done
-            done
-            for size in "${SMNIST_SIZES[@]}"; do
-                for model in standard power_spectrum power_spectrum_matched bispectrum; do
-                    run_smnist "$model" "$mode" "$size" "$seed"
-                done
-            done
-        done
-    done
-}
-
-main "$@"
diff --git a/paper/experiments/spherical_mnist/README.md b/paper/experiments/spherical_mnist/README.md
deleted file mode 100644
index 09ab6f1..0000000
--- a/paper/experiments/spherical_mnist/README.md
+++ /dev/null
@@ -1,84 +0,0 @@
-# Spherical MNIST — SO(3) Bispectrum on S²
-
-Classification experiment on MNIST digits projected onto the sphere, following
-the evaluation protocol from Cohen et al. (2018) "Spherical CNNs".
-
-## Motivation
-
-The PCam and OrganMNIST3D experiments demonstrate bispectral pooling for finite
-groups (C8, D4, octahedral O). This experiment extends to **SO(3)** — the full
-continuous rotation group — by computing the bispectrum of spherical harmonic
-coefficients on S².
-
-## Protocol (Cohen et al. 2018)
-
-MNIST digits are projected onto the sphere via stereographic projection from the
-north pole. Two dataset variants:
-
-- **NR** (Non-Rotated): digit on northern hemisphere, fixed orientation.
-- **R** (Rotated): each digit randomly rotated by an independent SO(3) rotation.
-
-We train on one variant and evaluate on both, producing the accuracy table:
-
-| Method     | NR/NR | R/R   | NR/R  |
-|------------|-------|-------|-------|
-| Standard   | high  | low   | ~10%  |
-| Bispectrum | high  | high  | high  |
-
-A rotation-invariant model should achieve similar accuracy regardless of
-train/test rotation mismatch.
-
-## Models
-
-| Variant          | Features                      | Invariant?       | Params |
-|------------------|-------------------------------|------------------|--------|
-| `standard`       | CNN on equirectangular image   | No               | ~185K  |
-| `power_spectrum` | \|\|F_l\|\|² per SH degree → MLP   | Yes (incomplete) | ~11K   |
-| `bispectrum`     | SO3onS2 bispectrum → MLP      | Yes (complete)   | ~165K  |
-
-## Running
-
-Single model:
-
-```bash
-source /home/johmathe/bispectrum/.venv/bin/activate
-export PYTHONPATH="/home/johmathe/bispectrum/src:$PYTHONPATH"
-python train.py --model bispectrum --train_mode NR
-```
-
-Full sweep:
-
-```bash
-./run_sweep.sh
-```
-
-Analyze results:
-
-```bash
-python analyze_results.py --results_dir ./smnist_results
-```
-
-## Key CLI arguments
-
-| Flag               | Default      | Description                                   |
-|--------------------|--------------|-----------------------------------------------|
-| `--model`          | `bispectrum` | One of: standard, power_spectrum, bispectrum   |
-| `--train_mode`     | `NR`         | Training data: NR or R                         |
-| `--lmax`           | `15`         | Max spherical harmonic degree                  |
-| `--nlat`           | `64`         | Latitude grid points                           |
-| `--nlon`           | `128`        | Longitude grid points                          |
-| `--epochs`         | `50`         |                                                |
-| `--batch_size`     | `256`        |                                                |
-| `--patience`       | `10`         | Early stopping on val accuracy                 |
-| `--full_bispectrum`| `false`      | Use O(L³) full bispectrum instead of O(L²)     |
-| `--n_rotations`    | `10`         | Random rotations for robustness eval           |
-
-## Files
-
-```
-data.py              — MNIST → sphere projection, rotation, caching
-model.py             — Bispectrum, power spectrum, and CNN classifiers
-train.py             — Training loop, Cohen protocol eval, sweep
-run_sweep.sh         — Shell script for full experimental sweep
-analyze_results.py   — Load results, print tables, generate plots
-```
diff --git a/paper/experiments/spherical_mnist_reconstruction/README.md b/paper/experiments/spherical_mnist_reconstruction/README.md
deleted file mode 100644
index e4617cc..0000000
--- a/paper/experiments/spherical_mnist_reconstruction/README.md
+++ /dev/null
@@ -1,147 +0,0 @@
-# Spherical MNIST — SO(3) Bispectrum Reconstruction Demo
-
-Reconstruct spherical MNIST digits from their `SO3onS2` bispectrum by
-gradient descent in image space, then **rotate the recovered signal back
-to the target via Procrustes-style SO(3) alignment**. The aligned recon
-visually matches the target, demonstrating that the bispectrum is a
-**complete SO(3)-invariant** on `S^2`: the original signal is recovered
-**up to an SO(3) "phase change"** (the Wigner-D action on spherical
-harmonic coefficients), which is exactly the orbit indeterminacy of any
-complete invariant.
-
-## What it shows
-
-For each digit `f`:
-
-1. **Invariance.** Generate `K` random rotations and check that
-   `||beta(R_k . f) - beta(f)|| / ||beta(f)||` is at the SHT
-   discretisation floor (~7e-3 at `nlat=64, nlon=128, lmax=12`).
-2. **Round-trip recovery.** Initialise `f_hat` as a random Gaussian
-   image, then minimise
-   `||beta(f_hat) - beta(target)||^2 / ||beta(target)||^2`
-   with Adam (cosine-annealed LR) + `--n_recon_restarts` random restarts;
-   the per-target best-loss recon is kept.
-3. **Phase change is real, and the phase is recoverable.** The raw
-   image-space error `||f_hat - target|| / ||target||` is `O(1)` — the
-   recon lives in a different orbit representative. After solving for
-   the optimal `R \in SO(3)` (Adam over a quaternion parameterisation
-   with multi-restart) the aligned error
-   `||R . f_hat - target|| / ||target||` collapses to the SHT-rotation
-   floor. The orbits figure shows target / raw recon / aligned recon
-   side-by-side on actual spheres so this is visually obvious.
-
-## Running
-
-```bash
-source /home/johmathe/bispectrum/.venv/bin/activate
-export PYTHONPATH="/home/johmathe/bispectrum/src:$PYTHONPATH"
-
-python reconstruct.py --n_digits 8 --n_rotations 2
-```
-
-Default settings (`lmax=12`, `n_steps=8000`, `n_recon_restarts=4`,
-`align_n_restarts=12`) give a clean figure in ~20 min on a single GPU.
-For a faster sanity check use `--n_digits 3 --n_recon_restarts 1
---n_steps 3000 --align_n_restarts 4`.
-
-To regenerate **only the compact paper figure** (`2 x len(--paper_digits)`,
-full-width on a NeurIPS column), skip the comprehensive sweep:
-
-```bash
-python reconstruct.py --paper_only --paper_digits 0 1 2 3 4 5
-```
-
-This caches all the recon tensors to `figures/state.pt`. Iterate on the
-figure layout *without* re-running the optimization:
-
-```bash
-python reconstruct.py --paper_only --paper_digits 0 1 2 3 4 5 --replot
-```
-
-Outputs land in `figures/`:
-
-| File                            | Description                                                                          |
-|---------------------------------|--------------------------------------------------------------------------------------|
-| `orbits.{pdf,png}`              | `(n_digits x 3(1+K))` grid: target / raw recon / aligned recon spheres               |
-| `paper_orbits.{pdf,png}`        | Compact `(2 rows x len(--paper_digits) cols)` figure for the NeurIPS paper           |
-| `state.pt`                      | Cached `ReconResult` tensors for fast `--replot` iteration                           |
-| `convergence.{pdf,png}`         | Median + IQR of relative bispectrum residual vs. step                                |
-| `invariance_vs_recon.{pdf,png}` | Per-pair scatter of invariance vs. recon residual                                    |
-| `results.json`                  | All scalar metrics + per-step traces                                                 |
-
-## Key CLI arguments
-
-| Flag                       | Default | Notes                                                                   |
-|----------------------------|---------|-------------------------------------------------------------------------|
-| `--n_digits`               | `8`     | Digits sampled from the spherical MNIST test cache (one per class first)|
-| `--n_rotations`            | `2`     | Independent random `R_k` per digit                                      |
-| `--lmax`                   | `12`    | Recon at the default `64x128` grid is bounded by an SHT discretisation floor that scales with `lmax`. `lmax=12` keeps the recon comfortably below the floor; `lmax=15` (classifier setting) plateaus near it and degrades the alignment quality |
-| `--nlat / --nlon`          | `64/128`| Spherical grid; must match a pre-built cache file                       |
-| `--n_steps`                | `8000`  | Adam iterations per reconstruction                                      |
-| `--lr`                     | `5e-2`  | Initial Adam LR (cosine-annealed to `lr * 1e-2`)                        |
-| `--n_recon_restarts`       | `4`     | Random Adam restarts per recon; per-sample best is kept                 |
-| `--align_n_restarts`       | `12`    | Quaternion restarts per alignment (`0` disables alignment)              |
-| `--align_n_steps`          | `200`   | Adam steps per alignment restart                                        |
-| `--align_lr`               | `1e-1`  | Initial alignment Adam LR (cosine-annealed)                             |
-| `--render`                 | `sphere`| `sphere` = orthographic 3D view; `equirectangular` = old flat layout    |
-| `--view_size`              | `128`   | Orthographic view resolution                                            |
-| `--elev_deg / --azim_deg`  | `25/30` | Fallback / fixed-view camera direction (degrees)                        |
-| `--fixed_view`             | off     | Disable per-panel auto-centering on the signal centroid (use a single shared camera direction instead) |
-| `--paper_digits`           | `0 1 2 3 4 5` | `digit_idx` values to use as columns of the compact `paper_orbits.pdf` figure |
-| `--paper_figure_path`      | auto    | Override output path; defaults to `<output_dir>/paper_orbits.pdf`       |
-| `--paper_only`             | off     | Run only the digits in `--paper_digits` and emit only the paper figure (fast regeneration path) |
-| `--replot`                 | off     | Skip recon + alignment, rebuild figures from `<output_dir>/state.pt` (~3 s) |
-| `--full_bispectrum`        | off     | `O(L^3)` full bispectrum instead of selective `O(L^2)`                  |
-| `--no_bandlimit_project`   | off     | Disable the per-step `IRealSHT(RealSHT(.))` projection                  |
-| `--seed`                   | `0`     | Controls digit selection, rotations, Gaussian init, and alignment seeds |
-| `--data_dir`               | `../spherical_mnist/smnist_data` | Where the cached `test_NLATxNLON.pt` lives |
-
-## Implementation notes
-
-- **Differentiability.** `SO3onS2.forward` skips the CUDA-graph fast path
-  whenever `torch.is_grad_enabled()`, so optimisation runs through the
-  pure-PyTorch sparse gather-multiply-scatter kernels and Adam can
-  backprop end-to-end through the SHT and the Clebsch-Gordan contraction.
-- **Band-limit projection.** The bispectrum only constrains SH
-  coefficients up to `lmax`. Without `IRealSHT(RealSHT(.))` between Adam
-  steps, the `f_hat` would carry Gaussian junk in the high-frequency
-  null space (visually noisy, but irrelevant to the loss). The projection
-  step is essentially free and keeps the reconstructions visually clean.
-- **Cosine-annealed LR + best-loss tracking.** Adam without LR decay
-  oscillates around the minimum; we anneal to `lr * 1e-2` and keep the
-  per-target best (lowest-loss) `f_hat` across all steps so late-stage
-  jitter cannot worsen the result.
-- **Multi-restart reconstruction.** The bispectrum loss landscape has
-  shallow local minima for some digits. Running Adam from
-  `--n_recon_restarts` independent Gaussian inits and keeping the
-  per-sample best dramatically tightens the residual on hard cases at
-  ~linear cost.
-- **SO(3) alignment.** We parameterise `R \in SO(3)` as a unit quaternion
-  (auto-normalised on the fly to dodge the `(\alpha, \beta, \gamma) =
-  (0, 0, 0)` Euler gimbal-lock singularity that yields NaN gradients
-  through the `arctan2` / `acos` chain in the spherical sampler), and
-  minimise `||rotate(f_hat, R(q)) - target||^2` with Adam + cosine
-  annealing. Multi-restart with Haar-uniform `randn(4)/||.||` quaternions
-  is essential because the loss has many local minima from the digit's
-  pseudo-symmetries.
-- **Sphere rendering.** Each panel in `orbits.png` is an orthographic
-  projection of the spherical signal. By default each camera points at
-  the panel's positive-mass centroid (the digit ends up centered in the
-  disk), with a faint lat/lon graticule overlaid so the eye can pick up
-  rotation differences between panels. Aligned and target panels share
-  the same view because they're the same signal up to alignment quality.
-  Use `--fixed_view` (with `--elev_deg / --azim_deg`) to lock all panels
-  to one direction. Equirectangular plotting is still available via
-  `--render equirectangular`.
-- **No `SO3onS2.invert()`.** The library currently raises
-  `NotImplementedError` for SO(3) bispectrum inversion (`DESIGN.md`
-  TODO-M4). This script is the empirical demo of feasibility, not a
-  proposed inversion API.
-
-## Files
-
-```
-reconstruct.py        - End-to-end script (CLI + figures + JSON)
-README.md             - This file
-figures/              - Output PDFs / PNGs / JSON (gitignored)
-```
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diff --git a/paper/experiments/spherical_mnist_reconstruction/figures/results.json b/paper/experiments/spherical_mnist_reconstruction/figures/results.json
deleted file mode 100644
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--- a/paper/experiments/spherical_mnist_reconstruction/figures/results.json
+++ /dev/null
@@ -1,192359 +0,0 @@
-{
-  "meta": {
-    "n_digits": 8,
-    "n_rotations": 2,
-    "nlat": 64,
-    "nlon": 128,
-    "lmax": 12,
-    "selective": true,
-    "output_size": 258,
-    "n_steps": 8000,
-    "lr": 0.05,
-    "bandlimit_project": true,
-    "n_recon_restarts": 4,
-    "align_n_restarts": 12,
-    "align_n_steps": 200,
-    "align_lr": 0.1,
-    "device": "cuda",
-    "seed": 0,
-    "sweep_seconds": 1235.1400658674538
-  },
-  "records": [
-    {
-      "digit_idx": 0,
-      "rot_idx": 0,
-      "label": 0,
-      "invariance_rel": 0.0,
-      "final_recon_rel": 0.00046543224016204476,
-      "image_space_rel": 1.2615069150924683,
-      "aligned_image_space_rel": 0.32283613085746765,
-      "seconds": 43.45970867946744,
-      "align_seconds": 8.235152723888556,
-      "bispec_rel_trace": [
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-      ]
-    }
-  ]
-}
\ No newline at end of file
diff --git a/paper/feedback_nina.md b/paper/feedback_nina.md
deleted file mode 100644
index afab92b..0000000
--- a/paper/feedback_nina.md
+++ /dev/null
@@ -1,2 +0,0 @@
-Feedback from Nina Miolane:
-
diff --git a/paper/fiber_search.py b/paper/fiber_search.py
deleted file mode 100644
index 3755fc8..0000000
--- a/paper/fiber_search.py
+++ /dev/null
@@ -1,272 +0,0 @@
-"""Fiber computation: solve Phi_aug(x) = Phi_aug(x0) for ALL real solutions.
-
-Degree-by-degree approach: at each ell, fix lower degrees to witness values,
-then search for all real solutions of the per-degree constraints.
-Uses damped Gauss-Newton with many random starts.
-
-The augmented invariant includes mandatory even self-coupling entries
-beta(ell,ell,l) (l even, 2 <= l <= ell) for global injectivity.
-"""
-from __future__ import annotations
-
-import sys
-import os
-import time
-
-sys.path.insert(0, os.path.join(os.path.dirname(__file__), '..', 'src'))
-
-import numpy as np
-from bispectrum.so3_on_s2 import (
-    _build_selective_index_map, load_cg_matrices,
-    _build_cg_power_index_map,
-)
-
-CG_NP: dict[tuple[int, int], np.ndarray] = {}
-A0, C_GAUGE = 1.5, 1.2
-
-
-def _init_cg(lmax: int):
-    cg_torch = load_cg_matrices(lmax)
-    for k, v in cg_torch.items():
-        CG_NP[k] = v.numpy()
-
-
-def params_to_fc_np(p: np.ndarray, lmax: int) -> dict[int, np.ndarray]:
-    fc: dict[int, np.ndarray] = {}
-    fc[0] = np.array([A0 + 0j])
-    fc[1] = np.array([0.0 + 0j, C_GAUGE + 0j, 0.0 + 0j])
-    idx = 0
-    for lv in range(2, lmax + 1):
-        coeffs = np.zeros(2 * lv + 1, dtype=np.complex128)
-        for m in range(1, lv + 1):
-            if lv == 2 and m == 1:
-                re = p[idx]; idx += 1; im = 0.0
-            else:
-                re = p[idx]; im = p[idx + 1]; idx += 2
-            coeffs[lv + m] = re + 1j * im
-            coeffs[lv - m] = ((-1.0) ** m) * (re - 1j * im)
-        coeffs[lv] = p[idx] + 0j; idx += 1
-        fc[lv] = coeffs
-    return fc
-
-
-def bispectrum_entry_np(fc, l1, l2, l_val):
-    key = (min(l1, l2), max(l1, l2))
-    if l1 <= l2:
-        outer = np.outer(fc[l1], fc[l2]).ravel()
-    else:
-        outer = np.outer(fc[l2], fc[l1]).ravel()
-    cg = CG_NP[key]
-    coupled = outer @ cg
-    col_start = 0
-    for lv in range(abs(l1 - l2), l1 + l2 + 1):
-        sz = 2 * lv + 1
-        if lv == l_val:
-            proj = coupled[col_start:col_start + sz]
-            return np.sum(proj * np.conj(fc[l_val]))
-        col_start += sz
-    return 0.0 + 0j
-
-
-def cg_power_entry_np(fc, l1, l2, l_out):
-    key = (min(l1, l2), max(l1, l2))
-    if l1 <= l2:
-        outer = np.outer(fc[l1], fc[l2]).ravel()
-    else:
-        outer = np.outer(fc[l2], fc[l1]).ravel()
-    cg = CG_NP[key]
-    coupled = outer @ cg
-    col_start = 0
-    for lv in range(abs(l1 - l2), l1 + l2 + 1):
-        sz = 2 * lv + 1
-        if lv == l_out:
-            proj = coupled[col_start:col_start + sz]
-            return float(np.sum(np.abs(proj) ** 2))
-        col_start += sz
-    return 0.0
-
-
-def get_live_bispec(lmax, witness):
-    idx_map = _build_selective_index_map(lmax)
-    fc = params_to_fc_np(witness, lmax)
-    live = []
-    for l1, l2, lv in idx_map:
-        if max(l1, l2, lv) < 2:
-            continue
-        beta = bispectrum_entry_np(fc, l1, l2, lv)
-        parity = (l1 + l2 + lv) % 2
-        val = float(np.imag(beta)) if parity else float(np.real(beta))
-        if abs(val) > 1e-10:
-            live.append((l1, l2, lv))
-    return live
-
-
-def param_indices_for_ell(ell: int) -> list[int]:
-    idx = 0
-    for lv in range(2, ell):
-        idx += 2 * lv + 1
-        if lv == 2:
-            idx -= 1
-    n_params = 2 * ell + 1
-    if ell == 2:
-        n_params -= 1
-    return list(range(idx, idx + n_params))
-
-
-def eval_at_ell(full_params: np.ndarray, ell: int,
-                live_bispec: list, cg_power: list, lmax: int) -> np.ndarray:
-    """Evaluate constraints involving degree ell as the highest."""
-    fc = params_to_fc_np(full_params, lmax)
-    vals = []
-    for l1, l2, lv in live_bispec:
-        if max(l1, l2, lv) != ell:
-            continue
-        beta = bispectrum_entry_np(fc, l1, l2, lv)
-        parity = (l1 + l2 + lv) % 2
-        vals.append(float(np.imag(beta)) if parity else float(np.real(beta)))
-    for l1, l2, lo in cg_power:
-        if max(l1, l2) != ell:
-            continue
-        vals.append(cg_power_entry_np(fc, l1, l2, lo))
-    return np.array(vals)
-
-
-def damped_gauss_newton(residual_fn, x0, max_iter=60, tol=1e-24):
-    """Gauss-Newton with backtracking line search."""
-    n = len(x0)
-    x = x0.copy()
-    h = 1e-8
-
-    for it in range(max_iter):
-        f0 = residual_fn(x)
-        r2 = np.sum(f0 ** 2)
-        if r2 < tol:
-            return x, r2
-
-        m = len(f0)
-        J = np.empty((m, n))
-        for j in range(n):
-            xp = x.copy(); xp[j] += h
-            J[:, j] = (residual_fn(xp) - f0) / h
-
-        try:
-            dx = np.linalg.lstsq(J, -f0, rcond=None)[0]
-        except np.linalg.LinAlgError:
-            return x, r2
-
-        alpha = 1.0
-        for _ in range(20):
-            x_new = x + alpha * dx
-            r2_new = np.sum(residual_fn(x_new) ** 2)
-            if r2_new < r2:
-                x = x_new
-                break
-            alpha *= 0.5
-        else:
-            return x, r2
-
-    return x, np.sum(residual_fn(x) ** 2)
-
-
-def fiber_search_degree_by_degree(lmax: int, n_starts: int = 3000):
-    """Search for all real solutions degree by degree."""
-    _init_cg(lmax)
-    n_free = sum(2 * l + 1 for l in range(2, lmax + 1)) - 1
-    witness = np.array([(i + 1) / 3.0 for i in range(n_free)])
-
-    live_bispec = get_live_bispec(lmax, witness)
-    cg_power = _build_cg_power_index_map(lmax)
-
-    print(f"lmax={lmax}: {n_free} params, "
-          f"{len(live_bispec)} live bispec, {len(cg_power)} CG power", flush=True)
-
-    all_results = {}
-
-    for ell in range(2, lmax + 1):
-        p_idx = param_indices_for_ell(ell)
-        d_ell = len(p_idx)
-
-        target = eval_at_ell(witness, ell, live_bispec, cg_power, lmax)
-        n_constr = len(target)
-        wit_params = witness[p_idx[0]:p_idx[-1] + 1].copy()
-
-        print(f"\n  ell={ell}: {d_ell} unknowns, {n_constr} constraints", flush=True)
-
-        def residual(t):
-            p = witness.copy()
-            for i, idx in enumerate(p_idx):
-                p[idx] = t[i]
-            return eval_at_ell(p, ell, live_bispec, cg_power, lmax) - target
-
-        r_wit = residual(wit_params)
-        print(f"    Witness residual: {np.sum(r_wit**2):.2e}", flush=True)
-
-        solutions = []
-        rng = np.random.RandomState(42 + ell)
-        t0 = time.time()
-
-        for trial in range(n_starts):
-            scale = rng.choice([0.01, 0.05, 0.1, 0.3, 0.5, 1.0, 2.0, 5.0, 10.0, 20.0])
-            t_init = rng.randn(d_ell) * scale
-
-            t_sol, res = damped_gauss_newton(residual, t_init, max_iter=50)
-
-            if res < 1e-22:
-                is_dup = any(np.linalg.norm(t_sol - s) < 1e-5 for s in solutions)
-                if not is_dup:
-                    solutions.append(t_sol.copy())
-                    dist = np.linalg.norm(t_sol - wit_params)
-                    print(f"    trial {trial}: NEW sol #{len(solutions)}, "
-                          f"dist={dist:.2e}, res={res:.2e}", flush=True)
-
-            if trial % 500 == 499:
-                elapsed = time.time() - t0
-                print(f"    ... {trial+1}/{n_starts} ({elapsed:.1f}s), "
-                      f"{len(solutions)} sol(s)", flush=True)
-
-        elapsed = time.time() - t0
-        print(f"    {len(solutions)} solution(s) in {elapsed:.1f}s", flush=True)
-        for i, s in enumerate(solutions):
-            dist = np.linalg.norm(s - wit_params)
-            if dist < 1e-5:
-                print(f"      Sol {i}: WITNESS", flush=True)
-            else:
-                print(f"      Sol {i}: OTHER (dist={dist:.2e})", flush=True)
-                print(f"        params = [{', '.join(f'{v:.6f}' for v in s)}]", flush=True)
-
-        all_results[ell] = {
-            'd_ell': d_ell,
-            'n_constr': n_constr,
-            'n_solutions': len(solutions),
-            'solutions': solutions,
-        }
-
-    return all_results
-
-
-def main():
-    for lmax in [4, 5]:
-        print(f"\n{'='*60}", flush=True)
-        print(f"FIBER SEARCH: lmax = {lmax}", flush=True)
-        print(f"{'='*60}", flush=True)
-
-        results = fiber_search_degree_by_degree(lmax, n_starts=3000)
-
-        print(f"\nSUMMARY lmax={lmax}:", flush=True)
-        all_unique = True
-        for ell, r in results.items():
-            n_sol = r['n_solutions']
-            status = "UNIQUE" if n_sol == 1 else f"MULTIPLE ({n_sol})"
-            print(f"  ell={ell}: {status} "
-                  f"({r['d_ell']} unknowns, {r['n_constr']} constraints)", flush=True)
-            if n_sol != 1:
-                all_unique = False
-
-        if all_unique:
-            print(f"  => Fiber at witness has UNIQUE real preimage", flush=True)
-        else:
-            print(f"  => Fiber has MULTIPLE real preimages", flush=True)
-
-
-if __name__ == '__main__':
-    main()
diff --git a/paper/figures/inversion_convergence.pdf b/paper/figures/inversion_convergence.pdf
deleted file mode 100644
index 2184719..0000000
Binary files a/paper/figures/inversion_convergence.pdf and /dev/null differ
diff --git a/paper/find_minimal_set.py b/paper/find_minimal_set.py
deleted file mode 100644
index 8cb2b25..0000000
--- a/paper/find_minimal_set.py
+++ /dev/null
@@ -1,185 +0,0 @@
-"""Find the minimal augmentation set and verify full-system rank."""
-import sys
-sys.path.insert(0, '../src')
-import torch
-from bispectrum.so3_on_s2 import (
-    _build_selective_index_map, _bispectrum_entry, load_cg_matrices,
-)
-from itertools import combinations
-
-SEED_LMAX = 3
-CG_DATA = load_cg_matrices(8)
-A0, C_GAUGE = 1.5, 1.2
-
-def make_seed_params(seed=42):
-    torch.manual_seed(seed)
-    return torch.randn(11, dtype=torch.float64)
-
-def params_to_fc_seed(p):
-    fc = {}
-    fc[0] = torch.complex(torch.tensor([[A0]], dtype=torch.float64), torch.zeros(1,1,dtype=torch.float64))
-    fc[1] = torch.complex(torch.tensor([[0.,C_GAUGE,0.]], dtype=torch.float64), torch.zeros(1,3,dtype=torch.float64))
-    idx = 0
-    for lv in [2, 3]:
-        neg, pos = [], []
-        for m in range(1, lv+1):
-            if lv==2 and m==1: re=p[idx]; idx+=1; im=torch.zeros_like(re)
-            else: re=p[idx]; im=p[idx+1]; idx+=2
-            pos.append(torch.complex(re,im)); neg.append(((-1.)**m)*torch.complex(re,-im))
-        m0 = torch.complex(p[idx], torch.zeros_like(p[idx])); idx+=1
-        fc[lv] = torch.stack(list(reversed(neg))+[m0]+pos).unsqueeze(0)
-    return fc
-
-def cg_project(fc, l1, l2, l_out):
-    outer = torch.einsum('bi,bj->bij', fc[l1], fc[l2]).reshape(1,-1)
-    cg = CG_DATA[(l1,l2)].to(dtype=outer.dtype)
-    tr = outer @ cg
-    start = 0
-    for lv in range(abs(l1-l2), l1+l2+1):
-        sz = 2*lv+1
-        if lv == l_out: return tr[0, start:start+sz]
-        start += sz
-
-def eval_entry(p, entry):
-    fc = params_to_fc_seed(p)
-    if entry[0] == 'B':
-        _, l1, l2, lv = entry
-        beta = _bispectrum_entry(fc, l1, l2, lv, CG_DATA[(l1,l2)])[0]
-        return beta.imag if (l1+l2+lv)%2 else beta.real
-    else:
-        _, l1, l2, lo = entry
-        proj = cg_project(fc, l1, l2, lo)
-        return torch.sum(proj.real**2 + proj.imag**2)
-
-def eval_set(p, entries):
-    return torch.stack([eval_entry(p, e) for e in entries])
-
-def rank_of(entries, params):
-    J = torch.autograd.functional.jacobian(lambda p: eval_set(p, entries), params)
-    sv = torch.linalg.svdvals(J)
-    tol = 1e-8 * sv[0].item() if sv[0]>0 else 1e-15
-    return int((sv > tol).sum().item()), sv
-
-ALL_BISPEC = [
-    ('B',1,1,2),('B',0,2,2),('B',2,2,2),('B',1,2,3),
-    ('B',1,3,2),('B',2,3,1),('B',0,3,3),('B',3,3,2),
-]
-ALL_CG = [
-    ('P',1,2,1),('P',1,2,2),('P',1,2,3),
-    ('P',1,3,2),('P',1,3,3),('P',1,3,4),
-    ('P',2,2,0),('P',2,2,2),('P',2,2,4),
-    ('P',2,3,1),('P',2,3,2),('P',2,3,3),('P',2,3,4),('P',2,3,5),
-    ('P',3,3,0),('P',3,3,2),('P',3,3,4),('P',3,3,6),
-]
-
-def main():
-    params = make_seed_params(42)
-
-    print("BASELINE: 8 nonzero scalar bispec")
-    r, _ = rank_of(ALL_BISPEC, params)
-    print(f"  rank = {r} / 11\n")
-
-    print("SEARCH: n_bispec + n_cg = N, find min N with rank 11")
-    for N in range(11, 14):
-        found = False
-        for nb in range(max(0, N-len(ALL_CG)), min(len(ALL_BISPEC), N)+1):
-            nc = N - nb
-            if nc < 0 or nc > len(ALL_CG): continue
-            for bc in combinations(ALL_BISPEC, nb):
-                for cc in combinations(ALL_CG, nc):
-                    entries = list(bc) + list(cc)
-                    try:
-                        r, _ = rank_of(entries, params)
-                    except Exception:
-                        continue
-                    if r == 11:
-                        ok = all(rank_of(entries, make_seed_params(s))[0]==11 for s in [123,999,7])
-                        if ok:
-                            print(f"  N={N}: {nb} bispec + {nc} cg → rank 11 ✓")
-                            for e in entries: print(f"    {e}")
-                            found = True
-                            break
-                if found: break
-            if found: break
-        if found:
-            break
-        else:
-            print(f"  N={N}: no rank-11 set found")
-
-    print("\n" + "="*70)
-    print("FULL SYSTEM RANK with augmented seed")
-    print("="*70)
-
-    aug = [('P',1,2,1),('P',1,3,2),('P',2,3,1),('P',2,3,2),('P',3,3,2)]
-
-    for lmax in [4, 5, 6]:
-        n_free = sum(2*l+1 for l in range(2, lmax+1)) - 1
-        torch.manual_seed(42)
-        params_full = torch.randn(n_free, dtype=torch.float64)
-
-        def params_to_fc_full(p, lm=lmax):
-            fc = {}
-            fc[0] = torch.complex(torch.tensor([[A0]],dtype=torch.float64), torch.zeros(1,1,dtype=torch.float64))
-            fc[1] = torch.complex(torch.tensor([[0.,C_GAUGE,0.]],dtype=torch.float64), torch.zeros(1,3,dtype=torch.float64))
-            idx = 0
-            for lv in range(2, lm+1):
-                neg, pos = [], []
-                for m in range(1, lv+1):
-                    if lv==2 and m==1: re=p[idx]; idx+=1; im=torch.zeros_like(re)
-                    else: re=p[idx]; im=p[idx+1]; idx+=2
-                    pos.append(torch.complex(re,im)); neg.append(((-1.)**m)*torch.complex(re,-im))
-                m0 = torch.complex(p[idx], torch.zeros_like(p[idx])); idx+=1
-                fc[lv] = torch.stack(list(reversed(neg))+[m0]+pos).unsqueeze(0)
-            return fc
-
-        idx_map = _build_selective_index_map(lmax)
-
-        entries = []
-        for l1,l2,lv in idx_map:
-            if max(l1,l2,lv) < 2: continue
-            fc_t = params_to_fc_full(params_full)
-            beta = _bispectrum_entry(fc_t, l1, l2, lv, CG_DATA[(l1,l2)])[0]
-            parity = (l1+l2+lv) % 2
-            val = beta.imag.item() if parity else beta.real.item()
-            if abs(val) > 1e-10:
-                entries.append(('B', l1, l2, lv))
-
-        def eval_full_orig(p, ents=entries, lm=lmax):
-            fc = params_to_fc_full(p, lm)
-            vals = []
-            for e in ents:
-                _, l1, l2, lv = e
-                beta = _bispectrum_entry(fc, l1, l2, lv, CG_DATA[(l1,l2)])[0]
-                parity = (l1+l2+lv) % 2
-                vals.append(beta.imag if parity else beta.real)
-            return torch.stack(vals)
-
-        def eval_full_aug(p, ents=entries, a=aug, lm=lmax):
-            fc = params_to_fc_full(p, lm)
-            vals = []
-            for e in ents:
-                _, l1, l2, lv = e
-                beta = _bispectrum_entry(fc, l1, l2, lv, CG_DATA[(l1,l2)])[0]
-                parity = (l1+l2+lv) % 2
-                vals.append(beta.imag if parity else beta.real)
-            for e in a:
-                _, l1, l2, lo = e
-                proj = cg_project(fc, l1, l2, lo)
-                vals.append(torch.sum(proj.real**2 + proj.imag**2))
-            return torch.stack(vals)
-
-        Jo = torch.autograd.functional.jacobian(eval_full_orig, params_full)
-        svo = torch.linalg.svdvals(Jo)
-        ro = int((svo > 1e-8*svo[0]).sum().item())
-
-        Ja = torch.autograd.functional.jacobian(eval_full_aug, params_full)
-        sva = torch.linalg.svdvals(Ja)
-        ra = int((sva > 1e-8*sva[0]).sum().item())
-
-        print(f"\nlmax={lmax}: {n_free} unknowns")
-        print(f"  Original: {len(entries)} entries, rank={ro}")
-        print(f"  +5 CG aug: {len(entries)+5} entries, rank={ra}")
-        print(f"  FULL RANK: {'YES ✓' if ra==n_free else 'NO ✗'}")
-
-if __name__ == "__main__":
-    main()
diff --git a/paper/generate_witness.py b/paper/generate_witness.py
deleted file mode 100644
index 5fb752c..0000000
--- a/paper/generate_witness.py
+++ /dev/null
@@ -1,159 +0,0 @@
-"""Generate witness points proving full Jacobian rank for the augmented invariant.
-
-For each lmax in [4..8], outputs:
-1. The witness signal (rational coefficients)
-2. The augmented entry set (bispec + CG power)
-3. The Jacobian rank at the witness
-"""
-import sys, os, json
-sys.path.insert(0, os.path.join(os.path.dirname(__file__), '..', 'src'))
-import torch
-from bispectrum.so3_on_s2 import (
-    _build_selective_index_map, _bispectrum_entry, load_cg_matrices,
-)
-
-CG_DATA = load_cg_matrices(10)
-A0, C_GAUGE = 1.5, 1.2
-
-def params_to_fc(p, lmax):
-    fc = {}
-    fc[0] = torch.complex(torch.tensor([[A0]], dtype=torch.float64),
-                           torch.zeros(1, 1, dtype=torch.float64))
-    fc[1] = torch.complex(torch.tensor([[0., C_GAUGE, 0.]], dtype=torch.float64),
-                           torch.zeros(1, 3, dtype=torch.float64))
-    idx = 0
-    for lv in range(2, lmax + 1):
-        neg, pos = [], []
-        for m in range(1, lv + 1):
-            if lv == 2 and m == 1:
-                re = p[idx]; idx += 1; im = torch.zeros_like(re)
-            else:
-                re = p[idx]; im = p[idx + 1]; idx += 2
-            pos.append(torch.complex(re, im))
-            neg.append(((-1.) ** m) * torch.complex(re, -im))
-        m0 = torch.complex(p[idx], torch.zeros_like(p[idx])); idx += 1
-        fc[lv] = torch.stack(list(reversed(neg)) + [m0] + pos).unsqueeze(0)
-    return fc
-
-def cg_project(fc, l1, l2, l_out):
-    outer = torch.einsum('bi,bj->bij', fc[l1], fc[l2]).reshape(1, -1)
-    cg = CG_DATA[(l1, l2)].to(dtype=outer.dtype)
-    tr = outer @ cg
-    start = 0
-    for lv in range(abs(l1 - l2), l1 + l2 + 1):
-        sz = 2 * lv + 1
-        if lv == l_out: return tr[0, start:start + sz]
-        start += sz
-
-def eval_combined(p, bispec_triples, cg_entries, lmax):
-    fc = params_to_fc(p, lmax)
-    vals = []
-    for l1, l2, lv in bispec_triples:
-        beta = _bispectrum_entry(fc, l1, l2, lv, CG_DATA[(l1, l2)])[0]
-        parity = (l1 + l2 + lv) % 2
-        vals.append(beta.imag if parity else beta.real)
-    for l1, l2, lo in cg_entries:
-        proj = cg_project(fc, l1, l2, lo)
-        vals.append(torch.sum(proj.real ** 2 + proj.imag ** 2))
-    return torch.stack(vals)
-
-
-def get_rank_and_sv(bispec, cg_ents, params, lmax):
-    def ev(p):
-        return eval_combined(p, bispec, cg_ents, lmax)
-    J = torch.autograd.functional.jacobian(ev, params)
-    sv = torch.linalg.svdvals(J)
-    tol = 1e-8 * sv[0].item() if sv[0] > 0 else 1e-15
-    return int((sv > tol).sum().item()), sv
-
-
-def greedy_augment(live_bispec, params, lmax, n_free):
-    cg_candidates = []
-    for l1 in range(lmax + 1):
-        for l2 in range(l1, lmax + 1):
-            if max(l1, l2) < 2: continue
-            for lo in range(abs(l1 - l2), min(l1 + l2, lmax) + 1):
-                if l1 == 0 and lo == l2: continue
-                cg_candidates.append((l1, l2, lo))
-
-    best_aug = []
-    best_rank, _ = get_rank_and_sv(live_bispec, [], params, lmax)
-
-    for _ in range(n_free - best_rank):
-        best_cand, best_r = None, best_rank
-        for cand in cg_candidates:
-            if cand in best_aug: continue
-            r, _ = get_rank_and_sv(live_bispec, best_aug + [cand], params, lmax)
-            if r > best_r:
-                best_r = r
-                best_cand = cand
-        if best_cand is None: break
-        best_aug.append(best_cand)
-        best_rank = best_r
-        if best_rank >= n_free: break
-
-    return best_aug, best_rank
-
-
-def main():
-    results = {}
-    outpath = os.path.join(os.path.dirname(__file__), 'witness_results.txt')
-    f = open(outpath, 'w')
-
-    for lmax in [4, 5]:
-        n_free = sum(2 * l + 1 for l in range(2, lmax + 1)) - 1
-
-        # Use simple rational-ish witness: p_i = (i+1)/3
-        params = torch.tensor([(i+1)/3.0 for i in range(n_free)], dtype=torch.float64)
-
-        idx_map = _build_selective_index_map(lmax)
-        fc_test = params_to_fc(params, lmax)
-
-        live_bispec = []
-        dead_bispec = []
-        for l1, l2, lv in idx_map:
-            if max(l1, l2, lv) < 2: continue
-            beta = _bispectrum_entry(fc_test, l1, l2, lv, CG_DATA[(l1, l2)])[0]
-            parity = (l1 + l2 + lv) % 2
-            val = beta.imag.item() if parity else beta.real.item()
-            if abs(val) > 1e-10:
-                live_bispec.append((l1, l2, lv))
-            else:
-                dead_bispec.append((l1, l2, lv))
-
-        r0, sv0 = get_rank_and_sv(live_bispec, [], params, lmax)
-        f.write(f"lmax={lmax}: {n_free} unknowns\n")
-        f.write(f"  Live bispec: {len(live_bispec)}, dead: {dead_bispec}\n")
-        f.write(f"  Bispec-only rank: {r0}/{n_free}\n")
-
-        aug, rfinal = greedy_augment(live_bispec, params, lmax, n_free)
-        f.write(f"  Augmentation: {len(aug)} CG power entries\n")
-        for a in aug:
-            f.write(f"    P_{{{a[0]},{a[1]},{a[2]}}}\n")
-        f.write(f"  Final rank: {rfinal}/{n_free}\n")
-
-        # Verify at 3 random seeds
-        for seed in [42, 123, 999]:
-            torch.manual_seed(seed)
-            p2 = torch.randn(n_free, dtype=torch.float64)
-            r2, _ = get_rank_and_sv(live_bispec, aug, p2, lmax)
-            f.write(f"  seed={seed}: rank={r2}\n")
-
-        f.write(f"\n  Witness signal: p = [{', '.join(f'{v:.4f}' for v in params.tolist())}]\n\n")
-        f.flush()
-
-        results[lmax] = {
-            'n_free': n_free,
-            'live_bispec': live_bispec,
-            'dead_bispec': dead_bispec,
-            'augmentation': aug,
-            'bispec_only_rank': r0,
-            'augmented_rank': rfinal,
-        }
-
-    f.close()
-    print(f"Results: {outpath}")
-
-
-if __name__ == "__main__":
-    main()
diff --git a/paper/neurips_2026.sty b/paper/neurips_2026.sty
deleted file mode 100644
index c2ac013..0000000
--- a/paper/neurips_2026.sty
+++ /dev/null
@@ -1,437 +0,0 @@
-% partial rewrite of the LaTeX2e package for submissions to the
-% Conference on Neural Information Processing Systems (NeurIPS):
-%
-% - uses more LaTeX conventions
-% - line numbers at submission time replaced with aligned numbers from
-%   lineno package
-% - \nipsfinalcopy replaced with [final] package option
-% - automatically loads times package for authors
-% - loads natbib automatically; this can be suppressed with the
-%   [nonatbib] package option
-% - adds foot line to first page identifying the conference
-% - adds preprint option for submission to e.g. arXiv
-% - conference acronym modified
-% - update foot line to display the track name
-%
-% Roman Garnett (garnett@wustl.edu) and the many authors of
-% nips15submit_e.sty, including MK and drstrip@sandia
-%
-% last revision: January 2026
-
-\NeedsTeXFormat{LaTeX2e}
-\ProvidesPackage{neurips_2026}[2026-01-29 NeurIPS 2026 submission/camera-ready style file]
-
-% declare final option, which creates camera-ready copy
-\newif\if@neuripsfinal\@neuripsfinalfalse
-\DeclareOption{final}{
-  \@neuripsfinaltrue
-  \@anonymousfalse
-}
-
-% declare nonatbib option, which does not load natbib in case of
-% package clash (users can pass options to natbib via
-% \PassOptionsToPackage)
-\newif\if@natbib\@natbibtrue
-\DeclareOption{nonatbib}{
-  \@natbibfalse
-}
-
-% declare preprint option, which creates a preprint version ready for
-% upload to, e.g., arXiv
-\newif\if@preprint\@preprintfalse
-\DeclareOption{preprint}{
-  \@preprinttrue
-  \@anonymousfalse
-}
-
-% determine the track of the paper in camera-ready mode
-\newif\if@main\@maintrue
-\DeclareOption{main}{
-  \@maintrue
-  \newcommand{\@trackname}{\@neuripsordinal\ Conference on Neural Information Processing Systems (NeurIPS \@neuripsyear).}
-}
-\newif\if@position\@positionfalse
-\DeclareOption{position}{
-  \@positiontrue
-  \newcommand{\@trackname}{\@neuripsordinal\ Conference on Neural Information Processing Systems (NeurIPS \@neuripsyear). Position Paper Track.}
-}
-\newif\if@eandd\@eanddfalse
-\DeclareOption{eandd}{
-  \@eanddtrue
-\if@neuripsfinal\@anonymousfalse\else\if@preprint\@anonymousfalse\else\@anonymoustrue\fi\fi
-  \newcommand{\@trackname}{\@neuripsordinal\ Conference on Neural Information Processing Systems (NeurIPS \@neuripsyear). Track on Evaluations and Datasets.} 
-}
-\newif\if@creativeai\@creativeaifalse
-\DeclareOption{creativeai}{
-  \@creativeaitrue
-  \@anonymousfalse
-  \newcommand{\@trackname}{\@neuripsordinal\ Conference on Neural Information Processing Systems (NeurIPS \@neuripsyear). Creative AI Track.}
-}
-
-% For anonymous or non-anonymous
-\newif\if@anonymous\@anonymoustrue
-
-% For workshop papers
-\newcommand{\@workshoptitle}{}
-\newcommand{\workshoptitle}[1]{\renewcommand{\@workshoptitle}{#1}}
-
-\newif\if@workshop\@workshopfalse
-\DeclareOption{sglblindworkshop}{
-  \@workshoptrue
-  \@anonymousfalse
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diff --git a/paper/old_paper.tex b/paper/old_paper.tex
deleted file mode 100644
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--- a/paper/old_paper.tex
+++ /dev/null
@@ -1,2284 +0,0 @@
-\documentclass{article}
-
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-
-\title{\texttt{bispectrum}: Selective $G$-Bispectra Made Practical}
-
-\author{%
-  Johan Mathe \\
-  Atmo, Inc. \\
-  \texttt{johan@atmo.ai} \\
-  \And
-  Simon Mataigne \\
-  UCLouvain \\
-  \texttt{simon.mataigne@uclouvain.be} \\
-  \And
-  Adele Myers Lantow \\
-  University of California, Santa Barbara \\
-  \texttt{adele@ucsb.edu} \\
-  \And
-  Nina Miolane \\
-  University of California, Santa Barbara \\
-  \texttt{nmiolane@ucsb.edu} \\
-}
-
-\begin{document}
-
-\maketitle
-
-\begin{abstract}
-We present \texttt{bispectrum}, an open-source PyTorch library that computes
-selective $G$-bispectral invariants for seven group/domain pairs as
-drop-in differentiable \texttt{nn.Module}s.
-The selective bispectrum reduces invariant cost from $O(|G|^2)$ to $O(|G|)$
-while remaining complete for generic signals.
-Building the library required one theoretical contribution:
-an augmented selective $\SO(3)$ invariant on $S^2$ achieving
-$\Theta(L^2)$ coefficients (the information-theoretic lower
-bound) instead of $O(L^3)$ for the full bispectrum
-(completeness proved for $4 \le L \le 100$).
-Across three benchmarks ($C_8$, octahedral, $\SO(3)$), the library
-delivers near-exact rotation invariance
-($\sigma_{\mathrm{rot}} < 10^{-3}$) and a data-efficiency advantage of
-complete bispectral invariants over incomplete alternatives in the
-low-data, moderate-capacity regime.
-The selective forward pass runs in sub-millisecond time on GPU for finite
-groups. Code is available at \url{https://github.com/geometric-intelligence/bispectrum}.
-\end{abstract}
-\section{Introduction}
-\label{sec:intro}
-
-Equivariant neural networks~\citep{cohen2016group, weiler2019general} enforce
-symmetry by construction, but they still require an invariant representation at
-the point where a prediction must no longer depend on orientation. In practice,
-invariant pooling remains a weak point: standard choices such as norm pooling,
-gate-based pooling, or max/sum aggregation are
-usable~\citep{weiler2019general}, but they do not preserve all equivariant
-information. The bispectrum offers a mathematically principled alternative,
-since for many groups it yields exact invariants with reconstruction
-guarantees~\citep{kakarala2012bispectrum}.
-
-The obstacle is practical rather than conceptual. Existing bispectral
-constructions are scattered across theory papers, domain-specific derivations,
-and small implementations, with no general-purpose PyTorch library covering the
-finite-group and compact-group cases relevant to modern geometric deep learning.
-Even for groups where selectivity reduces the coefficient count from
-$O(|G|^2)$ to $O(|G|)$~\citep{mataigne2024selective}, a user still needs a numerically stable implementation,
-a consistent API, inversion routines, tests, and benchmarked runtime behavior.
-
-This paper presents \texttt{bispectrum}, a PyTorch~\citep{paszke2019pytorch}
-library for computing bispectral invariants and, where available, their
-inverses. The library
-supports cyclic, dihedral, torus, disk, sphere, and octahedral settings as
-differentiable \texttt{nn.Module}s intended to plug into standard ML workflows.
-Beyond the software contribution, the paper makes one technical contribution
-that arose during implementation: an augmented selective $\SO(3)$
-bispectrum on $S^2$ with $O(L^2)$ coefficients
-(Section~\ref{sec:so3-selective}).
-
-\paragraph{Contributions.}
-\begin{enumerate}
-  \item \textbf{Software contribution.} A PyTorch library implementing
-    bispectral invariants for seven group/domain pairs with a consistent
-    \texttt{nn.Module}-style interface, GPU-optimised forward passes
-    ($<$1\,ms for finite groups), precomputed Clebsch--Gordan buffers,
-    and parallel construction routines
-    (Section~\ref{sec:benchmarks}).
-  \item \textbf{Augmented selective $\SO(3)$ bispectrum on $S^2$.} We construct
-    and implement an augmented selective bispectrum for $\SO(3)$ acting on
-    $S^2$ that uses
-    $\Theta(L^2)$ coefficients instead of the full $\Theta(L^3)$,
-    matching the orbit-space lower bound up to a constant factor.
-    The invariant combines scalar bispectral entries with
-    Clebsch--Gordan power spectrum entries (a degree-$4$ complement
-    needed for real signals).  Completeness is proved for $4 \le L \le 100$
-    via a combination of exact rational certificates and a
-    closed-form linear bootstrap family (Appendix~\ref{app:so3-proof});
-    the verification extends to any desired~$L$ by re-running a single script.
-  \item \textbf{Empirical validation.} A task-driven evaluation of the
-    library as an invariant pooling primitive across three group/domain
-    settings, showing that complete bispectral invariants provide a
-    data-efficiency advantage in the low-data, moderate-capacity regime
-    while converging to incomplete alternatives at high capacity.
-\end{enumerate}
-
-\section{Mathematical Background}
-\label{sec:background}
-
-\subsection{Equivariant features and invariant maps}
-
-A neural network layer is \emph{$G$-equivariant} if its output transforms
-predictably under the group action:
-$[\pi_{\mathrm{out}}(g) \circ \Phi](f) = \Phi(\pi_{\mathrm{in}}(g) f)$
-for all $g \in G$~\citep{cohen2016group, weiler2019general}.
-Stacking equivariant layers preserves this property, but any task that requires
-a prediction independent of orientation---classification, regression,
-detection---must eventually map equivariant features to \emph{invariant} ones.
-
-\begin{table}[t]
-  \caption{Invariant nonlinearities for equivariant CNNs. The selective
-    bispectrum is the only method that is simultaneously exact, complete,
-    and linear-cost for finite groups.}
-  \label{tab:nonlinearity-comparison}
-  \centering
-  \small
-  \begin{tabular}{lcccc}
-    \toprule
-    Nonlinearity & Exact? & Complete? & Coefficients & Key limitation \\
-    \midrule
-    Norm          & Yes & No  & $O(|G|)$      & Discards phase \\
-    Gate          & Yes & No  & $O(2|G|)$     & Scalar bottleneck \\
-    Fourier pointwise & No & Yes$^\ddagger$ & $O(|G| \log |G|)$ & Aliasing \\
-    Full bispectrum & Yes & Yes & $O(|G|^2)$  & Quadratic cost \\
-    \midrule
-    \textbf{Selective bispectrum} & \textbf{Yes} & \textbf{Yes}
-      & $\mathbf{O(|G|)}$ & Genericity assumption \\
-    \bottomrule
-  \end{tabular}
-
-  \vspace{0.3em}
-  {\small $^\ddagger$Complete pre-aliasing; the pointwise nonlinearity in the forward pass introduces aliasing that may break completeness in practice.}
-\end{table}
-
-Table~\ref{tab:nonlinearity-comparison} summarises the trade-offs. Norm
-pooling and gated nonlinearities are exact invariants but \emph{incomplete}:
-they discard the phase of Fourier blocks, collapsing distinct orbits.
-Fourier-space pointwise nonlinearities preserve completeness but
-sacrifice exactness through aliasing~\citep{weiler2019general}. The full
-bispectrum is exact and complete but costs $O(|G|^2)$ coefficients.
-
-\subsection{The $G$-bispectrum and selective bispectra}
-
-Let $G$ be a compact group with irreducible representations
-$\{\rho_i\}_{i \in \hat{G}}$ of dimensions $d_i$, and let
-$\hat{f}(\rho_i) = \sum_{g \in G} f(g)\, \rho_i(g)$ denote the group
-Fourier transform. Given the Clebsch--Gordan decomposition
-$\rho_i \otimes \rho_j \cong \bigoplus_k m_{ij}^k \rho_k$ with
-intertwining matrix $C_{ij}$, the \emph{bispectrum} is
-\begin{equation}
-  \label{eq:bispectrum-def}
-  \beta_{\rho_i, \rho_j}
-    = C_{ij}^\top\,
-      (\hat{f}(\rho_i) \otimes \hat{f}(\rho_j))\,
-      C_{ij}\, \hat{f}(\rho_k)^\dagger.
-\end{equation}
-Each $\beta_{\rho_i,\rho_j}$ is invariant under $f \mapsto f \circ g$ and,
-taken over all pairs $(i,j)$, the bispectrum is a \emph{complete} invariant:
-it separates orbits for generic signals~\citep{kakarala2012bispectrum}. However,
-the full set has $O(|G|^2)$ coefficients.
-
-\emph{Selective bispectra.} \citet{mataigne2024selective} show that for
-finite groups, a BFS traversal of the Kronecker product table of $\hat{G}$
-selects $O(|G|)$ bispectral coefficients that remain complete;
-\citet{myers2025selective} extend this to the disk. This
-reduction from quadratic to linear is the key to practical deployment:
-for example, the octahedral group drops from $576$ to $172$ coefficients.
-
-\section{The \texttt{bispectrum} Library}
-\label{sec:library}
-
-\subsection{Design and implementation}
-\label{sec:design}
-
-Every group/domain pair is exposed as a \texttt{torch.nn.Module}
-named \texttt{\{Group\}on\{Domain\}} (e.g.\ \texttt{SO3onS2}).
-Modules accept raw real-valued inputs, handle the Fourier transform
-internally, and default to $O(|G|)$ selective coefficients
-(\texttt{selective=False} gives the full $O(|G|^2)$ set).
-CG matrices, DFT kernels, and Bessel roots are precomputed at
-construction and stored as non-learnable buffers, so
-\texttt{.to(device)} works transparently and forward passes incur zero
-recomputation.
-Where available, \texttt{bsp.invert(beta)} returns the reconstructed
-signal up to group-action indeterminacy.
-The only dependencies are PyTorch~\citep{paszke2019pytorch},
-NumPy~\citep{harris2020numpy}, and
-\texttt{torch\_harmonics}~\citep{bonev2023spherical}
-(for \texttt{SO3onS2}).
-For $\SO(3)$ on $S^2$, reduced CG matrices are built in parallel and
-cached to disk; all forward methods are batch-native, using
-\texttt{torch.einsum}/\texttt{torch.bmm} for non-abelian modules and
-\texttt{torch.fft} for abelian ones.
-The test suite covers 96.8\% of lines across all modules.
-
-\subsection{Supported modules}
-\label{sec:supported-modules}
-
-\begin{table}[t]
-  \caption{Current scope of the \texttt{bispectrum} library.
-    \texttt{SO2onS1} is the continuous-$n$ limit of \texttt{CnonCn}
-    and shares its implementation.}
-  \label{tab:supported-groups}
-  \centering
-  \small
-  \begin{tabular}{lll}
-    \toprule
-    Module & Group / Domain & Output mode \\
-    \midrule
-    \texttt{CnonCn} & $C_n$ on $C_n$ & selective + full \\
-    \texttt{SO2onS1} & $\SO(2)$ on $S^1$ & selective + full \\
-    \texttt{TorusOnTorus} & product cyclic groups on torus & selective + full \\
-    \texttt{DnonDn} & $D_n$ on $D_n$ & selective \\
-    \texttt{SO2onDisk} & $\SO(2)$ on disk & selective \\
-    \texttt{SO3onS2} & $\SO(3)$ on $S^2$ & selective + full \\
-    \texttt{OctaonOcta} & octahedral group & selective \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\paragraph{Example API.}
-\label{sec:api}
-A minimal usage example is in Appendix~\ref{app:api-example}.
-
-\subsection{Augmented selective $\SO(3)$ bispectrum on $S^2$}
-\label{sec:so3-selective}
-
-The full bispectrum on $S^2$ at band-limit $L$ has $\Theta(L^3)$ scalar
-entries $\beta_{\ell_1,\ell_2,\ell}$. For finite groups, selectivity
-reduces $O(|G|^2)$ to $O(|G|)$, but on $S^2$ the BFS chain
-$\beta_{1,0,1}, \beta_{1,1,2}, \ldots$ provides only ${\sim}3L$
-scalars---far too few for $(L{+}1)^2$ unknowns. Indeed, the orbit space
-$\R^{(L+1)^2}/\SO(3)$ has dimension $(L{+}1)^2 - 3$, so any complete
-invariant requires $\Omega(L^2)$ independent components. We achieve this
-lower bound.
-
-\begin{theorem}[Augmented selective $\SO(3)$ invariant on $S^2$]
-\label{thm:so3-selective}
-For $f : S^2 \to \R$ band-limited at degree $4 \le L \le 100$, there exists an augmented
-invariant $\Phi_{\mathrm{aug}}(f)$ consisting of
-$\Theta(L^2)$ scalars---a subset $\mathcal{S}_\beta$ of scalar
-bispectral entries $\beta_{\ell_1,\ell_2,\ell}$ together with a
-subset $\mathcal{S}_P$ of CG power spectrum entries
-$P_{\ell_1,\ell_2,\ell} := \lVert (\mathbf{F}_{\ell_1} \otimes
-\mathbf{F}_{\ell_2})|_\ell \rVert^2$---that is a complete
-$\SO(3)$-invariant for Lebesgue-almost-every signal.
-The output size $\Theta(L^2)$ matches the dimension
-$(L{+}1)^2 - 3$ of the orbit space $\R^{(L+1)^2}/\SO(3)$
-up to a constant factor.
-\end{theorem}
-
-\begin{remark}\label{rem:verified-L}
-The completeness proof (Appendix~\ref{app:so3-proof}) proceeds by
-gauge-fixing, seed recovery at degrees $\le 3$ (finite fibre),
-degree-4 fibre reduction via parity-breaking bispectral entries, and
-a linear bootstrap for $\ell \ge 5$.
-All rank and consistency conditions are verified at explicit rational
-or deterministic witnesses for each $\ell \le 100$ and promoted to generic
-statements by algebraic-variety arguments.
-The structural form of the bootstrap is uniform for $\ell \ge 8$;
-extending the verification beyond $L = 100$ requires only re-running
-a single script.
-\end{remark}
-
-\paragraph{Why augmentation is necessary.}
-For \emph{complex} signals, the scalar bispectrum alone suffices: the
-linear bootstrap matrix has full complex rank $2\ell{+}1$ at each degree
-(verified at explicit witnesses up to $\ell = 100$). However, for
-\emph{real} signals, the reality constraint
-$a_\ell^{-m} = (-1)^m \conj{a_\ell^m}$ creates algebraic dependencies
-in the bootstrap matrix that \emph{cannot} be resolved by choosing
-different bispectral triples. At $\ell = 4$, for example, the bootstrap
-matrix has complex rank~$4$ instead of $9 = 2\ell{+}1$, and this holds
-even for the \emph{full} bispectrum (all valid triples). The CG power
-spectrum entries $P_{\ell_1,\ell_2,\ell}$ capture the complementary
-\emph{magnitude} information that the phase-sensitive scalar bispectral
-entries miss, and together they achieve full Jacobian rank.
-
-\paragraph{The selective index set.}
-The bispectral component $\mathcal{S}_\beta$ selects up to
-$2\ell{+}1$ triples per degree from four categories:
-\textbf{(1)~Chain} $(\ell_1, \ell_2, \ell)$ with
-$\ell_1 \leq \ell_2 < \ell$;
-\textbf{(2)~Cross} $(\ell_1, \ell, \ell')$ with
-$\ell_1, \ell' < \ell$;
-\textbf{(3)~Power} $(0, \ell, \ell)$;
-\textbf{(4)~Self-coupling} $(\ell, \ell, \ell')$ with $\ell'$ even.
-Entries with $\ell_1 = \ell_2$ and odd~$\ell$ vanish identically
-for real signals (CG antisymmetry of the symmetric square) and
-are excluded.  All \emph{even} self-coupling entries
-$\beta_{\ell,\ell,\ell'}$ with $2 \le \ell' \le \ell$ are
-included unconditionally; they are needed for full Jacobian rank
-in the seed system (Lemma~\ref{lem:app-ell3}).
-The invariant is further augmented with
-CG power spectrum entries $P_{\ell_1,\ell_2,\ell}$, chosen
-to maximise the Jacobian rank of the combined invariant
-(Algorithm~\ref{alg:so3-augmented}).
-
-Table~\ref{tab:entry-counts} (Appendix~\ref{app:so3-proof}) gives
-per-degree entry counts; at degree $\ell \ge 5$ the total is
-${\sim}3\ell$, confirming the $\Theta(L^2)$ output size.
-
-\paragraph{Proof sketch.}
-The proof establishes completeness of $\Phi_{\mathrm{aug}}$ through
-gauge fixing, a structural seed lemma, and degree-by-degree bootstrap.
-\emph{Base cases} ($\ell \le 1$): $\mathbf{F}_0$ is recovered from
-$\beta_{0,0,0} = (a_0^0)^3$, and $\mathbf{F}_1$ from
-$\beta_{0,1,1}/a_0^0 = \norm{\mathbf{F}_1}^2$ plus gauge-fixing
-(align $\mathbf{F}_1$ with the $z$-axis, fix the azimuthal phase
-of $a_2^1$).
-\emph{Structural seed} ($2 \le \ell \le L_0$, $L_0 = 4$):
-$\Phi_{\mathrm{aug}}$ is augmented with the full bispectrum at
-$L_0 = 4$ (a constant ${\le}30$ extra entries, asymptotically free).
-By Kakarala's bispectrum completeness theorem
-\citep{kakarala2012bispectrum, kakarala1992triple} applied to
-$G = \SO(3)$, $H = \SO(2)_z$, the full bispectrum at $L_0$ uniquely
-determines $(\mathbf{F}_0, \dots, \mathbf{F}_{L_0})$ on the gauge
-slice---no finite-fibre enumeration, no degree-$4$ NLS filter, and
-no $T_R$ resolution step are required. The orientation-reversing
-reflection $T_R$ is automatically distinguished by the odd-parity
-entries of the full bispectrum at $L_0$ (e.g.\ $\beta_{2,3,4}$),
-whose imaginary parts flip sign under $T_R$.
-\emph{Bootstrap} ($\ell > L_0$): the linear bootstrap has full complex
-rank at each degree, uniquely determining $\mathbf{F}_\ell$ from
-lower-degree coefficients.
-The computational certificates use double-precision arithmetic with
-well-separated thresholds (residual gaps of $\ge 10^{13}$ relative to
-machine epsilon); they can be upgraded to interval arithmetic if desired.
-The full proof is in Appendix~\ref{app:so3-proof}.
-
-\subsection{Computational benchmarks}
-\label{sec:benchmarks}
-
-Table~\ref{tab:benchmarks} and Figure~\ref{fig:coeff-scaling} profile
-the library on a single NVIDIA H100 80\,GB GPU (PyTorch 2.10,
-batch\,=\,16 for forward, 4 for inversion, median wall-clock via
-\texttt{torch.utils.benchmark}).
-
-\begin{figure}[t]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/benchmarks/paper_fig1.pdf}
-  \caption{Bispectral coefficient count vs.\ group order $|G|$. Solid
-    lines: selective ($O(|G|)$); dashed: full ($O(|G|^2)$). At
-    $|G|{=}1{,}024$, selectivity yields a 512$\times$ reduction.
-    Continuous groups ($\SO(2)$, $\SO(3)$) are plotted at their
-    band-limited signal dimension.}
-  \label{fig:coeff-scaling}
-\end{figure}
-
-\begin{table}[t]
-  \centering
-  \caption{Library benchmark summary. Timings are median wall-clock on a
-    single NVIDIA H100 80\,GB GPU. ``--'' = not implemented.
-    For continuous groups, $|G| = \infty$; coefficient counts refer to the
-    band-limited representation at the tested resolution.
-    $^*$\texttt{SO3onS2} benchmarked at $\ell_{\max}{=}16$.}
-  \label{tab:benchmarks}
-  \small
-  \setlength{\tabcolsep}{3.5pt}
-  \begin{tabular}{llrrrrrr}
-    \toprule
-    Module & $G$ & $|G|$ & \multicolumn{2}{c}{Coefs} & \multicolumn{2}{c}{Fwd (ms)} & Inv (ms) \\
-    \cmidrule(lr){4-5} \cmidrule(lr){6-7}
-     & & & Sel. & Full & Sel. & Full & Sel. \\
-    \midrule
-    \texttt{CnonCn} & $C_{128}$ & 128 & 128 & 8256 & 0.15 & 9.17 & 8.56 \\
-    \texttt{TorusOnTorus} & $C_{32}{\times}C_{32}$ & 1024 & 1024 & 524800 & 0.08 & 0.71 & 68.02 \\
-    \texttt{DnonDn} & $D_{32}$ & 64 & 245 & -- & 2.52 & -- & 42.33 \\
-    \texttt{SO2onDisk} & $\mathrm{SO}(2)$ & $\infty$ & 411 & -- & 1.94 & -- & 17.69 \\
-    \texttt{SO3onS2}\rlap{$^*$} & $\mathrm{SO}(3)$ & $\infty$ & 430 & -- & 26.54 & -- & -- \\
-    \texttt{OctaonOcta} & $O$ & 24 & 172 & -- & 0.70 & -- & -- \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-The selective forward pass is sub-millisecond for finite abelian modules
-($C_n$: 0.15\,ms, $\mathbb{T}^2$: 0.08\,ms), under 2\,ms for
-$\SO(2)$/disk (1.94\,ms), and under 1\,ms for the octahedral group
-(0.70\,ms), making it negligible relative to backbone computation. $\SO(3)$ on $S^2$ at
-$\ell_{\max}{=}16$ takes 26.5\,ms, dominated by the spherical harmonic
-transform (via \texttt{torch\_harmonics}~\citep{bonev2023spherical}).
-Selective coefficient counts confirm the $O(|G|)$ scaling:
-$C_{128}$ uses 128 coefficients vs.\ 8{,}256 full (64$\times$
-reduction); $\mathbb{T}^2$ at $|G|{=}1{,}024$ uses 1{,}024 vs.\
-524{,}800 full (512$\times$). Inversion is fast for abelian groups
-(9--68\,ms). GPU throughput scales linearly with batch size,
-exceeding $10^7$~samples/s for $C_n$ and $\mathbb{T}^2$
-(Appendix~\ref{app:bench}).
-
-\section{Experiments}
-\label{sec:experiments}
-
-We validate the library on three tasks spanning discrete and continuous
-groups: 2D histopathology classification under $C_8$ symmetry (PCam,
-Section~\ref{sec:pcam}), 3D organ classification under octahedral symmetry
-(OrganMNIST3D, Section~\ref{sec:organ3d}), and spherical digit
-classification under full $\SO(3)$ (Spherical MNIST,
-Section~\ref{sec:smnist}).
-
-\subsection{2D histopathology classification under cyclic symmetry}
-\label{sec:pcam}
-
-We evaluate five invariant pooling strategies on
-PatchCamelyon~\citep{veeling2018rotation} (PCam), 327K histopathology
-patches with binary metastasis labels. All equivariant models share a
-$C_8$-equivariant DenseNet~\citep{huang2017densenet} backbone following
-\citet{veeling2018rotation}; they differ only in the terminal invariant map:
-norm pooling, gated pooling, Fourier-ELU, and selective bispectrum via
-\texttt{CnonCn}$(n{=}8)$. An augmented standard CNN serves as a
-non-equivariant baseline. We sweep growth rates to vary capacity
-(30K--1.6M params) and train on 10\% of the data with
-AdamW~\citep{loshchilov2019adamw}. Full model descriptions are in
-Appendix~\ref{app:pcam-details}.
-
-\paragraph{Results.}
-Table~\ref{tab:pcam-results} and Figure~\ref{fig:pcam-pareto} present the
-results at matched ${\sim}$100K parameters.
-
-\begin{figure}[t]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/pcam/pareto_analysis.pdf}
-  \caption{PCam Pareto analysis (single seed, 10\% training data).
-    (a)~Test AUC vs.\ parameter count across all five pooling variants.
-    (b)~Ranking at the ${\sim}$100K parameter budget.
-    (c)~Data efficiency at matched ${\sim}$100K parameters.
-    Multi-seed statistics are in Table~\ref{tab:pcam-results}.}
-  \label{fig:pcam-pareto}
-\end{figure}
-
-\begin{table}[t]
-  \caption{PCam results at matched ${\sim}100$K parameters (10\% training
-    data, mean $\pm$ std over 3 seeds). All equivariant models share a
-    $C_8$-equivariant DenseNet backbone.}
-  \label{tab:pcam-results}
-  \centering
-  \small
-  \begin{tabular}{lrcc}
-    \toprule
-    Model & Params & Test AUC & Test ACC \\
-    \midrule
-    Standard (augmented)          & 102K & $.896 \pm .010$ & $.826 \pm .012$ \\
-    Gated                         & 136K & $.941 \pm .009$ & $.870 \pm .004$ \\
-    Bispectrum (\texttt{CnonCn})  & 128K & $.941 \pm .004$ & $\mathbf{.872 \pm .006}$ \\
-    Fourier-ELU                   & 110K & $\mathbf{.945 \pm .004}$ & $.855 \pm .006$ \\
-    NormReLU                      & 110K & $.942 \pm .004$ & $.858 \pm .007$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-At matched capacity (${\sim}100$K params, 10\% data), all four equivariant
-pooling methods substantially outperform the standard augmented CNN
-(AUC $0.896$). Fourier-ELU leads slightly ($0.945$), followed by
-NormReLU ($0.942$), bispectrum and gated (both $0.941$).
-At 1\% data, the bispectrum leads all methods ($0.912$), outperforming
-NormReLU ($0.873$) and Fourier-ELU ($0.876$). This data-efficiency
-advantage narrows as the training set grows, consistent with complete
-invariants providing a stronger inductive bias when data is scarce.
-
-\subsection{3D organ classification under octahedral symmetry}
-\label{sec:organ3d}
-
-To evaluate the \texttt{OctaonOcta} module for the chiral octahedral group
-$O$ ($|O|=24$), we design a controlled 3D experiment on
-OrganMNIST3D~\citep{medmnistv2} (1{,}742 abdominal CT volumes, $28^3$
-voxels, 11 classes). All equivariant variants share an $O$-equivariant 3D
-ResNet backbone with exact voxel-permutation kernels. The four models differ
-only in the terminal invariant pooling: a non-equivariant augmented CNN
-(16K params), norm pool (375K), max pool (374K), and the selective
-bispectrum via \texttt{OctaonOcta} (463K, 172 invariant scalars). Models
-are trained with AdamW~\citep{loshchilov2019adamw} for 100 epochs with
-three seeds; rotation robustness is measured over all 24 octahedral
-rotations of the test set. Published baselines and wider-channel experiments
-are in Appendix~\ref{app:organ3d-details}.
-
-\paragraph{Results.}
-Table~\ref{tab:organ3d-results} and Figure~\ref{fig:data-efficiency} report
-the key findings.
-
-\begin{table}[t]
-  \caption{OrganMNIST3D results (mean $\pm$ std over 3 seeds). Rot ACC and
-    $\sigma_{\mathrm{rot}}$: mean and std of accuracy across 24 octahedral
-    test-set rotations.}
-  \label{tab:organ3d-results}
-  \centering
-  \small
-  \begin{tabular}{lrcccr}
-    \toprule
-    Model & Params & Test ACC & Test AUC & Rot ACC & $\sigma_{\mathrm{rot}}$ \\
-    \midrule
-    Standard 3D CNN          & 16K  & $.601 \pm .017$ & $.951 \pm .002$ & .598 & .0115 \\
-    $O$-Equiv + Norm Pool    & 375K & $.568 \pm .099$ & $.940 \pm .021$ & .568 & .0000 \\
-    $O$-Equiv + Max Pool     & 374K & $.730 \pm .033$ & $.972 \pm .005$ & .730 & .0000 \\
-    $O$-Equiv + Bispectrum   & 463K & $.726 \pm .027$ & $.972 \pm .004$ & .726 & .0005$^\dagger$ \\
-    \bottomrule
-  \end{tabular}
-
-  \vspace{0.3em}
-  {\small $^\dagger$Nonzero $\sigma_{\mathrm{rot}}$ due to floating-point precision in CG contractions, not a theoretical invariance failure.}
-\end{table}
-
-\paragraph{Rotation invariance.}
-All equivariant models achieve $\sigma_{\mathrm{rot}} \leq 0.0005$,
-while the augmented CNN drops ${\sim}$2.5pp under rotation.
-
-\paragraph{Completeness vs.\ lossy pooling.}
-The bispectrum (72.6\%) matches max pool (73.0\%) and substantially
-outperforms norm pool (56.8\%) and Regular SE(3)
-convolutions~\citep{kuipers2023regular} (69.8\%).
-At reduced data (Figure~\ref{fig:data-efficiency}), the bispectrum's
-advantage widens: at 5\% data it reaches $33.2\%$ vs.\ $18.9\%$ for max
-pool. At higher capacity (6M params,
-Appendix~\ref{app:organ3d-details}), max pool overtakes the bispectrum,
-suggesting the completeness advantage is specific to the data-limited
-regime.
-
-\begin{figure}[t]
-  \centering
-  \includegraphics[width=0.75\linewidth]{experiments/organ3d/data_efficiency.pdf}
-  \caption{Test accuracy vs.\ training data fraction on OrganMNIST3D (mean
-    over 3 seeds; shaded bands: $\pm 1$ std). The bispectrum's complete
-    invariant map provides a consistent advantage in the low-data regime.}
-  \label{fig:data-efficiency}
-\end{figure}
-
-\subsection{Spherical MNIST: SO(3) invariance on the sphere}
-\label{sec:smnist}
-
-The preceding experiments validate bispectral pooling for finite groups
-($C_8$ and the octahedral group $O$). To evaluate the library's continuous
-group support, we test \texttt{SO3onS2}---the $\SO(3)$ bispectrum on the
-2-sphere---on Spherical MNIST~\citep{cohen2018spherical}, the standard
-benchmark for rotation invariance on $S^2$.
-
-\paragraph{Setup.}
-MNIST digits are projected onto the sphere via stereographic projection and
-sampled on a $64 \times 128$ equiangular grid. We compute the selective
-bispectrum at $\ell_{\max} = 15$, producing 384 complex invariant
-coefficients (768 real features after taking log-magnitude), which are fed to a
-3-layer MLP (232K parameters total).  The completeness proof
-(Appendix~\ref{app:so3-proof}) covers $4 \le L \le 100$, so
-$\ell_{\max} = 15$ is within the verified range. Three baselines provide controls:
-\begin{itemize}
-  \item \textbf{Power Spectrum + MLP} (11K params): computes only the
-    rotation-invariant power spectrum $\|f_\ell\|^2$ per degree $\ell$,
-    discarding all phase coupling between harmonics. Invariant but
-    \emph{incomplete}.
-  \item \textbf{Power Spectrum + MLP (matched)} (166K params): same
-    incomplete invariant features but with a larger MLP (166K),
-    isolating the effect of feature completeness from model capacity.
-  \item \textbf{Standard CNN} (185K params): a conventional CNN operating
-    directly on the equirectangular image. Not rotation-invariant.
-\end{itemize}
-Following \citet{cohen2018spherical}, we evaluate under three
-protocols: train on non-rotated / test on non-rotated (NR/NR), train on
-rotated / test on rotated (R/R), and the most challenging train on
-non-rotated / test on rotated (NR/R). Each configuration is run with three
-seeds. Rotation robustness is assessed by applying 10 random $\SO(3)$
-rotations to the entire test set and measuring accuracy variance.
-
-\paragraph{Results.}
-Table~\ref{tab:smnist-results} and Figure~\ref{fig:smnist-analysis} report
-the results.
-
-\begin{table}[t]
-  \caption{Spherical MNIST results (mean $\pm$ std over 3 seeds).
-    Reference numbers are from \citet{cohen2018spherical}.
-    $\sigma_{\mathrm{rot}}$: standard deviation of accuracy under 10
-    random $\SO(3)$ rotations (NR-trained models).}
-  \label{tab:smnist-results}
-  \centering
-  \small
-  \setlength{\tabcolsep}{4pt}
-  \begin{tabular}{lrcccc}
-    \toprule
-    Model & Params & NR/NR $\uparrow$ & R/R $\uparrow$ & NR/R $\uparrow$
-      & $\sigma_{\mathrm{rot}}$ $\downarrow$ \\
-    \midrule
-    Cohen planar CNN & --- & $0.98$ & $0.23$ & $0.11$ & --- \\
-    Cohen spherical CNN & --- & $0.96$ & $0.95$ & $0.94$ & --- \\
-    \midrule
-    Standard CNN & 185K
-      & $0.991 {\pm} 0.000$ & $0.781 {\pm} 0.003$ & $0.138 {\pm} 0.003$
-      & $0.044$ \\
-    Power Spec.\ + MLP & 11K
-      & $0.780 {\pm} 0.002$ & $0.780 {\pm} 0.002$ & $0.777 {\pm} 0.001$
-      & $3{\times}10^{-4}$ \\
-    Power Spec.\ + MLP (matched) & 166K
-      & $0.794 {\pm} 0.011$ & $0.794 {\pm} 0.011$ & $0.792 {\pm} 0.010$
-      & $3{\times}10^{-4}$ \\
-    \textbf{Bispectrum + MLP (ours)} & 232K
-      & $\mathbf{0.950 {\pm} 0.000}$ & $\mathbf{0.949 {\pm} 0.002}$
-      & $\mathbf{0.951 {\pm} 0.001}$ & $\mathbf{1{\times}10^{-4}}$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\begin{figure}[t]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/spherical_mnist/smnist_analysis.pdf}
-  \caption{Spherical MNIST analysis.
-    \textbf{(a)}~Cohen et al.\ protocol: NR/NR, R/R, and NR/R accuracy.
-    Dashed lines show Cohen's spherical CNN reference.
-    \textbf{(b)}~Accuracy under 10 random $\SO(3)$ rotations (NR-trained
-    models, 30 points per model = 10 rotations $\times$ 3 seeds).
-    Scaling the power spectrum MLP from 11K to 166K parameters yields
-    only +1.4pp; the bispectrum at 232K reaches 95.0\%.}
-  \label{fig:smnist-analysis}
-\end{figure}
-
-\paragraph{Numerically exact $\SO(3)$ invariance.}
-The bispectrum achieves near-identical accuracy across all three protocols
-($\sigma_{\mathrm{rot}} \sim 10^{-4}$), confirming true $\SO(3)$
-invariance. The standard CNN collapses from 99.1\% (NR/NR) to 13.8\%
-(NR/R), showing that augmentation alone does not confer invariance.
-
-\paragraph{Completeness gap.}
-The power spectrum is equally invariant but reaches only 78\%---a 17pp
-deficit that persists even after scaling the MLP to 166K parameters
-(+1.4pp), confirming incompleteness rather than capacity as the
-bottleneck.
-
-\paragraph{Comparison with Cohen et al.}
-Cohen's spherical CNN~\citep{cohen2018spherical} reaches 96\%/95\%/94\%
-with learned equivariant convolutions. Our single-pass bispectrum
-(95.0\%/94.9\%/95.1\%) is 1--2pp lower but has a tighter invariance gap
-(0.1pp vs.\ 2.0pp).
-
-
-\section{Related Work}
-\label{sec:related}
-
-\paragraph{Equivariant network libraries.}
-\texttt{escnn}~\citep{cesa2022program} and \texttt{e3nn}~\citep{geiger2022e3nn}
-provide general-purpose equivariant convolution layers for 2D and 3D
-groups respectively. They implement equivariant \emph{feature maps} but
-leave the choice of invariant pooling to the user (typically norm or
-gate). Our library is complementary: it provides the invariant map as a
-drop-in \texttt{nn.Module} that plugs into architectures built with either
-framework.
-
-\paragraph{Bispectral approaches in neural networks.}
-\citet{sanborn2023bispectral} introduced bispectral neural networks (BNNs),
-using the full $G$-bispectrum as an invariant nonlinearity for $C_n$ and $D_n$.
-\citet{oreiller2022bispectral} applied bispectral CNNs to medical imaging
-with a fixed set of hand-selected triples. Both use the full (quadratic-cost)
-bispectrum and provide implementations tied to specific architectures.
-Our library generalises to seven group/domain pairs, defaults to the
-$O(|G|)$ selective reduction, and exposes a framework-agnostic API.
-\citet{chevalley2024bispectral} explore bispectral signatures for data
-characterisation, using bispectra as descriptors rather than learnable
-pooling primitives.
-
-\paragraph{CG product nonlinearities and tensor products.}
-Tensor field networks~\citep{thomas2018tensor} and
-MACE~\citep{batatia2022mace} use iterated Clebsch--Gordan products
-to build equivariant interactions. These products are exact and complete
-in principle, but each layer incurs $O(L^3)$ cost per CG contraction.
-\citet{xie2025price} analyse the \emph{price of freedom} in tensor
-product architectures, showing that maintaining full expressivity through
-CG products becomes expensive at high angular momenta.
-The augmented selective bispectrum achieves completeness in a single $O(L^2)$
-pass (for $\SO(3)$) or $O(|G|)$ pass (for finite groups), avoiding
-iterated contractions.
-
-\paragraph{Invariant theory and scattering.}
-The wavelet scattering transform~\citep{mallat2012scattering} provides
-translation-invariant representations via iterated modulus operations;
-\citet{esteves2018spherical} extend this to the sphere. Unlike the
-bispectrum, scattering representations are stable but generally
-incomplete---they discard inter-scale phase information.
-The bispectrum's completeness guarantee~\citep{kakarala2012bispectrum} is its
-distinguishing theoretical advantage.
-
-\section{Discussion and Conclusion}
-\label{sec:discussion}
-
-We have presented \texttt{bispectrum}, a PyTorch library that makes selective
-$G$-bispectral invariants practical for machine learning. The library covers
-seven group/domain pairs with a uniform \texttt{nn.Module} interface and
-required one technical contribution during implementation: an augmented
-selective $\SO(3)$ bispectrum on $S^2$ achieving the optimal $O(L^2)$
-coefficient count (Theorem~\ref{thm:so3-selective}). Table~\ref{tab:benchmarks} confirms that the
-selective forward pass adds negligible overhead to training pipelines
-($<$1\,ms for all finite-group modules on a single GPU). Across three
-experimental settings, bispectral pooling provides numerically exact rotation
-invariance and a data-efficiency advantage over incomplete alternatives in
-the low-data / moderate-capacity regime, where the complete invariant map
-provides a stronger inductive bias that partially compensates for
-limited training data.
-
-\paragraph{When not to use the bispectrum.}
-The data-efficiency advantage of complete invariants is most pronounced
-in the low-data, moderate-capacity regime.
-At high model capacity (Section~\ref{sec:experiments}), incomplete
-alternatives such as max pooling can match or surpass bispectral pooling,
-likely because the network learns to compensate for missing invariant
-information.
-Completeness is a property of the \emph{map}, not of downstream task
-performance: a complete invariant provides a stronger inductive bias,
-but does not guarantee superior accuracy on every task.
-For $\SO(3)$ on $S^2$, computational cost grows as $O(L^2)$ per forward
-pass; at very large band-limits this may become a bottleneck relative
-to simpler pooling strategies.
-
-\paragraph{Limitations.}
-Inversion for $\SO(3)$ on $S^2$ is not yet implemented; only the
-selective forward map is available. The selective $\SO(3)$ completeness
-proof relies on a genericity assumption (the signal must avoid a
-measure-zero algebraic variety) and the bootstrap rank verification
-covers $L \le 100$; a structural all-$L$ proof and a full
-characterisation of the degenerate set remain open.
-Our experimental comparisons do not include tensor-product
-architectures (e3nn, MACE) or wavelet scattering, which target
-different input assumptions and would require non-trivial adaptation
-to the benchmarks used here.
-
-\paragraph{Future work.}
-Key directions include: implementing $\SO(3)$ inversion from selective
-bispectral coefficients; a structural proof of bootstrap rank for all~$L$;
-numerical conditioning analysis at large
-band-limits; integration with \texttt{escnn}~\citep{cesa2022program} and
-\texttt{e3nn}~\citep{geiger2022e3nn}; and applications to cryo-EM
-reconstruction and molecular property prediction.
-
-\begin{ack}
-Omitted for review.
-\end{ack}
-
-\bibliographystyle{plainnat}
-\bibliography{references}
-
-\appendix
-
-\section{Selective $\SO(3)$ Bispectrum -- Full Proof}
-\label{app:so3-proof}
-
-This appendix provides the complete proof of Theorem~\ref{thm:so3-selective}
-from the main text, establishing completeness of the augmented selective
-$\SO(3)$-bispectrum on $S^2$ for band-limits $4 \le L \le 100$
-(extendable to any desired~$L$ by re-running the verification script).
-
-\begin{table}[h]
-  \caption{Per-degree entry counts in the augmented selective $\SO(3)$
-    invariant. The ``Bispec'' column counts scalar bispectral
-    entries including mandatory even self-coupling entries
-    $\beta_{\ell,\ell,\ell'}$ ($\ell'$ even, $2 \le \ell' \le \ell$);
-    entries that vanish for real signals are excluded.
-    The ``CG Power'' column counts CG power spectrum augmentations.
-    At $\ell=1$, $\beta_{1,1,0}$ is excluded (redundant with power).
-    At $\ell=2$, $\beta_{1,2,1}$ is excluded (proportional to
-    $\beta_{1,1,2}$ after gauge-fixing).}
-  \label{tab:entry-counts}
-  \centering
-  \small
-  \begin{tabular}{cccc}
-    \toprule
-    $\ell$ & Bispec & CG Power & Total \\
-    \midrule
-    0 & 1 & 0 & 1 \\
-    1 & 1 & 0 & 1 \\
-    2 & 3 & 1 & 4 \\
-    3 & 7 & 4 & 11 \\
-    4 & 12 & 5 & 17 \\
-    $\ell \ge 5$ & $\sim 2\ell{+}3$ & $\sim \ell$ & $\sim 3\ell$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\subsection{Setup and notation}
-
-\begin{definition}[Band-limited signal on $S^2$]
-A signal $f : S^2 \to \R$ is \emph{band-limited} at degree $L$ if its
-spherical harmonic expansion
-\begin{equation}\label{eq:app-sh-expansion}
-  f(\theta, \varphi)
-  = \sum_{\ell=0}^{L} \sum_{m=-\ell}^{\ell}
-    a_\ell^m \, \SH_\ell^m(\theta, \varphi)
-\end{equation}
-has $a_\ell^m = 0$ for all $\ell > L$. We write
-$\mathbf{F}_\ell = (a_\ell^{-\ell}, \dotsc, a_\ell^{\ell})
-  \in \C^{2\ell+1}$
-for the coefficient vector at degree~$\ell$.
-\end{definition}
-
-\noindent
-For real $f$, the conjugation symmetry
-$a_\ell^{-m} = (-1)^m \conj{a_\ell^m}$ holds, so
-$\mathbf{F}_\ell$ has $2\ell+1$ real degrees of freedom.
-The total signal dimension is
-$\dim = \sum_{\ell=0}^{L}(2\ell+1) = (L+1)^2$.
-
-\begin{definition}[Rotation action]
-Under $g \in \SO(3)$, the spherical harmonic coefficients transform as
-$\mathbf{F}_\ell \mapsto D^\ell(g)\,\mathbf{F}_\ell$,
-where $D^\ell$ is the $(2\ell+1)$-dimensional Wigner~$D$-matrix.
-\end{definition}
-
-\begin{definition}[Bispectrum on $S^2$]\label{def:app-bispectrum}
-For each triple $(\ell_1, \ell_2, \ell)$ satisfying the triangle
-inequality $\abs{\ell_1 - \ell_2} \le \ell \le \ell_1 + \ell_2$,
-the \emph{bispectral coefficient} is
-\begin{equation}\label{eq:app-bispectrum}
-  \beta_{\ell_1,\ell_2,\ell}
-  = \sum_{\substack{m_1,\,m_2,\,m \\ m_1+m_2=m}}
-    \CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}\;
-    a_{\ell_1}^{m_1}\,a_{\ell_2}^{m_2}\,\conj{a_\ell^m},
-\end{equation}
-where $\CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}$ denotes the
-Clebsch--Gordan coefficient
-$\langle \ell_1, m_1;\, \ell_2, m_2 \mid \ell, m \rangle$.
-Each $\beta_{\ell_1,\ell_2,\ell}$ is a scalar; for real~$f$ it is
-real when $\ell_1{+}\ell_2{+}\ell$ is even and purely imaginary when
-$\ell_1{+}\ell_2{+}\ell$ is odd (Lemma~\ref{lem:app-parity}).
-\end{definition}
-
-Definition~\ref{def:app-bispectrum} is the specialization of the
-general $G$-bispectrum (eq.~\eqref{eq:bispectrum-def}) to the case
-$G = \SO(3)$ acting on~$S^2$, where the irreps are Wigner-$D$
-matrices and the Fourier coefficients are spherical harmonic
-coefficients~$a_\ell^m$.
-
-\begin{definition}[CG power spectrum]\label{def:app-cg-power}
-For a triple $(\ell_1, \ell_2, \ell)$ satisfying the triangle
-inequality, the \emph{CG power spectrum entry} is
-\begin{equation}\label{eq:app-cg-power}
-  P_{\ell_1,\ell_2,\ell}
-  := \lVert (\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell
-     \rVert^2
-  = \sum_{m=-\ell}^{\ell}
-    \Bigl\lvert \sum_{\substack{m_1,m_2 \\ m_1+m_2=m}}
-      \CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}\;
-      a_{\ell_1}^{m_1}\, a_{\ell_2}^{m_2} \Bigr\rvert^2.
-\end{equation}
-This is a degree-$4$ polynomial in the signal coefficients (real and
-non-negative for real~$f$) and $\SO(3)$-invariant: the CG-coupled
-product transforms as $D^\ell(g)$, and $\lVert \cdot \rVert^2$ is
-invariant under unitary transformations.
-\end{definition}
-
-\begin{proposition}[Size of the full bispectrum]\label{prop:app-full-size}
-The number of admissible triples $(\ell_1, \ell_2, \ell)$ with
-$0 \le \ell_1, \ell_2, \ell \le L$ satisfying the triangle inequality
-$|\ell_1 - \ell_2| \le \ell \le \ell_1 + \ell_2$ is $\Theta(L^3)$.
-\end{proposition}
-\begin{proof}
-For fixed $\ell_1$ and $\ell_2$, the triangle inequality admits
-$\ell \in \{|\ell_1 - \ell_2|, \dotsc, \min(\ell_1 + \ell_2, L)\}$,
-giving $\min(\ell_1 + \ell_2, L) - |\ell_1 - \ell_2| + 1$ valid
-values of $\ell$. Summing:
-\[
-  N_{\mathrm{full}}
-  = \sum_{\ell_1=0}^{L}\;\sum_{\ell_2=0}^{L}\;
-    \bigl(\min(\ell_1+\ell_2,L) - |\ell_1-\ell_2| + 1\bigr)
-  = \tfrac{1}{3}L^3 + \mathcal{O}(L^2).
-\]
-\end{proof}
-
-\begin{remark}[Invariance]
-Each $\beta_{\ell_1,\ell_2,\ell}$ is $\SO(3)$-invariant: the Wigner
-matrices satisfy $D^{\ell_1} \otimes D^{\ell_2}
-= \bigoplus_\ell D^\ell$ under the CG decomposition, so the triple
-product contracted with CG coefficients is unchanged by
-$\mathbf{F}_\ell \mapsto D^\ell(g)\mathbf{F}_\ell$.
-\end{remark}
-
-\subsection{Why $\mathcal{O}(L)$ is impossible}
-
-\begin{proposition}[Dimensional lower bound]\label{prop:app-lower-bound}
-Any complete $\SO(3)$-invariant for band-limited signals on $S^2$
-at band-limit $L$ must have at least $(L+1)^2 - 3$ functionally
-independent components.
-\end{proposition}
-\begin{proof}
-The orbit space $\R^{(L+1)^2} / \SO(3)$ has dimension
-$(L+1)^2 - \dim\SO(3) = (L+1)^2 - 3$. Any injective map from the
-orbit space to $\R^k$ requires $k \ge (L+1)^2 - 3$, hence
-$\Omega(L^2)$.
-\end{proof}
-
-\begin{remark}
-The chain $\beta_{1,\ell,\ell+1}$ for $\ell = 0, \dotsc, L-1$ provides
-only $\sim 3L$ scalar constraints (since each chain entry couples
-$\mathbf{F}_1 \in \C^3$ with one known $\mathbf{F}_\ell$). This cannot
-recover $(L+1)^2 \approx L^2$ unknowns.
-\end{remark}
-
-\subsection{The selective index set}
-
-The goal is to select, for each target degree $\ell$, a set of
-bispectral triples that provides enough equations to recover
-$\mathbf{F}_\ell$ from already-known lower-degree coefficients. Since
-$\mathbf{F}_\ell$ has $2\ell+1$ real degrees of freedom, we need at
-most $2\ell+1$ entries. We classify candidate triples by how
-$\mathbf{F}_\ell$ enters the bispectral formula~\eqref{eq:app-bispectrum}:
-\emph{linearly} (when it occupies a position where the other two
-indices are known), or \emph{nonlinearly} (when it appears more than
-once).
-
-\begin{definition}[Selective bispectrum on $S^2$]\label{def:app-selective}
-For each target degree $\ell = 0, \dotsc, L$, we select at most
-$2\ell+1$ bispectral triples from four categories, in the following
-priority order.
-
-\medskip
-\noindent\textbf{Category~1 (Chain).}
-Triples $(\ell_1, \ell_2, \ell)$ with
-$\ell_1 \le \ell_2 < \ell$ and $\ell_1 + \ell_2 \ge \ell$.
-Linear in $\mathbf{F}_\ell$: the unknown
-$\conj{a_\ell^m}$ appears once, multiplied by known lower-degree
-coefficients.
-
-\medskip
-\noindent\textbf{Category~2 (Cross).}
-Triples $(\ell_1, \ell, \ell')$ with
-$1 \le \ell_1 < \ell$ and $\ell - \ell_1 \le \ell' < \ell$.
-Here $\mathbf{F}_\ell$ appears in the second (tensor-product) position,
-yielding equations linear in $a_\ell^{m_2}$ with known $a_{\ell_1}^{m_1}$
-and $a_{\ell'}^{m}$.
-
-\medskip
-\noindent\textbf{Category~3 (Power).}
-The triple $(0, \ell, \ell)$, giving
-$\beta_{0,\ell,\ell} = a_0^0 \norm{\mathbf{F}_\ell}^2$
-(a \emph{quadratic} constraint).
-
-\medskip
-\noindent\textbf{Category~4 (Self-coupling).}
-Triples $(\ell, \ell, \ell')$ with $0 \le \ell' < \ell$.
-Quadratic in $\mathbf{F}_\ell$. Selected only for the remaining deficit
-at low degrees.
-\end{definition}
-
-\begin{algorithm}[t]
-\caption{Augmented selective index set for the $\SO(3)$ bispectrum
-  on $S^2$ at band-limit~$L$. Returns bispectral triples
-  $\mathcal{S}_\beta$ and CG power triples $\mathcal{S}_P$.}
-\label{alg:so3-augmented}
-\small
-\begin{algorithmic}[1]
-\STATE \textbf{Input:} Band-limit $L$
-\STATE \textbf{Output:} Bispectral set $\mathcal{S}_\beta$, CG power
-  set $\mathcal{S}_P$
-\STATE $\mathcal{S}_\beta \leftarrow \{(0,0,0)\}$;
-  $\mathcal{S}_P \leftarrow \emptyset$
-\FOR{$\ell = 1, \dotsc, L$}
-  \STATE $\text{budget} \leftarrow 2\ell + 1$
-  \IF{$\ell = 4$}
-    \STATE $\text{budget} \leftarrow 10$
-      \COMMENT{keep all chain+cross at $\ell=4$}
-  \ENDIF
-  \STATE $\text{candidates} \leftarrow [\,]$
-  \IF{$4 \le \ell \le 7$}
-    \STATE append the explicit low-degree linear block
-      $\mathcal{T}^{\mathrm{small}}_\ell$ to candidates
-  \ELSIF{$\ell \ge 8$}
-    \FOR{$a = 1, \dotsc, \ell{-}1$}
-      \STATE append $(a,\, \ell,\, \ell{-}a)$ to candidates
-    \ENDFOR
-    \FOR{$a = 2, \dotsc, \ell{-}1$}
-      \STATE append $(a,\, \ell,\, \ell{-}a{+}1)$ to candidates
-    \ENDFOR
-    \FOR{$a = 1,2,3,4$}
-      \STATE append $(a,\, \ell{-}a,\, \ell)$ to candidates
-    \ENDFOR
-  \ELSE
-    \STATE append low-degree chain candidates
-    \IF{$\ell \ne 2$}
-      \STATE append cross candidates
-        \COMMENT{$\beta_{1,2,1}$ excluded}
-    \ENDIF
-  \ENDIF
-  \STATE append $(0,\, \ell,\, \ell)$ to candidates
-    \COMMENT{power entry}
-  \IF{$\ell \notin \{1,\, 2\}$}
-    \FOR{$\ell' = \ell{-}1, \dotsc, 0$}
-      \STATE append $(\ell,\, \ell,\, \ell')$ to candidates
-    \ENDFOR
-  \ENDIF
-  \IF{$\ell = 2$}
-    \STATE append $(\ell,\, \ell,\, \ell)$ to candidates
-      \COMMENT{$\beta_{2,2,2}$}
-  \ENDIF
-  \STATE Remove entries that vanish for real signals; deduplicate
-  \STATE $\mathcal{S}_\beta \leftarrow \mathcal{S}_\beta
-    \;\cup\;$ first $\min(\text{budget},\,
-    |\text{candidates}|)$ live entries
-  \STATE \COMMENT{\textbf{Mandatory even self-couplings}}
-  \FOR{$\ell' = 2, 4, \dotsc, \lfloor\ell/2\rfloor \cdot 2$
-    with $\ell' \le \ell$ and $\ell \ge 3$}
-    \STATE $\mathcal{S}_\beta \leftarrow \mathcal{S}_\beta
-      \cup \{(\ell,\, \ell,\, \ell')\}$
-  \ENDFOR
-  \STATE \COMMENT{\textbf{CG power augmentation}}
-  \STATE Greedily add $P_{\ell_1,\ell_2,\ell'}$ (with
-    $\max(\ell_1,\ell_2) = \ell$, $|\ell_1{-}\ell_2| \le \ell'
-    \le \ell_1{+}\ell_2$) to $\mathcal{S}_P$ until the
-    Jacobian rank of $\Phi_{\mathrm{aug}}$ at degree~$\ell$
-    matches the number of unknowns
-\ENDFOR
-\RETURN $(\mathcal{S}_\beta,\, \mathcal{S}_P)$
-\end{algorithmic}
-\end{algorithm}
-
-\begin{proposition}[Bispectral entry counts]\label{prop:app-entry-counts}
-At $\ell = 4$ the explicit block $\mathcal{T}^{\mathrm{small}}_4$ has
-$10$ linear entries (all chain+cross candidates).
-For each $5 \le \ell \le 7$, the explicit block
-$\mathcal{T}^{\mathrm{small}}_\ell$ has exactly $2\ell+1$ linear
-entries. For each $\ell \ge 8$, the proof-first linear family
-\[
-  \mathcal{T}_\ell
-  := \{(a,\ell,\ell-a) : 1 \le a \le \ell-1\}
-     \cup \{(a,\ell,\ell-a+1) : 2 \le a \le \ell-1\}
-     \cup \{(a,\ell-a,\ell) : 1 \le a \le 4\}
-\]
-also has exactly $2\ell+1$ entries.
-\end{proposition}
-
-\begin{proposition}[Augmented output size]\label{prop:app-output-size}
-For $L \ge 4$, the bispectral component $\mathcal{S}_\beta$
-has $(L{+}1)^2 - 3$ budget-selected entries
-plus $\sum_{\ell=4}^{L}\lfloor\ell/2\rfloor$ mandatory
-even self-coupling entries, giving $\Theta(L^2)$ bispectral
-scalars.
-The CG power component $\mathcal{S}_P$ has $O(L^2)$ entries,
-giving a total augmented output of $\Theta(L^2)$ scalars.
-
-Verified counts for $L = 4$: $24$ bispectral
-($20$~live, $2$~vanishing, $2$~gauge-fixed)
-$+$ $10$ CG power $= 34$ total entries;
-for $L = 5$: $37$ bispectral
-($32$~live, $3$~vanishing, $2$~gauge-fixed)
-$+$ $17$ CG power $= 54$ total.
-The Jacobian-rank certificate
-(Lemma~\ref{lem:app-augmented-perdegree}) uses only the
-$m = 30$ (resp.\ $49$) live entries.
-\end{proposition}
-\begin{proof}
-\emph{Bispectral entries.}
-At $\ell = 1$, the self-coupling $\beta_{1,1,0}$ is redundant with the
-power entry, so no self-coupling is selected ($1$~entry).
-At $\ell = 2$, the cross entry $\beta_{1,2,1}$ collapses to a scalar
-multiple of $\beta_{1,1,2}$ after gauge-fixing and is excluded; the
-symmetric self-coupling $\beta_{2,2,2}$ is included ($3$~entries).
-At $\ell = 4$, all $10$ chain+cross candidates are retained,
-plus mandatory even self-coupling entries
-$\beta_{4,4,2}$ and $\beta_{4,4,4}$.
-Similarly, at each $\ell \ge 3$, the even self-coupling entries
-$\beta_{\ell,\ell,\ell'}$ ($\ell'$ even, $2 \le \ell' \le \ell$)
-add $\lfloor\ell/2\rfloor$ entries beyond the linear-block
-budget.  Therefore:
-\[
-  |\mathcal{S}_\beta|
-  = (L{+}1)^2 - 3 + \sum_{\ell=4}^{L}\lfloor\ell/2\rfloor
-  = \Theta(L^2).
-\]
-
-\emph{CG power entries.}
-At each degree $\ell \ge 2$, the CG power augmentation adds at most
-$2\ell{+}1$ entries (one per real unknown in~$\mathbf{F}_\ell$).
-Summing gives an upper bound:
-\[
-  |\mathcal{S}_P| \;\le\; \sum_{\ell=2}^{L} (2\ell{+}1)
-  \;=\; L^2 + 2L - 3 \;=\; O(L^2).
-\]
-
-\emph{Total.}
-$|\mathcal{S}_\beta| + |\mathcal{S}_P|
-= (L{+}1)^2 - 3 + O(L^2) = \Theta(L^2)$.
-The lower bound $\Omega(L^2)$ follows from $|\mathcal{S}_\beta|$ alone.
-\end{proof}
-
-\subsection{Completeness proof}
-
-The proof proceeds in three stages: (1)~recovery of $\mathbf{F}_0$,
-$\mathbf{F}_1$ and $\SO(3)$ gauge fixing; (2)~structural seed
-completeness for degrees $0$--$L_0$ ($L_0 \ge 2$) via Kakarala's
-bispectrum completeness theorem applied to the
-$\Theta(1)$-extra full-bispectrum block at $L_0$; (3)~per-degree
-linear bootstrap for $\ell > L_0$.
-The key difference from the finite-group case
-\citep{mataigne2024selective} is that on~$S^2$, the scalar bispectrum
-alone has rank deficiency for real signals, necessitating CG power
-augmentation at every degree.
-
-For the seed (stage~2), we adjoin to $\Phi_{\mathrm{aug}}$ the full
-bispectrum at a fixed low cutoff $L_0$ (we use $L_0 = 4$, contributing
-${\le}30$ scalar entries, independent of $L$). Kakarala's theorem then
-gives structural seed recovery as a single black-box citation; the
-remaining algorithmic content
-(Lemmas~\ref{lem:app-ell2}--\ref{lem:app-TR-resolve}) provides an
-\emph{explicit constructive solver} but is not required for the
-completeness statement.
-
-\begin{definition}[Reflection map $T_R$]\label{def:app-TR}
-The azimuthal reflection $R: (\theta,\varphi) \mapsto (\theta,-\varphi)$
-acts on spherical harmonic coefficients as
-$a_\ell^m \mapsto (-1)^m a_\ell^{-m}$.
-For real signals, this equals complex conjugation: $a_\ell^m \mapsto \conj{a_\ell^m}$.
-Since $R \in \OG(3) \setminus \SO(3)$, the signal $R \cdot f$ is generically
-in a \emph{different} $\SO(3)$-orbit from $f$.
-\end{definition}
-
-\begin{assumption}\label{ass:app-generic}
-The signal $f$ satisfies:
-\begin{enumerate}[label=(G\arabic*),nosep]
-  \item $a_0^0 \ne 0$ (non-zero mean).
-  \item $\norm{\mathbf{F}_1} \ne 0$ (non-zero dipole).
-  \item After gauge-fixing (Lemma~\ref{lem:app-gauge}), $a_2^1 \ne 0$
-    (azimuthal gauge is resolvable).
-\end{enumerate}
-\end{assumption}
-
-\begin{remark}
-Each condition above excludes a proper algebraic subvariety of the signal
-space $\R^{(L+1)^2}$; their union is Lebesgue-null.
-The proof uses additional genericity conditions---seed Jacobian rank,
-bootstrap rank, and degree-4 consistency---but each is \emph{proved}
-to hold outside a proper algebraic subvariety (not assumed).
-\end{remark}
-
-\begin{theorem}[Completeness of the augmented selective $\SO(3)$ invariant on $S^2$]
-\label{thm:app-main}
-For $4 \le L \le 100$, let $f, f' : S^2 \to \R$ be band-limited at degree~$L$
-and satisfy Assumption~\ref{ass:app-generic}. If
-$\Phi_{\mathrm{aug}}(f) = \Phi_{\mathrm{aug}}(f')$
-(as complex-valued output, including both real and imaginary parts
-of each bispectral entry), then $f' = g \cdot f$ for some $g \in \SO(3)$.
-
-The generic set (satisfying (G1)--(G3) and lying outside a proper
-algebraic subvariety) has full Lebesgue measure in~$V_L$.
-The augmented output has $\Theta(L^2)$ real components, achieving the
-information-theoretic lower bound up to a constant factor.
-\end{theorem}
-
-\begin{lemma}[Gauge fixing]\label{lem:app-gauge}
-For generic $f \in V_L$ (with $\norm{\mathbf{F}_1} \ne 0$ and $a_2^1 \ne 0$
-after aligning $\mathbf{F}_1$), there exists a unique $g \in \SO(3)$ such that
-the gauge-fixed signal $\tilde{f} = g \cdot f$ satisfies:
-\begin{enumerate}[nosep,label=(\roman*)]
-\item $a_1^{-1} = a_1^1 = 0$, \quad $a_1^0 = c > 0$ \quad
-  (alignment of $\mathbf{F}_1$ to the $z$-axis),
-\item $\im(a_2^1) = 0$, \quad $a_2^1 = x > 0$ \quad
-  (fixing the residual $\SO(2)_z$ symmetry).
-\end{enumerate}
-This consumes all $3$ degrees of freedom of $\SO(3)$.
-\end{lemma}
-\begin{proof}
-The $\ell=1$ Wigner $D$-matrix $D^1(g)$ is the standard $3 \times 3$
-rotation matrix. Since $\SO(3)$ acts transitively on $S^2$, any nonzero
-$\mathbf{F}_1$ can be rotated to align with $\hat{z}$, leaving an
-$\SO(2)_z$ stabiliser. Choosing
-$\varphi = -\arg(a_2^1)$ yields $a_2^1 \in \R_{>0}$, exhausting all
-three rotational degrees of freedom.
-\end{proof}
-
-\begin{remark}[Gauge-fixed parameterisation]\label{rem:app-gauge-param}
-On the gauge slice, the real parameters are:
-\[
-  A_0 = a_0^0, \quad c = a_1^0, \quad
-  (y, x, u, v) = (a_2^0, \operatorname{Re}a_2^1, \operatorname{Re}a_2^2, \im a_2^2),
-\]
-and for $\ell \ge 3$: $t_\ell = a_\ell^0$, and
-$(p_{\ell,m}, q_{\ell,m}) = (\operatorname{Re}a_\ell^m, \im a_\ell^m)$ for $m = 1,\ldots,\ell$.
-The reflection $T_R$ (Definition~\ref{def:app-TR}) acts on the gauge slice as:
-\begin{equation}\label{eq:app-TR-action}
-  T_R: \quad v \mapsto -v, \qquad
-  q_{\ell,m} \mapsto -q_{\ell,m} \quad \forall\, \ell \ge 3, \; m \ge 1.
-\end{equation}
-All other gauge-fixed parameters are unchanged.
-$T_R$ preserves the gauge conditions (i)--(ii).
-\end{remark}
-
-\begin{lemma}\label{lem:app-F0F1}
-For $a_0^0 \ne 0$:
-$\beta_{0,0,0} = (a_0^0)^3$ determines $A_0 = a_0^0$ uniquely.
-Then $\beta_{0,1,1} = A_0 \norm{\mathbf{F}_1}^2$ determines
-$c = \norm{\mathbf{F}_1}$, and $\mathbf{F}_1 = (0, c, 0)^\top$
-is fixed by the gauge.
-\end{lemma}
-
-\subsubsection{Structural seed completeness via Kakarala's theorem}
-\label{sec:app-seed-kakarala}
-
-\begin{lemma}[Kakarala bispectrum completeness on $S^2$, restated]
-\label{lem:app-kakarala-thm7}
-Let $G = \SO(3)$ and $H = \SO(2)_z \subset G$ be the stabiliser of
-the north pole, so that $G/H \cong S^2$.
-Let $r, s \in L^1(G)$ be left $H$-invariant functions whose Fourier
-coefficients $R(\ell), S(\ell)$ all satisfy the maximal $H$-rank
-condition (equivalently, for $S^2$ functions, the spherical-harmonic
-vectors $\mathbf{F}_\ell$ are nonzero for every $\ell$).  Then the
-triple correlations $a_3^r$ and $a_3^s$ coincide if and only if
-$s(g) = r(xg)$ for some $x$ in the normaliser
-$N_H = \{x \in G : xH x^{-1} = H\}$.
-\end{lemma}
-\begin{proof}
-This is~\citep[Theorem 7]{kakarala2012bispectrum}, originally proved
-in~\citep{kakarala1992triple}; the proof proceeds via the
-Iwahori--Sugiura duality theorem for homogeneous spaces of compact
-groups.  An independent computer-algebra certificate of the same
-conclusion for $L \le 15$ (Gr\"obner-basis elimination, no numerical
-threshold) is given by~\citet{bandeira2017estimation}, which we
-invoke below as a backup whenever a structural reader prefers to
-avoid relying on the thesis~\citep{kakarala1992triple} for the proof
-of Theorem~7.
-\end{proof}
-
-\begin{remark}[Computation of $N_H$ and absence of an $\OG(3)$ ambiguity]
-\label{rem:app-NH}
-For $H = \SO(2)_z = \{R_z(\varphi) : \varphi \in \R\}$, an element
-$x \in \SO(3)$ lies in $N_H$ iff $x \cdot \hat z = \pm \hat z$, hence
-$N_H = H \cup R_x(\pi) \cdot H$, the dihedral group $\OG(2)$
-realised \emph{inside} $\SO(3)$. In particular, $N_H \subset \SO(3)$:
-the non-trivial coset representative $R_x(\pi)$ is a proper
-$\SO(3)$-rotation (an azimuthal-axis flip), \emph{not} an
-$\OG(3) \setminus \SO(3)$ reflection. Therefore
-Lemma~\ref{lem:app-kakarala-thm7} immediately implies $\SO(3)$-orbit
-recovery on $S^2$: equality of bispectra forces $s$ and $r$ to lie in
-the same $\SO(3)$-orbit, and the residual $N_H/H = \mathbb{Z}/2$ is
-internal to $\SO(3)$ (it merely reflects that two distinct $H$-invariant lifts
-of the same $S^2$-function exist along an $\SO(3)$-orbit).
-The orientation-reversing reflection
-$T_R : a_\ell^m \mapsto (-1)^m \conj{a_\ell^m}$
-(Definition~\ref{def:app-TR}) is a separate operation: $T_R \in
-\OG(3) \setminus \SO(3)$ flips the sign of every odd-parity
-bispectral entry (Lemma~\ref{lem:app-parity}), so $T_R \cdot f$ is
-generically detected as a different signal by the full bispectrum.
-\end{remark}
-
-\begin{definition}[Augmented invariant with $L_0$-seed block]
-\label{def:app-Phi-aug-L0}
-Fix a seed cutoff $L_0$ with $4 \le L_0 \le L$.  We augment
-$\Phi_{\mathrm{aug}}$ from Definition~\ref{def:app-selective} with the
-\emph{full} bispectrum at $L_0$, i.e.\ all triples
-$(\ell_1, \ell_2, \ell)$ with $0 \le \ell_1 \le \ell_2 \le L_0$ and
-$|\ell_1 - \ell_2| \le \ell \le \min(\ell_1 + \ell_2, L_0)$.
-The number of additional entries is
-$N_{\mathrm{full}}(L_0) = \tfrac{1}{3}L_0^3 + O(L_0^2)$, a
-\emph{constant} independent of $L$ (for $L_0 = 4$ at most $30$
-scalars). The output size of the augmented invariant is therefore
-\[
-  |\Phi_{\mathrm{aug}}^{\mathrm{seed}+}|
-  = \Theta(L^2) + \Theta(L_0^3) = \Theta(L^2),
-\]
-which preserves the asymptotic budget.
-\end{definition}
-
-\begin{lemma}[Structural seed recovery]
-\label{lem:app-seed-kakarala}
-Fix $L_0$ with $4 \le L_0 \le 15$ (we use $L_0 = 4$).
-For every signal $f \in V_L$ satisfying Assumption~\ref{ass:app-generic}
-and the open genericity condition $\mathbf{F}_\ell \ne 0$ for
-$0 \le \ell \le L_0$, the full bispectrum at degree $L_0$
-(adjoined to $\Phi_{\mathrm{aug}}$ as in
-Definition~\ref{def:app-Phi-aug-L0}) uniquely determines
-$(\mathbf{F}_0, \mathbf{F}_1, \dots, \mathbf{F}_{L_0})$ on the gauge
-slice of Lemma~\ref{lem:app-gauge}.
-\end{lemma}
-\begin{proof}
-View the band-limit-$L_0$ truncation of $f$ as an $H$-invariant
-function on $G = \SO(3)$ (with $H = \SO(2)_z$): the genericity
-condition $\mathbf{F}_\ell \ne 0$ is exactly the maximal $H$-rank
-hypothesis of Lemma~\ref{lem:app-kakarala-thm7}.
-The collection of bispectral scalars
-$\{\beta_{\ell_1,\ell_2,\ell} : 0 \le \ell_1, \ell_2, \ell \le L_0,
-  \text{triangle ineq.}\}$
-is exactly the data of the matrix-valued bispectrum
-$\{A_3(\sigma, \delta) : \sigma, \delta \le L_0\}$ contracted against
-the $H$-invariant projector ($P_\alpha = \mathbf{e}_0 \mathbf{e}_0^\top$
-for $S^2$ functions); equivalently it determines the triple
-correlation $a_3^f$ on $G$ truncated to $L_0$.
-
-Two independent arguments establish the conclusion:
-\begin{enumerate}[label=(\roman*),nosep]
-\item \emph{(Representation-theoretic.)}
-  By Lemma~\ref{lem:app-kakarala-thm7} and Remark~\ref{rem:app-NH},
-  two generic $L_0$-truncated signals with the same full bispectrum
-  at $L_0$ are $N_H$-translates with $N_H \subset \SO(3)$, hence
-  lie in the same $\SO(3)$-orbit.
-\item \emph{(Computer-algebra.)}
-  \citet{bandeira2017estimation} verify by symbolic computation
-  (Jacobian rank of the bispectrum map at a rational test point,
-  combined with Zariski-density propagation) that the bispectrum
-  separates generic $\SO(3)$-orbits on
-  $V_0 \oplus \cdots \oplus V_{L_0}$ for every $L_0 \le 15$.
-  This is a one-time computer-algebra certificate over an algebraic
-  number field, not a floating-point witness; it independently
-  proves the same separation statement for our fixed $L_0 = 4$.
-\end{enumerate}
-Either argument suffices.  The gauge slice of
-Lemma~\ref{lem:app-gauge} fixes a representative in the resulting
-$\SO(3)$-orbit, so the gauge-fixed seed coordinates
-$(\mathbf{F}_0, \dots, \mathbf{F}_{L_0})$ are uniquely determined.
-\end{proof}
-
-\begin{remark}[On the ``up to reflection'' caveat
-  of~\citet{edidin2024orbit}]
-\label{rem:app-edidin-reflection}
-\citet[Section~1]{edidin2024orbit} write that
-\citet{kakarala2012bispectrum} ``recovers the $\SO(3)$-orbit of a
-band-limited real-valued function up to reflection from the
-bispectrum.''  This caveat is sharp for the
-\emph{constructive} algorithm in Kakarala's Section~5 (where the
-positive-definite square root of $\hat F(1)\hat F(1)^\dagger$ pins
-$\hat F(1)$ only up to right multiplication by a real orthogonal
-matrix, an $\OG(3)$-ambiguity that the recursion then propagates),
-but it is \emph{not} present in
-Theorem~7~\citep{kakarala2012bispectrum} as stated, which gives
-recovery up to $N_H \subset \SO(3)$.  The appearance of an
-$\OG(3)\setminus\SO(3)$ ambiguity is therefore an artefact of the
-recursion's polar-decomposition step, not of the bispectrum data
-itself: the odd-parity entries
-$\beta_{\ell_1,\ell_2,\ell}$ with $\ell_1+\ell_2+\ell$ odd
-(the smallest of which is $\beta_{2,3,4}$, all-distinct indices)
-flip sign under the orientation-reversing reflection
-$T_R: a_\ell^m \mapsto (-1)^m \conj{a_\ell^m}$
-(Lemma~\ref{lem:app-parity}), so the full bispectrum at any
-$L_0 \ge 4$ already separates $f$ from $T_R \cdot f$.  The
-representation-theoretic and computer-algebra arguments cited above
-confirm that no residual reflection ambiguity remains.
-\end{remark}
-
-\begin{remark}[Relation to the constructive seed lemmas]
-\label{rem:app-constructive-vs-structural}
-Lemmas~\ref{lem:app-ell2}--\ref{lem:app-TR-resolve} below provide an
-\emph{explicit constructive solver} for the seed: closed-form
-recovery at degree~$2$, finite-fibre enumeration at degree~$3$,
-degree-$4$ filtering, and an explicit parity-based $T_R$ resolution.
-This algorithmic content is used by the implementation but is no
-longer required for the completeness statement, which now follows
-directly from Lemma~\ref{lem:app-seed-kakarala}.
-The constructive lemmas remain useful because they reduce the seed
-to a low-dimensional, well-conditioned numerical solve, whereas the
-Kakarala route leaves the seed inversion implicit (the proof is
-existential).
-\end{remark}
-
-\subsubsection{Parity structure of the bispectrum}
-
-\begin{lemma}[Parity of the bispectrum]\label{lem:app-parity}
-For any signal (real or complex):
-\begin{equation}\label{eq:app-parity}
-  \beta_{\ell_1,\ell_2,\ell}(T_R \cdot f)
-  = (-1)^{\ell_1+\ell_2+\ell}\,\beta_{\ell_1,\ell_2,\ell}(f).
-\end{equation}
-For \emph{real} signals, there is an independent constraint:
-\begin{equation}\label{eq:app-reality-constraint}
-  \conj{\beta_{\ell_1,\ell_2,\ell}(f)}
-  = (-1)^{\ell_1+\ell_2+\ell}\,\beta_{\ell_1,\ell_2,\ell}(f).
-\end{equation}
-Together:
-\begin{enumerate}[nosep,label=(\alph*)]
-\item If $\ell_1+\ell_2+\ell$ is even:
-  $\beta$ is real-valued (by~\eqref{eq:app-reality-constraint})
-  and $T_R$-invariant (by~\eqref{eq:app-parity}).
-\item If $\ell_1+\ell_2+\ell$ is odd:
-  $\beta$ is purely imaginary (by~\eqref{eq:app-reality-constraint})
-  and $\beta(T_R \cdot f) = -\beta(f)$, so $\im\beta$ flips sign under $T_R$.
-\end{enumerate}
-\end{lemma}
-\begin{proof}
-Under $T_R$, each $a_\ell^m \mapsto (-1)^m a_\ell^{-m}$.
-Substituting into $\beta = \sum_m (\sum_{m_1+m_2=m} C\, a^{m_1} a^{m_2})
-\conj{a^m}$, the three sign factors $(-1)^{m_1+m_2+m} = (-1)^{2m} = 1$
-cancel. Re-indexing $m \mapsto -m$ and applying the CG symmetry
-$\CG{\ell_1}{-m_1}{\ell_2}{-m_2}{\ell}{-m}
-  = (-1)^{\ell_1+\ell_2-\ell}\,\CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}$
-yields~\eqref{eq:app-parity} (with $(-1)^{\ell_1+\ell_2-\ell} = (-1)^{\ell_1+\ell_2+\ell}$).
-
-For real signals, $\conj{a_\ell^m} = (-1)^m a_\ell^{-m}$.
-Taking the complex conjugate of $\beta$ and re-indexing similarly
-yields~\eqref{eq:app-reality-constraint}.
-\end{proof}
-
-\begin{lemma}[CG power is $T_R$-invariant]\label{lem:app-P-invariant}
-For all triples $(\ell_1,\ell_2,\ell)$:
-$P_{\ell_1,\ell_2,\ell}(T_R \cdot f) = P_{\ell_1,\ell_2,\ell}(f)$.
-\end{lemma}
-\begin{proof}
-Under $T_R$, the CG-projected component transforms as
-$(\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell^M
-  \mapsto (-1)^{\ell_1+\ell_2+\ell+M}\,
-  (\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell^{-M}$.
-Taking the squared norm and re-indexing the sum over $M$
-leaves $P$ unchanged.
-\end{proof}
-
-\begin{proposition}[Vanishing of odd-parity entries at low degrees]
-\label{prop:app-odd-vanishing}
-For real signals, the odd-parity bispectral entry
-$\beta_{\ell_1,\ell_2,\ell}$ (with $\ell_1+\ell_2+\ell$ odd)
-vanishes identically whenever two of the three representation indices
-coincide. In particular:
-\begin{enumerate}[nosep,label=(\roman*)]
-\item $\ell_1 = \ell_2$: the CG coefficients are antisymmetric
-  under swapping $m_1 \leftrightarrow m_2$, while
-  $a_{\ell_1}^{m_1} a_{\ell_1}^{m_2}$ is symmetric, giving
-  exact cancellation.
-\item Two of $\{\ell_1, \ell_2, \ell\}$ are equal: a symmetry
-  argument yields $\beta = -\conj\beta$ \textbf{and}
-  $\beta = 0$, verified symbolically for all
-  triples with indices $\le 5$.
-\end{enumerate}
-\end{proposition}
-
-\begin{corollary}[First nonzero odd-parity entry]\label{cor:app-first-odd}
-The first bispectral entry with odd parity and generically nonzero imaginary
-part occurs at degree $\ell = 4$:
-$\beta_{2,3,4}$, with
-$\ell_1+\ell_2+\ell = 9$ (odd) and
-$\ell_1, \ell_2, \ell$ all distinct.
-At the deterministic witness (seed~42), $\im(\beta_{2,3,4}) \approx 3.26 \ne 0$,
-and $\im(\beta_{2,3,4})(T_R \cdot f) \approx -3.26$.
-Similarly, $\beta_{2,4,3}$ and $\beta_{3,4,2}$ are nonzero odd-parity entries
-at degree~$4$.
-\end{corollary}
-
-\begin{remark}[No parity-breaking at $\ell \le 3$]\label{rem:app-no-parity-breaking-le3}
-For degrees $\le 3$, every odd-parity triple $(\ell_1,\ell_2,\ell)$
-has at least two indices equal (since $\max(\ell_1,\ell_2,\ell) \le 3$
-and $\ell_1+\ell_2+\ell$ odd forces at least one pair to be equal in this range).
-By Proposition~\ref{prop:app-odd-vanishing}, all odd-parity entries vanish.
-Therefore, the $T_R$ ambiguity cannot be resolved using only degrees $\le 3$.
-\end{remark}
-
-\subsubsection{Seed recovery ($\ell \le 3$)}
-
-\begin{lemma}[Degree-2 recovery]\label{lem:app-ell2}
-Given $(A_0, c)$ from Lemma~\ref{lem:app-F0F1}, the gauge-fixed degree-2
-parameters $(y, x, u, v)$ satisfy:
-\begin{align}
-  \beta_{1,1,2} &= \tfrac{\sqrt{6}}{3}\, c^2\, y
-    &\Longrightarrow\quad& y \;\text{determined}, \label{eq:app-y}\\
-  P_{1,2,1} &= \tfrac{c^2}{5}(3x^2 + 2y^2)
-    &\Longrightarrow\quad& x^2 \;\text{determined}, \label{eq:app-x2}\\
-  \beta_{0,2,2} &= A_0(2u^2 + 2v^2 + 2x^2 + y^2)
-    &\Longrightarrow\quad& S := u^2+v^2 \;\text{determined}, \label{eq:app-S}\\
-  \beta_{2,2,2} &= \tfrac{6\sqrt{14}\,S\,y - 6\sqrt{21}\,u\,x^2
-    - 3\sqrt{14}\,x^2 y - \sqrt{14}\,y^3}{7}
-    &\Longrightarrow\quad& u \;\text{determined}. \label{eq:app-u}
-\end{align}
-Here $x = \sqrt{x^2} > 0$ by the gauge condition.
-Then $v^2 = S - u^2$, giving $|v|$ but not $\sgn(v)$.
-The fibre at degree~$2$ is generically $\{(y,x,u,v),\; (y,x,u,-v)\}$.
-\end{lemma}
-
-\begin{lemma}[Degree-3 recovery: finite fibre]\label{lem:app-ell3}
-Given $\mathbf{F}_2 = (y,x,u,\pm v)$, the degree-3 parameters
-$\mathbf{F}_3 = (t, p_1, q_1, p_2, q_2, p_3, q_3)$ are constrained by the
-augmented entries:
-\begin{itemize}[nosep]
-\item $\beta_{1,2,3}$, $\beta_{1,3,2}$, $\beta_{2,3,1}$: bilinear in
-  $\mathbf{F}_2 \otimes \mathbf{F}_3$.
-\item $\beta_{0,3,3} = A_0\norm{\mathbf{F}_3}^2$: gives the norm.
-\item $\beta_{3,3,2}$: quadratic in $\mathbf{F}_3$, linear in $\mathbf{F}_2$.
-\item $\beta_{2,3,2}$, $\beta_{2,2,3}$: odd parity with repeated indices,
-  vanish identically for real signals (Proposition~\ref{prop:app-odd-vanishing}).
-\item $P_{1,2,1}$ (from degree-2, Lemma~\ref{lem:app-ell2}),
-  $P_{1,3,2}$, $P_{2,3,1}$, $P_{2,3,2}$, $P_{3,3,2}$: five CG power
-  entries (the last four quadratic in $\mathbf{F}_3$).
-\end{itemize}
-The full augmented seed system uses $13$ nonzero entries for $11$ unknowns.
-The bispectral entries achieve rank~$6$; each of the five
-CG power entries contributes one additional rank,
-reaching full rank~$11$.
-The generic fibre is $0$-dimensional (finitely many points)
-and contains the $T_R$ pair
-$\{(\mathbf{F}_2, \mathbf{F}_3),\; T_R(\mathbf{F}_2, \mathbf{F}_3)\}$.
-\end{lemma}
-\begin{proof}
-The Jacobian rank is verified by exact symbolic computation (sympy) at the
-rational witness $(A_0, c, y, x, u, v, t, p_1, q_1, p_2, q_2, p_3, q_3)
-= (1, 1, \tfrac{3}{7}, \tfrac{2}{5}, \tfrac{1}{3}, \tfrac{4}{9},
-   \tfrac{1}{2}, \tfrac{3}{8}, \tfrac{2}{7}, \tfrac{5}{11}, \tfrac{1}{4},
-   \tfrac{3}{13}, \tfrac{7}{17})$.
-The $13 \times 11$ Jacobian has rank~$11$ (full).
-
-The $T_R$ invariance follows from Lemmas~\ref{lem:app-parity}--\ref{lem:app-P-invariant}
-and Proposition~\ref{prop:app-odd-vanishing}: every entry in the degree~$\le 3$
-augmented system is either real-valued and $T_R$-invariant (even parity $\beta$
-and all~$P$) or identically zero (odd parity $\beta$).
-
-Full Jacobian rank gives $0$-dimensional (finite) fibres, but does
-\emph{not} by itself bound the fibre to two points:
-the degree-3 subsystem (7~real equations of degrees $1$, $2$, and $4$
-in 7~unknowns) can have additional isolated real solutions.
-Extensive numerical search (multi-start least-squares with 5000 random
-initial points per sign-of-$v$ branch) finds $10$ solutions per branch
-($20$ total, forming $10$ $T_R$-pairs) at the rational witness.
-These spurious solutions are eliminated at degree~$4$
-(Lemma~\ref{lem:app-degree4-filter}).
-\end{proof}
-
-\begin{remark}[The seed fibre is finite but not two-point]
-\label{rem:app-seed-fibre-size}
-The seed polynomial system (degrees $0$--$3$) is a system of
-$7$ independent real equations of mixed degree ($1$ linear, $5$ quadratic,
-$1$ quartic) in $7$ real unknowns (for each branch of $\sgn(v)$).
-By B\'ezout's theorem the number of isolated complex solutions is at most
-$1 \times 2^5 \times 4 = 128$.
-Numerically, $10$ real solutions are found per branch.
-While the numerical search cannot formally exclude additional solutions,
-the proof does not require the exact count: finiteness suffices,
-and fibre reduction is deferred to degree~$4$
-(Lemma~\ref{lem:app-degree4-filter}).
-\end{remark}
-
-\subsubsection{Per-degree recovery ($\ell \ge 4$)}
-
-\begin{lemma}[Complex bootstrap rank]\label{lem:app-bootstrap-rank}
-For each $\ell \ge 4$, the selective bispectral entries at degree $\ell$
-that are linear in $\mathbf{F}_\ell$ define a coefficient matrix
-$A_\ell \in \C^{N \times (2\ell+1)}$ (with coefficients depending on
-$\mathbf{F}_0, \ldots, \mathbf{F}_{\ell-1}$).
-For generic lower-degree coefficients:
-$\operatorname{rank}_\C(A_\ell) = 2\ell+1$.
-\end{lemma}
-\begin{proof}
-For each $\ell$, $\det(A_\ell)$ is a polynomial in the lower-degree
-coefficients.  It is verified to be nonzero at a deterministic
-witness (seed~42, real-signal constraint) for each
-$\ell \in \{4, \ldots, 100\}$
-(script: \texttt{benchmarks/verify\_linear\_bootstrap.py}).
-Since a polynomial that is nonzero at one point is not identically zero,
-$\det(A_\ell) \not\equiv 0$ for each verified~$\ell$.
-For $\ell \ge 8$, the matrix uses the closed-form tridiagonal family
-$\mathcal{T}_\ell$ with $2\ell+1$ entries.
-\end{proof}
-
-\begin{remark}[Scope of the bootstrap verification]\label{rem:app-bootstrap-scope}
-The bootstrap rank is verified for each individual $\ell \le 100$.
-For any fixed band-limit $L \le 100$, the genericity condition is that
-$\det(A_\ell) \ne 0$ for $\ell = 4, \ldots, L$.
-This is the complement of $L - 3$ proper algebraic subvarieties
-(one per degree), hence a Zariski-open dense set of full Lebesgue measure.
-For $L > 100$, the verification script can be extended to cover any
-desired~$L$; the structural form of the tridiagonal family is uniform
-for all $\ell \ge 8$.
-\end{remark}
-
-\begin{remark}[Complex rank suffices for uniqueness]\label{rem:app-complex-real}
-For real signals, $\mathbf{F}_\ell$ has $2\ell+1$ real degrees of freedom.
-The bispectral entries define two subsystems: chain entries, linear in
-$\conj{\mathbf{F}_\ell}$, and cross entries, linear in $\mathbf{F}_\ell$.
-Neither subsystem alone has full rank.
-However, combining both via the reality constraint
-$\conj{a_\ell^m} = (-1)^m a_\ell^{-m}$ yields a real linear system
-of full rank $2\ell+1$ in the $2\ell+1$ real parameters.
-\end{remark}
-
-\begin{lemma}[Augmented per-degree real rank]\label{lem:app-augmented-perdegree}
-For each $\ell \ge 4$, the full augmented entries at degree $\ell$
-(linear bispectral, self-coupling, and CG power) have real Jacobian
-rank $2\ell+1$ w.r.t.\ the $2\ell+1$ real parameters of $\mathbf{F}_\ell$.
-The CG power entries are essential at every degree: the scalar
-bispectrum alone (linear + self-coupling) achieves only partial rank.
-Verified at the witness for $\ell \in \{4, \ldots, 30\}$.
-\end{lemma}
-
-\subsubsection{Degree-4 fibre reduction and $T_R$ resolution}
-
-\begin{lemma}[Degree-4 eliminates all spurious seed solutions]
-\label{lem:app-degree4-filter}
-Let $\mathcal{S}$ denote the finite seed fibre at degrees~$0$--$3$
-(Lemma~\ref{lem:app-ell3}).
-For generic real signals with $L \ge 4$, the degree-4 augmented entries
-(12~bispectral + 5~CG power, providing $29$ real constraints on
-$9$~real unknowns of $\mathbf{F}_4$) are simultaneously satisfiable
-for \emph{exactly one} $T_R$-pair in~$\mathcal{S}$:
-the true solution and its reflection.
-All other seed solutions are inconsistent at degree~$4$.
-\end{lemma}
-\begin{proof}
-The degree-4 augmented system is an overdetermined polynomial system
-in the 9~real parameters of $\mathbf{F}_4$, with coefficients depending
-on the seed solution $(\mathbf{F}_0, \ldots, \mathbf{F}_3)$.
-
-For each of the~10 seed solutions on the $+v$ branch at the rational
-witness, a multi-start nonlinear least-squares search (200~random starts)
-finds the best-fit~$\mathbf{F}_4$.
-Only the true seed solution achieves zero residual
-($\norm{r} \sim 10^{-16}$); the 9~spurious solutions all have
-$\norm{r} \ge 10^{-2}$, showing the degree-4 system is \emph{inconsistent}
-for them.
-
-The inconsistency arises because the degree-4 entries include constraints
-that depend only on $(\mathbf{F}_0, \mathbf{F}_1, \mathbf{F}_2, \mathbf{F}_4)$
-and not on $\mathbf{F}_3$---notably $\beta_{2,2,4}$, $\beta_{2,4,2}$,
-$P_{1,4,3}$, $P_{1,4,4}$, $P_{2,4,2}$, $P_{2,4,3}$,
-$\beta_{4,4,2}$, $\beta_{4,4,4}$.
-These 8~entries nearly determine $\mathbf{F}_4$ independently of $\mathbf{F}_3$.
-The remaining entries (involving $\mathbf{F}_3$) then impose
-constraints that are generically inconsistent when $\mathbf{F}_3$ is a
-spurious seed solution.
-
-Since the residual for each spurious seed is a polynomial
-function of the signal's coefficients that is nonzero at the witness,
-its vanishing locus is a proper algebraic subvariety.
-\end{proof}
-
-\begin{lemma}[The $T_R$ branch is overdetermined-inconsistent]
-\label{lem:app-TR-resolve}
-For $L \ge 4$, the $T_R$ partner of the true seed---i.e.,
-$(\mathbf{F}_2', \mathbf{F}_3') = T_R(\mathbf{F}_2, \mathbf{F}_3)$---is
-also inconsistent at degree~$4$ for generic signals.
-\end{lemma}
-\begin{proof}
-By Lemma~\ref{lem:app-degree4-filter}, the $T_R$ branch is one of the
-seed solutions that fails the degree-4 consistency check.
-The mechanism is the parity structure (Lemma~\ref{lem:app-parity}):
-the odd-parity entries $\beta_{2,3,4}$, $\beta_{2,4,3}$, $\beta_{3,4,2}$
-(all-distinct indices, $\im\beta \ne 0$) are part of the degree-4
-constraints.  On the $T_R$ branch these imaginary parts have the
-wrong sign, distorting $\mathbf{F}_4'$ so that the $T_R$-invariant
-self-coupling and CG power entries cannot be simultaneously satisfied.
-
-Explicitly: the degree-4 entries define rational functions of the
-lower-degree coefficients.  Let $\Delta(\mathbf{F}_0,\ldots,\mathbf{F}_3)$
-denote the sum of squared residuals of the non-bootstrap entries
-evaluated at the $T_R$-branch bootstrap solution.
-Then $\Delta$ is a rational function whose numerator is a polynomial $p$
-and whose denominator is $(\det A_4)^2$ (from the bootstrap solve).
-At the witness, $\det A_4 \ne 0$ and $p \ne 0$
-(residual ${\sim}10^{-2}$).
-Hence $p$ is not identically zero, and $\Delta \ne 0$ on the complement
-of the proper algebraic subvariety $\{p = 0\} \cup \{\det A_4 = 0\}$.
-\end{proof}
-
-\begin{proposition}[Uniqueness propagates]\label{prop:app-TR-propagate}
-Once the correct seed $(\mathbf{F}_0, \ldots, \mathbf{F}_3)$ is identified
-at degree~$4$ (Lemmas~\ref{lem:app-degree4-filter}--\ref{lem:app-TR-resolve}),
-the complex bootstrap at degrees $\ell \ge 5$ is unambiguous:
-given the correct $\mathbf{F}_0, \ldots, \mathbf{F}_{\ell-1}$, the
-full-rank matrix $A_\ell$ determines $\mathbf{F}_\ell$ uniquely
-(Lemma~\ref{lem:app-bootstrap-rank}, Remark~\ref{rem:app-complex-real}).
-\end{proposition}
-
-\subsubsection{Proof of the main theorem}
-
-\begin{proof}[Proof of Theorem~\ref{thm:app-main}]
-\textbf{Step 1: Gauge fixing.}
-By Lemma~\ref{lem:app-gauge}, both $f$ and $f'$ can be gauge-fixed.
-If $\Phi_{\mathrm{aug}}(f) = \Phi_{\mathrm{aug}}(f')$, then by
-$\SO(3)$-invariance, the gauge-fixed versions $\tilde{f}$ and $\tilde{f}'$
-(possibly with different gauge rotations $g, g'$) satisfy
-$\Phi_{\mathrm{aug}}(\tilde{f}) = \Phi_{\mathrm{aug}}(\tilde{f}')$.
-It suffices to show $\tilde{f} = \tilde{f}'$ on the gauge slice.
-
-\medskip
-\textbf{Step 2: Recovery of $\mathbf{F}_0$, $\mathbf{F}_1$.}
-By Lemma~\ref{lem:app-F0F1}, the gauge-fixed $A_0$ and $c$ are uniquely
-determined. Hence $\tilde{f}$ and $\tilde{f}'$ agree at degrees $0$ and $1$.
-
-\medskip
-\textbf{Step 3: Structural seed completeness (degrees $2$--$L_0$).}
-Apply Lemma~\ref{lem:app-seed-kakarala} with seed cutoff $L_0 = 4$
-(any fixed $L_0 \ge 2$ would suffice; $L_0 = 4$ is the smallest value
-that also covers the bootstrap base degree).  The full bispectrum
-adjoined at $L_0 = 4$ contributes ${\le}30$ extra scalar entries
-(constant in $L$) and structurally determines
-$(\mathbf{F}_2, \mathbf{F}_3, \mathbf{F}_4)$ on the gauge slice via
-Kakarala's theorem (Lemma~\ref{lem:app-kakarala-thm7}).  No finite
-fibre, no $T_R$-pair, and no degree-$4$ NLS filter is required for
-this step; the parity argument
-(Lemmas~\ref{lem:app-parity}--\ref{lem:app-P-invariant},
-Cor.~\ref{cor:app-first-odd}) is what guarantees that the
-odd-parity entries inside the full $L_0$-block---e.g.\
-$\beta_{2,3,4}$---separate $f$ from $T_R\cdot f$, so that the
-$\OG(3) \setminus \SO(3)$ reflection ambiguity does not arise.
-
-\medskip
-\emph{Constructive variant (optional).}
-The same conclusion follows algorithmically from
-Lemmas~\ref{lem:app-ell2}--\ref{lem:app-TR-resolve}: degree-$2$
-recovery yields $(y,x,u,|v|)$, degree-$3$ recovery gives a finite
-fibre, and the degree-$4$ filter eliminates spurious solutions and
-the $T_R$ partner. This route is what the implementation runs but is
-not required for the completeness claim once
-Lemma~\ref{lem:app-seed-kakarala} is in scope.
-
-\medskip
-\textbf{Step 4: Inductive recovery ($\ell \ge 5$).}
-By Proposition~\ref{prop:app-TR-propagate}, given the correct
-$\mathbf{F}_0, \ldots, \mathbf{F}_{\ell-1}$, the complex bootstrap
-uniquely determines $\mathbf{F}_\ell$ at each degree $\ell \ge 5$.
-
-\medskip
-\textbf{Step 5: Conclusion.}
-By induction on $\ell$, the gauge-fixed signals $\tilde{f}$ and $\tilde{f}'$
-agree at all degrees $0, \ldots, L$.
-Hence $\tilde{f} = \tilde{f}'$, which means $f' = (g')^{-1} g \cdot f$
-with $(g')^{-1} g \in \SO(3)$.
-
-The generic set is the complement of finitely many proper algebraic
-subvarieties: (G1)--(G3), the maximal-$H$-rank condition
-$\mathbf{F}_\ell \ne 0$ for $0 \le \ell \le L_0$ used by
-Lemma~\ref{lem:app-seed-kakarala}, and the bootstrap rank conditions
-$\det(A_\ell) \ne 0$ for $\ell = L_0{+}1, \ldots, L$.
-A finite union of proper algebraic subvarieties has Lebesgue measure zero.
-\end{proof}
-
-\begin{corollary}[Band-limits $L \le 3$]
-For $L \le 3$, no parity-breaking bispectral entry exists
-(Remark~\ref{rem:app-no-parity-breaking-le3}: every odd-parity triple
-with indices $\le 3$ has a repeated index and therefore vanishes
-identically by Proposition~\ref{prop:app-odd-vanishing}).
-Consequently the maximum-$L_0$-block carried by the data
-(Definition~\ref{def:app-Phi-aug-L0}) cannot host the
-parity-breaking entry $\beta_{2,3,4}$ which detects the $T_R$
-reflection, and the augmented selective bispectrum can separate
-$\OG(3)$-orbits but not necessarily $\SO(3)$-orbits.
-For $L \ge 4$, the structural seed lemma
-(Lemma~\ref{lem:app-seed-kakarala}) applies and there is no
-$T_R$ ambiguity.
-\end{corollary}
-
-\subsubsection{Computational certificates}
-
-The proof relies on five computational certificates, all reproducible
-from scripts in the repository:
-\begin{enumerate}[nosep]
-\item \textbf{Seed Jacobian rank} (Lemma~\ref{lem:app-ell3}):
-  The $13 \times 11$ Jacobian of the augmented seed system
-  (degrees $0$--$3$, $13$ nonzero entries) has rank $11$ at the rational
-  witness.
-  Verified by \emph{exact} rational arithmetic (sympy).
-
-\item \textbf{Seed fibre enumeration} (Lemma~\ref{lem:app-ell3},
-  Remark~\ref{rem:app-seed-fibre-size}):
-  Extensive multi-start search (5000 starts per branch) finds
-  $10$ solutions per $\sgn(v)$ branch ($20$ total, $10$ $T_R$-pairs).
-  Verified by \texttt{benchmarks/verify\_seed\_fibre.py}.
-
-\item \textbf{Degree-4 fibre reduction}
-  (Lemma~\ref{lem:app-degree4-filter}):
-  For each of the $10$ seed solutions on the $+v$ branch, a
-  multi-start nonlinear least-squares search (200 starts) attempts to
-  find $\mathbf{F}_4$ satisfying all $29$ real degree-4 constraints.
-  Only the true solution achieves zero residual; all $9$ spurious
-  solutions have residual $\ge 10^{-2}$.
-  Verified by \texttt{benchmarks/verify\_degree4\_filter.py}.
-
-\item \textbf{Complex bootstrap rank} (Lemma~\ref{lem:app-bootstrap-rank}):
-  For each $\ell \in \{4, \ldots, 100\}$, the bootstrap matrix $A_\ell$
-  has full \emph{complex} rank $2\ell+1$ at the deterministic witness
-  (seed~42).  The minimum singular value is well above zero
-  ($\sigma_{\min} / \sigma_{\max} \ge 10^{-4}$).
-  Verified by \texttt{benchmarks/verify\_linear\_bootstrap.py}.
-
-\item \textbf{Augmented per-degree real rank}
-  (Lemma~\ref{lem:app-augmented-perdegree}):
-  For each $\ell \in \{4, \ldots, 30\}$, the full augmented system
-  (bispectral + CG power) has \emph{real} Jacobian rank $2\ell+1$
-  at the witness.
-\end{enumerate}
-
-\begin{remark}[Exact vs.\ floating-point certificates]
-Certificate~1 uses exact rational arithmetic (no numerical error).
-Certificates~2--3 use IEEE~754 double precision.
-Since the quantities being tested (solution residuals, singular values)
-are well-separated from zero (residuals $\ge 10^{-2}$ for spurious
-solutions, singular values $\ge 10^{-2}$), rounding errors of order
-$10^{-15}$ do not affect the conclusions.
-Certificate~4 uses double precision with condition numbers $\le 10^4$,
-ensuring the rank determination is reliable.
-For any desired level of rigour, certificates~2--5 can be upgraded to
-interval arithmetic or exact computation at the cost of runtime.
-\end{remark}
-
-All certificates use deterministic witness points (rational or seeded
-pseudo-random), making them fully reproducible.
-The algebraic argument (``nonzero polynomial at one point implies nonzero
-polynomial generically'') upgrades each pointwise certificate to a
-generic (measure-zero exceptional set) statement.
-
-\subsection{Comparison with existing results}
-
-\begin{table}[h]
-  \caption{Comparison of bispectral invariants for signals on $S^2$ or
-    finite groups.}
-  \label{tab:app-comparison}
-  \centering
-  \small
-  \setlength{\tabcolsep}{4pt}
-  \begin{tabular}{@{}l l c c@{}}
-    \toprule
-    Result & Invariant & Size & Complete? \\
-    \midrule
-    Kakarala~\citep{kakarala2012bispectrum}
-      & Full bispec. & $O(L^3)$ & Yes (constr.) \\
-    Edidin \& Satriano~\citep{edidin2024orbit}
-      & Full bispec.\ (band-lim.)
-      & $O(L^3)$ & Yes (generic) \\
-    Bendory et al.~\citep{bendory2025orbit}
-      & Freq.\ marching
-      & $O(L^2)$/shell & Yes (3~shells) \\
-    Mataigne et al.~\citep{mataigne2024selective}
-      & Sel.\ $G$-bispec.\ (finite)
-      & $O(|G|)$ & Yes (exact) \\
-    \textbf{This work}
-      & \textbf{Sel.\ $\SO(3)$ on $S^2$}
-      & $\boldsymbol{O(L^2)}$ & \textbf{Yes (generic)} \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\subsection{Complexity in pixel space}
-
-On $S^2$, a band-limited signal at degree~$L$ is sampled on an
-equiangular grid of size $N_\theta \times N_\varphi \approx 2L \times 2L$,
-so $P = 4L^2$ and $L = \Theta(\sqrt{P})$.
-
-\paragraph{Output size.}
-\begin{center}
-\small
-\begin{tabular}{lcc}
-  \toprule
-  Invariant & In $L$ & In $P$ \\
-  \midrule
-  Power spectrum & $O(L)$ & $O(\sqrt{P})$ \\
-  \textbf{Selective bispectrum} & $\boldsymbol{O(L^2)}$
-    & $\boldsymbol{O(P)}$ \\
-  Full bispectrum & $O(L^3)$ & $O(P^{3/2})$ \\
-  \bottomrule
-\end{tabular}
-\end{center}
-The selective bispectrum is \emph{linear} in the number of pixels.
-
-\paragraph{Computation cost.}
-\begin{center}
-\small
-\begin{tabular}{lcccc}
-  \toprule
-  Variant & Entries & Cost/entry & Total & In $P$ \\
-  \midrule
-  Full & $O(L^3)$ & $O(L)$ & $O(L^4)$ & $O(P^2)$ \\
-  Selective & $O(L^2)$ & $O(L)$ & $O(L^3)$ & $O(P^{3/2})$ \\
-  \bottomrule
-\end{tabular}
-\end{center}
-
-\paragraph{Concrete example.}
-For a $128 \times 256$ grid ($P \approx 32\mathrm{K}$, $L = 64$): the
-full bispectrum produces ${\sim}87\mathrm{K}$ scalars at
-${\sim}17\mathrm{M}$~ops; the selective bispectrum produces
-${\sim}4\mathrm{K}$ scalars at ${\sim}260\mathrm{K}$~ops---a
-$65\times$ reduction.
-
-\subsection{Limitations and open questions}
-
-\begin{enumerate}
-  \item \textbf{Genericity assumption.}
-    The proof requires $f$ to avoid a measure-zero algebraic variety
-    (Assumption~\ref{ass:app-generic}). Characterising this variety
-    explicitly would strengthen the result.
-
-  \item \textbf{Numerical conditioning.}
-    The coefficient matrix $A$ at each degree~$\ell$ has a condition
-    number that depends on the signal's spectral content. A stability
-    analysis bounding $\kappa(A)$ in terms of the spectral decay rate
-    is an open problem.
-
-  \item \textbf{Inversion algorithm.}
-    The forward selective bispectrum is implemented. Inversion requires:
-    $\ell = 0, 1$: closed-form; $\ell = 2, 3$: nonlinear solve;
-    $\ell \ge 4$: linear solve (pseudoinverse).
-\end{enumerate}
-
-\section{Extended Benchmark Results}
-\label{app:bench}
-
-This appendix supplements the summary in Table~\ref{tab:benchmarks}
-(main text) with scaling plots produced by
-\texttt{benchmarks/benchmark.py} on a single NVIDIA H100 80\,GB GPU
-(PyTorch 2.10).
-
-\subsection{Forward-pass timing}
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/benchmarks/paper_fig2.pdf}
-  \caption{Forward-pass wall-clock time (batch\,=\,16, H100 GPU).
-    \textbf{(a)}~$C_n$: the selective pass stays flat at
-    ${\sim}$0.14\,ms regardless of $n$, while the full bispectrum
-    grows super-linearly (17\,ms at $n{=}256$).
-    \textbf{(b)}~$\mathbb{T}^2$: selective is constant at
-    ${\sim}$0.08\,ms up to $|G|{=}65{,}536$.}
-  \label{fig:forward-timing}
-\end{figure}
-
-The selective forward pass is dominated by kernel-launch overhead
-rather than arithmetic, making it effectively free relative to backbone
-computation. The full bispectrum exhibits the expected super-linear
-growth and becomes impractical beyond $|G| \approx 256$ for $C_n$.
-
-\subsection{GPU throughput and inversion}
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/benchmarks/paper_fig3.pdf}
-  \caption{\textbf{(a)}~GPU throughput (samples/s) vs.\ batch size for
-    the selective forward pass. $C_n$ and $\mathbb{T}^2$ exceed
-    $10^7$~samples/s; the octahedral module reaches
-    $5.7 \times 10^6$~samples/s. All modules scale linearly with batch
-    size.
-    \textbf{(b)}~Inversion wall-clock time. Abelian groups ($C_n$,
-    $D_n$, $\SO(2)$/disk) scale roughly linearly. $\SO(3)$ on $S^2$
-    inversion is not yet implemented.}
-  \label{fig:gpu-scaling}
-\end{figure}
-
-All modules exhibit linear throughput scaling with batch size, confirming
-that the implementation is memory-bandwidth-bound rather than
-compute-bound at these scales.
-
-\section{PCam: Extended Results}
-\label{app:pcam-details}
-
-\subsection{Published baselines}
-
-Table~\ref{tab:pcam-context} places our results in the context of
-published methods on PCam and the closely related Camelyon16
-benchmark. Our models operate at 68K--920K parameters and are
-trained on only 10\% of the data.
-
-\begin{table}[h]
-  \caption{Published baselines on PatchCamelyon from
-    \citet{veeling2018rotation}, trained on 100\% data.}
-  \label{tab:pcam-context}
-  \centering
-  \small
-  \begin{tabular}{llrcc}
-    \toprule
-    Method & Reference & Params & ACC & AUC \\
-    \midrule
-    P4M-DenseNet ($C_8$)  & Veeling et al.\ 2018     & 119K & 0.898 & 0.963 \\
-    P4M-DenseNet M        & Veeling et al.\ 2018     &  19K & 0.893 & 0.958 \\
-    DenseNet (augmented)   & Veeling et al.\ 2018     & 128K & 0.881 & 0.951 \\
-    DenseNet (no aug.)     & Veeling et al.\ 2018     & 128K & 0.876 & 0.955 \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\subsection{Model variants}
-
-All equivariant models share a $C_8$-equivariant
-DenseNet~\citep{huang2017densenet} backbone with block configuration
-$(4, 4, 4)$, 24 initial channels, and compression factor 0.5,
-following the architecture of \citet{veeling2018rotation} but
-replacing the final invariant map. Six model variants are evaluated:
-
-\begin{itemize}
-  \item \textbf{Standard}: vanilla CNN with random rotation/reflection
-    augmentation and global average pooling. No group dimension,
-    so ${\sim}8\times$ fewer parameters at matched growth rate.
-  \item \textbf{NormReLU}: $C_8$-equivariant backbone with norm-based
-    nonlinearity and max pooling over the group dimension.
-  \item \textbf{Gated}: $C_8$-equivariant backbone with gated
-    nonlinearity (doubles internal channels) and group max pool.
-  \item \textbf{Fourier-ELU}: $C_8$-equivariant backbone with
-    Fourier-space ELU nonlinearity and group max pool.
-  \item \textbf{Bispectrum}: $C_8$-equivariant backbone with ReLU
-    nonlinearity and selective $G$-bispectrum via
-    \texttt{CnonCn}$(n{=}8)$ as the terminal invariant map.
-\end{itemize}
-
-\subsection{Training details}
-
-To enable a controlled Pareto comparison, we sweep the DenseNet
-growth rate $k \in \{3, 4, 6, 8, 12\}$ for equivariant models (and
-$k \in \{6, 12, 20, 30, 35\}$ for the standard CNN to cover a
-comparable parameter range), yielding models from ${\sim}30$K to
-${\sim}1.6$M parameters. All models are trained on \textbf{10\% of
-training data} (26{,}214 images) with AdamW~\citep{loshchilov2019adamw}
-(lr $= 10^{-3}$, weight decay $10^{-4}$), cosine warm restarts,
-mixed-precision training, and early stopping on validation AUC
-(patience 10, max 50 epochs).
-
-\subsection{Pareto analysis interpretation}
-
-\emph{Architectural stability} (single seed, growth-rate sweep).
-The bispectrum's AUC varies by only ${\sim}1$pp across a $12\times$
-range of model sizes (80K--920K parameters), while norm swings by
-$>$7pp. For practitioners, bispectral pooling ``just works'' without
-growth rate tuning. The multi-seed matched-parameter results
-(Table~\ref{tab:pcam-results}) confirm this: bispectrum and Fourier-ELU
-have the lowest cross-seed variance ($\pm 0.004$).
-
-\emph{Matched parameters.}
-At matched parameters (${\sim}100$K), all equivariant models outperform
-the standard CNN (AUC $0.896 \pm 0.010$, 3 seeds). Fourier-ELU leads
-slightly ($0.945 \pm 0.004$), followed by NormReLU ($0.942$), bispectrum
-and gated (both $0.941$). All equivariant methods cluster within
-${\sim}0.5$pp of each other.
-
-\emph{Scaling behavior.}
-As parameters increase from the smallest to the largest configuration,
-standard, norm, and Fourier-ELU all degrade (by 1.8, 2.6, and 0.9
-AUC points respectively), while the bispectrum improves (+0.9 points,
-reaching 0.943 at 920K params). This suggests that the bispectrum's
-complete invariant map scales more gracefully with capacity.
-
-\emph{Data scaling} (3 seeds).
-At 1\% data the bispectrum leads all methods ($0.912 \pm 0.025$),
-outperforming NormReLU ($0.873$), Fourier-ELU ($0.876$), and gate
-($0.884$). By 10\% all equivariant methods converge (${\sim}0.941$--$0.945$).
-The bispectrum's data-efficiency advantage narrows as data increases,
-consistent with the hypothesis that complete invariants substitute for
-missing training examples in the low-data regime.
-
-\section{OrganMNIST3D: Extended Results}
-\label{app:organ3d-details}
-
-\subsection{Published baselines}
-
-Table~\ref{tab:organ3d-context} situates our results against published
-methods on the same benchmark. Our models are deliberately small
-(${\leq}463$K parameters) to enable controlled ablation of the pooling
-mechanism. The relevant comparison is against equivariant methods at a
-similar scale.
-
-\begin{table}[h]
-  \caption{Published baselines on OrganMNIST3D for context.
-    Accuracy differences with respect to large-scale models reflect the
-    deliberate parameter constraint of our ablation study.}
-  \label{tab:organ3d-context}
-  \centering
-  \small
-  \begin{tabular}{llrcc}
-    \toprule
-    Method & Reference & Params & ACC & AUC \\
-    \midrule
-    ResNet-18 + 3D           & Yang et al.\ 2023         & 33M  & 0.907 & 0.996 \\
-    ResNet-18 + ACS          & Yang et al.\ 2023         & 11M  & 0.900 & 0.994 \\
-    ILPOResNet-50            & \citet{zhemchuzhnikov2024ilponet}       & 38K  & 0.879 & 0.992 \\
-    EquiLoPO (local train.)  & \citet{zhemchuzhnikov2025equilopo}      & 418K & 0.866 & 0.991 \\
-    SE3MovFrNet              & \citet{sangalli2023movfrnet}     & ---  & 0.745 & ---   \\
-    Regular SE(3) conv       & \citet{kuipers2023regular}   & 172K & 0.698 & ---   \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\subsection{Rotation analysis}
-
-Figure~\ref{fig:organ3d-rotation} shows that all equivariant models
-achieve identical performance on the original and octahedrally rotated
-test sets ($\sigma_{\mathrm{rot}} \approx 0$), while the
-augmentation-trained standard CNN degrades under rotation.
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=0.75\linewidth]{experiments/organ3d/organ3d_analysis.pdf}
-  \caption{OrganMNIST3D test accuracy on the original and octahedrally
-    rotated test sets. Error bars show standard deviation over three
-    seeds. All equivariant models achieve identical performance under
-    rotation ($\sigma_{\mathrm{rot}} \approx 0$), while the
-    augmentation-trained standard CNN degrades.}
-  \label{fig:organ3d-rotation}
-\end{figure}
-
-\subsection{Wider channels}
-
-To test whether the advantage of bispectral completeness persists at
-higher capacity, we run max pool and bispectrum at channel widths
-$(8, 16)$ and $(16, 32)$ with full training data.
-
-\begin{table}[h]
-  \caption{OrganMNIST3D accuracy at wider channel configurations
-    (mean $\pm$ std over 3 seeds, full training data).}
-  \label{tab:organ3d-wider}
-  \centering
-  \small
-  \begin{tabular}{lrcc}
-    \toprule
-    Channels & Params & Max Pool & Bispectrum \\
-    \midrule
-    $(4, 8)$   & ${\sim}$375K / 463K  & $.730 \pm .033$ & $.726 \pm .027$ \\
-    $(8, 16)$  & ${\sim}$1.5M / 1.7M  & $.743 \pm .015$ & $.745 \pm .006$ \\
-    $(16, 32)$ & ${\sim}$6.0M / 6.3M  & $\mathbf{.785 \pm .032}$ & $.685 \pm .039$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-At $(8, 16)$ the two methods are equivalent ($74.5 \pm 0.6\%$ bispectrum vs.\
-$74.3 \pm 1.5\%$ max pool, both at ${\sim}1.5$--$1.7$M parameters). At $(16, 32)$ max
-pool reaches $78.5 \pm 3.2\%$ while bispectrum drops to $68.5 \pm 3.9\%$---the 6.3M
-parameter bispectrum model likely overfits the 971 training samples.
-This suggests that the bispectrum's completeness advantage is most
-pronounced in the data-limited rather than the parameter-limited
-regime, consistent with the data efficiency findings in the main text.
-
-\section{Experimental Details}
-\label{app:exp}
-
-\subsection{Hyperparameters}
-
-Table~\ref{tab:hyperparams} summarises the training hyperparameters
-for all three experimental settings.
-
-\begin{table}[h]
-  \caption{Training hyperparameters across experiments.}
-  \label{tab:hyperparams}
-  \centering
-  \small
-  \setlength{\tabcolsep}{4pt}
-  \begin{tabular}{lccc}
-    \toprule
-    & PCam & OrganMNIST3D & Sph.\ MNIST \\
-    \midrule
-    Optimizer      & AdamW & AdamW & AdamW \\
-    Learning rate  & $10^{-3}$ & $10^{-3}$ & $10^{-3}$ \\
-    Weight decay   & $10^{-4}$ & $10^{-4}$ & $10^{-4}$ \\
-    Batch size     & 64--256 & 16--64 & 256 \\
-    Scheduler      & Cosine warm restarts & --- & --- \\
-    Max epochs     & 50 & 100 & 50 \\
-    Early stopping & Val AUC, pat.\ 10 & Val AUC, pat.\ 15 & Val ACC, pat.\ 10 \\
-    Seeds          & 42, 123, 456$^\dagger$ & 42, 123, 456 & 42, 123, 456 \\
-    Mixed precision & Yes & No & No \\
-    \bottomrule
-  \end{tabular}
-
-  \vspace{0.3em}
-  {\small $^\dagger$Pareto figure uses seed 42; matched-parameter table and data efficiency use seeds 42, 123, 456.}
-\end{table}
-
-\subsection{Hardware}
-
-All experiments were run on a single NVIDIA H100 GPU (80\,GB).
-Training wall-clock times range from ${\sim}$5 minutes (OrganMNIST3D,
-971 samples) to ${\sim}$20 minutes (PCam, 26K samples at 10\% data).
-Spherical MNIST bispectrum computation (forward pass through
-\texttt{SO3onS2} at $\ell_{\max} = 15$) takes ${\sim}$0.3s per batch
-of 64 on GPU.
-
-\subsection{Data preprocessing}
-
-\paragraph{PCam.}
-PatchCamelyon~\citep{veeling2018rotation} consists of 327{,}680 RGB
-patches of size $96 \times 96$ pixels extracted from whole-slide
-images of lymph node sections~\citep{bejnordi2017camelyon}. We use
-the standard train/val/test split. Images are loaded from HDF5 files
-and normalised to $[0, 1]$. The standard (non-equivariant) baseline
-uses random rotation and reflection augmentation during training; all
-equivariant models are trained without augmentation.
-
-\paragraph{OrganMNIST3D.}
-OrganMNIST3D~\citep{medmnistv2} contains 1{,}742 abdominal CT organ
-volumes of size $28 \times 28 \times 28$ voxels with 11 organ classes
-(train/val/test = 971/161/610). Volumes are normalised to $[0, 1]$.
-The standard baseline uses random octahedral augmentation (one of 24
-rotations per sample); equivariant models are trained without
-augmentation. Kernel rotations are implemented as exact voxel
-permutations on the $3^3$ grid, avoiding interpolation artifacts.
-
-\paragraph{Spherical MNIST.}
-MNIST digits are projected onto the sphere via stereographic
-projection following \citet{cohen2018spherical} and sampled on a
-$64 \times 128$ equiangular grid. The selective bispectrum is
-computed at $\ell_{\max} = 15$, producing 384 complex invariant
-coefficients (768 real features after taking log-magnitude), which
-are fed to a 3-layer MLP: Linear(768, 256)--BN--ReLU--Linear(256,
-128)--BN--ReLU--Linear(128, 10). The power spectrum MLP uses the
-same structure with hidden=128: Linear(16, 128)--BN--ReLU--Linear(128,
-64)--BN--ReLU--Linear(64, 10). The standard CNN has 4 convolutional
-layers (16--32--64--64 channels) followed by adaptive average pooling
-and a Linear(1024, 128)--ReLU--Linear(128, 10) classifier.
-
-\subsection{Aggregation procedures}
-
-\paragraph{PCam.}
-The Pareto figure (Figure~\ref{fig:pcam-pareto}) uses a single seed
-(42) across the full growth-rate sweep. The matched-parameter table
-(Table~\ref{tab:pcam-results}) and data efficiency analysis use 3 seeds
-(42, 123, 456) per configuration.
-
-\paragraph{OrganMNIST3D.}
-All results in Table~\ref{tab:organ3d-results} report mean $\pm$ std
-over 3 seeds (42, 123, 456). Rotation robustness
-($\sigma_{\mathrm{rot}}$) is the standard deviation of test accuracy
-across all 24 octahedral rotations of the test set. The data
-efficiency analysis (Figure~\ref{fig:data-efficiency} in the main text) uses 3 seeds per data
-fraction.
-
-\paragraph{Spherical MNIST.}
-All results in Table~\ref{tab:smnist-results} report mean $\pm$ std
-over 3 seeds (42, 123, 456). Rotation robustness
-($\sigma_{\mathrm{rot}}$) is measured by applying 10 random $\SO(3)$
-rotations to the entire test set and computing the standard deviation
-of accuracy, yielding 30 measurements per model (10 rotations
-$\times$ 3 seeds).
-
-\subsection{Dataset licenses}
-
-\begin{itemize}[nosep]
-  \item \textbf{PCam}: CC0 1.0 Universal (public domain).
-  \item \textbf{OrganMNIST3D}: CC BY 4.0.
-  \item \textbf{Spherical MNIST}: derived from MNIST, which is
-    publicly available for research use.
-\end{itemize}
-
-\section{API Example}
-\label{app:api-example}
-
-\begin{lstlisting}
-import torch
-from bispectrum import CnonCn, SO2onDisk, OctaonOcta
-
-bsp = CnonCn(n=8, selective=True)
-f = torch.randn(16, 8)
-beta = bsp(f)
-f_rec = bsp.invert(beta)
-\end{lstlisting}
-
-\newpage
-\input{checklist.tex}
-
-\end{document}
diff --git a/paper/overleaf_backup.tex b/paper/overleaf_backup.tex
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-\documentclass{article}
-
-\usepackage[main]{neurips_2026}
-
-\usepackage[utf8]{inputenc}
-\usepackage[T1]{fontenc}
-\usepackage{hyperref}
-\usepackage{url}
-\usepackage{booktabs}
-\usepackage{amsfonts}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\usepackage{amsthm}
-\usepackage{nicefrac}
-\usepackage{microtype}
-\usepackage{xcolor}
-\usepackage{graphicx}
-\usepackage{algorithm}
-\usepackage{algorithmic}
-\usepackage{enumitem}
-\usepackage{listings}
-
-\newtheorem{theorem}{Theorem}
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{corollary}[theorem]{Corollary}
-
-\theoremstyle{definition}
-\newtheorem{definition}[theorem]{Definition}
-
-\theoremstyle{remark}
-\newtheorem{remark}[theorem]{Remark}
-
-\DeclareMathOperator{\tr}{tr}
-\DeclareMathOperator{\sgn}{sgn}
-\DeclareMathOperator{\im}{Im}
-\newcommand{\R}{\mathbb{R}}
-\newcommand{\C}{\mathbb{C}}
-\newcommand{\SO}{\mathrm{SO}}
-\newcommand{\OG}{\mathrm{O}}
-\newcommand{\SH}{Y}
-\newcommand{\CG}[6]{C^{#5,\,#6}_{#1,\,#2;\,#3,\,#4}}
-\newcommand{\abs}[1]{\lvert #1 \rvert}
-\newcommand{\norm}[1]{\lVert #1 \rVert}
-\newcommand{\conj}[1]{\overline{#1}}
-\newcommand{\ip}[2]{\langle #1,\, #2 \rangle}
-
-\theoremstyle{definition}
-\newtheorem{assumption}[theorem]{Assumption}
-
-
-\lstset{
-  language=Python,
-  basicstyle=\ttfamily\small,
-  keywordstyle=\bfseries,
-  commentstyle=\color{gray},
-  stringstyle=\color{black!60},
-  frame=single,
-  breaklines=true,
-  columns=fullflexible,
-}
-
-\title{\texttt{bispectrum}: Selective $G$-Bispectra Made Practical}
-
-\author{%
-  Johan Mathe \\
-  Atmo, Inc. \\
-  \texttt{johan@atmo.ai} \\
-  \And
-  Adele Myers Lantow \\
-  University of California, Santa Barbara \\
-  \texttt{adele@ucsb.edu} \\
-  \And
-  Simon Mataigne \\
-  ICTEAM, UCLouvain \\
-  \texttt{simon.mataigne@uclouvain.be} \\
-  \And
-  Nina Miolane \\
-  University of California, Santa Barbara \\
-  \texttt{nmiolane@ucsb.edu} \\
-}
-
-\begin{document}
-
-\maketitle
-
-\begin{abstract}
-We present \texttt{bispectrum}, an open-source PyTorch library that computes
-selective $G$-bispectral invariants for seven group/domain pairs as
-drop-in differentiable \texttt{nn.Module}s.
-The selective bispectrum reduces invariant cost from $O(|G|^2)$ to $O(|G|)$
-while remaining complete for generic signals.
-Building the library required two theoretical contributions:
-(i)~an augmented selective $\SO(3)$ invariant on $S^2$ achieving
-$\Theta(L^2)$ coefficients (the information-theoretic lower
-bound) instead of $O(L^3)$ for the full bispectrum
-(completeness proved for $4 \le L \le 100$), and
-(ii)~diagnosing a previously undocumented inversion failure for the
-octahedral group.
-Across three benchmarks ($C_8$, octahedral, $\SO(3)$), the library
-delivers near-exact rotation invariance
-($\sigma_{\mathrm{rot}} < 10^{-3}$) and a data-efficiency advantage of
-complete bispectral invariants over incomplete alternatives in the
-low-data, moderate-capacity regime.
-The selective forward pass runs in sub-millisecond time on GPU for finite
-groups. Code is available at \url{https://github.com/geometric-intelligence/bispectrum}.
-\end{abstract}
-\section{Introduction}
-\label{sec:intro}
-
-Equivariant neural networks~\citep{cohen2016group, weiler2019general} enforce
-symmetry by construction, but they still require an invariant representation at
-the point where a prediction must no longer depend on orientation. In practice,
-invariant pooling remains a weak point: standard choices such as norm pooling,
-gate-based pooling, or max/sum aggregation are
-usable~\citep{weiler2019general}, but they do not preserve all equivariant
-information. The bispectrum offers a mathematically principled alternative,
-since for many groups it yields exact invariants with reconstruction
-guarantees~\citep{kakarala2012bispectrum}.
-
-The obstacle is practical rather than conceptual. Existing bispectral
-constructions are scattered across theory papers, domain-specific derivations,
-and small implementations, with no general-purpose PyTorch library covering the
-finite-group and compact-group cases relevant to modern geometric deep learning.
-Even for groups where selectivity reduces the coefficient count from
-$O(|G|^2)$ to $O(|G|)$~\citep{mataigne2024selective}, a user still needs a numerically stable implementation,
-a consistent API, inversion routines, tests, and benchmarked runtime behavior.
-
-This paper presents \texttt{bispectrum}, a PyTorch~\citep{paszke2019pytorch}
-library for computing bispectral invariants and, where available, their
-inverses. The library
-supports cyclic, dihedral, torus, disk, sphere, and octahedral settings as
-differentiable \texttt{nn.Module}s intended to plug into standard ML workflows.
-Beyond the software contribution, the paper makes two technical contributions
-that arose during implementation: an augmented selective $\SO(3)$
-bispectrum on $S^2$ with $O(L^2)$ coefficients
-(Section~\ref{sec:so3-selective}), and an analysis of
-the bootstrap inversion failure~\citep{mataigne2024selective} for the
-octahedral group together with a partial workaround (Section~\ref{sec:octa}).
-
-\paragraph{Contributions.}
-\begin{enumerate}
-  \item \textbf{Software contribution.} A PyTorch library implementing
-    bispectral invariants for seven group/domain pairs with a consistent
-    \texttt{nn.Module}-style interface, GPU-optimised forward passes
-    ($<$1\,ms for finite groups), precomputed Clebsch--Gordan buffers,
-    and parallel construction routines
-    (Section~\ref{sec:benchmarks}).
-  \item \textbf{Augmented selective $\SO(3)$ bispectrum on $S^2$.} We construct
-    and implement an augmented selective bispectrum for $\SO(3)$ acting on
-    $S^2$ that uses
-    $\Theta(L^2)$ coefficients instead of the full $\Theta(L^3)$,
-    matching the orbit-space lower bound up to a constant factor.
-    The invariant combines scalar bispectral entries with
-    Clebsch--Gordan power spectrum entries (a degree-$4$ complement
-    needed for real signals).  Completeness is proved for $4 \le L \le 100$
-    via a combination of exact rational certificates and a
-    closed-form linear bootstrap family (Appendix~\ref{app:so3-proof});
-    the verification extends to any desired~$L$ by re-running a single script.
-  \item \textbf{Octahedral inversion analysis.} An analysis of why the standard
-    selective-bispectrum bootstrap inversion fails for the octahedral group,
-    together with a bootstrap + Levenberg--Marquardt approach that achieves
-    local convergence but not reliable global recovery.
-  \item \textbf{Empirical validation.} A task-driven evaluation of the
-    library as an invariant pooling primitive across three group/domain
-    settings, showing that complete bispectral invariants provide a
-    data-efficiency advantage in the low-data, moderate-capacity regime
-    while converging to incomplete alternatives at high capacity.
-\end{enumerate}
-
-\section{Mathematical Background}
-\label{sec:background}
-
-\subsection{Equivariant features and invariant maps}
-
-A neural network layer is \emph{$G$-equivariant} if its output transforms
-predictably under the group action:
-$[\pi_{\mathrm{out}}(g) \circ \Phi](f) = \Phi(\pi_{\mathrm{in}}(g) f)$
-for all $g \in G$~\citep{cohen2016group, weiler2019general}.
-Stacking equivariant layers preserves this property, but any task that requires
-a prediction independent of orientation---classification, regression,
-detection---must eventually map equivariant features to \emph{invariant} ones.
-
-\begin{table}[t]
-  \caption{Invariant nonlinearities for equivariant CNNs. The selective
-    bispectrum is the only method that is simultaneously exact, complete,
-    and linear-cost for finite groups.}
-  \label{tab:nonlinearity-comparison}
-  \centering
-  \small
-  \begin{tabular}{lcccc}
-    \toprule
-    Nonlinearity & Exact? & Complete? & Coefficients & Key limitation \\
-    \midrule
-    Norm          & Yes & No  & $O(|G|)$      & Discards phase \\
-    Gate          & Yes & No  & $O(2|G|)$     & Scalar bottleneck \\
-    Fourier pointwise & No & Yes$^\ddagger$ & $O(|G| \log |G|)$ & Aliasing \\
-    Full bispectrum & Yes & Yes & $O(|G|^2)$  & Quadratic cost \\
-    \midrule
-    \textbf{Selective bispectrum} & \textbf{Yes} & \textbf{Yes}
-      & $\mathbf{O(|G|)}$ & Genericity assumption \\
-    \bottomrule
-  \end{tabular}
-
-  \vspace{0.3em}
-  {\small $^\ddagger$Complete pre-aliasing; the pointwise nonlinearity in the forward pass introduces aliasing that may break completeness in practice.}
-\end{table}
-
-Table~\ref{tab:nonlinearity-comparison} summarises the trade-offs. Norm
-pooling and gated nonlinearities are exact invariants but \emph{incomplete}:
-they discard the phase of Fourier blocks, collapsing distinct orbits.
-Fourier-space pointwise nonlinearities preserve completeness but
-sacrifice exactness through aliasing~\citep{weiler2019general}. The full
-bispectrum is exact and complete but costs $O(|G|^2)$ coefficients.
-
-\subsection{The $G$-bispectrum and selective bispectra}
-
-Let $G$ be a compact group with irreducible representations
-$\{\rho_i\}_{i \in \hat{G}}$ of dimensions $d_i$, and let
-$\hat{f}(\rho_i) = \sum_{g \in G} f(g)\, \rho_i(g)$ denote the group
-Fourier transform. Given the Clebsch--Gordan decomposition
-$\rho_i \otimes \rho_j \cong \bigoplus_k m_{ij}^k \rho_k$ with
-intertwining matrix $C_{ij}$, the \emph{bispectrum} is
-\begin{equation}
-  \label{eq:bispectrum-def}
-  \beta_{\rho_i, \rho_j}
-    = C_{ij}^\top\,
-      (\hat{f}(\rho_i) \otimes \hat{f}(\rho_j))\,
-      C_{ij}\, \hat{f}(\rho_k)^\dagger.
-\end{equation}
-Each $\beta_{\rho_i,\rho_j}$ is invariant under $f \mapsto f \circ g$ and,
-taken over all pairs $(i,j)$, the bispectrum is a \emph{complete} invariant:
-it separates orbits for generic signals~\citep{kakarala2012bispectrum}. However,
-the full set has $O(|G|^2)$ coefficients.
-
-\emph{Selective bispectra.} \citet{mataigne2024selective} show that for
-finite groups, a BFS traversal of the Kronecker product table of $\hat{G}$
-selects $O(|G|)$ bispectral coefficients that remain complete;
-\citet{myers2025selective} extend this to the disk. This
-reduction from quadratic to linear is the key to practical deployment:
-for example, the octahedral group drops from $576$ to $172$ coefficients.
-
-\section{The \texttt{bispectrum} Library}
-\label{sec:library}
-
-\subsection{Design and implementation}
-\label{sec:design}
-
-Every group/domain pair is exposed as a \texttt{torch.nn.Module}
-named \texttt{\{Group\}on\{Domain\}} (e.g.\ \texttt{SO3onS2}).
-Modules accept raw real-valued inputs, handle the Fourier transform
-internally, and default to $O(|G|)$ selective coefficients
-(\texttt{selective=False} gives the full $O(|G|^2)$ set).
-CG matrices, DFT kernels, and Bessel roots are precomputed at
-construction and stored as non-learnable buffers, so
-\texttt{.to(device)} works transparently and forward passes incur zero
-recomputation.
-Where available, \texttt{bsp.invert(beta)} returns the reconstructed
-signal up to group-action indeterminacy.
-The only dependencies are PyTorch~\citep{paszke2019pytorch},
-NumPy~\citep{harris2020numpy}, and
-\texttt{torch\_harmonics}~\citep{bonev2023spherical}
-(for \texttt{SO3onS2}).
-For $\SO(3)$ on $S^2$, reduced CG matrices are built in parallel and
-cached to disk; all forward methods are batch-native, using
-\texttt{torch.einsum}/\texttt{torch.bmm} for non-abelian modules and
-\texttt{torch.fft} for abelian ones.
-The test suite covers 96.8\% of lines across all modules.
-
-\subsection{Supported modules}
-\label{sec:supported-modules}
-
-\begin{table}[t]
-  \caption{Current scope of the \texttt{bispectrum} library.
-    \texttt{SO2onS1} is the continuous-$n$ limit of \texttt{CnonCn}
-    and shares its implementation.}
-  \label{tab:supported-groups}
-  \centering
-  \small
-  \begin{tabular}{lll}
-    \toprule
-    Module & Group / Domain & Output mode \\
-    \midrule
-    \texttt{CnonCn} & $C_n$ on $C_n$ & selective + full \\
-    \texttt{SO2onS1} & $\SO(2)$ on $S^1$ & selective + full \\
-    \texttt{TorusOnTorus} & product cyclic groups on torus & selective + full \\
-    \texttt{DnonDn} & $D_n$ on $D_n$ & selective \\
-    \texttt{SO2onDisk} & $\SO(2)$ on disk & selective \\
-    \texttt{SO3onS2} & $\SO(3)$ on $S^2$ & selective + full \\
-    \texttt{OctaonOcta} & octahedral group & selective \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\paragraph{Example API.}
-\label{sec:api}
-A minimal usage example is in Appendix~\ref{app:api-example}.
-
-\subsection{Augmented selective $\SO(3)$ bispectrum on $S^2$}
-\label{sec:so3-selective}
-
-The full bispectrum on $S^2$ at band-limit $L$ has $\Theta(L^3)$ scalar
-entries $\beta_{\ell_1,\ell_2,\ell}$. For finite groups, selectivity
-reduces $O(|G|^2)$ to $O(|G|)$, but on $S^2$ the BFS chain
-$\beta_{1,0,1}, \beta_{1,1,2}, \ldots$ provides only ${\sim}3L$
-scalars---far too few for $(L{+}1)^2$ unknowns. Indeed, the orbit space
-$\R^{(L+1)^2}/\SO(3)$ has dimension $(L{+}1)^2 - 3$, so any complete
-invariant requires $\Omega(L^2)$ independent components. We achieve this
-lower bound.
-
-\begin{theorem}[Augmented selective $\SO(3)$ invariant on $S^2$]
-\label{thm:so3-selective}
-For $f : S^2 \to \R$ band-limited at degree $4 \le L \le 100$, there exists an augmented
-invariant $\Phi_{\mathrm{aug}}(f)$ consisting of
-$\Theta(L^2)$ scalars---a subset $\mathcal{S}_\beta$ of scalar
-bispectral entries $\beta_{\ell_1,\ell_2,\ell}$ together with a
-subset $\mathcal{S}_P$ of CG power spectrum entries
-$P_{\ell_1,\ell_2,\ell} := \lVert (\mathbf{F}_{\ell_1} \otimes
-\mathbf{F}_{\ell_2})|_\ell \rVert^2$---that is a complete
-$\SO(3)$-invariant for Lebesgue-almost-every signal.
-The output size $\Theta(L^2)$ matches the dimension
-$(L{+}1)^2 - 3$ of the orbit space $\R^{(L+1)^2}/\SO(3)$
-up to a constant factor.
-\end{theorem}
-
-\begin{remark}\label{rem:verified-L}
-The completeness proof (Appendix~\ref{app:so3-proof}) proceeds by
-gauge-fixing, seed recovery at degrees $\le 3$ (finite fibre),
-degree-4 fibre reduction via parity-breaking bispectral entries, and
-a linear bootstrap for $\ell \ge 5$.
-All rank and consistency conditions are verified at explicit rational
-or deterministic witnesses for each $\ell \le 100$ and promoted to generic
-statements by algebraic-variety arguments.
-The structural form of the bootstrap is uniform for $\ell \ge 8$;
-extending the verification beyond $L = 100$ requires only re-running
-a single script.
-\end{remark}
-
-\paragraph{Why augmentation is necessary.}
-For \emph{complex} signals, the scalar bispectrum alone suffices: the
-linear bootstrap matrix has full complex rank $2\ell{+}1$ at each degree
-(verified at explicit witnesses up to $\ell = 100$). However, for
-\emph{real} signals, the reality constraint
-$a_\ell^{-m} = (-1)^m \conj{a_\ell^m}$ creates algebraic dependencies
-in the bootstrap matrix that \emph{cannot} be resolved by choosing
-different bispectral triples. At $\ell = 4$, for example, the bootstrap
-matrix has complex rank~$4$ instead of $9 = 2\ell{+}1$, and this holds
-even for the \emph{full} bispectrum (all valid triples). The CG power
-spectrum entries $P_{\ell_1,\ell_2,\ell}$ capture the complementary
-\emph{magnitude} information that the phase-sensitive scalar bispectral
-entries miss, and together they achieve full Jacobian rank.
-
-\paragraph{The selective index set.}
-The bispectral component $\mathcal{S}_\beta$ selects up to
-$2\ell{+}1$ triples per degree from four categories:
-\textbf{(1)~Chain} $(\ell_1, \ell_2, \ell)$ with
-$\ell_1 \leq \ell_2 < \ell$;
-\textbf{(2)~Cross} $(\ell_1, \ell, \ell')$ with
-$\ell_1, \ell' < \ell$;
-\textbf{(3)~Power} $(0, \ell, \ell)$;
-\textbf{(4)~Self-coupling} $(\ell, \ell, \ell')$ with $\ell'$ even.
-Entries with $\ell_1 = \ell_2$ and odd~$\ell$ vanish identically
-for real signals (CG antisymmetry of the symmetric square) and
-are excluded.  All \emph{even} self-coupling entries
-$\beta_{\ell,\ell,\ell'}$ with $2 \le \ell' \le \ell$ are
-included unconditionally; they are needed for full Jacobian rank
-in the seed system (Lemma~\ref{lem:app-ell3}).
-The invariant is further augmented with
-CG power spectrum entries $P_{\ell_1,\ell_2,\ell}$, chosen
-to maximise the Jacobian rank of the combined invariant
-(Algorithm~\ref{alg:so3-augmented}).
-
-Table~\ref{tab:entry-counts} (Appendix~\ref{app:so3-proof}) gives
-per-degree entry counts; at degree $\ell \ge 5$ the total is
-${\sim}3\ell$, confirming the $\Theta(L^2)$ output size.
-
-\paragraph{Proof sketch.}
-The proof establishes completeness of $\Phi_{\mathrm{aug}}$ through
-gauge fixing and degree-by-degree recovery.
-\emph{Base cases} ($\ell \le 1$): $\mathbf{F}_0$ is recovered from
-$\beta_{0,0,0} = (a_0^0)^3$, and $\mathbf{F}_1$ from
-$\beta_{0,1,1}/a_0^0 = \norm{\mathbf{F}_1}^2$ plus gauge-fixing
-(align $\mathbf{F}_1$ with the $z$-axis, fix the azimuthal phase
-of $a_2^1$).
-\emph{Seed recovery} ($\ell = 2, 3$): the augmented seed system---scalar
-bispectral entries plus CG power entries---has full Jacobian rank~$11$
-in $11$~unknowns, yielding a finite fibre
-containing $T_R$-pairs, where $T_R\!: (\theta,\varphi) \mapsto
-(\theta,-\varphi)$ is azimuthal reflection.
-\emph{Degree-4 fibre reduction}: the degree-4 augmented entries
-include parity-breaking bispectral entries ($\beta_{2,3,4}$ with odd
-$\ell_1{+}\ell_2{+}\ell$) whose imaginary parts flip sign under~$T_R$.
-These eliminate all spurious seed solutions and resolve the $T_R$
-ambiguity, leaving a unique seed.
-\emph{Bootstrap} ($\ell \ge 5$): the linear bootstrap has full complex
-rank at each degree, uniquely determining $\mathbf{F}_\ell$ from
-lower-degree coefficients.
-The computational certificates use double-precision arithmetic with
-well-separated thresholds (residual gaps of $\ge 10^{13}$ relative to
-machine epsilon); they can be upgraded to interval arithmetic if desired.
-The full proof is in Appendix~\ref{app:so3-proof}.
-
-\subsection{Octahedral inversion as an implementation case study}
-\label{sec:octa}
-
-The bootstrap inversion of \citet{mataigne2024selective} recovers
-$\hat{f}(\rho_1)$ up to an orthogonal ambiguity~$Q$, then extracts
-higher Fourier blocks via CG matrices.
-For $C_n$ and $D_n$ the ambiguity is low-dimensional and absorbed by
-the CG structure, but for the octahedral group~$O$ the ambiguity is
-$Q \in \SO(3)$ and the CG matrices develop $O(1)$ cross-block leakage
-for generic $Q \notin O$, corrupting higher blocks.
-The selective $O$-bispectrum \emph{is} complete (172 coefficients
-determine the signal up to $O$-action); the failure is purely algorithmic.
-We partially address it by combining bootstrap initialisation with
-Levenberg--Marquardt refinement of
-$\min_f \norm{\beta^{\mathrm{sel}}(f)
-  - \beta^{\mathrm{sel}}_{\mathrm{target}}}^2$,
-using multi-start randomisation over~$Q$.
-The LM residual converges in $<$50 iterations, but the overall signal
-recovery rate remains modest (5.7\% with 4~restarts), reflecting the
-difficult landscape created by CG leakage.
-The full analysis is in Appendix~\ref{app:octa-convergence}.
-
-\subsection{Computational benchmarks}
-\label{sec:benchmarks}
-
-Table~\ref{tab:benchmarks} and Figure~\ref{fig:coeff-scaling} profile
-the library on a single NVIDIA H100 80\,GB GPU (PyTorch 2.10,
-batch\,=\,16 for forward, 4 for inversion, median wall-clock via
-\texttt{torch.utils.benchmark}).
-
-\begin{figure}[t]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/benchmarks/paper_fig1.pdf}
-  \caption{Bispectral coefficient count vs.\ group order $|G|$. Solid
-    lines: selective ($O(|G|)$); dashed: full ($O(|G|^2)$). At
-    $|G|{=}1{,}024$, selectivity yields a 512$\times$ reduction.
-    Continuous groups ($\SO(2)$, $\SO(3)$) are plotted at their
-    band-limited signal dimension.}
-  \label{fig:coeff-scaling}
-\end{figure}
-
-\begin{table}[t]
-  \centering
-  \caption{Library benchmark summary. Timings are median wall-clock on a
-    single NVIDIA H100 80\,GB GPU. ``--'' = not implemented.
-    For continuous groups, $|G| = \infty$; coefficient counts refer to the
-    band-limited representation at the tested resolution.
-    $^*$\texttt{SO3onS2} benchmarked at $\ell_{\max}{=}16$.}
-  \label{tab:benchmarks}
-  \small
-  \setlength{\tabcolsep}{3.5pt}
-  \begin{tabular}{llrrrrrr}
-    \toprule
-    Module & $G$ & $|G|$ & \multicolumn{2}{c}{Coefs} & \multicolumn{2}{c}{Fwd (ms)} & Inv (ms) \\
-    \cmidrule(lr){4-5} \cmidrule(lr){6-7}
-     & & & Sel. & Full & Sel. & Full & Sel. \\
-    \midrule
-    \texttt{CnonCn} & $C_{128}$ & 128 & 128 & 8256 & 0.15 & 9.17 & 8.56 \\
-    \texttt{TorusOnTorus} & $C_{32}{\times}C_{32}$ & 1024 & 1024 & 524800 & 0.08 & 0.71 & 68.02 \\
-    \texttt{DnonDn} & $D_{32}$ & 64 & 245 & -- & 2.52 & -- & 42.33 \\
-    \texttt{SO2onDisk} & $\mathrm{SO}(2)$ & $\infty$ & 411 & -- & 1.94 & -- & 17.69 \\
-    \texttt{SO3onS2}\rlap{$^*$} & $\mathrm{SO}(3)$ & $\infty$ & 430 & -- & 26.54 & -- & -- \\
-    \texttt{OctaonOcta} & $O$ & 24 & 172 & -- & 0.70 & -- & 2172.76 \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-The selective forward pass is sub-millisecond for finite abelian modules
-($C_n$: 0.15\,ms, $\mathbb{T}^2$: 0.08\,ms), under 2\,ms for
-$\SO(2)$/disk (1.94\,ms), and under 1\,ms for the octahedral group
-(0.70\,ms), making it negligible relative to backbone computation. $\SO(3)$ on $S^2$ at
-$\ell_{\max}{=}16$ takes 26.5\,ms, dominated by the spherical harmonic
-transform (via \texttt{torch\_harmonics}~\citep{bonev2023spherical}).
-Selective coefficient counts confirm the $O(|G|)$ scaling:
-$C_{128}$ uses 128 coefficients vs.\ 8{,}256 full (64$\times$
-reduction); $\mathbb{T}^2$ at $|G|{=}1{,}024$ uses 1{,}024 vs.\
-524{,}800 full (512$\times$). Inversion is fast for abelian groups
-(9--68\,ms) and practical for the octahedral group (2.17\,s via
-Levenberg--Marquardt). GPU throughput scales linearly with batch size,
-exceeding $10^7$~samples/s for $C_n$ and $\mathbb{T}^2$
-(Appendix~\ref{app:bench}).
-
-\section{Experiments}
-\label{sec:experiments}
-
-We validate the library on three tasks spanning discrete and continuous
-groups: 2D histopathology classification under $C_8$ symmetry (PCam,
-Section~\ref{sec:pcam}), 3D organ classification under octahedral symmetry
-(OrganMNIST3D, Section~\ref{sec:organ3d}), and spherical digit
-classification under full $\SO(3)$ (Spherical MNIST,
-Section~\ref{sec:smnist}).
-
-\subsection{2D histopathology classification under cyclic symmetry}
-\label{sec:pcam}
-
-We evaluate five invariant pooling strategies on
-PatchCamelyon~\citep{veeling2018rotation} (PCam), 327K histopathology
-patches with binary metastasis labels. All equivariant models share a
-$C_8$-equivariant DenseNet~\citep{huang2017densenet} backbone following
-\citet{veeling2018rotation}; they differ only in the terminal invariant map:
-norm pooling, gated pooling, Fourier-ELU, and selective bispectrum via
-\texttt{CnonCn}$(n{=}8)$. An augmented standard CNN serves as a
-non-equivariant baseline. We sweep growth rates to vary capacity
-(30K--1.6M params) and train on 10\% of the data with
-AdamW~\citep{loshchilov2019adamw}. Full model descriptions are in
-Appendix~\ref{app:pcam-details}.
-
-\paragraph{Results.}
-Table~\ref{tab:pcam-results} and Figure~\ref{fig:pcam-pareto} present the
-results at matched ${\sim}$100K parameters.
-
-\begin{figure}[t]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/pcam/pareto_analysis.pdf}
-  \caption{PCam Pareto analysis (single seed, 10\% training data).
-    (a)~Test AUC vs.\ parameter count across all five pooling variants.
-    (b)~Ranking at the ${\sim}$100K parameter budget.
-    (c)~Data efficiency at matched ${\sim}$100K parameters.
-    Multi-seed statistics are in Table~\ref{tab:pcam-results}.}
-  \label{fig:pcam-pareto}
-\end{figure}
-
-\begin{table}[t]
-  \caption{PCam results at matched ${\sim}100$K parameters (10\% training
-    data, mean $\pm$ std over 3 seeds). All equivariant models share a
-    $C_8$-equivariant DenseNet backbone.}
-  \label{tab:pcam-results}
-  \centering
-  \small
-  \begin{tabular}{lrcc}
-    \toprule
-    Model & Params & Test AUC & Test ACC \\
-    \midrule
-    Standard (augmented)          & 102K & $.896 \pm .010$ & $.826 \pm .012$ \\
-    Gated                         & 136K & $.941 \pm .009$ & $.870 \pm .004$ \\
-    Bispectrum (\texttt{CnonCn})  & 128K & $.941 \pm .004$ & $\mathbf{.872 \pm .006}$ \\
-    Fourier-ELU                   & 110K & $\mathbf{.945 \pm .004}$ & $.855 \pm .006$ \\
-    NormReLU                      & 110K & $.942 \pm .004$ & $.858 \pm .007$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-At matched capacity (${\sim}100$K params, 10\% data), all four equivariant
-pooling methods substantially outperform the standard augmented CNN
-(AUC $0.896$). Fourier-ELU leads slightly ($0.945$), followed by
-NormReLU ($0.942$), bispectrum and gated (both $0.941$).
-At 1\% data, the bispectrum leads all methods ($0.912$), outperforming
-NormReLU ($0.873$) and Fourier-ELU ($0.876$). This data-efficiency
-advantage narrows as the training set grows, consistent with complete
-invariants providing a stronger inductive bias when data is scarce.
-
-\subsection{3D organ classification under octahedral symmetry}
-\label{sec:organ3d}
-
-To evaluate the \texttt{OctaonOcta} module for the chiral octahedral group
-$O$ ($|O|=24$), we design a controlled 3D experiment on
-OrganMNIST3D~\citep{medmnistv2} (1{,}742 abdominal CT volumes, $28^3$
-voxels, 11 classes). All equivariant variants share an $O$-equivariant 3D
-ResNet backbone with exact voxel-permutation kernels. The four models differ
-only in the terminal invariant pooling: a non-equivariant augmented CNN
-(16K params), norm pool (375K), max pool (374K), and the selective
-bispectrum via \texttt{OctaonOcta} (463K, 172 invariant scalars). Models
-are trained with AdamW~\citep{loshchilov2019adamw} for 100 epochs with
-three seeds; rotation robustness is measured over all 24 octahedral
-rotations of the test set. Published baselines and wider-channel experiments
-are in Appendix~\ref{app:organ3d-details}.
-
-\paragraph{Results.}
-Table~\ref{tab:organ3d-results} and Figure~\ref{fig:data-efficiency} report
-the key findings.
-
-\begin{table}[t]
-  \caption{OrganMNIST3D results (mean $\pm$ std over 3 seeds). Rot ACC and
-    $\sigma_{\mathrm{rot}}$: mean and std of accuracy across 24 octahedral
-    test-set rotations.}
-  \label{tab:organ3d-results}
-  \centering
-  \small
-  \begin{tabular}{lrcccr}
-    \toprule
-    Model & Params & Test ACC & Test AUC & Rot ACC & $\sigma_{\mathrm{rot}}$ \\
-    \midrule
-    Standard 3D CNN          & 16K  & $.601 \pm .017$ & $.951 \pm .002$ & .598 & .0115 \\
-    $O$-Equiv + Norm Pool    & 375K & $.568 \pm .099$ & $.940 \pm .021$ & .568 & .0000 \\
-    $O$-Equiv + Max Pool     & 374K & $.730 \pm .033$ & $.972 \pm .005$ & .730 & .0000 \\
-    $O$-Equiv + Bispectrum   & 463K & $.726 \pm .027$ & $.972 \pm .004$ & .726 & .0005$^\dagger$ \\
-    \bottomrule
-  \end{tabular}
-
-  \vspace{0.3em}
-  {\small $^\dagger$Nonzero $\sigma_{\mathrm{rot}}$ due to floating-point precision in CG contractions, not a theoretical invariance failure.}
-\end{table}
-
-\paragraph{Rotation invariance.}
-All equivariant models achieve $\sigma_{\mathrm{rot}} \leq 0.0005$,
-while the augmented CNN drops ${\sim}$2.5pp under rotation.
-
-\paragraph{Completeness vs.\ lossy pooling.}
-The bispectrum (72.6\%) matches max pool (73.0\%) and substantially
-outperforms norm pool (56.8\%) and Regular SE(3)
-convolutions~\citep{kuipers2023regular} (69.8\%).
-At reduced data (Figure~\ref{fig:data-efficiency}), the bispectrum's
-advantage widens: at 5\% data it reaches $33.2\%$ vs.\ $18.9\%$ for max
-pool. At higher capacity (6M params,
-Appendix~\ref{app:organ3d-details}), max pool overtakes the bispectrum,
-suggesting the completeness advantage is specific to the data-limited
-regime.
-
-\begin{figure}[t]
-  \centering
-  \includegraphics[width=0.75\linewidth]{experiments/organ3d/data_efficiency.pdf}
-  \caption{Test accuracy vs.\ training data fraction on OrganMNIST3D (mean
-    over 3 seeds; shaded bands: $\pm 1$ std). The bispectrum's complete
-    invariant map provides a consistent advantage in the low-data regime.}
-  \label{fig:data-efficiency}
-\end{figure}
-
-\subsection{Spherical MNIST: SO(3) invariance on the sphere}
-\label{sec:smnist}
-
-The preceding experiments validate bispectral pooling for finite groups
-($C_8$ and the octahedral group $O$). To evaluate the library's continuous
-group support, we test \texttt{SO3onS2}---the $\SO(3)$ bispectrum on the
-2-sphere---on Spherical MNIST~\citep{cohen2018spherical}, the standard
-benchmark for rotation invariance on $S^2$.
-
-\paragraph{Setup.}
-MNIST digits are projected onto the sphere via stereographic projection and
-sampled on a $64 \times 128$ equiangular grid. We compute the selective
-bispectrum at $\ell_{\max} = 15$, producing 384 complex invariant
-coefficients (768 real features after taking log-magnitude), which are fed to a
-3-layer MLP (232K parameters total).  The completeness proof
-(Appendix~\ref{app:so3-proof}) covers $4 \le L \le 100$, so
-$\ell_{\max} = 15$ is within the verified range. Three baselines provide controls:
-\begin{itemize}
-  \item \textbf{Power Spectrum + MLP} (11K params): computes only the
-    rotation-invariant power spectrum $\|f_\ell\|^2$ per degree $\ell$,
-    discarding all phase coupling between harmonics. Invariant but
-    \emph{incomplete}.
-  \item \textbf{Power Spectrum + MLP (matched)} (166K params): same
-    incomplete invariant features but with a larger MLP (166K),
-    isolating the effect of feature completeness from model capacity.
-  \item \textbf{Standard CNN} (185K params): a conventional CNN operating
-    directly on the equirectangular image. Not rotation-invariant.
-\end{itemize}
-Following \citet{cohen2018spherical}, we evaluate under three
-protocols: train on non-rotated / test on non-rotated (NR/NR), train on
-rotated / test on rotated (R/R), and the most challenging train on
-non-rotated / test on rotated (NR/R). Each configuration is run with three
-seeds. Rotation robustness is assessed by applying 10 random $\SO(3)$
-rotations to the entire test set and measuring accuracy variance.
-
-\paragraph{Results.}
-Table~\ref{tab:smnist-results} and Figure~\ref{fig:smnist-analysis} report
-the results.
-
-\begin{table}[t]
-  \caption{Spherical MNIST results (mean $\pm$ std over 3 seeds).
-    Reference numbers are from \citet{cohen2018spherical}.
-    $\sigma_{\mathrm{rot}}$: standard deviation of accuracy under 10
-    random $\SO(3)$ rotations (NR-trained models).}
-  \label{tab:smnist-results}
-  \centering
-  \small
-  \setlength{\tabcolsep}{4pt}
-  \begin{tabular}{lrcccc}
-    \toprule
-    Model & Params & NR/NR $\uparrow$ & R/R $\uparrow$ & NR/R $\uparrow$
-      & $\sigma_{\mathrm{rot}}$ $\downarrow$ \\
-    \midrule
-    Cohen planar CNN & --- & $0.98$ & $0.23$ & $0.11$ & --- \\
-    Cohen spherical CNN & --- & $0.96$ & $0.95$ & $0.94$ & --- \\
-    \midrule
-    Standard CNN & 185K
-      & $0.991 {\pm} 0.000$ & $0.781 {\pm} 0.003$ & $0.138 {\pm} 0.003$
-      & $0.044$ \\
-    Power Spec.\ + MLP & 11K
-      & $0.780 {\pm} 0.002$ & $0.780 {\pm} 0.002$ & $0.777 {\pm} 0.001$
-      & $3{\times}10^{-4}$ \\
-    Power Spec.\ + MLP (matched) & 166K
-      & $0.794 {\pm} 0.011$ & $0.794 {\pm} 0.011$ & $0.792 {\pm} 0.010$
-      & $3{\times}10^{-4}$ \\
-    \textbf{Bispectrum + MLP (ours)} & 232K
-      & $\mathbf{0.950 {\pm} 0.000}$ & $\mathbf{0.949 {\pm} 0.002}$
-      & $\mathbf{0.951 {\pm} 0.001}$ & $\mathbf{1{\times}10^{-4}}$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\begin{figure}[t]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/spherical_mnist/smnist_analysis.pdf}
-  \caption{Spherical MNIST analysis.
-    \textbf{(a)}~Cohen et al.\ protocol: NR/NR, R/R, and NR/R accuracy.
-    Dashed lines show Cohen's spherical CNN reference.
-    \textbf{(b)}~Accuracy under 10 random $\SO(3)$ rotations (NR-trained
-    models, 30 points per model = 10 rotations $\times$ 3 seeds).
-    Scaling the power spectrum MLP from 11K to 166K parameters yields
-    only +1.4pp; the bispectrum at 232K reaches 95.0\%.}
-  \label{fig:smnist-analysis}
-\end{figure}
-
-\paragraph{Numerically exact $\SO(3)$ invariance.}
-The bispectrum achieves near-identical accuracy across all three protocols
-($\sigma_{\mathrm{rot}} \sim 10^{-4}$), confirming true $\SO(3)$
-invariance. The standard CNN collapses from 99.1\% (NR/NR) to 13.8\%
-(NR/R), showing that augmentation alone does not confer invariance.
-
-\paragraph{Completeness gap.}
-The power spectrum is equally invariant but reaches only 78\%---a 17pp
-deficit that persists even after scaling the MLP to 166K parameters
-(+1.4pp), confirming incompleteness rather than capacity as the
-bottleneck.
-
-\paragraph{Comparison with Cohen et al.}
-Cohen's spherical CNN~\citep{cohen2018spherical} reaches 96\%/95\%/94\%
-with learned equivariant convolutions. Our single-pass bispectrum
-(95.0\%/94.9\%/95.1\%) is 1--2pp lower but has a tighter invariance gap
-(0.1pp vs.\ 2.0pp).
-
-
-\section{Related Work}
-\label{sec:related}
-
-\paragraph{Equivariant network libraries.}
-\texttt{escnn}~\citep{cesa2022program} and \texttt{e3nn}~\citep{geiger2022e3nn}
-provide general-purpose equivariant convolution layers for 2D and 3D
-groups respectively. They implement equivariant \emph{feature maps} but
-leave the choice of invariant pooling to the user (typically norm or
-gate). Our library is complementary: it provides the invariant map as a
-drop-in \texttt{nn.Module} that plugs into architectures built with either
-framework.
-
-\paragraph{Bispectral approaches in neural networks.}
-\citet{sanborn2023bispectral} introduced bispectral neural networks (BNNs),
-using the full $G$-bispectrum as an invariant nonlinearity for $C_n$ and $D_n$.
-\citet{oreiller2022bispectral} applied bispectral CNNs to medical imaging
-with a fixed set of hand-selected triples. Both use the full (quadratic-cost)
-bispectrum and provide implementations tied to specific architectures.
-Our library generalizes to seven group/domain pairs, defaults to the
-$O(|G|)$ selective reduction, and exposes a framework-agnostic API.
-\citet{chevalley2024bispectral} explore bispectral signatures for data
-characterisation, using bispectra as descriptors rather than learnable
-pooling primitives.
-
-\paragraph{CG product nonlinearities and tensor products.}
-Tensor field networks~\citep{thomas2018tensor} and
-MACE~\citep{batatia2022mace} use iterated Clebsch--Gordan products
-to build equivariant interactions. These products are exact and complete
-in principle, but each layer incurs $O(L^3)$ cost per CG contraction.
-\citet{xie2025price} analyse the \emph{price of freedom} in tensor
-product architectures, showing that maintaining full expressivity through
-CG products becomes expensive at high angular momenta.
-The augmented selective bispectrum achieves completeness in a single $O(L^2)$
-pass (for $\SO(3)$) or $O(|G|)$ pass (for finite groups), avoiding
-iterated contractions.
-
-\paragraph{Invariant theory and scattering.}
-The wavelet scattering transform~\citep{mallat2012scattering} provides
-translation-invariant representations via iterated modulus operations;
-\citet{esteves2018spherical} extend this to the sphere. Unlike the
-bispectrum, scattering representations are stable but generally
-incomplete---they discard inter-scale phase information.
-The bispectrum's completeness guarantee~\citep{kakarala2012bispectrum} is its
-distinguishing theoretical advantage.
-
-\section{Discussion and Conclusion}
-\label{sec:discussion}
-
-We have presented \texttt{bispectrum}, a PyTorch library that makes selective
-$G$-bispectral invariants practical for machine learning. The library covers
-seven group/domain pairs with a uniform \texttt{nn.Module} interface and
-required two technical contributions during implementation: an augmented
-selective $\SO(3)$ bispectrum on $S^2$ achieving the optimal $O(L^2)$
-coefficient count (Theorem~\ref{thm:so3-selective}), and an analysis of
-the octahedral bootstrap inversion failure with a partial Levenberg--Marquardt
-workaround (Section~\ref{sec:octa}). Table~\ref{tab:benchmarks} confirms that the
-selective forward pass adds negligible overhead to training pipelines
-($<$1\,ms for all finite-group modules on a single GPU). Across three
-experimental settings, bispectral pooling provides numerically exact rotation
-invariance and a data-efficiency advantage over incomplete alternatives in
-the low-data / moderate-capacity regime, where the complete invariant map
-provides a stronger inductive bias that partially compensates for
-limited training data.
-
-\paragraph{When not to use the bispectrum.}
-The data-efficiency advantage of complete invariants is most pronounced
-in the low-data, moderate-capacity regime.
-At high model capacity (Section~\ref{sec:experiments}), incomplete
-alternatives such as max pooling can match or surpass bispectral pooling,
-likely because the network learns to compensate for missing invariant
-information.
-Completeness is a property of the \emph{map}, not of downstream task
-performance: a complete invariant provides a stronger inductive bias,
-but does not guarantee superior accuracy on every task.
-For $\SO(3)$ on $S^2$, computational cost grows as $O(L^2)$ per forward
-pass; at very large band-limits this may become a bottleneck relative
-to simpler pooling strategies.
-
-\paragraph{Limitations.}
-Inversion for $\SO(3)$ on $S^2$ is not yet implemented; only the
-selective forward map is available. The selective $\SO(3)$ completeness
-proof relies on a genericity assumption (the signal must avoid a
-measure-zero algebraic variety) and the bootstrap rank verification
-covers $L \le 100$; a structural all-$L$ proof and a full
-characterisation of the degenerate set remain open.
-The octahedral inversion achieves only a 5.7\% signal recovery rate.
-Our experimental comparisons do not include tensor-product
-architectures (e3nn, MACE) or wavelet scattering, which target
-different input assumptions and would require non-trivial adaptation
-to the benchmarks used here.
-
-\paragraph{Future work.}
-Key directions include: implementing $\SO(3)$ inversion from selective
-bispectral coefficients; a structural proof of bootstrap rank for all~$L$;
-numerical conditioning analysis at large
-band-limits; integration with \texttt{escnn}~\citep{cesa2022program} and
-\texttt{e3nn}~\citep{geiger2022e3nn}; and applications to cryo-EM
-reconstruction and molecular property prediction.
-
-\begin{ack}
-Omitted for review.
-\end{ack}
-
-\bibliographystyle{plainnat}
-\bibliography{references}
-
-\appendix
-
-\section{Selective $\SO(3)$ Bispectrum -- Full Proof}
-\label{app:so3-proof}
-
-This appendix provides the complete proof of Theorem~\ref{thm:so3-selective}
-from the main text, establishing completeness of the augmented selective
-$\SO(3)$-bispectrum on $S^2$ for band-limits $4 \le L \le 100$
-(extendable to any desired~$L$ by re-running the verification script).
-
-\begin{table}[h]
-  \caption{Per-degree entry counts in the augmented selective $\SO(3)$
-    invariant. The ``Bispec'' column counts scalar bispectral
-    entries including mandatory even self-coupling entries
-    $\beta_{\ell,\ell,\ell'}$ ($\ell'$ even, $2 \le \ell' \le \ell$);
-    entries that vanish for real signals are excluded.
-    The ``CG Power'' column counts CG power spectrum augmentations.
-    At $\ell=1$, $\beta_{1,1,0}$ is excluded (redundant with power).
-    At $\ell=2$, $\beta_{1,2,1}$ is excluded (proportional to
-    $\beta_{1,1,2}$ after gauge-fixing).}
-  \label{tab:entry-counts}
-  \centering
-  \small
-  \begin{tabular}{cccc}
-    \toprule
-    $\ell$ & Bispec & CG Power & Total \\
-    \midrule
-    0 & 1 & 0 & 1 \\
-    1 & 1 & 0 & 1 \\
-    2 & 3 & 1 & 4 \\
-    3 & 7 & 4 & 11 \\
-    4 & 12 & 5 & 17 \\
-    $\ell \ge 5$ & $\sim 2\ell{+}3$ & $\sim \ell$ & $\sim 3\ell$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\subsection{Setup and notation}
-
-\begin{definition}[Band-limited signal on $S^2$]
-A signal $f : S^2 \to \R$ is \emph{band-limited} at degree $L$ if its
-spherical harmonic expansion
-\begin{equation}\label{eq:app-sh-expansion}
-  f(\theta, \varphi)
-  = \sum_{\ell=0}^{L} \sum_{m=-\ell}^{\ell}
-    a_\ell^m \, \SH_\ell^m(\theta, \varphi)
-\end{equation}
-has $a_\ell^m = 0$ for all $\ell > L$. We write
-$\mathbf{F}_\ell = (a_\ell^{-\ell}, \dotsc, a_\ell^{\ell})
-  \in \C^{2\ell+1}$
-for the coefficient vector at degree~$\ell$.
-\end{definition}
-
-\noindent
-For real $f$, the conjugation symmetry
-$a_\ell^{-m} = (-1)^m \conj{a_\ell^m}$ holds, so
-$\mathbf{F}_\ell$ has $2\ell+1$ real degrees of freedom.
-The total signal dimension is
-$\dim = \sum_{\ell=0}^{L}(2\ell+1) = (L+1)^2$.
-
-\begin{definition}[Rotation action]
-Under $g \in \SO(3)$, the spherical harmonic coefficients transform as
-$\mathbf{F}_\ell \mapsto D^\ell(g)\,\mathbf{F}_\ell$,
-where $D^\ell$ is the $(2\ell+1)$-dimensional Wigner~$D$-matrix.
-\end{definition}
-
-\begin{definition}[Bispectrum on $S^2$]\label{def:app-bispectrum}
-For each triple $(\ell_1, \ell_2, \ell)$ satisfying the triangle
-inequality $\abs{\ell_1 - \ell_2} \le \ell \le \ell_1 + \ell_2$,
-the \emph{bispectral coefficient} is
-\begin{equation}\label{eq:app-bispectrum}
-  \beta_{\ell_1,\ell_2,\ell}
-  = \sum_{\substack{m_1,\,m_2,\,m \\ m_1+m_2=m}}
-    \CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}\;
-    a_{\ell_1}^{m_1}\,a_{\ell_2}^{m_2}\,\conj{a_\ell^m},
-\end{equation}
-where $\CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}$ denotes the
-Clebsch--Gordan coefficient
-$\langle \ell_1, m_1;\, \ell_2, m_2 \mid \ell, m \rangle$.
-Each $\beta_{\ell_1,\ell_2,\ell}$ is a scalar; for real~$f$ it is
-real when $\ell_1{+}\ell_2{+}\ell$ is even and purely imaginary when
-$\ell_1{+}\ell_2{+}\ell$ is odd (Lemma~\ref{lem:app-parity}).
-\end{definition}
-
-Definition~\ref{def:app-bispectrum} is the specialization of the
-general $G$-bispectrum (eq.~\eqref{eq:bispectrum-def}) to the case
-$G = \SO(3)$ acting on~$S^2$, where the irreps are Wigner-$D$
-matrices and the Fourier coefficients are spherical harmonic
-coefficients~$a_\ell^m$.
-
-\begin{definition}[CG power spectrum]\label{def:app-cg-power}
-For a triple $(\ell_1, \ell_2, \ell)$ satisfying the triangle
-inequality, the \emph{CG power spectrum entry} is
-\begin{equation}\label{eq:app-cg-power}
-  P_{\ell_1,\ell_2,\ell}
-  := \lVert (\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell
-     \rVert^2
-  = \sum_{m=-\ell}^{\ell}
-    \Bigl\lvert \sum_{\substack{m_1,m_2 \\ m_1+m_2=m}}
-      \CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}\;
-      a_{\ell_1}^{m_1}\, a_{\ell_2}^{m_2} \Bigr\rvert^2.
-\end{equation}
-This is a degree-$4$ polynomial in the signal coefficients (real and
-non-negative for real~$f$) and $\SO(3)$-invariant: the CG-coupled
-product transforms as $D^\ell(g)$, and $\lVert \cdot \rVert^2$ is
-invariant under unitary transformations.
-\end{definition}
-
-\begin{proposition}[Size of the full bispectrum]\label{prop:app-full-size}
-The number of admissible triples $(\ell_1, \ell_2, \ell)$ with
-$0 \le \ell_1, \ell_2, \ell \le L$ satisfying the triangle inequality
-$|\ell_1 - \ell_2| \le \ell \le \ell_1 + \ell_2$ is $\Theta(L^3)$.
-\end{proposition}
-\begin{proof}
-For fixed $\ell_1$ and $\ell_2$, the triangle inequality admits
-$\ell \in \{|\ell_1 - \ell_2|, \dotsc, \min(\ell_1 + \ell_2, L)\}$,
-giving $\min(\ell_1 + \ell_2, L) - |\ell_1 - \ell_2| + 1$ valid
-values of $\ell$. Summing:
-\[
-  N_{\mathrm{full}}
-  = \sum_{\ell_1=0}^{L}\;\sum_{\ell_2=0}^{L}\;
-    \bigl(\min(\ell_1+\ell_2,L) - |\ell_1-\ell_2| + 1\bigr)
-  = \tfrac{1}{3}L^3 + \mathcal{O}(L^2).
-\]
-\end{proof}
-
-\begin{remark}[Invariance]
-Each $\beta_{\ell_1,\ell_2,\ell}$ is $\SO(3)$-invariant: the Wigner
-matrices satisfy $D^{\ell_1} \otimes D^{\ell_2}
-= \bigoplus_\ell D^\ell$ under the CG decomposition, so the triple
-product contracted with CG coefficients is unchanged by
-$\mathbf{F}_\ell \mapsto D^\ell(g)\mathbf{F}_\ell$.
-\end{remark}
-
-\subsection{Why $\mathcal{O}(L)$ is impossible}
-
-\begin{proposition}[Dimensional lower bound]\label{prop:app-lower-bound}
-Any complete $\SO(3)$-invariant for band-limited signals on $S^2$
-at band-limit $L$ must have at least $(L+1)^2 - 3$ functionally
-independent components.
-\end{proposition}
-\begin{proof}
-The orbit space $\R^{(L+1)^2} / \SO(3)$ has dimension
-$(L+1)^2 - \dim\SO(3) = (L+1)^2 - 3$. Any injective map from the
-orbit space to $\R^k$ requires $k \ge (L+1)^2 - 3$, hence
-$\Omega(L^2)$.
-\end{proof}
-
-\begin{remark}
-The chain $\beta_{1,\ell,\ell+1}$ for $\ell = 0, \dotsc, L-1$ provides
-only $\sim 3L$ scalar constraints (since each chain entry couples
-$\mathbf{F}_1 \in \C^3$ with one known $\mathbf{F}_\ell$). This cannot
-recover $(L+1)^2 \approx L^2$ unknowns.
-\end{remark}
-
-\subsection{The selective index set}
-
-The goal is to select, for each target degree $\ell$, a set of
-bispectral triples that provides enough equations to recover
-$\mathbf{F}_\ell$ from already-known lower-degree coefficients. Since
-$\mathbf{F}_\ell$ has $2\ell+1$ real degrees of freedom, we need at
-most $2\ell+1$ entries. We classify candidate triples by how
-$\mathbf{F}_\ell$ enters the bispectral formula~\eqref{eq:app-bispectrum}:
-\emph{linearly} (when it occupies a position where the other two
-indices are known), or \emph{nonlinearly} (when it appears more than
-once).
-
-\begin{definition}[Selective bispectrum on $S^2$]\label{def:app-selective}
-For each target degree $\ell = 0, \dotsc, L$, we select at most
-$2\ell+1$ bispectral triples from four categories, in the following
-priority order.
-
-\medskip
-\noindent\textbf{Category~1 (Chain).}
-Triples $(\ell_1, \ell_2, \ell)$ with
-$\ell_1 \le \ell_2 < \ell$ and $\ell_1 + \ell_2 \ge \ell$.
-Linear in $\mathbf{F}_\ell$: the unknown
-$\conj{a_\ell^m}$ appears once, multiplied by known lower-degree
-coefficients.
-
-\medskip
-\noindent\textbf{Category~2 (Cross).}
-Triples $(\ell_1, \ell, \ell')$ with
-$1 \le \ell_1 < \ell$ and $\ell - \ell_1 \le \ell' < \ell$.
-Here $\mathbf{F}_\ell$ appears in the second (tensor-product) position,
-yielding equations linear in $a_\ell^{m_2}$ with known $a_{\ell_1}^{m_1}$
-and $a_{\ell'}^{m}$.
-
-\medskip
-\noindent\textbf{Category~3 (Power).}
-The triple $(0, \ell, \ell)$, giving
-$\beta_{0,\ell,\ell} = a_0^0 \norm{\mathbf{F}_\ell}^2$
-(a \emph{quadratic} constraint).
-
-\medskip
-\noindent\textbf{Category~4 (Self-coupling).}
-Triples $(\ell, \ell, \ell')$ with $0 \le \ell' < \ell$.
-Quadratic in $\mathbf{F}_\ell$. Selected only for the remaining deficit
-at low degrees.
-\end{definition}
-
-\begin{algorithm}[t]
-\caption{Augmented selective index set for the $\SO(3)$ bispectrum
-  on $S^2$ at band-limit~$L$. Returns bispectral triples
-  $\mathcal{S}_\beta$ and CG power triples $\mathcal{S}_P$.}
-\label{alg:so3-augmented}
-\small
-\begin{algorithmic}[1]
-\STATE \textbf{Input:} Band-limit $L$
-\STATE \textbf{Output:} Bispectral set $\mathcal{S}_\beta$, CG power
-  set $\mathcal{S}_P$
-\STATE $\mathcal{S}_\beta \leftarrow \{(0,0,0)\}$;
-  $\mathcal{S}_P \leftarrow \emptyset$
-\FOR{$\ell = 1, \dotsc, L$}
-  \STATE $\text{budget} \leftarrow 2\ell + 1$
-  \IF{$\ell = 4$}
-    \STATE $\text{budget} \leftarrow 10$
-      \COMMENT{keep all chain+cross at $\ell=4$}
-  \ENDIF
-  \STATE $\text{candidates} \leftarrow [\,]$
-  \IF{$4 \le \ell \le 7$}
-    \STATE append the explicit low-degree linear block
-      $\mathcal{T}^{\mathrm{small}}_\ell$ to candidates
-  \ELSIF{$\ell \ge 8$}
-    \FOR{$a = 1, \dotsc, \ell{-}1$}
-      \STATE append $(a,\, \ell,\, \ell{-}a)$ to candidates
-    \ENDFOR
-    \FOR{$a = 2, \dotsc, \ell{-}1$}
-      \STATE append $(a,\, \ell,\, \ell{-}a{+}1)$ to candidates
-    \ENDFOR
-    \FOR{$a = 1,2,3,4$}
-      \STATE append $(a,\, \ell{-}a,\, \ell)$ to candidates
-    \ENDFOR
-  \ELSE
-    \STATE append low-degree chain candidates
-    \IF{$\ell \ne 2$}
-      \STATE append cross candidates
-        \COMMENT{$\beta_{1,2,1}$ excluded}
-    \ENDIF
-  \ENDIF
-  \STATE append $(0,\, \ell,\, \ell)$ to candidates
-    \COMMENT{power entry}
-  \IF{$\ell \notin \{1,\, 2\}$}
-    \FOR{$\ell' = \ell{-}1, \dotsc, 0$}
-      \STATE append $(\ell,\, \ell,\, \ell')$ to candidates
-    \ENDFOR
-  \ENDIF
-  \IF{$\ell = 2$}
-    \STATE append $(\ell,\, \ell,\, \ell)$ to candidates
-      \COMMENT{$\beta_{2,2,2}$}
-  \ENDIF
-  \STATE Remove entries that vanish for real signals; deduplicate
-  \STATE $\mathcal{S}_\beta \leftarrow \mathcal{S}_\beta
-    \;\cup\;$ first $\min(\text{budget},\,
-    |\text{candidates}|)$ live entries
-  \STATE \COMMENT{\textbf{Mandatory even self-couplings}}
-  \FOR{$\ell' = 2, 4, \dotsc, \lfloor\ell/2\rfloor \cdot 2$
-    with $\ell' \le \ell$ and $\ell \ge 3$}
-    \STATE $\mathcal{S}_\beta \leftarrow \mathcal{S}_\beta
-      \cup \{(\ell,\, \ell,\, \ell')\}$
-  \ENDFOR
-  \STATE \COMMENT{\textbf{CG power augmentation}}
-  \STATE Greedily add $P_{\ell_1,\ell_2,\ell'}$ (with
-    $\max(\ell_1,\ell_2) = \ell$, $|\ell_1{-}\ell_2| \le \ell'
-    \le \ell_1{+}\ell_2$) to $\mathcal{S}_P$ until the
-    Jacobian rank of $\Phi_{\mathrm{aug}}$ at degree~$\ell$
-    matches the number of unknowns
-\ENDFOR
-\RETURN $(\mathcal{S}_\beta,\, \mathcal{S}_P)$
-\end{algorithmic}
-\end{algorithm}
-
-\begin{proposition}[Bispectral entry counts]\label{prop:app-entry-counts}
-At $\ell = 4$ the explicit block $\mathcal{T}^{\mathrm{small}}_4$ has
-$10$ linear entries (all chain+cross candidates).
-For each $5 \le \ell \le 7$, the explicit block
-$\mathcal{T}^{\mathrm{small}}_\ell$ has exactly $2\ell+1$ linear
-entries. For each $\ell \ge 8$, the proof-first linear family
-\[
-  \mathcal{T}_\ell
-  := \{(a,\ell,\ell-a) : 1 \le a \le \ell-1\}
-     \cup \{(a,\ell,\ell-a+1) : 2 \le a \le \ell-1\}
-     \cup \{(a,\ell-a,\ell) : 1 \le a \le 4\}
-\]
-also has exactly $2\ell+1$ entries.
-\end{proposition}
-
-\begin{proposition}[Augmented output size]\label{prop:app-output-size}
-For $L \ge 4$, the bispectral component $\mathcal{S}_\beta$
-has $(L{+}1)^2 - 3$ budget-selected entries
-plus $\sum_{\ell=4}^{L}\lfloor\ell/2\rfloor$ mandatory
-even self-coupling entries, giving $\Theta(L^2)$ bispectral
-scalars.
-The CG power component $\mathcal{S}_P$ has $O(L^2)$ entries,
-giving a total augmented output of $\Theta(L^2)$ scalars.
-
-Verified counts for $L = 4$: $24$ bispectral
-($20$~live, $2$~vanishing, $2$~gauge-fixed)
-$+$ $10$ CG power $= 34$ total entries;
-for $L = 5$: $37$ bispectral
-($32$~live, $3$~vanishing, $2$~gauge-fixed)
-$+$ $17$ CG power $= 54$ total.
-The Jacobian-rank certificate
-(Lemma~\ref{lem:app-augmented-perdegree}) uses only the
-$m = 30$ (resp.\ $49$) live entries.
-\end{proposition}
-\begin{proof}
-\emph{Bispectral entries.}
-At $\ell = 1$, the self-coupling $\beta_{1,1,0}$ is redundant with the
-power entry, so no self-coupling is selected ($1$~entry).
-At $\ell = 2$, the cross entry $\beta_{1,2,1}$ collapses to a scalar
-multiple of $\beta_{1,1,2}$ after gauge-fixing and is excluded; the
-symmetric self-coupling $\beta_{2,2,2}$ is included ($3$~entries).
-At $\ell = 4$, all $10$ chain+cross candidates are retained,
-plus mandatory even self-coupling entries
-$\beta_{4,4,2}$ and $\beta_{4,4,4}$.
-Similarly, at each $\ell \ge 3$, the even self-coupling entries
-$\beta_{\ell,\ell,\ell'}$ ($\ell'$ even, $2 \le \ell' \le \ell$)
-add $\lfloor\ell/2\rfloor$ entries beyond the linear-block
-budget.  Therefore:
-\[
-  |\mathcal{S}_\beta|
-  = (L{+}1)^2 - 3 + \sum_{\ell=4}^{L}\lfloor\ell/2\rfloor
-  = \Theta(L^2).
-\]
-
-\emph{CG power entries.}
-At each degree $\ell \ge 2$, the CG power augmentation adds at most
-$2\ell{+}1$ entries (one per real unknown in~$\mathbf{F}_\ell$).
-Summing gives an upper bound:
-\[
-  |\mathcal{S}_P| \;\le\; \sum_{\ell=2}^{L} (2\ell{+}1)
-  \;=\; L^2 + 2L - 3 \;=\; O(L^2).
-\]
-
-\emph{Total.}
-$|\mathcal{S}_\beta| + |\mathcal{S}_P|
-= (L{+}1)^2 - 3 + O(L^2) = \Theta(L^2)$.
-The lower bound $\Omega(L^2)$ follows from $|\mathcal{S}_\beta|$ alone.
-\end{proof}
-
-\subsection{Completeness proof}
-
-The proof proceeds in four stages: (1)~recovery of $\mathbf{F}_0$,
-$\mathbf{F}_1$ and $\SO(3)$ gauge fixing; (2)~seed recovery at degrees
-$2$--$3$ with a parity analysis of the bispectrum; (3)~fibre reduction
-at degree~$4$ via parity-breaking entries; (4)~per-degree linear
-bootstrap for $\ell \ge 5$.
-The key difference from the finite-group case
-\citep{mataigne2024selective} is that on~$S^2$, the scalar bispectrum
-alone has rank deficiency for real signals, necessitating CG power
-augmentation at every degree.
-
-\begin{definition}[Reflection map $T_R$]\label{def:app-TR}
-The azimuthal reflection $R: (\theta,\varphi) \mapsto (\theta,-\varphi)$
-acts on spherical harmonic coefficients as
-$a_\ell^m \mapsto (-1)^m a_\ell^{-m}$.
-For real signals, this equals complex conjugation: $a_\ell^m \mapsto \conj{a_\ell^m}$.
-Since $R \in \OG(3) \setminus \SO(3)$, the signal $R \cdot f$ is generically
-in a \emph{different} $\SO(3)$-orbit from $f$.
-\end{definition}
-
-\begin{assumption}\label{ass:app-generic}
-The signal $f$ satisfies:
-\begin{enumerate}[label=(G\arabic*),nosep]
-  \item $a_0^0 \ne 0$ (non-zero mean).
-  \item $\norm{\mathbf{F}_1} \ne 0$ (non-zero dipole).
-  \item After gauge-fixing (Lemma~\ref{lem:app-gauge}), $a_2^1 \ne 0$
-    (azimuthal gauge is resolvable).
-\end{enumerate}
-\end{assumption}
-
-\begin{remark}
-Each condition above excludes a proper algebraic subvariety of the signal
-space $\R^{(L+1)^2}$; their union is Lebesgue-null.
-The proof uses additional genericity conditions---seed Jacobian rank,
-bootstrap rank, and degree-4 consistency---but each is \emph{proved}
-to hold outside a proper algebraic subvariety (not assumed).
-\end{remark}
-
-\begin{theorem}[Completeness of the augmented selective $\SO(3)$ invariant on $S^2$]
-\label{thm:app-main}
-For $4 \le L \le 100$, let $f, f' : S^2 \to \R$ be band-limited at degree~$L$
-and satisfy Assumption~\ref{ass:app-generic}. If
-$\Phi_{\mathrm{aug}}(f) = \Phi_{\mathrm{aug}}(f')$
-(as complex-valued output, including both real and imaginary parts
-of each bispectral entry), then $f' = g \cdot f$ for some $g \in \SO(3)$.
-
-The generic set (satisfying (G1)--(G3) and lying outside a proper
-algebraic subvariety) has full Lebesgue measure in~$V_L$.
-The augmented output has $\Theta(L^2)$ real components, achieving the
-information-theoretic lower bound up to a constant factor.
-\end{theorem}
-
-\begin{lemma}[Gauge fixing]\label{lem:app-gauge}
-For generic $f \in V_L$ (with $\norm{\mathbf{F}_1} \ne 0$ and $a_2^1 \ne 0$
-after aligning $\mathbf{F}_1$), there exists a unique $g \in \SO(3)$ such that
-the gauge-fixed signal $\tilde{f} = g \cdot f$ satisfies:
-\begin{enumerate}[nosep,label=(\roman*)]
-\item $a_1^{-1} = a_1^1 = 0$, \quad $a_1^0 = c > 0$ \quad
-  (alignment of $\mathbf{F}_1$ to the $z$-axis),
-\item $\im(a_2^1) = 0$, \quad $a_2^1 = x > 0$ \quad
-  (fixing the residual $\SO(2)_z$ symmetry).
-\end{enumerate}
-This consumes all $3$ degrees of freedom of $\SO(3)$.
-\end{lemma}
-\begin{proof}
-The $\ell=1$ Wigner $D$-matrix $D^1(g)$ is the standard $3 \times 3$
-rotation matrix. Since $\SO(3)$ acts transitively on $S^2$, any nonzero
-$\mathbf{F}_1$ can be rotated to align with $\hat{z}$, leaving an
-$\SO(2)_z$ stabiliser. Choosing
-$\varphi = -\arg(a_2^1)$ yields $a_2^1 \in \R_{>0}$, exhausting all
-three rotational degrees of freedom.
-\end{proof}
-
-\begin{remark}[Gauge-fixed parameterisation]\label{rem:app-gauge-param}
-On the gauge slice, the real parameters are:
-\[
-  A_0 = a_0^0, \quad c = a_1^0, \quad
-  (y, x, u, v) = (a_2^0, \operatorname{Re}a_2^1, \operatorname{Re}a_2^2, \im a_2^2),
-\]
-and for $\ell \ge 3$: $t_\ell = a_\ell^0$, and
-$(p_{\ell,m}, q_{\ell,m}) = (\operatorname{Re}a_\ell^m, \im a_\ell^m)$ for $m = 1,\ldots,\ell$.
-The reflection $T_R$ (Definition~\ref{def:app-TR}) acts on the gauge slice as:
-\begin{equation}\label{eq:app-TR-action}
-  T_R: \quad v \mapsto -v, \qquad
-  q_{\ell,m} \mapsto -q_{\ell,m} \quad \forall\, \ell \ge 3, \; m \ge 1.
-\end{equation}
-All other gauge-fixed parameters are unchanged.
-$T_R$ preserves the gauge conditions (i)--(ii).
-\end{remark}
-
-\begin{lemma}\label{lem:app-F0F1}
-For $a_0^0 \ne 0$:
-$\beta_{0,0,0} = (a_0^0)^3$ determines $A_0 = a_0^0$ uniquely.
-Then $\beta_{0,1,1} = A_0 \norm{\mathbf{F}_1}^2$ determines
-$c = \norm{\mathbf{F}_1}$, and $\mathbf{F}_1 = (0, c, 0)^\top$
-is fixed by the gauge.
-\end{lemma}
-
-\subsubsection{Parity structure of the bispectrum}
-
-\begin{lemma}[Parity of the bispectrum]\label{lem:app-parity}
-For any signal (real or complex):
-\begin{equation}\label{eq:app-parity}
-  \beta_{\ell_1,\ell_2,\ell}(T_R \cdot f)
-  = (-1)^{\ell_1+\ell_2+\ell}\,\beta_{\ell_1,\ell_2,\ell}(f).
-\end{equation}
-For \emph{real} signals, there is an independent constraint:
-\begin{equation}\label{eq:app-reality-constraint}
-  \conj{\beta_{\ell_1,\ell_2,\ell}(f)}
-  = (-1)^{\ell_1+\ell_2+\ell}\,\beta_{\ell_1,\ell_2,\ell}(f).
-\end{equation}
-Together:
-\begin{enumerate}[nosep,label=(\alph*)]
-\item If $\ell_1+\ell_2+\ell$ is even:
-  $\beta$ is real-valued (by~\eqref{eq:app-reality-constraint})
-  and $T_R$-invariant (by~\eqref{eq:app-parity}).
-\item If $\ell_1+\ell_2+\ell$ is odd:
-  $\beta$ is purely imaginary (by~\eqref{eq:app-reality-constraint})
-  and $\beta(T_R \cdot f) = -\beta(f)$, so $\im\beta$ flips sign under $T_R$.
-\end{enumerate}
-\end{lemma}
-\begin{proof}
-Under $T_R$, each $a_\ell^m \mapsto (-1)^m a_\ell^{-m}$.
-Substituting into $\beta = \sum_m (\sum_{m_1+m_2=m} C\, a^{m_1} a^{m_2})
-\conj{a^m}$, the three sign factors $(-1)^{m_1+m_2+m} = (-1)^{2m} = 1$
-cancel. Re-indexing $m \mapsto -m$ and applying the CG symmetry
-$\CG{\ell_1}{-m_1}{\ell_2}{-m_2}{\ell}{-m}
-  = (-1)^{\ell_1+\ell_2-\ell}\,\CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}$
-yields~\eqref{eq:app-parity} (with $(-1)^{\ell_1+\ell_2-\ell} = (-1)^{\ell_1+\ell_2+\ell}$).
-
-For real signals, $\conj{a_\ell^m} = (-1)^m a_\ell^{-m}$.
-Taking the complex conjugate of $\beta$ and re-indexing similarly
-yields~\eqref{eq:app-reality-constraint}.
-\end{proof}
-
-\begin{lemma}[CG power is $T_R$-invariant]\label{lem:app-P-invariant}
-For all triples $(\ell_1,\ell_2,\ell)$:
-$P_{\ell_1,\ell_2,\ell}(T_R \cdot f) = P_{\ell_1,\ell_2,\ell}(f)$.
-\end{lemma}
-\begin{proof}
-Under $T_R$, the CG-projected component transforms as
-$(\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell^M
-  \mapsto (-1)^{\ell_1+\ell_2+\ell+M}\,
-  (\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell^{-M}$.
-Taking the squared norm and re-indexing the sum over $M$
-leaves $P$ unchanged.
-\end{proof}
-
-\begin{proposition}[Vanishing of odd-parity entries at low degrees]
-\label{prop:app-odd-vanishing}
-For real signals, the odd-parity bispectral entry
-$\beta_{\ell_1,\ell_2,\ell}$ (with $\ell_1+\ell_2+\ell$ odd)
-vanishes identically whenever two of the three representation indices
-coincide. In particular:
-\begin{enumerate}[nosep,label=(\roman*)]
-\item $\ell_1 = \ell_2$: the CG coefficients are antisymmetric
-  under swapping $m_1 \leftrightarrow m_2$, while
-  $a_{\ell_1}^{m_1} a_{\ell_1}^{m_2}$ is symmetric, giving
-  exact cancellation.
-\item Two of $\{\ell_1, \ell_2, \ell\}$ are equal: a symmetry
-  argument yields $\beta = -\conj\beta$ \textbf{and}
-  $\beta = 0$, verified symbolically for all
-  triples with indices $\le 5$.
-\end{enumerate}
-\end{proposition}
-
-\begin{corollary}[First nonzero odd-parity entry]\label{cor:app-first-odd}
-The first bispectral entry with odd parity and generically nonzero imaginary
-part occurs at degree $\ell = 4$:
-$\beta_{2,3,4}$, with
-$\ell_1+\ell_2+\ell = 9$ (odd) and
-$\ell_1, \ell_2, \ell$ all distinct.
-At the deterministic witness (seed~42), $\im(\beta_{2,3,4}) \approx 3.26 \ne 0$,
-and $\im(\beta_{2,3,4})(T_R \cdot f) \approx -3.26$.
-Similarly, $\beta_{2,4,3}$ and $\beta_{3,4,2}$ are nonzero odd-parity entries
-at degree~$4$.
-\end{corollary}
-
-\begin{remark}[No parity-breaking at $\ell \le 3$]
-For degrees $\le 3$, every odd-parity triple $(\ell_1,\ell_2,\ell)$
-has at least two indices equal (since $\max(\ell_1,\ell_2,\ell) \le 3$
-and $\ell_1+\ell_2+\ell$ odd forces at least one pair to be equal in this range).
-By Proposition~\ref{prop:app-odd-vanishing}, all odd-parity entries vanish.
-Therefore, the $T_R$ ambiguity cannot be resolved using only degrees $\le 3$.
-\end{remark}
-
-\subsubsection{Seed recovery ($\ell \le 3$)}
-
-\begin{lemma}[Degree-2 recovery]\label{lem:app-ell2}
-Given $(A_0, c)$ from Lemma~\ref{lem:app-F0F1}, the gauge-fixed degree-2
-parameters $(y, x, u, v)$ satisfy:
-\begin{align}
-  \beta_{1,1,2} &= \tfrac{\sqrt{6}}{3}\, c^2\, y
-    &\Longrightarrow\quad& y \;\text{determined}, \label{eq:app-y}\\
-  P_{1,2,1} &= \tfrac{c^2}{5}(3x^2 + 2y^2)
-    &\Longrightarrow\quad& x^2 \;\text{determined}, \label{eq:app-x2}\\
-  \beta_{0,2,2} &= A_0(2u^2 + 2v^2 + 2x^2 + y^2)
-    &\Longrightarrow\quad& S := u^2+v^2 \;\text{determined}, \label{eq:app-S}\\
-  \beta_{2,2,2} &= \tfrac{6\sqrt{14}\,S\,y - 6\sqrt{21}\,u\,x^2
-    - 3\sqrt{14}\,x^2 y - \sqrt{14}\,y^3}{7}
-    &\Longrightarrow\quad& u \;\text{determined}. \label{eq:app-u}
-\end{align}
-Here $x = \sqrt{x^2} > 0$ by the gauge condition.
-Then $v^2 = S - u^2$, giving $|v|$ but not $\sgn(v)$.
-The fibre at degree~$2$ is generically $\{(y,x,u,v),\; (y,x,u,-v)\}$.
-\end{lemma}
-
-\begin{lemma}[Degree-3 recovery: finite fibre]\label{lem:app-ell3}
-Given $\mathbf{F}_2 = (y,x,u,\pm v)$, the degree-3 parameters
-$\mathbf{F}_3 = (t, p_1, q_1, p_2, q_2, p_3, q_3)$ are constrained by the
-augmented entries:
-\begin{itemize}[nosep]
-\item $\beta_{1,2,3}$, $\beta_{1,3,2}$, $\beta_{2,3,1}$: bilinear in
-  $\mathbf{F}_2 \otimes \mathbf{F}_3$.
-\item $\beta_{0,3,3} = A_0\norm{\mathbf{F}_3}^2$: gives the norm.
-\item $\beta_{3,3,2}$: quadratic in $\mathbf{F}_3$, linear in $\mathbf{F}_2$.
-\item $\beta_{2,3,2}$, $\beta_{2,2,3}$: odd parity with repeated indices,
-  vanish identically for real signals (Proposition~\ref{prop:app-odd-vanishing}).
-\item $P_{1,2,1}$ (from degree-2, Lemma~\ref{lem:app-ell2}),
-  $P_{1,3,2}$, $P_{2,3,1}$, $P_{2,3,2}$, $P_{3,3,2}$: five CG power
-  entries (the last four quadratic in $\mathbf{F}_3$).
-\end{itemize}
-The full augmented seed system uses $13$ nonzero entries for $11$ unknowns.
-The bispectral entries achieve rank~$6$; each of the five
-CG power entries contributes one additional rank,
-reaching full rank~$11$.
-The generic fibre is $0$-dimensional (finitely many points)
-and contains the $T_R$ pair
-$\{(\mathbf{F}_2, \mathbf{F}_3),\; T_R(\mathbf{F}_2, \mathbf{F}_3)\}$.
-\end{lemma}
-\begin{proof}
-The Jacobian rank is verified by exact symbolic computation (sympy) at the
-rational witness $(A_0, c, y, x, u, v, t, p_1, q_1, p_2, q_2, p_3, q_3)
-= (1, 1, \tfrac{3}{7}, \tfrac{2}{5}, \tfrac{1}{3}, \tfrac{4}{9},
-   \tfrac{1}{2}, \tfrac{3}{8}, \tfrac{2}{7}, \tfrac{5}{11}, \tfrac{1}{4},
-   \tfrac{3}{13}, \tfrac{7}{17})$.
-The $13 \times 11$ Jacobian has rank~$11$ (full).
-
-The $T_R$ invariance follows from Lemmas~\ref{lem:app-parity}--\ref{lem:app-P-invariant}
-and Proposition~\ref{prop:app-odd-vanishing}: every entry in the degree~$\le 3$
-augmented system is either real-valued and $T_R$-invariant (even parity $\beta$
-and all~$P$) or identically zero (odd parity $\beta$).
-
-Full Jacobian rank gives $0$-dimensional (finite) fibres, but does
-\emph{not} by itself bound the fibre to two points:
-the degree-3 subsystem (7~real equations of degrees $1$, $2$, and $4$
-in 7~unknowns) can have additional isolated real solutions.
-Extensive numerical search (multi-start least-squares with 5000 random
-initial points per sign-of-$v$ branch) finds $10$ solutions per branch
-($20$ total, forming $10$ $T_R$-pairs) at the rational witness.
-These spurious solutions are eliminated at degree~$4$
-(Lemma~\ref{lem:app-degree4-filter}).
-\end{proof}
-
-\begin{remark}[The seed fibre is finite but not two-point]
-\label{rem:app-seed-fibre-size}
-The seed polynomial system (degrees $0$--$3$) is a system of
-$7$ independent real equations of mixed degree ($1$ linear, $5$ quadratic,
-$1$ quartic) in $7$ real unknowns (for each branch of $\sgn(v)$).
-By B\'ezout's theorem the number of isolated complex solutions is at most
-$1 \times 2^5 \times 4 = 128$.
-Numerically, $10$ real solutions are found per branch.
-While the numerical search cannot formally exclude additional solutions,
-the proof does not require the exact count: finiteness suffices,
-and fibre reduction is deferred to degree~$4$
-(Lemma~\ref{lem:app-degree4-filter}).
-\end{remark}
-
-\subsubsection{Per-degree recovery ($\ell \ge 4$)}
-
-\begin{lemma}[Complex bootstrap rank]\label{lem:app-bootstrap-rank}
-For each $\ell \ge 4$, the selective bispectral entries at degree $\ell$
-that are linear in $\mathbf{F}_\ell$ define a coefficient matrix
-$A_\ell \in \C^{N \times (2\ell+1)}$ (with coefficients depending on
-$\mathbf{F}_0, \ldots, \mathbf{F}_{\ell-1}$).
-For generic lower-degree coefficients:
-$\operatorname{rank}_\C(A_\ell) = 2\ell+1$.
-\end{lemma}
-\begin{proof}
-For each $\ell$, $\det(A_\ell)$ is a polynomial in the lower-degree
-coefficients.  It is verified to be nonzero at a deterministic
-witness (seed~42, real-signal constraint) for each
-$\ell \in \{4, \ldots, 100\}$
-(script: \texttt{benchmarks/verify\_linear\_bootstrap.py}).
-Since a polynomial that is nonzero at one point is not identically zero,
-$\det(A_\ell) \not\equiv 0$ for each verified~$\ell$.
-For $\ell \ge 8$, the matrix uses the closed-form tridiagonal family
-$\mathcal{T}_\ell$ with $2\ell+1$ entries.
-\end{proof}
-
-\begin{remark}[Scope of the bootstrap verification]\label{rem:app-bootstrap-scope}
-The bootstrap rank is verified for each individual $\ell \le 100$.
-For any fixed band-limit $L \le 100$, the genericity condition is that
-$\det(A_\ell) \ne 0$ for $\ell = 4, \ldots, L$.
-This is the complement of $L - 3$ proper algebraic subvarieties
-(one per degree), hence a Zariski-open dense set of full Lebesgue measure.
-For $L > 100$, the verification script can be extended to cover any
-desired~$L$; the structural form of the tridiagonal family is uniform
-for all $\ell \ge 8$.
-\end{remark}
-
-\begin{remark}[Complex rank suffices for uniqueness]\label{rem:app-complex-real}
-For real signals, $\mathbf{F}_\ell$ has $2\ell+1$ real degrees of freedom.
-The bispectral entries define two subsystems: chain entries, linear in
-$\conj{\mathbf{F}_\ell}$, and cross entries, linear in $\mathbf{F}_\ell$.
-Neither subsystem alone has full rank.
-However, combining both via the reality constraint
-$\conj{a_\ell^m} = (-1)^m a_\ell^{-m}$ yields a real linear system
-of full rank $2\ell+1$ in the $2\ell+1$ real parameters.
-\end{remark}
-
-\begin{lemma}[Augmented per-degree real rank]\label{lem:app-augmented-perdegree}
-For each $\ell \ge 4$, the full augmented entries at degree $\ell$
-(linear bispectral, self-coupling, and CG power) have real Jacobian
-rank $2\ell+1$ w.r.t.\ the $2\ell+1$ real parameters of $\mathbf{F}_\ell$.
-The CG power entries are essential at every degree: the scalar
-bispectrum alone (linear + self-coupling) achieves only partial rank.
-Verified at the witness for $\ell \in \{4, \ldots, 30\}$.
-\end{lemma}
-
-\subsubsection{Degree-4 fibre reduction and $T_R$ resolution}
-
-\begin{lemma}[Degree-4 eliminates all spurious seed solutions]
-\label{lem:app-degree4-filter}
-Let $\mathcal{S}$ denote the finite seed fibre at degrees~$0$--$3$
-(Lemma~\ref{lem:app-ell3}).
-For generic real signals with $L \ge 4$, the degree-4 augmented entries
-(12~bispectral + 5~CG power, providing $29$ real constraints on
-$9$~real unknowns of $\mathbf{F}_4$) are simultaneously satisfiable
-for \emph{exactly one} $T_R$-pair in~$\mathcal{S}$:
-the true solution and its reflection.
-All other seed solutions are inconsistent at degree~$4$.
-\end{lemma}
-\begin{proof}
-The degree-4 augmented system is an overdetermined polynomial system
-in the 9~real parameters of $\mathbf{F}_4$, with coefficients depending
-on the seed solution $(\mathbf{F}_0, \ldots, \mathbf{F}_3)$.
-
-For each of the~10 seed solutions on the $+v$ branch at the rational
-witness, a multi-start nonlinear least-squares search (200~random starts)
-finds the best-fit~$\mathbf{F}_4$.
-Only the true seed solution achieves zero residual
-($\norm{r} \sim 10^{-16}$); the 9~spurious solutions all have
-$\norm{r} \ge 10^{-2}$, showing the degree-4 system is \emph{inconsistent}
-for them.
-
-The inconsistency arises because the degree-4 entries include constraints
-that depend only on $(\mathbf{F}_0, \mathbf{F}_1, \mathbf{F}_2, \mathbf{F}_4)$
-and not on $\mathbf{F}_3$---notably $\beta_{2,2,4}$, $\beta_{2,4,2}$,
-$P_{1,4,3}$, $P_{1,4,4}$, $P_{2,4,2}$, $P_{2,4,3}$,
-$\beta_{4,4,2}$, $\beta_{4,4,4}$.
-These 8~entries nearly determine $\mathbf{F}_4$ independently of $\mathbf{F}_3$.
-The remaining entries (involving $\mathbf{F}_3$) then impose
-constraints that are generically inconsistent when $\mathbf{F}_3$ is a
-spurious seed solution.
-
-Since the residual for each spurious seed is a polynomial
-function of the signal's coefficients that is nonzero at the witness,
-its vanishing locus is a proper algebraic subvariety.
-\end{proof}
-
-\begin{lemma}[The $T_R$ branch is overdetermined-inconsistent]
-\label{lem:app-TR-resolve}
-For $L \ge 4$, the $T_R$ partner of the true seed---i.e.,
-$(\mathbf{F}_2', \mathbf{F}_3') = T_R(\mathbf{F}_2, \mathbf{F}_3)$---is
-also inconsistent at degree~$4$ for generic signals.
-\end{lemma}
-\begin{proof}
-By Lemma~\ref{lem:app-degree4-filter}, the $T_R$ branch is one of the
-seed solutions that fails the degree-4 consistency check.
-The mechanism is the parity structure (Lemma~\ref{lem:app-parity}):
-the odd-parity entries $\beta_{2,3,4}$, $\beta_{2,4,3}$, $\beta_{3,4,2}$
-(all-distinct indices, $\im\beta \ne 0$) are part of the degree-4
-constraints.  On the $T_R$ branch these imaginary parts have the
-wrong sign, distorting $\mathbf{F}_4'$ so that the $T_R$-invariant
-self-coupling and CG power entries cannot be simultaneously satisfied.
-
-Explicitly: the degree-4 entries define rational functions of the
-lower-degree coefficients.  Let $\Delta(\mathbf{F}_0,\ldots,\mathbf{F}_3)$
-denote the sum of squared residuals of the non-bootstrap entries
-evaluated at the $T_R$-branch bootstrap solution.
-Then $\Delta$ is a rational function whose numerator is a polynomial $p$
-and whose denominator is $(\det A_4)^2$ (from the bootstrap solve).
-At the witness, $\det A_4 \ne 0$ and $p \ne 0$
-(residual ${\sim}10^{-2}$).
-Hence $p$ is not identically zero, and $\Delta \ne 0$ on the complement
-of the proper algebraic subvariety $\{p = 0\} \cup \{\det A_4 = 0\}$.
-\end{proof}
-
-\begin{proposition}[Uniqueness propagates]\label{prop:app-TR-propagate}
-Once the correct seed $(\mathbf{F}_0, \ldots, \mathbf{F}_3)$ is identified
-at degree~$4$ (Lemmas~\ref{lem:app-degree4-filter}--\ref{lem:app-TR-resolve}),
-the complex bootstrap at degrees $\ell \ge 5$ is unambiguous:
-given the correct $\mathbf{F}_0, \ldots, \mathbf{F}_{\ell-1}$, the
-full-rank matrix $A_\ell$ determines $\mathbf{F}_\ell$ uniquely
-(Lemma~\ref{lem:app-bootstrap-rank}, Remark~\ref{rem:app-complex-real}).
-\end{proposition}
-
-\subsubsection{Proof of the main theorem}
-
-\begin{proof}[Proof of Theorem~\ref{thm:app-main}]
-\textbf{Step 1: Gauge fixing.}
-By Lemma~\ref{lem:app-gauge}, both $f$ and $f'$ can be gauge-fixed.
-If $\Phi_{\mathrm{aug}}(f) = \Phi_{\mathrm{aug}}(f')$, then by
-$\SO(3)$-invariance, the gauge-fixed versions $\tilde{f}$ and $\tilde{f}'$
-(possibly with different gauge rotations $g, g'$) satisfy
-$\Phi_{\mathrm{aug}}(\tilde{f}) = \Phi_{\mathrm{aug}}(\tilde{f}')$.
-It suffices to show $\tilde{f} = \tilde{f}'$ on the gauge slice.
-
-\medskip
-\textbf{Step 2: Recovery of $\mathbf{F}_0$, $\mathbf{F}_1$.}
-By Lemma~\ref{lem:app-F0F1}, the gauge-fixed $A_0$ and $c$ are uniquely
-determined. Hence $\tilde{f}$ and $\tilde{f}'$ agree at degrees $0$ and $1$.
-
-\medskip
-\textbf{Step 3: Seed recovery (degrees 2--3).}
-By Lemma~\ref{lem:app-ell2}, the degree-2 parameters $(y, x, u, |v|)$
-are uniquely determined.
-By Lemma~\ref{lem:app-ell3}, the augmented seed system at degrees $0$--$3$
-has full Jacobian rank~$11$, giving a $0$-dimensional (finite) fibre
-$\mathcal{S} = \{s_1, \ldots, s_K, T_R(s_1), \ldots, T_R(s_K)\}$
-consisting of $T_R$-paired points.
-The gauge-fixed $\tilde{f}'$ agrees with one of these seed solutions.
-
-\medskip
-\textbf{Step 4: Degree-4 fibre reduction.}
-By Lemma~\ref{lem:app-degree4-filter}, the degree-4 augmented entries
-are simultaneously satisfiable for at most one $T_R$-pair in
-$\mathcal{S}$: the true solution $s_1 = (\mathbf{F}_2, \mathbf{F}_3)$ and
-$T_R(s_1)$.  All other seed solutions are eliminated.
-By Lemma~\ref{lem:app-TR-resolve}, the $T_R$ partner is itself inconsistent
-at degree~$4$ due to the parity-breaking entries
-$\beta_{2,3,4}, \beta_{2,4,3}, \beta_{3,4,2}$
-(whose imaginary parts flip sign under~$T_R$).
-This leaves the unique solution
-$(\mathbf{F}_2', \mathbf{F}_3') = (\mathbf{F}_2, \mathbf{F}_3)$,
-with $\mathbf{F}_4' = \mathbf{F}_4$ determined by the augmented system.
-
-\medskip
-\textbf{Step 5: Inductive recovery ($\ell \ge 5$).}
-By Proposition~\ref{prop:app-TR-propagate}, given the correct
-$\mathbf{F}_0, \ldots, \mathbf{F}_{\ell-1}$, the complex bootstrap
-uniquely determines $\mathbf{F}_\ell$ at each degree $\ell \ge 5$.
-
-\medskip
-\textbf{Step 6: Conclusion.}
-By induction on $\ell$, the gauge-fixed signals $\tilde{f}$ and $\tilde{f}'$
-agree at all degrees $0, \ldots, L$.
-Hence $\tilde{f} = \tilde{f}'$, which means $f' = (g')^{-1} g \cdot f$
-with $(g')^{-1} g \in \SO(3)$.
-
-The generic set is the complement of finitely many proper algebraic
-subvarieties: (G1)--(G3), the seed Jacobian rank condition, the
-degree-4 fibre-reduction consistency for each spurious seed solution
-(each is a polynomial condition verified nonzero at the witness),
-and the bootstrap rank conditions $\det(A_\ell) \ne 0$ for
-$\ell = 4, \ldots, L$.
-A finite union of proper algebraic subvarieties has Lebesgue measure zero.
-\end{proof}
-
-\begin{corollary}[Band-limits $L \le 3$]
-For $L \le 3$, the augmented selective bispectrum has finite fibres
-that include $T_R$-pairs (by Lemma~\ref{lem:app-ell3}), but without
-degree-4 entries, the spurious seed solutions and the $T_R$ ambiguity
-cannot be resolved.
-The invariant separates $\OG(3)$-orbits
-(where $\OG(3) = \SO(3) \cup T_R \cdot \SO(3)$) among the true
-$T_R$-pair, but the fibre may contain additional points.
-\end{corollary}
-
-\subsubsection{Computational certificates}
-
-The proof relies on five computational certificates, all reproducible
-from scripts in the repository:
-\begin{enumerate}[nosep]
-\item \textbf{Seed Jacobian rank} (Lemma~\ref{lem:app-ell3}):
-  The $13 \times 11$ Jacobian of the augmented seed system
-  (degrees $0$--$3$, $13$ nonzero entries) has rank $11$ at the rational
-  witness.
-  Verified by \emph{exact} rational arithmetic (sympy).
-
-\item \textbf{Seed fibre enumeration} (Lemma~\ref{lem:app-ell3},
-  Remark~\ref{rem:app-seed-fibre-size}):
-  Extensive multi-start search (5000 starts per branch) finds
-  $10$ solutions per $\sgn(v)$ branch ($20$ total, $10$ $T_R$-pairs).
-  Verified by \texttt{benchmarks/verify\_seed\_fibre.py}.
-
-\item \textbf{Degree-4 fibre reduction}
-  (Lemma~\ref{lem:app-degree4-filter}):
-  For each of the $10$ seed solutions on the $+v$ branch, a
-  multi-start nonlinear least-squares search (200 starts) attempts to
-  find $\mathbf{F}_4$ satisfying all $29$ real degree-4 constraints.
-  Only the true solution achieves zero residual; all $9$ spurious
-  solutions have residual $\ge 10^{-2}$.
-  Verified by \texttt{benchmarks/verify\_degree4\_filter.py}.
-
-\item \textbf{Complex bootstrap rank} (Lemma~\ref{lem:app-bootstrap-rank}):
-  For each $\ell \in \{4, \ldots, 100\}$, the bootstrap matrix $A_\ell$
-  has full \emph{complex} rank $2\ell+1$ at the deterministic witness
-  (seed~42).  The minimum singular value is well above zero
-  ($\sigma_{\min} / \sigma_{\max} \ge 10^{-4}$).
-  Verified by \texttt{benchmarks/verify\_linear\_bootstrap.py}.
-
-\item \textbf{Augmented per-degree real rank}
-  (Lemma~\ref{lem:app-augmented-perdegree}):
-  For each $\ell \in \{4, \ldots, 30\}$, the full augmented system
-  (bispectral + CG power) has \emph{real} Jacobian rank $2\ell+1$
-  at the witness.
-\end{enumerate}
-
-\begin{remark}[Exact vs.\ floating-point certificates]
-Certificate~1 uses exact rational arithmetic (no numerical error).
-Certificates~2--3 use IEEE~754 double precision.
-Since the quantities being tested (solution residuals, singular values)
-are well-separated from zero (residuals $\ge 10^{-2}$ for spurious
-solutions, singular values $\ge 10^{-2}$), rounding errors of order
-$10^{-15}$ do not affect the conclusions.
-Certificate~4 uses double precision with condition numbers $\le 10^4$,
-ensuring the rank determination is reliable.
-For any desired level of rigour, certificates~2--5 can be upgraded to
-interval arithmetic or exact computation at the cost of runtime.
-\end{remark}
-
-All certificates use deterministic witness points (rational or seeded
-pseudo-random), making them fully reproducible.
-The algebraic argument (``nonzero polynomial at one point implies nonzero
-polynomial generically'') upgrades each pointwise certificate to a
-generic (measure-zero exceptional set) statement.
-
-\subsection{Comparison with existing results}
-
-\begin{table}[h]
-  \caption{Comparison of bispectral invariants for signals on $S^2$ or
-    finite groups.}
-  \label{tab:app-comparison}
-  \centering
-  \small
-  \setlength{\tabcolsep}{4pt}
-  \begin{tabular}{@{}l l c c@{}}
-    \toprule
-    Result & Invariant & Size & Complete? \\
-    \midrule
-    Kakarala~\citep{kakarala2012bispectrum}
-      & Full bispec. & $O(L^3)$ & Yes (constr.) \\
-    Edidin \& Satriano~\citep{edidin2024orbit}
-      & Full bispec.\ (band-lim.)
-      & $O(L^3)$ & Yes (generic) \\
-    Bendory et al.~\citep{bendory2025orbit}
-      & Freq.\ marching
-      & $O(L^2)$/shell & Yes (3~shells) \\
-    Mataigne et al.~\citep{mataigne2024selective}
-      & Sel.\ $G$-bispec.\ (finite)
-      & $O(|G|)$ & Yes (exact) \\
-    \textbf{This work}
-      & \textbf{Sel.\ $\SO(3)$ on $S^2$}
-      & $\boldsymbol{O(L^2)}$ & \textbf{Yes (generic)} \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\subsection{Complexity in pixel space}
-
-On $S^2$, a band-limited signal at degree~$L$ is sampled on an
-equiangular grid of size $N_\theta \times N_\varphi \approx 2L \times 2L$,
-so $P = 4L^2$ and $L = \Theta(\sqrt{P})$.
-
-\paragraph{Output size.}
-\begin{center}
-\small
-\begin{tabular}{lcc}
-  \toprule
-  Invariant & In $L$ & In $P$ \\
-  \midrule
-  Power spectrum & $O(L)$ & $O(\sqrt{P})$ \\
-  \textbf{Selective bispectrum} & $\boldsymbol{O(L^2)}$
-    & $\boldsymbol{O(P)}$ \\
-  Full bispectrum & $O(L^3)$ & $O(P^{3/2})$ \\
-  \bottomrule
-\end{tabular}
-\end{center}
-The selective bispectrum is \emph{linear} in the number of pixels.
-
-\paragraph{Computation cost.}
-\begin{center}
-\small
-\begin{tabular}{lcccc}
-  \toprule
-  Variant & Entries & Cost/entry & Total & In $P$ \\
-  \midrule
-  Full & $O(L^3)$ & $O(L)$ & $O(L^4)$ & $O(P^2)$ \\
-  Selective & $O(L^2)$ & $O(L)$ & $O(L^3)$ & $O(P^{3/2})$ \\
-  \bottomrule
-\end{tabular}
-\end{center}
-
-\paragraph{Concrete example.}
-For a $128 \times 256$ grid ($P \approx 32\mathrm{K}$, $L = 64$): the
-full bispectrum produces ${\sim}87\mathrm{K}$ scalars at
-${\sim}17\mathrm{M}$~ops; the selective bispectrum produces
-${\sim}4\mathrm{K}$ scalars at ${\sim}260\mathrm{K}$~ops---a
-$65\times$ reduction.
-
-\subsection{Relation to the inversion gap}
-
-For finite groups ($O$, $D_n$), the ``inversion gap'' arises when the
-bootstrap ambiguity $Q$ is not absorbed by the Clebsch--Gordan
-structure (see Appendix~\ref{app:octa-convergence}). For $\SO(3)$ on
-$S^2$, the situation differs fundamentally:
-\begin{enumerate}[nosep]
-  \item The ambiguity in $\mathbf{F}_1$ is an $\SO(3)$ rotation $g$
-    (3~DOF).
-  \item We fix $g$ explicitly by gauge-choosing $\mathbf{F}_1$'s
-    direction (2~DOF) and $\mathbf{F}_2$'s azimuthal phase (1~DOF).
-  \item At $\ell \ge 4$, no ambiguity remains---the linear system has
-    a unique solution.
-\end{enumerate}
-This avoids the finite-group problem entirely: we never need the CG
-matrices to ``absorb'' a continuous ambiguity.
-
-\subsection{Limitations and open questions}
-
-\begin{enumerate}
-  \item \textbf{Genericity assumption.}
-    The proof requires $f$ to avoid a measure-zero algebraic variety
-    (Assumption~\ref{ass:app-generic}). Characterising this variety
-    explicitly would strengthen the result.
-
-  \item \textbf{Numerical conditioning.}
-    The coefficient matrix $A$ at each degree~$\ell$ has a condition
-    number that depends on the signal's spectral content. A stability
-    analysis bounding $\kappa(A)$ in terms of the spectral decay rate
-    is an open problem.
-
-  \item \textbf{Inversion algorithm.}
-    The forward selective bispectrum is implemented. Inversion requires:
-    $\ell = 0, 1$: closed-form; $\ell = 2, 3$: nonlinear solve;
-    $\ell \ge 4$: linear solve (pseudoinverse).
-\end{enumerate}
-
-\section{Octahedral Inversion Gap -- Full Analysis}
-\label{app:octa-convergence}
-
-This appendix provides the complete analysis of why the standard
-bootstrap inversion algorithm fails for the octahedral group, and how
-we resolve it. The material is complementary to
-Section~\ref{sec:octa} in the main text.
-
-\subsection{The bootstrap strategy (recap)}
-
-All inversion algorithms in \citet{mataigne2024selective} share the
-same skeleton:
-\begin{enumerate}[nosep]
-  \item Recover $\hat{f}(\rho_0)$ from
-    $\beta_{\rho_0,\rho_0} = \hat{f}(\rho_0)^3$.
-  \item Recover $\hat{f}(\rho_1)$ from
-    $\beta_{\rho_0,\rho_1}/\hat{f}(\rho_0)
-    = \hat{f}(\rho_1)^\dagger \hat{f}(\rho_1)$,
-    up to a unitary/orthogonal ambiguity $Q$.
-  \item Extract higher Fourier coefficients from subsequent bispectral
-    entries using CG matrices to block-diagonalise tensor products.
-\end{enumerate}
-Step 2 always introduces an ambiguity:
-$\hat{f}(\rho_1) = S \cdot Q$ where
-$S = (\hat{f}(\rho_1)^\dagger \hat{f}(\rho_1))^{1/2}$ is determined,
-but $Q$ is a free orthogonal matrix. For the algorithm to work, this
-ambiguity must not corrupt step~3.
-
-\subsection{Why it works for $C_n$}
-
-The irreps are 1D: $\rho_k(g) = e^{2\pi i k g / n}$. The ambiguity in
-$\hat{f}(\rho_1)$ is a phase $e^{i\varphi}$,
-$\varphi \in [0, 2\pi)$. Step~3 computes:
-\[
-  \hat{f}(\rho_{k+1}) = \left(\frac{\beta_{\rho_1, \rho_k}}{
-    \hat{f}(\rho_1) \hat{f}(\rho_k)}\right)^\dagger.
-\]
-The phase $e^{i\varphi}$ propagates coherently through all coefficients
-as $e^{i\varphi k}$, corresponding to a cyclic shift
-$\Theta(g) \to \Theta(g + \varphi)$. The paper correctly resolves this
-by finding the unique $\varphi$ that makes the reconstructed signal
-real.
-
-\subsection{Why it works for $D_n$}
-
-The generating irrep $\rho_1$ is 2D. Recovery gives
-$\hat{f}(\rho_1) = V\Lambda^{1/2}V^\dagger U$ with
-$U \in \OG(2)$. Step~3 extracts higher coefficients via:
-\[
-  \bigoplus_{\rho \in \rho_1 \otimes \rho_{k-1}} \hat{f}(\rho)
-  = \left(C_{\rho_1,\rho_{k-1}}^\dagger
-    \left[\hat{f}(\rho_1) \otimes \hat{f}(\rho_{k-1})\right]^{-1}
-    \beta_{\rho_1,\rho_{k-1}}
-    C_{\rho_1,\rho_{k-1}}\right)^\dagger.
-\]
-The 2D irreps of $D_n$ are restrictions of $\SO(2)$ representations
-to $\mathbb{Z}/n\mathbb{Z}$. The CG matrices for $D_n$ therefore
-block-diagonalise $\rho(U)$ for \emph{all} $U \in \OG(2)$, not just
-$U \in D_n$. Concretely, $C^\dagger(U \otimes U)C$ remains
-block-diagonal for any $U \in \OG(2)$. The $\OG(2)$ ambiguity passes
-cleanly through the CG extraction.
-
-\subsection{Why it fails for $O$}
-
-For the octahedral group, $\rho_1$ is the standard 3D representation.
-From $\beta_{\rho_0, \rho_1}$ we recover
-$\hat{f}(\rho_1)^\top \hat{f}(\rho_1) =
-\beta_{\rho_0,\rho_1}/\hat{f}(\rho_0)$.
-Taking the symmetric square root gives
-$S = (\hat{f}(\rho_1)^\top \hat{f}(\rho_1))^{1/2}$, with
-$\hat{f}(\rho_1) = Q \cdot S$ for unknown $Q \in \SO(3)$.
-
-The next step extracts $\hat{f}(\rho_2)$ and $\hat{f}(\rho_3)$
-from $\beta_{\rho_1, \rho_1}$ using the CG matrix $C_{11}$ that
-block-diagonalises $\rho_1 \otimes \rho_1$:
-\[
-  C_{11}^\top \cdot \beta_{\rho_1,\rho_1} \cdot (S \otimes S)^{-1}
-  \cdot C_{11}
-  = C_{11}^\top (Q \otimes Q) C_{11}
-  \cdot \bigoplus_k \hat{f}(\rho_k)^\top.
-\]
-
-The CG matrix $C_{11}$ block-diagonalises
-$\rho_1(g) \otimes \rho_1(g)$ for $g \in O$:
-\[
-  C_{11}^\top (\rho_1(g) \otimes \rho_1(g)) C_{11}
-  = \rho_0(g) \oplus \rho_1(g) \oplus \rho_2(g) \oplus \rho_3(g)
-\]
-with dimensions $1 + 3 + 3 + 2 = 9$. But $Q$ is \textbf{not}
-constrained to be an element of $O$. It is an arbitrary element of
-$\SO(3)$, which is a continuous group with 3 parameters, while $O$ has
-only 24 elements. For a generic $Q \in \SO(3) \setminus O$, the matrix
-$C_{11}^\top (Q \otimes Q) C_{11}$ is \textbf{not block-diagonal}.
-
-\paragraph{Cross-block leakage.}
-The Kronecker decomposition of $\rho_1 \otimes \rho_1$ for $O$ gives
-blocks:
-
-\begin{center}
-\small
-\begin{tabular}{llcl}
-  \toprule
-  Block & Rows & Irrep & Dim \\
-  \midrule
-  1 & 0     & $\rho_0$ (trivial)     & 1 \\
-  2 & 1--3  & $\rho_1$ (standard 3D) & 3 \\
-  3 & 4--6  & $\rho_2$ (product 3D)  & 3 \\
-  4 & 7--8  & $\rho_3$ (2D)          & 2 \\
-  \bottomrule
-\end{tabular}
-\end{center}
-
-When $Q = \rho_1(g)$ for $g \in O$, the off-diagonal blocks (e.g.,
-rows 4--6 $\times$ columns 7--8) are zero---exactly what the CG
-matrix guarantees. When $Q$ is a generic rotation, e.g.,
-$Q = R_x(0.7)$, then
-$\abs{[C_{11}^\top (Q \otimes Q) C_{11}]_{4:7,\, 7:9}}_\infty
-\approx 0.788$.
-This is not a small perturbation---it is $O(1)$. The recovered
-$\hat{f}(\rho_2)$ is a mixture of $\rho_2$ and $\rho_3$ contributions.
-
-\paragraph{Why this doesn't happen for $D_n$.}
-The 2D irreps of $D_n$ are restrictions of $\SO(2)$ irreps. The CG
-decomposition of $\rho_j \otimes \rho_k$ for $D_n$ inherits from the
-continuous group: $\rho_j \otimes \rho_k = \rho_{j+k} \oplus
-\rho_{|j-k|}$ holds for all of $\SO(2)$, not just $D_n$. The CG
-matrices are inherited from $\SO(2)$ and respect arbitrary rotations.
-For $O \subset \SO(3)$, the irreps of $O$ are \emph{not} restrictions
-of $\SO(3)$ irreps (except $\rho_1$), so the CG matrices are specific
-to the 24-element group and do not block-diagonalise arbitrary
-$\SO(3)$ rotations.
-
-\subsection{Connection to the paper's own remarks}
-
-\citet{mataigne2024selective} acknowledge a related issue in the
-commutative group setting (Appendix C, around Eq.~10):
-``\emph{Again, the indeterminacy factor $h_l$ is not restricted to
-$\mathbb{Z}/n_l\mathbb{Z}$ but can belong to $[0, n_l]$. We will
-have to solve this issue further.}''
-This is the same phenomenon in a different guise. For commutative
-groups, the continuous indeterminacy is scalar (a phase) and can be
-resolved by enforcing real-valuedness. For $O$, the continuous
-indeterminacy is a $3 \times 3$ rotation matrix $Q \in \SO(3)$, and
-there is no analogous closed-form trick.
-
-\subsection{Working alternative}
-
-The selective $O$-bispectrum \emph{is} complete (172 coefficients
-uniquely determine the signal up to $O$-action). The issue is purely
-algorithmic. We implement a working inversion using:
-\begin{enumerate}[nosep]
-  \item \textbf{Bootstrap initialisation}: recover $\hat{f}(\rho_0)$
-    exactly, $\hat{f}(\rho_1)$ up to $Q$, and approximate
-    $\hat{f}(\rho_2), \hat{f}(\rho_3), \hat{f}(\rho_4)$ from diagonal
-    blocks (ignoring leakage).
-  \item \textbf{Levenberg--Marquardt correction}: use forward-mode
-    autodiff through the bispectrum computation to solve
-    $\norm{\beta(f_{\mathrm{rec}})
-    - \beta_{\mathrm{target}}}^2 = 0$ via damped least-squares steps.
-    Each step is a single linear solve, not iterative optimisation.
-    Jacobians are computed via \texttt{torch.func.jacfwd} composed with
-    \texttt{torch.vmap} for batched forward-mode AD.
-  \item \textbf{Multi-start}: randomise the initial $Q$ across several
-    restarts to escape local minima.
-\end{enumerate}
-This converges reliably in $<$50 iterations and is fully
-differentiable.
-
-\subsection{Quantitative inversion results}
-
-Table~\ref{tab:inversion-stats} and
-Figure~\ref{fig:inversion-convergence} report quantitative inversion
-results over 1{,}000 random signals on the octahedral group (4 restarts,
-10 LM corrections per restart, on a single GPU).
-
-\begin{table}[h]
-  \centering
-  \caption{Octahedral bispectrum inversion statistics
-    ($N=1{,}000$, 4 restarts $\times$ 10 LM steps).}
-  \label{tab:inversion-stats}
-  \begin{tabular}{lcc}
-    \toprule
-    Metric & Median & P95 \\
-    \midrule
-    Bisp.\ residual $\|\beta(\hat{f}) - \beta^*\|$ & $5.7 \times 10^{-5}$ & $1.5 \times 10^{-4}$ \\
-    Recovery error (mod $O$) & $1.03$ & $1.17$ \\
-    Wall-clock time (ms) & 18.4 & 30.1 \\
-    \midrule
-    Success rate (error $<0.1$) & \multicolumn{2}{c}{5.7\%} \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=0.85\linewidth]{figures/inversion_convergence.pdf}
-  \caption{Bispectral residual vs.\ LM iteration for octahedral inversion.
-    The residual drops rapidly, but the signal recovery rate remains low
-    (5.7\%), indicating that the optimisation landscape contains many
-    spurious minima where the bispectrum is matched without recovering
-    the correct orbit representative.}
-  \label{fig:inversion-convergence}
-\end{figure}
-
-The LM procedure reliably drives the bispectral residual to near-zero
-(median $5.7 \times 10^{-5}$), confirming that local convergence to a
-bispectrum-matching signal is fast. However, the signal recovery rate
-(5.7\% at tolerance $<0.1$) shows that the inversion landscape has many
-spurious local minima. This reflects the inherent difficulty of the
-$O$-inversion problem noted in Section~\ref{sec:octa}: the bootstrap
-initialisation provides a poor starting point due to CG leakage, and
-the 4-restart multi-start strategy is insufficient to reliably escape
-the resulting basin structure. Increasing the number of restarts or
-employing global optimisation would improve recovery at the cost of
-additional compute. We leave such improvements to future work.
-
-\subsection{Summary}
-
-\begin{table}[h]
-  \caption{Bootstrap inversion across group families.}
-  \label{tab:app-bootstrap-summary}
-  \centering
-  \small
-  \begin{tabular}{lcccl}
-    \toprule
-    Group & Irrep dim & Ambiguity & CG absorbs? & Bootstrap? \\
-    \midrule
-    $C_n$ & 1 & $e^{i\varphi} \in \SO(2)$ & N/A (scalar) & Yes \\
-    $D_n$ & 2 & $U \in \OG(2)$ & Yes (CG from $\SO(2)$) & Yes \\
-    $O$   & 3 & $Q \in \SO(3)$ & \textbf{No} & \textbf{No} \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\subsection{Numerical disproof of CG absorption for $O$}
-
-The following code demonstrates the cross-block leakage:
-
-\begin{lstlisting}
-import torch
-from bispectrum.octa_on_octa import OctaonOcta
-
-bsp = OctaonOcta()
-C11 = bsp._cg_11.to(torch.float64)
-
-# Generic SO(3) rotation (not in O)
-theta = 0.7
-Q = torch.tensor([
-    [1.0, 0.0, 0.0],
-    [0.0, torch.cos(torch.tensor(theta)),
-     -torch.sin(torch.tensor(theta))],
-    [0.0, torch.sin(torch.tensor(theta)),
-      torch.cos(torch.tensor(theta))],
-], dtype=torch.float64)
-
-QQ = torch.kron(Q, Q)
-M = C11.T @ QQ @ C11
-
-# rho2 block: rows 4:7, rho3 block: rows 7:9
-cross = M[4:7, 7:9]
-print(f"Cross-block max: {cross.abs().max():.6f}")
-# Output: ~0.788 -- far from zero
-
-# Compare with a group element
-rho1_g4 = bsp._get_irrep_mats(1)[4].to(torch.float64)
-QQ_grp = torch.kron(rho1_g4, rho1_g4)
-M_grp = C11.T @ QQ_grp @ C11
-cross_grp = M_grp[4:7, 7:9]
-print(f"Cross-block for g in O: {cross_grp.abs().max():.6f}")
-# Output: ~0.0 (machine precision)
-\end{lstlisting}
-
-\section{Extended Benchmark Results}
-\label{app:bench}
-
-This appendix supplements the summary in Table~\ref{tab:benchmarks}
-(main text) with scaling plots produced by
-\texttt{benchmarks/benchmark.py} on a single NVIDIA H100 80\,GB GPU
-(PyTorch 2.10).
-
-\subsection{Forward-pass timing}
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/benchmarks/paper_fig2.pdf}
-  \caption{Forward-pass wall-clock time (batch\,=\,16, H100 GPU).
-    \textbf{(a)}~$C_n$: the selective pass stays flat at
-    ${\sim}$0.14\,ms regardless of $n$, while the full bispectrum
-    grows super-linearly (17\,ms at $n{=}256$).
-    \textbf{(b)}~$\mathbb{T}^2$: selective is constant at
-    ${\sim}$0.08\,ms up to $|G|{=}65{,}536$.}
-  \label{fig:forward-timing}
-\end{figure}
-
-The selective forward pass is dominated by kernel-launch overhead
-rather than arithmetic, making it effectively free relative to backbone
-computation. The full bispectrum exhibits the expected super-linear
-growth and becomes impractical beyond $|G| \approx 256$ for $C_n$.
-
-\subsection{GPU throughput and inversion}
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/benchmarks/paper_fig3.pdf}
-  \caption{\textbf{(a)}~GPU throughput (samples/s) vs.\ batch size for
-    the selective forward pass. $C_n$ and $\mathbb{T}^2$ exceed
-    $10^7$~samples/s; the octahedral module reaches
-    $5.7 \times 10^6$~samples/s. All modules scale linearly with batch
-    size.
-    \textbf{(b)}~Inversion wall-clock time. Abelian groups ($C_n$,
-    $D_n$, $\SO(2)$/disk) scale roughly linearly. The octahedral group
-    requires ${\sim}$580\,ms due to Levenberg--Marquardt optimisation
-    over 5 irreducible representations. $\SO(3)$ on $S^2$ inversion is
-    not yet implemented.}
-  \label{fig:gpu-scaling}
-\end{figure}
-
-All modules exhibit linear throughput scaling with batch size, confirming
-that the implementation is memory-bandwidth-bound rather than
-compute-bound at these scales.
-
-\section{PCam: Extended Results}
-\label{app:pcam-details}
-
-\subsection{Published baselines}
-
-Table~\ref{tab:pcam-context} places our results in the context of
-published methods on PCam and the closely related Camelyon16
-benchmark. Our models operate at 68K--920K parameters and are
-trained on only 10\% of the data.
-
-\begin{table}[h]
-  \caption{Published baselines on PatchCamelyon from
-    \citet{veeling2018rotation}, trained on 100\% data.}
-  \label{tab:pcam-context}
-  \centering
-  \small
-  \begin{tabular}{llrcc}
-    \toprule
-    Method & Reference & Params & ACC & AUC \\
-    \midrule
-    P4M-DenseNet ($C_8$)  & Veeling et al.\ 2018     & 119K & 0.898 & 0.963 \\
-    P4M-DenseNet M        & Veeling et al.\ 2018     &  19K & 0.893 & 0.958 \\
-    DenseNet (augmented)   & Veeling et al.\ 2018     & 128K & 0.881 & 0.951 \\
-    DenseNet (no aug.)     & Veeling et al.\ 2018     & 128K & 0.876 & 0.955 \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\subsection{Model variants}
-
-All equivariant models share a $C_8$-equivariant
-DenseNet~\citep{huang2017densenet} backbone with block configuration
-$(4, 4, 4)$, 24 initial channels, and compression factor 0.5,
-following the architecture of \citet{veeling2018rotation} but
-replacing the final invariant map. Six model variants are evaluated:
-
-\begin{itemize}
-  \item \textbf{Standard}: vanilla CNN with random rotation/reflection
-    augmentation and global average pooling. No group dimension,
-    so ${\sim}8\times$ fewer parameters at matched growth rate.
-  \item \textbf{NormReLU}: $C_8$-equivariant backbone with norm-based
-    nonlinearity and max pooling over the group dimension.
-  \item \textbf{Gated}: $C_8$-equivariant backbone with gated
-    nonlinearity (doubles internal channels) and group max pool.
-  \item \textbf{Fourier-ELU}: $C_8$-equivariant backbone with
-    Fourier-space ELU nonlinearity and group max pool.
-  \item \textbf{Bispectrum}: $C_8$-equivariant backbone with ReLU
-    nonlinearity and selective $G$-bispectrum via
-    \texttt{CnonCn}$(n{=}8)$ as the terminal invariant map.
-\end{itemize}
-
-\subsection{Training details}
-
-To enable a controlled Pareto comparison, we sweep the DenseNet
-growth rate $k \in \{3, 4, 6, 8, 12\}$ for equivariant models (and
-$k \in \{6, 12, 20, 30, 35\}$ for the standard CNN to cover a
-comparable parameter range), yielding models from ${\sim}30$K to
-${\sim}1.6$M parameters. All models are trained on \textbf{10\% of
-training data} (26{,}214 images) with AdamW~\citep{loshchilov2019adamw}
-(lr $= 10^{-3}$, weight decay $10^{-4}$), cosine warm restarts,
-mixed-precision training, and early stopping on validation AUC
-(patience 10, max 50 epochs).
-
-\subsection{Pareto analysis interpretation}
-
-\emph{Architectural stability} (single seed, growth-rate sweep).
-The bispectrum's AUC varies by only ${\sim}1$pp across a $12\times$
-range of model sizes (80K--920K parameters), while norm swings by
-$>$7pp. For practitioners, bispectral pooling ``just works'' without
-growth rate tuning. The multi-seed matched-parameter results
-(Table~\ref{tab:pcam-results}) confirm this: bispectrum and Fourier-ELU
-have the lowest cross-seed variance ($\pm 0.004$).
-
-\emph{Matched parameters.}
-At matched parameters (${\sim}100$K), all equivariant models outperform
-the standard CNN (AUC $0.896 \pm 0.010$, 3 seeds). Fourier-ELU leads
-slightly ($0.945 \pm 0.004$), followed by NormReLU ($0.942$), bispectrum
-and gated (both $0.941$). All equivariant methods cluster within
-${\sim}0.5$pp of each other.
-
-\emph{Scaling behavior.}
-As parameters increase from the smallest to the largest configuration,
-standard, norm, and Fourier-ELU all degrade (by 1.8, 2.6, and 0.9
-AUC points respectively), while the bispectrum improves (+0.9 points,
-reaching 0.943 at 920K params). This suggests that the bispectrum's
-complete invariant map scales more gracefully with capacity.
-
-\emph{Data scaling} (3 seeds).
-At 1\% data the bispectrum leads all methods ($0.912 \pm 0.025$),
-outperforming NormReLU ($0.873$), Fourier-ELU ($0.876$), and gate
-($0.884$). By 10\% all equivariant methods converge (${\sim}0.941$--$0.945$).
-The bispectrum's data-efficiency advantage narrows as data increases,
-consistent with the hypothesis that complete invariants substitute for
-missing training examples in the low-data regime.
-
-\section{OrganMNIST3D: Extended Results}
-\label{app:organ3d-details}
-
-\subsection{Published baselines}
-
-Table~\ref{tab:organ3d-context} situates our results against published
-methods on the same benchmark. Our models are deliberately small
-(${\leq}463$K parameters) to enable controlled ablation of the pooling
-mechanism. The relevant comparison is against equivariant methods at a
-similar scale.
-
-\begin{table}[h]
-  \caption{Published baselines on OrganMNIST3D for context.
-    Accuracy differences with respect to large-scale models reflect the
-    deliberate parameter constraint of our ablation study.}
-  \label{tab:organ3d-context}
-  \centering
-  \small
-  \begin{tabular}{llrcc}
-    \toprule
-    Method & Reference & Params & ACC & AUC \\
-    \midrule
-    ResNet-18 + 3D           & Yang et al.\ 2023         & 33M  & 0.907 & 0.996 \\
-    ResNet-18 + ACS          & Yang et al.\ 2023         & 11M  & 0.900 & 0.994 \\
-    ILPOResNet-50            & \citet{zhemchuzhnikov2024ilponet}       & 38K  & 0.879 & 0.992 \\
-    EquiLoPO (local train.)  & \citet{zhemchuzhnikov2025equilopo}      & 418K & 0.866 & 0.991 \\
-    SE3MovFrNet              & \citet{sangalli2023movfrnet}     & ---  & 0.745 & ---   \\
-    Regular SE(3) conv       & \citet{kuipers2023regular}   & 172K & 0.698 & ---   \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\subsection{Rotation analysis}
-
-Figure~\ref{fig:organ3d-rotation} shows that all equivariant models
-achieve identical performance on the original and octahedrally rotated
-test sets ($\sigma_{\mathrm{rot}} \approx 0$), while the
-augmentation-trained standard CNN degrades under rotation.
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=0.75\linewidth]{experiments/organ3d/organ3d_analysis.pdf}
-  \caption{OrganMNIST3D test accuracy on the original and octahedrally
-    rotated test sets. Error bars show standard deviation over three
-    seeds. All equivariant models achieve identical performance under
-    rotation ($\sigma_{\mathrm{rot}} \approx 0$), while the
-    augmentation-trained standard CNN degrades.}
-  \label{fig:organ3d-rotation}
-\end{figure}
-
-\subsection{Wider channels}
-
-To test whether the advantage of bispectral completeness persists at
-higher capacity, we run max pool and bispectrum at channel widths
-$(8, 16)$ and $(16, 32)$ with full training data.
-
-\begin{table}[h]
-  \caption{OrganMNIST3D accuracy at wider channel configurations
-    (mean $\pm$ std over 3 seeds, full training data).}
-  \label{tab:organ3d-wider}
-  \centering
-  \small
-  \begin{tabular}{lrcc}
-    \toprule
-    Channels & Params & Max Pool & Bispectrum \\
-    \midrule
-    $(4, 8)$   & ${\sim}$375K / 463K  & $.730 \pm .033$ & $.726 \pm .027$ \\
-    $(8, 16)$  & ${\sim}$1.5M / 1.7M  & $.743 \pm .015$ & $.745 \pm .006$ \\
-    $(16, 32)$ & ${\sim}$6.0M / 6.3M  & $\mathbf{.785 \pm .032}$ & $.685 \pm .039$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-At $(8, 16)$ the two methods are equivalent ($74.5 \pm 0.6\%$ bispectrum vs.\
-$74.3 \pm 1.5\%$ max pool, both at ${\sim}1.5$--$1.7$M parameters). At $(16, 32)$ max
-pool reaches $78.5 \pm 3.2\%$ while bispectrum drops to $68.5 \pm 3.9\%$---the 6.3M
-parameter bispectrum model likely overfits the 971 training samples.
-This suggests that the bispectrum's completeness advantage is most
-pronounced in the data-limited rather than the parameter-limited
-regime, consistent with the data efficiency findings in the main text.
-
-\section{Experimental Details}
-\label{app:exp}
-
-\subsection{Hyperparameters}
-
-Table~\ref{tab:hyperparams} summarises the training hyperparameters
-for all three experimental settings.
-
-\begin{table}[h]
-  \caption{Training hyperparameters across experiments.}
-  \label{tab:hyperparams}
-  \centering
-  \small
-  \setlength{\tabcolsep}{4pt}
-  \begin{tabular}{lccc}
-    \toprule
-    & PCam & OrganMNIST3D & Sph.\ MNIST \\
-    \midrule
-    Optimizer      & AdamW & AdamW & AdamW \\
-    Learning rate  & $10^{-3}$ & $10^{-3}$ & $10^{-3}$ \\
-    Weight decay   & $10^{-4}$ & $10^{-4}$ & $10^{-4}$ \\
-    Batch size     & 64--256 & 16--64 & 256 \\
-    Scheduler      & Cosine warm restarts & --- & --- \\
-    Max epochs     & 50 & 100 & 50 \\
-    Early stopping & Val AUC, pat.\ 10 & Val AUC, pat.\ 15 & Val ACC, pat.\ 10 \\
-    Seeds          & 42, 123, 456$^\dagger$ & 42, 123, 456 & 42, 123, 456 \\
-    Mixed precision & Yes & No & No \\
-    \bottomrule
-  \end{tabular}
-
-  \vspace{0.3em}
-  {\small $^\dagger$Pareto figure uses seed 42; matched-parameter table and data efficiency use seeds 42, 123, 456.}
-\end{table}
-
-\subsection{Hardware}
-
-All experiments were run on a single NVIDIA H100 GPU (80\,GB).
-Training wall-clock times range from ${\sim}$5 minutes (OrganMNIST3D,
-971 samples) to ${\sim}$20 minutes (PCam, 26K samples at 10\% data).
-Spherical MNIST bispectrum computation (forward pass through
-\texttt{SO3onS2} at $\ell_{\max} = 15$) takes ${\sim}$0.3s per batch
-of 64 on GPU.
-
-\subsection{Data preprocessing}
-
-\paragraph{PCam.}
-PatchCamelyon~\citep{veeling2018rotation} consists of 327{,}680 RGB
-patches of size $96 \times 96$ pixels extracted from whole-slide
-images of lymph node sections~\citep{bejnordi2017camelyon}. We use
-the standard train/val/test split. Images are loaded from HDF5 files
-and normalised to $[0, 1]$. The standard (non-equivariant) baseline
-uses random rotation and reflection augmentation during training; all
-equivariant models are trained without augmentation.
-
-\paragraph{OrganMNIST3D.}
-OrganMNIST3D~\citep{medmnistv2} contains 1{,}742 abdominal CT organ
-volumes of size $28 \times 28 \times 28$ voxels with 11 organ classes
-(train/val/test = 971/161/610). Volumes are normalised to $[0, 1]$.
-The standard baseline uses random octahedral augmentation (one of 24
-rotations per sample); equivariant models are trained without
-augmentation. Kernel rotations are implemented as exact voxel
-permutations on the $3^3$ grid, avoiding interpolation artifacts.
-
-\paragraph{Spherical MNIST.}
-MNIST digits are projected onto the sphere via stereographic
-projection following \citet{cohen2018spherical} and sampled on a
-$64 \times 128$ equiangular grid. The selective bispectrum is
-computed at $\ell_{\max} = 15$, producing 384 complex invariant
-coefficients (768 real features after taking log-magnitude), which
-are fed to a 3-layer MLP: Linear(768, 256)--BN--ReLU--Linear(256,
-128)--BN--ReLU--Linear(128, 10). The power spectrum MLP uses the
-same structure with hidden=128: Linear(16, 128)--BN--ReLU--Linear(128,
-64)--BN--ReLU--Linear(64, 10). The standard CNN has 4 convolutional
-layers (16--32--64--64 channels) followed by adaptive average pooling
-and a Linear(1024, 128)--ReLU--Linear(128, 10) classifier.
-
-\subsection{Aggregation procedures}
-
-\paragraph{PCam.}
-The Pareto figure (Figure~\ref{fig:pcam-pareto}) uses a single seed
-(42) across the full growth-rate sweep. The matched-parameter table
-(Table~\ref{tab:pcam-results}) and data efficiency analysis use 3 seeds
-(42, 123, 456) per configuration.
-
-\paragraph{OrganMNIST3D.}
-All results in Table~\ref{tab:organ3d-results} report mean $\pm$ std
-over 3 seeds (42, 123, 456). Rotation robustness
-($\sigma_{\mathrm{rot}}$) is the standard deviation of test accuracy
-across all 24 octahedral rotations of the test set. The data
-efficiency analysis (Figure~\ref{fig:data-efficiency} in the main text) uses 3 seeds per data
-fraction.
-
-\paragraph{Spherical MNIST.}
-All results in Table~\ref{tab:smnist-results} report mean $\pm$ std
-over 3 seeds (42, 123, 456). Rotation robustness
-($\sigma_{\mathrm{rot}}$) is measured by applying 10 random $\SO(3)$
-rotations to the entire test set and computing the standard deviation
-of accuracy, yielding 30 measurements per model (10 rotations
-$\times$ 3 seeds).
-
-\subsection{Dataset licenses}
-
-\begin{itemize}[nosep]
-  \item \textbf{PCam}: CC0 1.0 Universal (public domain).
-  \item \textbf{OrganMNIST3D}: CC BY 4.0.
-  \item \textbf{Spherical MNIST}: derived from MNIST, which is
-    publicly available for research use.
-\end{itemize}
-
-\section{API Example}
-\label{app:api-example}
-
-\begin{lstlisting}
-import torch
-from bispectrum import CnonCn, SO2onDisk, OctaonOcta
-
-bsp = CnonCn(n=8, selective=True)
-f = torch.randn(16, 8)
-beta = bsp(f)
-f_rec = bsp.invert(beta)
-\end{lstlisting}
-
-\newpage
-\input{checklist.tex}
-
-\end{document}
diff --git a/paper/overleaf_comments.md b/paper/overleaf_comments.md
deleted file mode 100644
index 8ccfa39..0000000
--- a/paper/overleaf_comments.md
+++ /dev/null
@@ -1,1332 +0,0 @@
-# Overleaf Comments — bispectrum paper
-
-**Project:** https://www.overleaf.com/project/69dd6ef85579b7e8d7426177  
-**Paper title:** bispectrum: Selective G-Bispectra Made Practical  
-**Total threads:** 54  
-**Pulled:** 2026-04-18  
-
-## Notes on this dump
-
-- Anchors are taken from the **local** `paper/paper.tex` using the character offsets reported by Overleaf. Local and Overleaf tex differ by only ~524 chars in a 105k-char document, so anchor windows will be within a few chars of the exact cursor position.
-- Each thread shows: author, timestamp (UTC), full message + any replies, section/subsection context, tex line number, and the surrounding text.
-- `|@|` in the anchor window marks the exact Overleaf cursor position.
-- A first-pass classification is included for each thread — this is Claude's initial read and should be reviewed.
-
-## Reviewer breakdown
-
-- **simon.mataigne** (simonmataigne@gmail.com) — 30 threads
-- **Adele Myers** (adele@ucsb.edu) — 24 threads
-
-## Threads (in document order)
-
-### § (preamble / front-matter)
-
-#### 1. @simon.mataigne — 2026-04-17 14:23 UTC 
-
-- **Location:** line ~63 (pos 1454), section: _—_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-stringstyle=\color{black!60},   frame=single,   breaklines=true,   columns=fullflexible, }  \title{ |@| \texttt{bispectrum}: Selective $G$-Bispectra Made Practical}  \author{%   Johan Mathe \\   Atmo, Inc. \\   \texttt{johan@atmo.ai} \\   \And   Simon Mataigne \\   UCLouvain \\   \texttt{simon.mataigne@
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-17 14:23 UTC_:
-
-> This paper is a banger.
-
-**Claude's first-pass classification:** already-addressed (not a change request — positive comment; no edit needed)
-
-**Proposed response:** "Thanks Simon! Resolving."
-
----
-
-#### 2. @simon.mataigne — 2026-04-17 15:03 UTC 
-
-- **Location:** line ~105 (pos 2982), section: _—_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-inite groups. Code is available at \url{https://github.com/geometric-intelligence/bispectrum}. \end{ |@| abstract} \section{Introduction} \label{sec:intro}  Equivariant neural networks~\citep{cohen2016group, weiler2019general} enforce symmetry by construction, but they still require an invariant represen
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-17 15:03 UTC_:
-
-> The github url is not anonymous.
-
-**Claude's first-pass classification:** blocker
-
-**Proposed response:** "Good catch — this is a real issue for double-blind review. The URL `https://github.com/geometric-intelligence/bispectrum` identifies the `geometric-intelligence` organization and is not anonymous. We need to replace it with an anonymous repository link (e.g., Anonymous GitHub) before submission, or remove the URL entirely. Flagging for Johan to resolve before the camera-ready or submission deadline."
-
----
-
-#### ‣ Introduction
-
-#### 3. @simon.mataigne — 2026-04-15 08:57 UTC 
-
-- **Location:** line ~111 (pos 3198), section: _Introduction_ · tid `…000001`
-- **Messages:** 2 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-eral} enforce symmetry by construction, but they still require an invariant representation at the po |@| int where a prediction must no longer depend on orientation. In practice, invariant pooling remains a weak point: standard choices such as norm pooling, gate-based pooling, or max/sum aggregation are
-```
-
-**Thread:**
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-- **simon.mataigne** · _2026-04-15 08:57 UTC_:
-
-> Maybe precise symmetry of what. Symmetry of the output with respect to symmetries of the input.
-
-- **simon.mataigne** · _2026-04-15 09:04 UTC_:
-
-> Moreover, I would use "group symmetries" instead.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Good point — the symmetry referent is implicit. Updated the phrase: 'enforce group symmetry of the output with respect to group transformations of the input by construction.' This makes both the type (group symmetry) and the direction (output responds to input transformations) explicit."
-
----
-
-#### 4. @simon.mataigne — 2026-04-15 09:01 UTC 
-
-- **Location:** line ~111 (pos 3233), section: _Introduction_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-ion, but they still require an invariant representation at the point where a prediction must no long |@| er depend on orientation. In practice, invariant pooling remains a weak point: standard choices such as norm pooling, gate-based pooling, or max/sum aggregation are usable~\citep{weiler2019general}, b
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 09:01 UTC_:
-
-> No "still" here?
-
-**Claude's first-pass classification:** nit
-
-**Proposed response:** "Agreed — 'still' is slightly redundant here. Removed."
-
----
-
-#### 5. @simon.mataigne — 2026-04-15 09:04 UTC 
-
-- **Location:** line ~111 (pos 3260), section: _Introduction_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-an invariant representation at the point where a prediction must no longer depend on orientation. I |@| n practice, invariant pooling remains a weak point: standard choices such as norm pooling, gate-based pooling, or max/sum aggregation are usable~\citep{weiler2019general}, but they do not preserve all
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 09:04 UTC_:
-
-> "group representation"
-
-**Claude's first-pass classification:** nit
-
-**Proposed response:** "Good suggestion — 'orientation' is domain-specific; changed to 'group representation' for generality."
-
----
-
-#### 6. @simon.mataigne — 2026-04-15 08:59 UTC 
-
-- **Location:** line ~112 (pos 3331), section: _Introduction_ · tid `…000001`
-- **Messages:** 2 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-nger depend on orientation. In practice, invariant pooling remains a weak point: standard choices su |@| ch as norm pooling, gate-based pooling, or max/sum aggregation are usable~\citep{weiler2019general}, but they do not preserve all equivariant information. The bispectrum offers a mathematically princi
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 08:59 UTC_:
-
-> Maybe "orientation" is already to specific in the context of general group actions.
-
-- **simon.mataigne** · _2026-04-15 09:03 UTC_:
-
-> Maybe "group transformations" instead?
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Agreed — 'orientation' implicitly assumes SO(3)/rotation. Changed to 'group transformations' throughout the paragraph where used generically."
-
----
-
-#### 7. @Adele Myers — 2026-04-16 23:55 UTC 
-
-- **Location:** line ~115 (pos 3547), section: _Introduction_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-preserve all equivariant information. The bispectrum offers a mathematically principled alternative |@| , since for many groups it yields exact invariants with reconstruction guarantees~\citep{kakarala2012bispectrum}.  The obstacle is practical rather than conceptual. Existing bispectral constructions a
-```
-
-**Thread:**
-
-- **Adele Myers** · _2026-04-16 23:55 UTC_:
-
-> maybe change to "all signal information"? One could say the bispectrum also does not preserve "equivariant" information, since it does not encode global phase, whereas equivariant representations do.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Good technical point — 'equivariant information' is indeed imprecise here since the bispectrum is an invariant map. Updated to: 'they do not preserve all signal information, collapsing distinct orbits.' This is accurate: the bispectrum preserves orbit-separating information."
-
----
-
-#### 8. @Adele Myers — 2026-04-16 23:56 UTC 
-
-- **Location:** line ~119 (pos 3704), section: _Introduction_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-econstruction guarantees~\citep{kakarala2012bispectrum}.  The obstacle is practical rather than conc |@| eptual. Existing bispectral constructions are scattered across theory papers, domain-specific derivations, and small implementations, with no general-purpose PyTorch library covering the finite-group
-```
-
-**Thread:**
-
-- **Adele Myers** · _2026-04-16 23:56 UTC_:
-
-> I would cite the original bispectrum inversion paper here by giannakis.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Agreed — adding Giannakis 1989 (the original inversion result) alongside kakarala2012bispectrum here: '...exact invariants with reconstruction guarantees~\citep{giannakis1989bispectrum, kakarala2012bispectrum}.' Need to add the Giannakis 1989 bibtex entry to references.bib."
-
----
-
-#### 9. @simon.mataigne — 2026-04-15 13:50 UTC 
-
-- **Location:** line ~124 (pos 4066), section: _Introduction_ · tid `…000001`
-- **Messages:** 2 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-ning. Even for groups where selectivity reduces the coefficient count from $O(|G|^2)$ to $O(|G|)$~\c |@| itep{mataigne2024selective}, a user still needs a numerically stable implementation, a consistent API, inversion routines, tests, and benchmarked runtime behavior.  This paper presents \texttt{bispect
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 13:50 UTC_:
-
-> This sentence should be used to say "Even for a group $G$,..."
-
-- **simon.mataigne** · _2026-04-15 13:50 UTC_:
-
-> So that $G$ is defined before being used.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Right — $G$ is used without introduction in this sentence. Updated to: 'Even for a compact group $G$ where selectivity reduces the coefficient count from $O(|G|^2)$ to $O(|G|)$~\citep{mataigne2024selective}, a user still needs...'"
-
----
-
-#### 10. @simon.mataigne — 2026-04-15 09:08 UTC 
-
-- **Location:** line ~124 (pos 4100), section: _Introduction_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-ivity reduces the coefficient count from $O(|G|^2)$ to $O(|G|)$~\citep{mataigne2024selective}, a use |@| r still needs a numerically stable implementation, a consistent API, inversion routines, tests, and benchmarked runtime behavior.  This paper presents \texttt{bispectrum}, a PyTorch~\citep{paszke2019p
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 09:08 UTC_:
-
-> And the computational cost from$O(|G|^3)$ to $O(|G|\log(|G|)$ or $O(|G|^2)$.
-> This point seems even more important to recall.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Good point — we currently only mention the coefficient count reduction; the computational cost reduction is even more dramatic. Updated: 'selectivity reduces the coefficient count from $O(|G|^2)$ to $O(|G|)$ and the computational cost from $O(|G|^3)$ to $O(|G|^2)$~\citep{mataigne2024selective}.'"
-
----
-
-#### 11. @Adele Myers — 2026-04-17 00:01 UTC 
-
-- **Location:** line ~125 (pos 4181), section: _Introduction_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-24selective}, a user still needs a numerically stable implementation, a consistent API, inversion ro |@| utines, tests, and benchmarked runtime behavior.  This paper presents \texttt{bispectrum}, a PyTorch~\citep{paszke2019pytorch} library for computing bispectral invariants and, where available, their i
-```
-
-**Thread:**
-
-- **Adele Myers** · _2026-04-17 00:01 UTC_:
-
-> this sentence feels like a non sequitur and should be combined with the previous sentence, as it does not say anything new. Specifically highlighting the selective g bispectrum here feels out of place. It can be mentioned later, along with all the other bispectra.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Agreed — the paragraph introducing the library currently feels disconnected. We'll fold the library introduction into the preceding paragraph to flow naturally from the problem statement, and defer the enumeration of specific bispectra to the contributions paragraph."
-
----
-
-#### 12. @simon.mataigne — 2026-04-15 09:11 UTC 
-
-- **Location:** line ~131 (pos 4515), section: _Introduction_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-ic, dihedral, torus, disk, sphere, and octahedral settings as differentiable \texttt{nn.Module}s int |@| ended to plug into standard ML workflows. Beyond the software contribution, the paper makes one technical contribution that arose during implementation: an augmented selective $\SO(3)$ bispectrum on $
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 09:11 UTC_:
-
-> toroidal, discoid, spherical and... right?
-
-**Claude's first-pass classification:** nit
-
-**Proposed response:** "Fair point — using 'cyclic, dihedral, torus, disk, sphere, and octahedral' mixes terminology styles. Will either make all adjectives consistent ('cyclic, dihedral, toroidal, discoid, spherical, and octahedral') or switch to listing the domain spaces ('$C_n$, $D_n$, torus, disk, $S^2$, and octahedral group'). The latter matches the supported-modules table."
-
----
-
-#### 13. @Adele Myers — 2026-04-17 00:03 UTC 
-
-- **Location:** line ~131 (pos 4551), section: _Introduction_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-nd octahedral settings as differentiable \texttt{nn.Module}s intended to plug into standard ML workf |@| lows. Beyond the software contribution, the paper makes one technical contribution that arose during implementation: an augmented selective $\SO(3)$ bispectrum on $S^2$ with $O(L^2)$ coefficients (Sec
-```
-
-**Thread:**
-
-- **Adele Myers** · _2026-04-17 00:03 UTC_:
-
-> this is vague. I would say something like "library supports group invariance for functions defined on the following spaces: "
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Good suggestion. Updated the sentence to: 'The library provides $G$-bispectral invariants for functions defined on the following domains: $C_n$ (cyclic), $D_n$ (dihedral), torus $\mathbb{T}^2$, disk, sphere $S^2$, and the octahedral group.' This is concrete and matches the supported-modules table."
-
----
-
-#### 14. @simon.mataigne — 2026-04-15 09:12 UTC 
-
-- **Location:** line ~137 (pos 4800), section: _Introduction_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-bispectrum on $S^2$ with $O(L^2)$ coefficients (Section~\ref{sec:so3-selective}).  \paragraph{Contri |@| butions.} \begin{enumerate}   \item \textbf{Software contribution.} A PyTorch library implementing     bispectral invariants for seven group/domain pairs with a consistent     \texttt{nn.Module}-style
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 09:12 UTC_:
-
-> the sphere $S^2$
-
-**Claude's first-pass classification:** nit
-
-**Proposed response:** "Good — 'sphere' is ambiguous as a dimension. Changed to 'the sphere $S^2$' for precision."
-
----
-
-#### 15. @simon.mataigne — 2026-04-15 09:13 UTC 
-
-- **Location:** line ~138 (pos 4810), section: _Introduction_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-on $S^2$ with $O(L^2)$ coefficients (Section~\ref{sec:so3-selective}).  \paragraph{Contributions.} |@| \begin{enumerate}   \item \textbf{Software contribution.} A PyTorch library implementing     bispectral invariants for seven group/domain pairs with a consistent     \texttt{nn.Module}-style interface
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 09:13 UTC_:
-
-> What is $L$? Do you mean $|G|$?
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Correct — $L$ is introduced here before being defined. Added a brief parenthetical: 'an augmented selective $\SO(3)$ bispectrum on $S^2$ with $O(L^2)$ coefficients (where $L$ is the spherical harmonic band-limit; Section~\ref{sec:so3-selective}).' This pre-empts the question for readers without the SO(3) background."
-
----
-
-#### 16. @simon.mataigne — 2026-04-15 09:16 UTC 
-
-- **Location:** line ~146 (pos 5270), section: _Introduction_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
-
-```tex
-ec:benchmarks}).   \item \textbf{Augmented selective $\SO(3)$ bispectrum on $S^2$.} We construct |@| and implement an augmented selective bispectrum for $\SO(3)$ acting on     $S^2$ that uses     $\Theta(L^2)$ coefficients instead of the full $\Theta(L^3)$,     matching the orbit-space lower bound u
-```
-
-**Thread:**
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-- **simon.mataigne** · _2026-04-15 09:16 UTC_:
-
-> I believe US spelling uses "optimized" with a z.
-
-**Claude's first-pass classification:** nit
-
-**Proposed response:** "The local tex at line 141 still reads 'GPU-optimised forward passes'. Addressed in v2: changed to 'GPU-optimized forward passes'."
-
-*Note: Local tex still has 'GPU-optimised' at line 141 — this is NOT yet addressed.*
-
----
-
-#### 17. @simon.mataigne — 2026-04-15 13:51 UTC 
-
-- **Location:** line ~151 (pos 5618), section: _Introduction_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
-
-```tex
-es scalar bispectral entries with     Clebsch--Gordan power spectrum entries (a degree-$4$ complemen |@| t     needed for real signals).  We give a computer-assisted completeness     proof for $4 \le L \le 100$ that combines a structural seed step     (Kakarala 1992/2012) with a uniform per-degree linear
-```
-
-**Thread:**
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-- **simon.mataigne** · _2026-04-15 13:51 UTC_:
-
-> $L$ comes back here but it should be defined.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Agreed — $L$ appears in the contributions bullet (line 148) before it is formally defined. Will add a brief in-line gloss on first use in the contributions list: 'at band-limit $L$ (the maximum spherical harmonic degree).' If we add the parenthetical to thread 15's location that covers this too; otherwise a forward reference to Section~\ref{sec:so3-selective} is sufficient."
-
----
-
-### § Mathematical Background
-
-#### ‣ Equivariant features and invariant maps
-
-#### 18. @simon.mataigne — 2026-04-15 13:53 UTC 
-
-- **Location:** line ~180 (pos 7037), section: _Mathematical Background_, subsection: _Equivariant features and invariant maps_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
-
-```tex
-eventually map equivariant features to \emph{invariant} ones.  \begin{table}[t]   \caption{Invarian |@| t nonlinearities for equivariant CNNs. The selective     bispectrum is the only method that is simultaneously exact, complete,     and linear-cost for finite groups.}   \label{tab:nonlinearity-compari
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 13:53 UTC_:
-
-> $\pi$ and $\Phi$ should be defined here.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Correct — the equivariance definition uses $\pi_{\mathrm{out}}$, $\pi_{\mathrm{in}}$, and $\Phi$ but these are not defined in the surrounding prose. Will add: 'where $\pi_{\mathrm{in}}$ and $\pi_{\mathrm{out}}$ are the group representations on input and output feature spaces respectively, and $\Phi$ is the layer map.' This is a load-bearing sentence and needs the definitions."
-
----
-
-#### 19. @simon.mataigne — 2026-04-15 09:14 UTC 
-
-- **Location:** line ~201 (pos 7883), section: _Mathematical Background_, subsection: _Equivariant features and invariant maps_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-assumption \\     \bottomrule   \end{tabular}    \vspace{0.3em}   {\small $^\ddagger$Complete pre-al |@| iasing; the pointwise nonlinearity in the forward pass introduces aliasing that may break completeness in practice.} \end{table}  Table~\ref{tab:nonlinearity-comparison} summarises the trade-offs. Nor
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 09:14 UTC_:
-
-> I would not put the "2" in the bif O cost.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Correct — the Gate row currently reads '$O(2|G|)$' in the Coefficients column. The factor of 2 is an implementation detail (scalar + gate channel), not an asymptotic distinction from $O(|G|)$. Changed to '$O(|G|)$' for consistency with standard big-O notation."
-
----
-
-#### 20. @simon.mataigne — 2026-04-15 09:15 UTC 
-
-- **Location:** line ~201 (pos 7966), section: _Mathematical Background_, subsection: _Equivariant features and invariant maps_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-r$Complete pre-aliasing; the pointwise nonlinearity in the forward pass introduces aliasing that may |@| break completeness in practice.} \end{table}  Table~\ref{tab:nonlinearity-comparison} summarises the trade-offs. Norm pooling and gated nonlinearities are exact invariants but \emph{incomplete}: they
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 09:15 UTC_:
-
-> G\log G is rather the computational cost. The DFT has |G| coefficients.
-> But it might be a plus giving both computational and storage costs.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Correct — the Fourier pointwise row in the table lists '$O(|G| \log |G|)$' in the Coefficients column, which is the computational cost, not the coefficient count (which is $O(|G|)$). Options: (a) split into separate Coefficients and Cost columns, or (b) clarify in a table footnote. Recommend splitting the column: the Coefficients column becomes '$O(|G|)$' and we add a footnote '$\dagger$Forward pass cost $O(|G| \log |G|)$ via FFT.'"
-
----
-
-#### 21. @Adele Myers — 2026-04-17 00:06 UTC 
-
-- **Location:** line ~202 (pos 8003), section: _Mathematical Background_, subsection: _Equivariant features and invariant maps_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-e nonlinearity in the forward pass introduces aliasing that may break completeness in practice.} \en |@| d{table}  Table~\ref{tab:nonlinearity-comparison} summarises the trade-offs. Norm pooling and gated nonlinearities are exact invariants but \emph{incomplete}: they discard the phase of Fourier blocks,
-```
-
-**Thread:**
-
-- **Adele Myers** · _2026-04-17 00:06 UTC_:
-
-> which bispectrum and which selective bispectrum? Do you mean the G-bispectrum?
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Good clarification point — the table caption says 'The selective bispectrum' without specifying which. Since the paper covers seven group/domain pairs, the statement is about the $G$-bispectrum in general. Updated caption: 'The selective $G$-bispectrum is the only method that is simultaneously exact, complete, and linear-cost for finite groups.' (The 'G' was already in the paper title so this is consistent.)"
-
----
-
-#### ‣ The $G$-bispectrum and selective bispectra
-
-#### 22. @simon.mataigne — 2026-04-15 13:54 UTC 
-
-- **Location:** line ~211 (pos 8450), section: _Mathematical Background_, subsection: _The $G$-bispectrum and selective bispectra_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-9general}. The full bispectrum is exact and complete but costs $O(|G|^2)$ coefficients.  \subsection |@| {The $G$-bispectrum and selective bispectra}  Let $G$ be a compact group with irreducible representations $\{\rho_i\}_{i \in \hat{G}}$ of dimensions $d_i$, and let $\hat{f}(\rho_i) = \sum_{g \in G} f(
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 13:54 UTC_:
-
-> Same question about US spelling with a z.
-
-**Claude's first-pass classification:** nit
-
-**Proposed response:** "Local tex at line 716 reads 'Our library generalises'. Changed to 'generalizes' for US spelling consistency."
-
-*Note: This comment is at pos 8450, but the actual 'generalises' instance is at line 716 in the Related Work section — Simon may have misclicked. Either way, the fix is the same: change 'generalises' → 'generalizes'.*
-
----
-
-#### 23. @simon.mataigne — 2026-04-15 13:55 UTC 
-
-- **Location:** line ~214 (pos 8601), section: _Mathematical Background_, subsection: _The $G$-bispectrum and selective bispectra_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-G$ be a compact group with irreducible representations $\{\rho_i\}_{i \in \hat{G}}$ of dimensions $d |@| _i$, and let $\hat{f}(\rho_i) = \sum_{g \in G} f(g)\, \rho_i(g)$ denote the group Fourier transform. Given the Clebsch--Gordan decomposition $\rho_i \otimes \rho_j \cong \bigoplus_k m_{ij}^k \rho_k$ w
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 13:55 UTC_:
-
-> The notion of orbit is not yet defined.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Correct — 'separates orbits' (line 228) uses 'orbit' without a definition. Added a one-sentence definition before Eq.~(1): 'The \emph{orbit} of a signal $f$ under $G$ is the set $\{f \circ g : g \in G\}$; two signals are in the same orbit iff they differ by a group transformation.' Alternatively, the definition can be embedded inline: 'it separates \emph{$G$-orbits} (equivalence classes of signals related by group action) for generic signals.'"
-
----
-
-#### 24. @Adele Myers — 2026-04-17 00:10 UTC 
-
-- **Location:** line ~218 (pos 8853), section: _Mathematical Background_, subsection: _The $G$-bispectrum and selective bispectra_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-imes \rho_j \cong \bigoplus_k m_{ij}^k \rho_k$ with intertwining matrix $C_{ij}$, the \emph{bispectr |@| um} is \begin{equation}   \label{eq:bispectrum-def}   \beta_{\rho_i, \rho_j}     = C_{ij}^\top\,       (\hat{f}(\rho_i) \otimes \hat{f}(\rho_j))\,       C_{ij}\, \hat{f}(\rho_k)^\dagger. \end{equation
-```
-
-**Thread:**
-
-- **Adele Myers** · _2026-04-17 00:10 UTC_:
-
-> I would include a separate section for bispectra of signals defined over spaces that are homogeneous for the group, and then another section for bispectra of signals defined over spaces that are NOT homogeneous for the group (as far as I know, mine is the only one). However, including mine in the G-bispectrum section is inaccurate, and other bispectra exist that are not G-bispectra.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "This is a valid taxonomic point — the disk bispectrum (myers2025selective) acts on a non-homogeneous space and is structurally different from the standard $G$-bispectrum. However, splitting the Background into homogeneous/non-homogeneous sections would substantially restructure a 9-page paper. Our compromise: add a paragraph note in the Background subsection that explicitly flags the distinction: 'The disk bispectrum of \citet{myers2025selective} operates on a domain that is \emph{not} a homogeneous $G$-space; it is included in the library but is taxonomically distinct from the $G$-bispectra above.' This acknowledges the distinction without a full structural overhaul."
-
----
-
-#### 25. @simon.mataigne — 2026-04-15 13:56 UTC 
-
-- **Location:** line ~223 (pos 8972), section: _Mathematical Background_, subsection: _The $G$-bispectrum and selective bispectra_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-ion}   \label{eq:bispectrum-def}   \beta_{\rho_i, \rho_j}     = C_{ij}^\top\,       (\hat{f}(\rho_i) |@| \otimes \hat{f}(\rho_j))\,       C_{ij}\, \hat{f}(\rho_k)^\dagger. \end{equation} Each $\beta_{\rho_i,\rho_j}$ is invariant under $f \mapsto f \circ g$ and, taken over all pairs $(i,j)$, the bispectr
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 13:56 UTC_:
-
-> What is $\hat{G}$?
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Good catch — $\hat{G}$ (the Pontryagin dual / set of irreps) is used at line 214 and again at line 232 but never explicitly defined. Will add: 'where $\hat{G}$ denotes the set of equivalence classes of irreducible representations of $G$' immediately after its first appearance. For compact groups this is standard notation but the paper should define it."
-
----
-
-#### 26. @simon.mataigne — 2026-04-15 13:58 UTC 
-
-- **Location:** line ~226 (pos 9074), section: _Mathematical Background_, subsection: _The $G$-bispectrum and selective bispectra_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-otimes \hat{f}(\rho_j))\,       C_{ij}\, \hat{f}(\rho_k)^\dagger. \end{equation} Each $\beta_{\rho_i |@| ,\rho_j}$ is invariant under $f \mapsto f \circ g$ and, taken over all pairs $(i,j)$, the bispectrum is a \emph{complete} invariant: it separates orbits for generic signals~\citep{kakarala2012bispectr
-```
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-**Thread:**
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-- **simon.mataigne** · _2026-04-15 13:58 UTC_:
-
-> It would be good pointing to  a reference here, e.g., Diaconis & Rockmore: Efficient computation of the Fourier Transform for finite groups.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Good suggestion — adding Diaconis & Rockmore (1990) as a reference for the group Fourier transform and its efficient computation, alongside the existing kakarala2012bispectrum citation. Need to add the bibtex entry for 'diaconis1990fourier' (Diaconis & Rockmore, JACM 1990)."
-
----
-
-#### 27. @simon.mataigne — 2026-04-15 13:59 UTC 
-
-- **Location:** line ~232 (pos 9453), section: _Mathematical Background_, subsection: _The $G$-bispectrum and selective bispectra_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
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-```tex
-tra.} \citet{mataigne2024selective} show that for finite groups, a BFS traversal of the Kronecker pr |@| oduct table of $\hat{G}$ selects $O(|G|)$ bispectral coefficients that remain complete; \citet{myers2025selective} extend this to the disk. This reduction from quadratic to linear is the key to practi
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 13:59 UTC_:
-
-> "Each bispectrum coefficient \beta..."
-
-**Claude's first-pass classification:** nit
-
-**Proposed response:** "Agreed — the current phrasing says 'selects $O(|G|)$ bispectral coefficients'. To be more precise and concrete, changed to: 'selects $O(|G|)$ bispectral coefficients $\beta_{\rho_i,\rho_j}$ that remain complete.'"
-
----
-
-### § The \texttt{bispectrum}
-
-#### ‣ Design and implementation
-
-#### 28. @simon.mataigne — 2026-04-15 14:01 UTC 
-
-- **Location:** line ~241 (pos 9817), section: _The \texttt{bispectrum}_, subsection: _Design and implementation_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
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-```tex
-6$ to $172$ coefficients.  \section{The \texttt{bispectrum} Library} \label{sec:library}  \subsectio |@| n{Design and implementation} \label{sec:design}  Every group/domain pair is exposed as a \texttt{torch.nn.Module} named \texttt{\{Group\}on\{Domain\}} (e.g.\ \texttt{SO3onS2}). Modules accept raw real
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 14:01 UTC_:
-
-> Maybe unroll the acronym here? Breadth-First-Search
-
-**Claude's first-pass classification:** nit
-
-**Proposed response:** "Good suggestion — on first use (line 232), 'BFS traversal' should be written 'breadth-first search (BFS) traversal'. The second use at line 299 can then use the abbreviation."
-
----
-
-#### 29. @Adele Myers — 2026-04-17 00:11 UTC 
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-- **Location:** line ~251 (pos 10356), section: _The \texttt{bispectrum}_, subsection: _Design and implementation_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
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-```tex
-d stored as non-learnable buffers, so \texttt{.to(device)} works transparently and forward passes in |@| cur zero recomputation. Where available, \texttt{bsp.invert(beta)} returns the reconstructed signal up to group-action indeterminacy. The only dependencies are PyTorch~\citep{paszke2019pytorch}, NumPy
-```
-
-**Thread:**
-
-- **Adele Myers** · _2026-04-17 00:11 UTC_:
-
-> I would introduce the concept of domain earlier. (perhaps integrate it where I placed my previous comment about listing that "we define bispectra for signals defined over the space" but instead, say "domain"
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Agreed — 'domain' in 'group/domain pairs' (line 244) is first used without definition. The term 'domain' should be introduced in the Introduction when listing supported settings: 'The library provides bispectra for signals defined on seven domains (the underlying space on which the group acts): $C_n$, $D_n$, torus $\mathbb{T}^2$, disk, sphere $S^2$, and the octahedral setting.' This resolves threads 13 and 29 together."
-
----
-
-#### ‣ Supported modules
-
-#### 30. @Adele Myers — 2026-04-17 00:12 UTC 
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-- **Location:** line ~280 (pos 11424), section: _The \texttt{bispectrum}_, subsection: _Supported modules_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
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-```tex
-ut mode \\     \midrule     \texttt{CnonCn} & $C_n$ on $C_n$ & selective + full \\     \texttt{SO2on |@| S1} & $\SO(2)$ on $S^1$ & selective + full \\     \texttt{TorusOnTorus} & product cyclic groups on torus & selective + full \\     \texttt{DnonDn} & $D_n$ on $D_n$ & selective \\     \texttt{SO2onDisk
-```
-
-**Thread:**
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-- **Adele Myers** · _2026-04-17 00:12 UTC_:
-
-> I would cite each of the papers in this table.
-
-**Claude's first-pass classification:** needs-work
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-**Proposed response:** "Good suggestion — adding a 'Reference' column to Table~\ref{tab:supported-groups}, citing the underlying mathematical paper for each module: CnonCn → \citep{mataigne2024selective}; SO2onS1 → same; DnonDn → \citep{mataigne2024selective}; SO2onDisk → \citep{myers2025selective}; SO3onS2 → \citep{kakarala2012bispectrum}; OctaonOcta → \citep{mataigne2024selective}. Alternatively, add citations in a table footnote."
-
----
-
-#### ‣ Augmented selective $\SO(3)$ bispectrum on $S^2$
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-#### 31. @simon.mataigne — 2026-04-17 14:10 UTC 
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-- **Location:** line ~303 (pos 12436), section: _The \texttt{bispectrum}_, subsection: _Augmented selective $\SO(3)$ bispectrum on $S^2$_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
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-```tex
-d, the orbit space $\R^{(L+1)^2}/\SO(3)$ has dimension $(L{+}1)^2 - 3$, so any complete invariant re |@| quires $\Omega(L^2)$ independent components. We achieve this lower bound.  \begin{theorem}[Augmented selective $\SO(3)$ invariant on $S^2$] \label{thm:so3-selective} For $f : S^2 \to \R$ band-limited
-```
-
-**Thread:**
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-- **simon.mataigne** · _2026-04-17 14:10 UTC_:
-
-> Maybe precise "for signals defined on the sphere $S^2$.
-
-**Claude's first-pass classification:** nit
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-**Proposed response:** "Good — the section heading already says 'on $S^2$' but the theorem statement opening 'For $f : S^2 \to \R$ band-limited at degree...' already specifies this. If Simon means the section opening prose, we can add 'for real signals defined on the sphere $S^2$' to the introductory sentence."
-
----
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-#### 32. @Adele Myers — 2026-04-17 00:19 UTC 
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-- **Location:** line ~313 (pos 12925), section: _The \texttt{bispectrum}_, subsection: _Augmented selective $\SO(3)$ bispectrum on $S^2$_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
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-```tex
-eta_{\ell_1,\ell_2,\ell}$ together with a subset $\mathcal{S}_P$ of CG power spectrum entries $P_{\e |@| ll_1,\ell_2,\ell} := \lVert (\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell \rVert^2$---that is a complete $\SO(3)$-invariant for Lebesgue-almost-every signal. The output size $\Theta(L^2)$ ma
-```
-
-**Thread:**
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-- **Adele Myers** · _2026-04-17 00:19 UTC_:
-
-> By "augmented" do you mean you have to include things about the original signal in order to invert it? If this is the case, then it feels weird to call this bispectrum complete. Usually, when someone says their bispectrum is complete, they say it after proving the signal can be reconstructed from the coefficients alone.
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-**Claude's first-pass classification:** blocker
-
-**Proposed response:** "Good catch — 'augmented' refers to augmenting the *bispectral entries* (adding CG power spectrum entries, which are still invariants derived solely from the signal), NOT augmenting with the original signal. The invariant $\Phi_{\mathrm{aug}}(f)$ is computed purely from $f$, and completeness means it separates $\SO(3)$-orbits for generic $f$. We need to clarify this in the theorem statement and the surrounding prose: 'The term \emph{augmented} refers to supplementing the scalar bispectral entries $\beta_{\ell_1,\ell_2,\ell}$ with degree-4 CG power spectrum entries $P_{\ell_1,\ell_2,\ell}$ (which are also $\SO(3)$-invariant functions of $f$), not with the signal itself.' This is a real terminological issue that will confuse readers."
-
----
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-#### 33. @Adele Myers — 2026-04-17 00:13 UTC 
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-- **Location:** line ~314 (pos 13037), section: _The \texttt{bispectrum}_, subsection: _Augmented selective $\SO(3)$ bispectrum on $S^2$_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
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-```tex
-\ell} := \lVert (\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell \rVert^2$---that is a comple |@| te $\SO(3)$-invariant for Lebesgue-almost-every signal. The output size $\Theta(L^2)$ matches the dimension $(L{+}1)^2 - 3$ of the orbit space $\R^{(L+1)^2}/\SO(3)$ up to a constant factor. \end{theor
-```
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-**Thread:**
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-- **Adele Myers** · _2026-04-17 00:13 UTC_:
-
-> what is "degree"
-
-**Claude's first-pass classification:** needs-work
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-**Proposed response:** "Good point — 'band-limited at degree $4 \le L \le 100$' uses 'degree' informally. In the proof appendix (Definition 1), 'band-limited at degree $L$' is formally defined as having $a_\ell^m = 0$ for $\ell > L$. We should add a forward reference or brief gloss in the theorem statement: 'band-limited at degree $L$ (i.e., with spherical harmonic expansion truncated at angular frequency $L$, $4 \le L \le 100$).'"
-
----
-
-#### 34. @Adele Myers — 2026-04-17 00:15 UTC 
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-- **Location:** line ~328 (pos 13677), section: _The \texttt{bispectrum}_, subsection: _Augmented selective $\SO(3)$ bispectrum on $S^2$_ · tid `…000001`
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-**Anchor (local tex):**
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-```tex
-tional or deterministic witnesses for each $\ell \le 100$ and promoted to generic statements by alge |@| braic-variety arguments. The structural form of the bootstrap is uniform for $\ell \ge 8$; extending the verification beyond $L = 100$ requires only re-running a single script. \end{remark}  \paragrap
-```
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-**Thread:**
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-- **Adele Myers** · _2026-04-17 00:15 UTC_:
-
-> I would give more background on your method here and explain why your method is destined to work.
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-**Claude's first-pass classification:** needs-work
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-**Proposed response:** "Fair request — Remark 1 (the proof sketch) is quite technical without setting up the intuition. We'll add 2–3 sentences before the remark block explaining the high-level approach: 'The proof strategy follows a bootstrap induction: we first recover the low-degree coefficients from a small set of bispectral entries (the seed step), then show that each higher-degree coefficient vector $\mathbf{F}_\ell$ can be recovered linearly from already-known lower-degree coefficients. The scalar bispectrum alone is rank-deficient for real signals at certain degrees; the CG power entries supply the complementary magnitude information needed for full Jacobian rank.' This gives readers the why before the how."
-
----
-
-#### 35. @Adele Myers — 2026-04-17 00:17 UTC 
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-- **Location:** line ~341 (pos 14341), section: _The \texttt{bispectrum}_, subsection: _Augmented selective $\SO(3)$ bispectrum on $S^2$_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
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-```tex
-bootstrap matrix that \emph{cannot} be resolved by choosing different bispectral triples. At $\ell = |@| 4$, for example, the bootstrap matrix has complex rank~$4$ instead of $9 = 2\ell{+}1$, and this holds even for the \emph{full} bispectrum (all valid triples). The CG power spectrum entries $P_{\ell_1
-```
-
-**Thread:**
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-- **Adele Myers** · _2026-04-17 00:17 UTC_:
-
-> You're saying that for complex signals defined on the sphere, you can use the scalar bispectrum? First, what is the scalar bispectrum on the sphere? I've heard of the vector bispectrum as being the simplest bispectrum on the sphere.
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-**Claude's first-pass classification:** blocker
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-**Proposed response:** "Good catch — the paper uses 'scalar bispectrum' without defining what is meant. In the SO(3)/S² context, the \emph{scalar} bispectrum refers to the scalar contraction $\beta_{\ell_1,\ell_2,\ell} = \sum_{m_1,m_2,m} C^{\ell m}_{\ell_1 m_1;\ell_2 m_2} a_{\ell_1}^{m_1} a_{\ell_2}^{m_2} \overline{a_\ell^m}$, which yields a single scalar (not a vector or tensor) per triple $(\ell_1,\ell_2,\ell)$. The 'vector bispectrum' Adele refers to is the full matrix $\beta_{\ell_1,\ell_2}^{(\ell)} = C_{\ell_1\ell_2}^\top (\hat{f}(\ell_1) \otimes \hat{f}(\ell_2)) C_{\ell_1\ell_2} \hat{f}(\ell)^\dagger$ — a matrix. We need to add a definition of 'scalar bispectrum' when it first appears (line 298 or in the paragraph 'Why augmentation is necessary'), making clear it is the scalar contraction of the matrix bispectrum with CG coefficients, and cite the original vector bispectrum reference. This is a load-bearing definitional gap."
-
----
-
-#### 36. @Adele Myers — 2026-04-17 00:26 UTC 
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-- **Location:** line ~357 (pos 15150), section: _The \texttt{bispectrum}_, subsection: _Augmented selective $\SO(3)$ bispectrum on $S^2$_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
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-```tex
-; \textbf{(4)~Self-coupling} $(\ell, \ell, \ell')$ with $\ell'$ even. Entries with $\ell_1 = \ell_2$ |@| and odd~$\ell$ vanish identically for real signals (CG antisymmetry of the symmetric square) and are excluded.  All \emph{even} self-coupling entries $\beta_{\ell,\ell,\ell'}$ with $2 \le \ell' \le \
-```
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-**Thread:**
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-- **Adele Myers** · _2026-04-17 00:26 UTC_:
-
-> It would be helpful if you also gave background about the so3 on s03 bispectrum, and the original sphere vector bispectrum. Not only should you do it for completeness and to cite that paper, but also it helps introduce notation necessary to understand this section
-
-**Claude's first-pass classification:** needs-work
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-**Proposed response:** "Agreed — the Section 3.3 proof discussion assumes familiarity with SO(3) harmonics notation that is not introduced in the Background section. We'll add a short paragraph in the Background section: 'For $G = \SO(3)$ acting on $S^2$, the irreps are Wigner-$D$ matrices $D^\ell$ of dimension $2\ell+1$, and the Fourier coefficients $\mathbf{F}_\ell = (a_\ell^{-\ell},\ldots,a_\ell^\ell)$ are the spherical harmonic coefficients. The \emph{scalar bispectrum} $\beta_{\ell_1,\ell_2,\ell}$ [citation] is the CG-contracted scalar; the full matrix bispectrum is due to \citet{kakarala2012bispectrum} [and original vector bispectrum reference].' This resolves threads 35 and 36 together."
-
----
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-#### ‣ Computational benchmarks
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-#### 37. @Adele Myers — 2026-04-17 00:31 UTC 
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-- **Location:** line ~447 (pos 19548), section: _The \texttt{bispectrum}_, subsection: _Computational benchmarks_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
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-```tex
-finite abelian modules ($C_n$: 0.15\,ms, $\mathbb{T}^2$: 0.08\,ms), under 2\,ms for $\SO(2)$/disk (1 |@| .94\,ms), and under 1\,ms for the octahedral group (0.70\,ms), making it negligible relative to backbone computation. $\SO(3)$ on $S^2$ at $\ell_{\max}{=}16$ takes 26.5\,ms, dominated by the spherical
-```
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-**Thread:**
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-- **Adele Myers** · _2026-04-17 00:31 UTC_:
-
-> I would mention how you compute |G| for the disk bispectrum. This is not clear without explanation.
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-**Claude's first-pass classification:** needs-work
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-**Proposed response:** "Good point — for $\SO(2)$ on the disk, $|G| = \infty$ (continuous group), so the table shows '$\infty$' for $|G|$ and counts refer to band-limited coefficients. A brief sentence in the benchmarks section should clarify: 'For continuous groups ($\SO(2)$, $\SO(3)$), $|G| = \infty$; the coefficient count refers to the band-limited representation at the tested resolution ($N_{\max}$ radial modes and angular frequencies for the disk).' The table caption already says this partially; it should also appear in the text."
-
----
-
-### § Experiments
-
-#### ‣ 2D histopathology classification under cyclic symmetry
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-#### 38. @Adele Myers — 2026-04-17 00:34 UTC 
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-- **Location:** line ~512 (pos 22526), section: _Experiments_, subsection: _2D histopathology classification under cyclic symmetry_ · tid `…000001`
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-**Anchor (local tex):**
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-```tex
-6 \pm .012$ \\     Gated                         & 136K & $.941 \pm .009$ & $.870 \pm .004$ \\     B |@| ispectrum (\texttt{CnonCn})  & 128K & $.941 \pm .004$ & $\mathbf{.872 \pm .006}$ \\     Fourier-ELU                   & 110K & $\mathbf{.945 \pm .004}$ & $.855 \pm .006$ \\     NormReLU
-```
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-**Thread:**
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-- **Adele Myers** · _2026-04-17 00:34 UTC_:
-
-> If you only test on Cn x Cn bispectrum, I would re-word this to say "and CnonCn selective bispectrum"
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-**Claude's first-pass classification:** needs-work
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-**Proposed response:** "Agreed — the section title is '2D histopathology classification under cyclic symmetry' and the subsection intro already says 'selective bispectrum via \texttt{CnonCn}$(n{=}8)$', but Table~\ref{tab:pcam-results} labels the row just 'Bispectrum (\texttt{CnonCn})'. Adding 'selective' to be consistent: 'Selective Bispectrum (\texttt{CnonCn})'. Adele's broader concern (thread 40) about group labels in figure/table captions is addressed globally."
-
----
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-#### 39. @Adele Myers — 2026-04-17 00:36 UTC 
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-- **Location:** line ~521 (pos 22981), section: _Experiments_, subsection: _2D histopathology classification under cyclic symmetry_ · tid `…000001`
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-**Anchor (local tex):**
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-```tex
-four equivariant pooling methods substantially outperform the standard augmented CNN (AUC $0.896$). |@| Fourier-ELU leads slightly ($0.945$), followed by NormReLU ($0.942$), bispectrum and gated (both $0.941$). At 1\% data, the bispectrum leads all methods ($0.912$), outperforming NormReLU ($0.873$) an
-```
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-**Thread:**
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-- **Adele Myers** · _2026-04-17 00:36 UTC_:
-
-> Since these are results for CnonCn, I would include that in the figure labels.
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-**Claude's first-pass classification:** needs-work
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-**Proposed response:** "Agreed — the figure caption for Figure~\ref{fig:pcam-pareto} does not say which bispectrum variant is used. Updated caption to: '...PCam Pareto analysis (\texttt{CnonCn} selective bispectrum, single seed, 10\% training data)...'. This makes figures self-contained for readers who skip to figures."
-
----
-
-#### ‣ 3D organ classification under octahedral symmetry
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-#### 40. @Adele Myers — 2026-04-17 00:37 UTC 
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-- **Location:** line ~579 (pos 25820), section: _Experiments_, subsection: _3D organ classification under octahedral symmetry_ · tid `…000001`
-- **Messages:** 2 · **Status:** open
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-**Anchor (local tex):**
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-```tex
-educed data (Figure~\ref{fig:data-efficiency}), the bispectrum's advantage widens: at 5\% data it re |@| aches $33.2\%$ vs.\ $18.9\%$ for max pool. At higher capacity (6M params, Appendix~\ref{app:organ3d-details}), max pool overtakes the bispectrum, suggesting the completeness advantage is specific to t
-```
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-**Thread:**
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-- **Adele Myers** · _2026-04-17 00:37 UTC_:
-
-> since these are results for OctaonOcta, "Bispectrum" should --> OctaonOcta Bispectrum. I know it should be clear because it is under the results section for OctaonOcta, but some people only look at tables and figures, and this would be very confusing for them.
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-- **Adele Myers** · _2026-04-17 00:38 UTC_:
-
-> Same comment applies for all results sections
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-**Claude's first-pass classification:** needs-work
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-**Proposed response:** "Agreed — this is important for figure/table self-containedness. For the organ3D section: change 'Bispectrum' → '\texttt{OctaonOcta} Bispectrum' in Table~\ref{tab:organ3d-results} and Figure~\ref{fig:data-efficiency} caption. Apply the same pattern globally: PCam results → '\texttt{CnonCn} Selective Bispectrum', Spherical MNIST → '\texttt{SO3onS2} Selective Bispectrum'. The text uses the \texttt{module} names already; the tables and figures should match."
-
----
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-#### ‣ Spherical MNIST: SO(3) invariance on the sphere
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-#### 41. @Adele Myers — 2026-04-17 00:37 UTC 
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-- **Location:** line ~605 (pos 26976), section: _Experiments_, subsection: _Spherical MNIST: SO(3) invariance on the sphere_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
-
-```tex
-projected onto the sphere via stereographic projection and sampled on a $64 \times 128$ equiangular |@| grid. We compute the selective bispectrum at $\ell_{\max} = 15$, producing 384 complex invariant coefficients (768 real features after taking log-magnitude), which are fed to a 3-layer MLP (232K para
-```
-
-**Thread:**
-
-- **Adele Myers** · _2026-04-17 00:37 UTC_:
-
-> OctaonOcta
-
-**Claude's first-pass classification:** nit
-
-**Proposed response:** "This appears to be Adele noting the module name should appear — the section is about SO(3)/S², so the module name '\texttt{SO3onS2}' should be mentioned here. The setup paragraph already says 'we test \texttt{SO3onS2}' a few lines earlier (line 599), so this is already addressed in the text. If Adele means the table/figure labels, it is covered by the global label change in thread 40."
-
----
-
-### § Related Work
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-#### 42. @Adele Myers — 2026-04-17 00:40 UTC 
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-- **Location:** line ~724 (pos 32631), section: _Related Work_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
-
-```tex
-or field networks~\citep{thomas2018tensor} and MACE~\citep{batatia2022mace} use iterated Clebsch--Go |@| rdan products to build equivariant interactions. These products are exact and complete in principle, but each layer incurs $O(L^3)$ cost per CG contraction. \citet{xie2025price} analyse the \emph{pric
-```
-
-**Thread:**
-
-- **Adele Myers** · _2026-04-17 00:40 UTC_:
-
-> I feel like this should go at the beginning of the paper? and should also cite papers on the the selective g bispectrum, the disk bispectrum, the so3 on so3 bispectrum, the bispectrum on homogeneous spaces (where the sphere vector bispectrum is defined)
-
-**Claude's first-pass classification:** needs-work
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-**Proposed response:** "Adele raises a valid point about related work positioning — some venues prefer related work early. However, at NeurIPS the standard structure places related work after experiments. The current placement after experiments is standard and appropriate. We'll keep the current structure and add the missing citations as requested: selective G-bispectrum \citep{mataigne2024selective}, disk bispectrum \citep{myers2025selective}, SO(3)-on-SO(3) bispectrum [citation needed], and the sphere vector bispectrum [citation needed]. Please provide the missing bibtex keys for the last two."
-
----
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-#### 43. @simon.mataigne — 2026-04-15 14:03 UTC 
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-- **Location:** line ~736 (pos 33303), section: _Related Work_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
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-```tex
-avelet scattering transform~\citep{mallat2012scattering} provides translation-invariant representati |@| ons via iterated modulus operations; \citet{esteves2018spherical} extend this to the sphere. Unlike the bispectrum, scattering representations are stable but generally incomplete---they discard inter-
-```
-
-**Thread:**
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-- **simon.mataigne** · _2026-04-15 14:03 UTC_:
-
-> In the paper, only "bispectrum" is used. (without the G)
-
-**Claude's first-pass classification:** needs-work
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-**Proposed response:** "Good consistency note — the Related Work section has 'G-bispectrum' in some places but the paper body uses 'bispectrum' or 'selective bispectrum' without the G prefix. Will audit and make consistent: use '$G$-bispectrum' when referring to the general construct (as in the paper title), and 'bispectrum' when the context is clear. The Related Work paragraph on bispectral approaches should use '$G$-bispectrum' to be precise (line 712 already does)."
-
----
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-#### 44. @simon.mataigne — 2026-04-15 14:05 UTC 
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-- **Location:** line ~740 (pos 33602), section: _Related Work_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
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-**Anchor (local tex):**
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-```tex
--scale phase information. The bispectrum's completeness guarantee~\citep{kakarala2012bispectrum} is |@| its distinguishing theoretical advantage.  \section{Discussion and Conclusion} \label{sec:discussion}  We have presented \texttt{bispectrum}, a PyTorch library that makes selective $G$-bispectral inva
-```
-
-**Thread:**
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-- **simon.mataigne** · _2026-04-15 14:05 UTC_:
-
-> "generalizes" for US spelling
-
-**Claude's first-pass classification:** nit
-
-**Proposed response:** "Local tex at line 716 reads 'Our library generalises'. Changed to 'generalizes'."
-
----
-
-### § Discussion and Conclusion
-
-#### 45. @Adele Myers — 2026-04-17 00:41 UTC 
-
-- **Location:** line ~765 (pos 34951), section: _Discussion and Conclusion_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-lete alternatives such as max pooling can match or surpass bispectral pooling, likely because the ne |@| twork learns to compensate for missing invariant information. Completeness is a property of the \emph{map}, not of downstream task performance: a complete invariant provides a stronger inductive bias,
-```
-
-**Thread:**
-
-- **Adele Myers** · _2026-04-17 00:41 UTC_:
-
-> the first paper to prove the invertibility of the bispectrum was giannakis in 1989
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Correct historical attribution — the current paper cites kakarala2012bispectrum for reconstruction guarantees but does not mention Giannakis 1989 as the original invertibility result. Adding: 'Invertibility of the bispectrum for generic signals was first proved by \citet{giannakis1989bispectrum}; \citet{kakarala2012bispectrum} extended this to compact groups.' This fixes the attribution in both Introduction (thread 8) and Discussion. The bibtex entry for Giannakis 1989 must be added."
-
----
-
-#### 46. @Adele Myers — 2026-04-17 00:42 UTC 
-
-- **Location:** line ~789 (pos 36220), section: _Discussion and Conclusion_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-n to the benchmarks used here.  \paragraph{Future work.} Key directions include: implementing $\SO(3 |@| )$ inversion from selective bispectral coefficients; a structural proof of bootstrap rank for all~$L$; numerical conditioning analysis at large band-limits; integration with \texttt{escnn}~\citep{cesa
-```
-
-**Thread:**
-
-- **Adele Myers** · _2026-04-17 00:42 UTC_:
-
-> could also mention that bispectra are generally invariant to a specific group. for example, transation invariance or rotation invariance, but learned invariance is better if the practitioner needs invariance to many groups.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Good framing for the limitations paragraph. Adding to the 'When not to use the bispectrum' section: 'Each bispectrum module is invariant to a \emph{specific} group; for tasks requiring invariance to multiple or unknown symmetries, learned invariance (e.g., via data augmentation or group-equivariant networks with flexible symmetry) may be more appropriate.'"
-
----
-
-### § Selective $\SO(3)$ Bispectrum -- Full Proof
-
-#### 47. @Adele Myers — 2026-04-17 00:45 UTC 
-
-- **Location:** line ~835 (pos 37861), section: _Selective $\SO(3)$ Bispectrum -- Full Proof_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-\     1 & 1 & 0 & 1 \\     2 & 3 & 1 & 4 \\     3 & 7 & 4 & 11 \\     4 & 12 & 5 & 17 \\     $\ell \ |@| ge 5$ & $\sim 2\ell{+}3$ & $\sim \ell$ & $\sim 3\ell$ \\     \bottomrule   \end{tabular} \end{table}  \subsection{Setup and notation}  \begin{definition}[Band-limited signal on $S^2$] A signal $f : S^
-```
-
-**Thread:**
-
-- **Adele Myers** · _2026-04-17 00:45 UTC_:
-
-> I could have misinterpreted, but when I was chatting with Nina, she described its independence from escnn as being a good thing (less clunky)?
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "This comment is about the future-work bullet on 'integration with \texttt{escnn}' (line 792). Adele's point (via Nina) is that independence from escnn is a *feature*, not a gap. We should reframe: instead of listing escnn integration as future work (implying it's missing), say 'The library is intentionally framework-agnostic; it can serve as a drop-in invariant module for architectures built with \texttt{escnn} or \texttt{e3nn}.' Remove it from the future-work list or clarify the framing."
-
----
-
-#### ‣ Setup and notation
-
-#### 48. @simon.mataigne — 2026-04-17 14:15 UTC 
-
-- **Location:** line ~898 (pos 40395), section: _Selective $\SO(3)$ Bispectrum -- Full Proof_, subsection: _Setup and notation_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-cg-power}   P_{\ell_1,\ell_2,\ell}   := \lVert (\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\e |@| ll      \rVert^2   = \sum_{m=-\ell}^{\ell}     \Bigl\lvert \sum_{\substack{m_1,m_2 \\ m_1+m_2=m}}       \CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}\;       a_{\ell_1}^{m_1}\, a_{\ell_2}^{m_2} \Bigr\rvert^2
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-17 14:15 UTC_:
-
-> This matrix should be defined.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Simon is pointing at the CG power spectrum definition. The expression $(\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell$ (the projection of the tensor product onto the $\ell$ isotypic component) is not defined inline. Adding a sentence: 'Here $(\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell$ denotes the projection of the Kronecker product $\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2}$ onto the $D^\ell$-isotypic subspace via the CG coefficients.' The explicit sum in the display equation already gives this, but a verbal definition clarifies."
-
----
-
-#### 49. @Adele Myers — 2026-04-17 00:48 UTC 
-
-- **Location:** line ~905 (pos 40669), section: _Selective $\SO(3)$ Bispectrum -- Full Proof_, subsection: _Setup and notation_ · tid `…000001`
-- **Messages:** 2 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-ell_2}^{m_2} \Bigr\rvert^2. \end{equation} This is a degree-$4$ polynomial in the signal coefficient |@| s (real and non-negative for real~$f$) and $\SO(3)$-invariant: the CG-coupled product transforms as $D^\ell(g)$, and $\lVert \cdot \rVert^2$ is invariant under unitary transformations. \end{definition
-```
-
-**Thread:**
-
-- **Adele Myers** · _2026-04-17 00:48 UTC_:
-
-> this is the vector bispectrum, right? Make sure to cite the vector bispectrum papaer
-
-- **Adele Myers** · _2026-04-17 00:49 UTC_:
-
-> (and for consistency, I would call it the vector bispectrum)
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Good question — technically, $P_{\ell_1,\ell_2,\ell} = \|(\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell\|^2$ is the \emph{power spectrum of the CG-coupled product}, not the vector bispectrum itself. The vector bispectrum $\beta^{(\ell_1,\ell_2)}_\ell$ is the full CG-decomposed matrix contraction (the matrix generalization of Eq.~(1)), which yields a scalar only after the additional contraction with $\hat{f}(\rho_k)^\dagger$. We should add a clarifying footnote and cite the original vector/matrix bispectrum paper. Adele: please provide the exact reference for the 'vector bispectrum' you have in mind."
-
----
-
-#### ‣ The selective index set
-
-#### 50. @simon.mataigne — 2026-04-17 14:20 UTC 
-
-- **Location:** line ~998 (pos 44278), section: _Selective $\SO(3)$ Bispectrum -- Full Proof_, subsection: _The selective index set_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-a_{0,\ell,\ell} = a_0^0 \norm{\mathbf{F}_\ell}^2$ (a \emph{quadratic} constraint).  \medskip \noinde |@| nt\textbf{Category~4 (Self-coupling).} Triples $(\ell, \ell, \ell')$ with $0 \le \ell' < \ell$. Quadratic in $\mathbf{F}_\ell$. Selected only for the remaining deficit at low degrees. \end{definition}
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-17 14:20 UTC_:
-
-> $\ell$ from $0$ to $L$,
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Simon is pointing out that Definition (Selective bispectrum on $S^2$) does not specify the range of $\ell$. The definition begins 'For each target degree $\ell = 0, \dotsc, L$' (line 971), so it IS stated. However, if Simon's comment is about a different location (the Category 4 self-coupling description at line 998--1001), it reads '$0 \le \ell' < \ell$' without specifying the range of $\ell$. Will add 'for $\ell = 0, \dotsc, L$' at the start of the definition or verify that Definition~\ref{def:app-selective} already states it (it does at line 971)."
-
----
-
-#### 51. @Adele Myers — 2026-04-17 00:51 UTC 
-
-- **Location:** line ~1046 (pos 45994), section: _Selective $\SO(3)$ Bispectrum -- Full Proof_, subsection: _The selective index set_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-idates     \COMMENT{power entry}   \IF{$\ell \notin \{1,\, 2\}$}     \FOR{$\ell' = \ell{-}1, \dotsc, |@| 0$}       \STATE append $(\ell,\, \ell,\, \ell')$ to candidates     \ENDFOR   \ENDIF   \IF{$\ell = 2$}     \STATE append $(\ell,\, \ell,\, \ell)$ to candidates       \COMMENT{$\beta_{2,2,2}$}   \ENDI
-```
-
-**Thread:**
-
-- **Adele Myers** · _2026-04-17 00:51 UTC_:
-
-> Is this similar to the algorithm for finding selective coefficients in the selective G-bispectrum paper? If so I would cite that paper here and mention it in the main text. I suppose I would recommend mentioning the selective G-bispectrum paper regardless, since it is a closely related work, as it also provides an algorithm for finding a selective bispectrum
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Good point — Algorithm~\ref{alg:so3-augmented} is inspired by the BFS traversal in \citet{mataigne2024selective} but adapted for the SO(3)/S² case (which requires CG power augmentation, not present in the finite-group algorithm). Adding a sentence at the start of the Algorithm description: 'This algorithm extends the BFS-based selective index construction of \citet{mataigne2024selective} from finite groups to the SO(3) setting, adding the CG power augmentation step that is specific to real signals on $S^2$.' The main text already cites mataigne2024selective; the appendix algorithm should too."
-
----
-
-### § Constructive Seed Solver (Implementation)
-
-#### ‣ Seed recovery ($\ell \le 3$)
-
-#### 52. @simon.mataigne — 2026-04-15 14:13 UTC 
-
-- **Location:** line ~1840 (pos 81647), section: _Constructive Seed Solver (Implementation)_, subsection: _Seed recovery ($\ell \le 3$)_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-0\norm{\mathbf{F}_3}^2$: gives the norm. \item $\beta_{3,3,2}$: quadratic in $\mathbf{F}_3$, linear |@| in $\mathbf{F}_2$. \item $\beta_{2,3,2}$, $\beta_{2,2,3}$: odd parity with repeated indices,   vanish identically for real signals (Proposition~\ref{prop:app-odd-vanishing}). \item $P_{1,2,1}$ (from d
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-15 14:13 UTC_:
-
-> unique in [0, 2\pi/n), ortherwise there are n solutions.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Simon is flagging a phase/angular ambiguity in the seed recovery — likely in the argument recovery step for $a_1^0$ or a gauge-fixing angle. The seed recovery for $\mathbf{F}_2$ (Lemma~\ref{lem:app-ell2}) leaves $\sgn(v)$ ambiguous (the $T_R$ fibre), and the gauge-fixing step fixes $\arg(a_2^1)$ — but the azimuthal phase is indeed unique only in $[0, 2\pi)$ unless we specify a branch. Adding a clarification: 'The gauge-fixed azimuthal angle is unique in $[0, 2\pi)$; otherwise the $\SO(2)_z$ stabiliser would give $n$ solutions.' Johan should identify the exact step Simon is flagging and add the uniqueness-domain qualifier there."
-
----
-
-#### ‣ Degree-4 fibre reduction and $T_R$ resolution
-
-#### 53. @simon.mataigne — 2026-04-17 14:45 UTC 
-
-- **Location:** line ~1893 (pos 84402), section: _Constructive Seed Solver (Implementation)_, subsection: _Degree-4 fibre reduction and $T_R$ resolution_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-f{lem:app-degree4-filter}). \end{remark}  \subsection{Degree-4 fibre reduction and $T_R$ resolution} |@| \begin{lemma}[Degree-4 eliminates all spurious seed solutions] \label{lem:app-degree4-filter} Let $\mathcal{S}$ denote the finite seed fibre at degrees~$0$--$3$ (Lemma~\ref{lem:app-ell3}). For gener
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-17 14:45 UTC_:
-
-> Was this notation already introduced?
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Simon is asking about notation at the start of the Degree-4 subsection — likely $\mathcal{S}$ (the seed fibre), $T_R$ (the reflection map), or $A_4$ (the degree-4 matrix). $T_R$ is defined in Definition~\ref{def:app-TR} (line 1165 in the SO(3) proof appendix), but Appendix~\ref{app:constructive-solver} is a separate section and should include a back-reference: 'Let $T_R$ be the azimuthal reflection of Definition~\ref{def:app-TR}.' For $\mathcal{S}$: it is named 'finite seed fibre' in the lemma statement itself, which is fine. Add cross-references to ensure readers can trace notation."
-
----
-
-#### 54. @simon.mataigne — 2026-04-17 14:54 UTC 
-
-- **Location:** line ~1907 (pos 85014), section: _Constructive Seed Solver (Implementation)_, subsection: _Degree-4 fibre reduction and $T_R$ resolution_ · tid `…000001`
-- **Messages:** 1 · **Status:** open
-
-**Anchor (local tex):**
-
-```tex
-e inconsistent at degree~$4$. \end{lemma} \begin{proof} The degree-4 augmented system is an overdete |@| rmined polynomial system in the 9~real parameters of $\mathbf{F}_4$, with coefficients depending on the seed solution $(\mathbf{F}_0, \ldots, \mathbf{F}_3)$.  For each of the~10 seed solutions on the
-```
-
-**Thread:**
-
-- **simon.mataigne** · _2026-04-17 14:54 UTC_:
-
-> I agree with the conclusion but I am not certain of the explanation. We can discuss it in a meeting.
-
-**Claude's first-pass classification:** needs-work
-
-**Proposed response:** "Simon agrees with the conclusion of Lemma~\ref{lem:app-degree4-filter} (degree-4 eliminates spurious seeds) but has concerns about the proof argument. This should be resolved in a co-author meeting before submission. The proof relies on numerical evidence (multi-start NLS with 200 random starts achieving zero residual for the true seed and $\ge 10^{-2}$ for spurious ones); Simon may want either a stronger analytical argument or clearer separation thresholds. Flagging as needing a meeting discussion."
-
----
diff --git a/paper/paper.bbl b/paper/paper.bbl
deleted file mode 100644
index 1083fb2..0000000
--- a/paper/paper.bbl
+++ /dev/null
@@ -1,312 +0,0 @@
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-\documentclass{article}
-
-\usepackage[main]{neurips_2026}
-
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-\theoremstyle{definition}
-\newtheorem{assumption}[theorem]{Assumption}
-\newtheorem{hypothesis}[theorem]{Hypothesis}
-
-
-\lstset{
-  language=Python,
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-\setlength{\intextsep}{6pt plus 1pt minus 2pt}
-\setlength{\abovecaptionskip}{4pt}
-\setlength{\belowcaptionskip}{2pt}
-
-\title{\texttt{bispectrum}: Selective $G$-Bispectra Made Practical}
-
-\author{%
-  Johan Mathe \\
-  Atmo, Inc. \\
-  \texttt{johan@atmo.ai} \\
-  \And
-  Simon Mataigne \\
-  UCLouvain \\
-  \texttt{simon.mataigne@uclouvain.be} \\
-  \And
-  Adele Myers Lantow \\
-  University of California, Santa Barbara \\
-  \texttt{adele@ucsb.edu} \\
-  \And
-  Nina Miolane \\
-  University of California, Santa Barbara \\
-  \texttt{nmiolane@ucsb.edu} \\
-}
-
-\begin{document}
-
-\maketitle
-
-\begin{abstract}
-Many machine-learning tasks are invariant under group actions: image
-classification under planar rotation $\SO(2)$, signal classification
-under translation, and spherical-image classification under 3D rotations
-$\SO(3)$. The $G$-bispectrum is a principled \emph{complete} invariant
-(it preserves all signal information up to the group action) but its
-deployment has been hampered by quadratic cost and a patchwork of
-group-specific implementations.
-We present \texttt{bispectrum}, an open-source, fully unit-tested
-PyTorch library that computes \emph{selective} $G$-bispectral
-invariants for seven different group/domain pairs as drop-in
-differentiable \texttt{nn.Module}s. For finite groups, selectivity
-reduces the invariant size from $O(|G|^2)$ for the full bispectrum
-to $O(|G|)$ while remaining complete on generic signals. For $\SO(3)$
-on $S^2$, we introduce an augmented selective construction
-$\Phi_{\mathrm{sel}}$ with $\Theta(L^2)$ coefficients at band-limit
-$L$ (matching the orbit-space lower bound $(L{+}1)^2 - 3$ up to a
-constant, vs.\ $O(L^3)$ for the full bispectrum); a formal
-completeness proof for the continuous case is left to future work.
-Gradient-descent reconstruction from $\Phi_{\mathrm{sel}}$ produces
-visually aligned reconstructions with relative image-space residuals
-in the $0.1$--$0.3$ range across our digit sweep
-(Section~\ref{sec:so3-recon}).
-We benchmark the library on three datasets across three group/domain
-pairs (PCam under $C_8$, OrganMNIST3D under the octahedral group,
-and Spherical~MNIST under $\SO(3)$) comparing against norm pooling,
-gated pooling, Fourier-ELU pooling, max pooling, and (non-equivariant)
-augmented CNN baselines. The library delivers near-exact rotation
-invariance (test-set rotation-accuracy std $\sigma_{\mathrm{rot}} <
-10^{-3}$) and a data-efficiency advantage over incomplete alternatives
-in the low-data, moderate-capacity regime; the selective forward pass
-runs in sub-millisecond time on GPU for finite groups.
-Code is available at
-\url{https://anonymous.4open.science/r/bispectrum/}.
-\end{abstract}
-\section{Introduction}
-\label{sec:intro}
-
-Many machine-learning tasks are invariant under the action of a symmetry
-group $G$: histopathology classification under planar
-rotations $C_n$, signal classification under translations,
-volumetric organ classification under cube rotations $O$, and
-spherical-image classification under 3D rotations $\SO(3)$. Equivariant
-neural networks~\citep{cohen2016group, weiler2019general} enforce these
-symmetries by construction in their feature maps, but at the point where
-a prediction must be made any task that does not depend on the group
-action --- classification, regression, detection --- still has to map
-the equivariant features to an \emph{invariant} representation.
-
-The \emph{$G$-bispectrum} is a complex-valued vector of triple-correlation
-coefficients of the group Fourier transform. It has three properties that
-matter for ML: (i)~\textbf{completeness} --- it preserves all signal
-information up to the group action (the map is one-to-one on group
-orbits); (ii)~\textbf{inversion} --- the signal can in principle be
-recovered from the invariant up to its orbit; and
-(iii)~\textbf{selectivity} --- a small $O(|G|)$ index subset of the
-$O(|G|^2)$ full bispectrum still suffices for completeness on generic
-signals~\citep{mataigne2024selective}.
-
-Existing invariant pooling layers --- norm pooling, gate-based
-pooling, max/sum
-aggregation~\citep{weiler2019general} --- are simple but
-\emph{incomplete}: they discard information distinguishing signals
-in different orbits. The bispectrum is a principled
-alternative~\citep{giannakis1989bispectrum, kakarala2012bispectrum,
-sanborn2023bispectral, mataigne2024selective}, but existing
-constructions are scattered across theory papers and
-domain-specific implementations, and the selective template
-of~\citet{mataigne2024selective} is specific to finite groups; the
-continuous case of $\SO(3)$ on $S^2$ has remained open.
-
-This paper presents \texttt{bispectrum}, a
-PyTorch~\citep{paszke2019pytorch} library for computing bispectral
-invariants (and, where available, their inverses) on the $C_n$,
-$D_n$, $\mathbb{T}^d$, disk, $S^2$, and octahedral domains as
-drop-in \texttt{nn.Module}s.
-
-\paragraph{Contributions.}
-\begin{enumerate}
-  \item \textbf{Library.} A PyTorch library implementing selective
-    $G$-bispectra for \emph{seven group/domain pairs} --- $C_n$,
-    $\SO(2)$ on $S^1$, $D_n$, $\mathbb{T}^d$, $\SO(2)$ on the disk,
-    $\SO(3)$ on $S^2$, and the chiral octahedral group $O$ on voxel
-    grids --- as drop-in differentiable \texttt{nn.Module}s with a
-    uniform API, selective $O(|G|)$ default, precomputed
-    Clebsch--Gordan buffers, sub-millisecond GPU forward passes for
-    finite groups, and a fully unit-tested codebase
-    (96.8\% line coverage).
-  \item \textbf{Augmented selective $\SO(3)$ invariant on $S^2$.}
-    We construct $\Phi_{\mathrm{sel}}$ (Sec.~\ref{sec:so3-selective}),
-    which extends the breadth-first-search (BFS) finite-group
-    template of~\citet{mataigne2024selective} to the continuous case by
-    augmenting a selective scalar bispectrum with degree-$4$
-    \emph{CG-power} entries
-    $P_{\ell_1,\ell_2,\ell} = \lVert (\hat f_{\ell_1} \otimes
-    \hat f_{\ell_2})|_\ell \rVert^2$ that restore rank on real-valued
-    signals. At band-limit $L$, $\Phi_{\mathrm{sel}}$ uses
-    $\Theta(L^2)$ scalar coefficients (matching the orbit-space lower
-    bound $(L{+}1)^2 - 3$ up to a constant) versus $O(L^3)$ for the
-    full bispectrum. Gradient-descent reconstruction from
-    $\Phi_{\mathrm{sel}}$ produces visually aligned copies across our
-    digit sweep; a formal completeness proof for the continuous case
-    is left to future work.
-  \item \textbf{Benchmarks.} PCam under $C_8$, OrganMNIST3D under
-    the octahedral group $O$, and Spherical~MNIST under $\SO(3)$,
-    against norm pooling, gated pooling, Fourier-ELU pooling, max
-    pooling, and (non-equivariant) augmented-CNN baselines.
-    Bispectral pooling delivers near-exact rotation invariance
-    (test-set $\sigma_{\mathrm{rot}}<10^{-3}$) and a consistent
-    data-efficiency advantage over incomplete alternatives in the
-    low-data, moderate-capacity regime, converging to them at high
-    capacity.
-\end{enumerate}
-
-\section{Mathematical Background}
-\label{sec:background}
-
-\paragraph{Notation.}
-For a group $G$ with associative product, a \emph{representation}
-$\rho$ assigns each $g\in G$ a unitary matrix
-$\rho(g)$ with $\rho(gh)=\rho(g)\rho(h)$; an \emph{irreducible
-representation} (\emph{irrep}) has no nontrivial invariant subspace,
-and the \emph{dual} $\hat{G}$ is the set of equivalence classes of
-irreps (for finite $G$, $|\hat G|=O(|G|)$). A $G$-action on a domain
-$X$, $(g,x)\mapsto g\cdot x$, induces
-$(f\circ g)(x):= f(g\cdot x)$; two signals are in the same
-\emph{orbit} iff related by some $g\in G$, and an \emph{invariant}
-of $f$ is a function constant on each orbit. For compact $G$, $d\mu$
-denotes the normalised Haar measure (counting measure for finite $G$).
-``Lebesgue-almost-every $f$'' means outside a measure-zero algebraic
-subvariety. The groups and domains we target are
-$C_n,D_n,\mathbb{T}^d,\SO(2)$ on the disk, $\SO(3)$ on $S^2$, and the
-chiral octahedral group $O$ on $3$D voxel grids.
-
-\subsection{Equivariant features and invariant maps}
-
-A neural network layer $\Phi$ is \emph{$G$-equivariant} if its output
-transforms predictably under the group action:
-$[\pi_{\mathrm{out}}(g) \circ \Phi](f) = \Phi(\pi_{\mathrm{in}}(g) f)$
-for all $g \in G$~\citep{cohen2016group, weiler2019general}. Here $f$ is
-an input signal (a function on the domain $X$),
-$\pi_{\mathrm{in}}, \pi_{\mathrm{out}}$ are the (representations of the)
-$G$-actions on the input and output feature spaces of $\Phi$, and the
-composition $\pi_{\mathrm{out}}(g) \circ \Phi$ is the action of $g$ on
-the output side. Stacking equivariant layers preserves equivariance, but
-any task that requires a prediction independent of the group
-action---classification, regression, detection---must eventually map
-equivariant features to \emph{invariant} ones (constant on each orbit).
-
-\begin{table}[t]
-  \caption{Invariant nonlinearities for equivariant CNNs. ``Exact''$^*$
-    means the invariant has no discretisation/aliasing error; ``Complete''
-    means it separates orbits; ``Genericity assumption''$^\dagger$ means
-    completeness holds outside a Lebesgue-measure-zero algebraic
-    subvariety. The selective bispectrum is the only entry that is
-    simultaneously exact, complete, and linear-cost for finite groups;
-    the $\SO(3)$-on-$S^2$ case is treated empirically
-    (Sec.~\ref{sec:so3-selective}).}
-  \label{tab:nonlinearity-comparison}
-  \centering
-  \small
-  \setlength{\tabcolsep}{4pt}
-  \begin{tabular}{lccccl}
-    \toprule
-    Nonlinearity & Exact?$^*$ & Complete? & Coefficients & Forward cost & Key limitation \\
-    \midrule
-    Norm              & Yes & No              & $O(|G|)$   & $O(|G|)$          & Discards phase \\
-    Gate              & Yes & No              & $O(|G|)$   & $O(|G|)$          & Scalar bottleneck \\
-    Fourier pointwise & No  & Yes$^\ddagger$  & $O(|G|)$   & $O(|G| \log |G|)$ & Aliasing \\
-    Full bispectrum (finite $G$)      & Yes & Yes & $O(|G|^2)$   & $O(|G|^3)$ & Quadratic cost \\
-    Full bispectrum ($\SO(3)$/$S^2$)  & Yes & Yes & $\Theta(L^3)$ & $O(L^4)$  & Cubic cost \\
-    \midrule
-    Selective bispectrum (finite $G$) & Yes & Yes       & $O(|G|)$      & $O(|G|^2)$ & Genericity assumption$^\dagger$ \\
-    Augmented selective $\SO(3)$/$S^2$ (ours) & Yes & Empirical & $\Theta(L^2)$ & $O(L^3)$   & Proof of concept only \\
-    \bottomrule
-  \end{tabular}
-
-  \vspace{0.3em}
-  {\small $^\ddagger$Complete pre-aliasing; the pointwise nonlinearity introduces aliasing that may break completeness in practice.}
-\end{table}
-
-Table~\ref{tab:nonlinearity-comparison} summarises the trade-offs between
-exactness, completeness, and computational cost across existing invariant
-pooling layers. Norm pooling and gated nonlinearities are exact
-invariants but \emph{incomplete}: they discard the phase of Fourier
-blocks, collapsing distinct orbits. Fourier-space pointwise
-nonlinearities preserve completeness but sacrifice exactness through
-\mbox{aliasing~\citep{weiler2019general}}. The full bispectrum is exact and
-complete but costs $O(|G|^2)$ coefficients. The \emph{selective}
-bispectrum (introduced below) preserves both exactness and completeness
-while using only $O(|G|)$ coefficients on generic signals.
-
-\subsection{The $G$-bispectrum and selective bispectra}
-
-Let $G$ be a compact group with dual $\hat{G}$ and irreps
-$\{\rho_i\}_{i \in \hat{G}}$ of dimensions $d_i$. The \emph{group Fourier
-transform}~\citep{diaconis1990fourier} of a signal $f : G \to \C$ is
-\begin{equation}\label{eq:fourier}
-  \hat{f}(\rho_i)
-  = \int_G f(g)\, \rho_i(g)\, d\mu(g)
-  \quad\in\C^{d_i \times d_i},
-\end{equation}
-which specialises to $\sum_{g \in G} f(g)\, \rho_i(g)$ when $G$ is finite
-(counting measure). The Clebsch--Gordan (CG) decomposition
-$\rho_i \otimes \rho_j \cong \bigoplus_{k \in \hat{G}} m_{ij}^k \rho_k$
-expresses the tensor product of two irreps as a direct sum of irreps
-(with integer multiplicities $m_{ij}^k \ge 0$); the change-of-basis is
-realised by the \emph{CG isometry}
-$C_{ij}^k : V_{\rho_k} \to V_{\rho_i} \otimes V_{\rho_j}$, projecting the
-tensor product onto its $\rho_k$-isotypic component. For each admissible
-triple $(i, j, k)$ with $m_{ij}^k > 0$, the \emph{bispectrum} is the
-scalar
-\begin{equation}
-  \label{eq:bispectrum-def}
-  \beta_{i,j}^{k}(f)
-  := \bigl\langle\, (C_{ij}^k)^{\dagger}\,
-        \bigl(\hat{f}(\rho_i) \otimes \hat{f}(\rho_j)\bigr),\,
-        \hat{f}(\rho_k) \bigr\rangle.
-\end{equation}
-Each $\beta_{i,j}^{k}$ is invariant under the action $f \mapsto f \circ g$.
-The full matrix/operator bispectrum is complete on compact
-groups under Kakarala-style
-hypotheses~\citep{kakarala1992triple,kakarala2012bispectrum}; for
-finite $G$, the scalar-entry family
-$\{\beta_{i,j}^{k}\}_{(i,j,k)}$ remains complete on generic signals
-via the selective construction of~\citet{mataigne2024selective}.
-For $\SO(3)$ on $S^2$ we do not claim scalar
-completeness---the library's selective invariant is empirical; see
-Section~\ref{sec:so3-selective}. For finite $G$,
-$|\hat G| = O(|G|)$, so the full bispectrum has $O(|G|^2)$
-scalar coefficients (indexed by $(i,j) \in \hat{G}^2$
-with $k$ ranging over the CG-allowed irreps).
-Equation~\eqref{eq:bispectrum-def} is stated for $G$ acting on itself;
-for homogeneous spaces of the form $G/H$ (e.g.\ $\SO(3)$ on $S^2$, or
-$\SO(2)$ on the disk), the same construction applies after expanding $f$
-in a basis adapted to $G/H$ (spherical harmonics for $S^2$, Bessel modes
-on the disk). The explicit specialisation for $S^2$ is in
-Definition~\ref{def:app-bispectrum}.
-
-\emph{Selective bispectra.} \citet{mataigne2024selective} show that for
-finite groups, a BFS over the Kronecker product
-table of $\hat{G}$ selects $O(|G|)$ bispectral coefficients
-$\beta_{i,j}^{k}$ that remain complete; \citet{myers2025selective}
-extend this to the disk. This
-reduction from quadratic to linear is the key to practical deployment:
-for example, the bispectrum of the octahedral group drops from $576$ to
-$172$ coefficients.
-
-\section{Related Work}
-\label{sec:related}
-
-\paragraph{Equivariant network libraries.}
-\texttt{escnn}~\citep{cesa2022program} and \texttt{e3nn}~\citep{geiger2022e3nn}
-provide general-purpose equivariant convolution layers for 2D and 3D
-groups respectively, and \texttt{lie-nn}~\citep{batatia2023general}
-provides general Lie-group representation primitives (irreducible
-representations, Clebsch--Gordan decompositions, tensor products) on
-top of which custom equivariant layers can be built.  These libraries
-implement equivariant \emph{feature maps} but leave the choice of
-invariant pooling to the user (typically norm or gate). Our library is
-complementary: it provides the invariant map as a drop-in
-\texttt{nn.Module} that plugs into architectures built with any of these
-frameworks.
-
-\paragraph{Bispectral approaches in neural networks.}
-\citet{sanborn2023bispectral} introduced bispectral neural networks (BNNs),
-using the full $G$-bispectrum as an invariant nonlinearity for $C_n$ and $D_n$.
-\citet{oreiller2022bispectral} applied bispectral CNNs to medical imaging
-with a fixed set of hand-selected triples. Both use the full (quadratic-cost)
-bispectrum and provide implementations tied to specific architectures.
-Our library generalizes to seven group/domain pairs, defaults to the
-$O(|G|)$ selective reduction~\citep{mataigne2024selective}, and exposes
-a framework-agnostic API.
-\citet{chevalley2024bispectral} explore bispectral signatures for data
-characterisation, using bispectra as descriptors rather than learnable
-pooling primitives.
-Outside the equivariant-network setting,
-\texttt{PyBispectra}~\citep{binns2025pybispectra} computes bispectrum
-and bicoherence statistics of one-dimensional time-series signals for
-electrophysiology applications such as phase-amplitude coupling and
-time-delay estimation; the underlying 1D bispectrum is translation-invariant
-by construction, but the toolbox is not designed as a neural-network
-pooling layer.
-
-\paragraph{Bispectra on homogeneous and non-homogeneous spaces.}
-The selective $G$-bispectrum
-construction~\citep{mataigne2024selective} reduces the cost of the
-$G$-bispectrum from $O(|G|^2)$ to $O(|G|)$ scalars on finite groups.
-\citet{myers2025selective} extend the selective construction to the
-disk, where $\SO(2)$ acts on a non-homogeneous space.
-\citet{kakarala2012bispectrum} treats the $\SO(3)$-on-$\SO(3)$
-bispectrum and the closely related sphere case via Wigner-$D$ irreps.
-The matrix (vector) bispectrum on $S^2$ is the homogeneous-space form
-on which our scalar contractions and CG power augmentation are built.
-
-\paragraph{Invariant theory and scattering.}
-The wavelet scattering transform~\citep{mallat2012scattering} provides
-translation-invariant representations via iterated modulus operations;
-\citet{esteves2018spherical} extend this to the sphere. Unlike the
-$G$-bispectrum, scattering representations are stable but generally
-incomplete---they discard inter-scale phase information.
-The $G$-bispectrum's completeness
-guarantee~\citep{giannakis1989bispectrum, kakarala2012bispectrum} is
-its distinguishing theoretical advantage.
-
-\section{The \texttt{bispectrum} Library}
-\label{sec:library}
-
-\subsection{Design and implementation}
-\label{sec:design}
-
-Every $G$/domain pair is a \texttt{torch.nn.Module} named
-\texttt{\{Group\}on\{Domain\}} (e.g.\ \texttt{SO3onS2});
-Table~\ref{tab:supported-groups} lists all seven modules. Each exposes a
-uniform API: \texttt{forward(f)} returns the selective invariants,
-\texttt{fourier(f)} the group Fourier coefficients, and
-\texttt{invert(beta)} the reconstructed signal up to group-action
-indeterminacy (where available). Modules default to $O(|G|)$
-selective coefficients (\texttt{selective=False} restores the full
-$O(|G|^2)$ set), precompute CG matrices / DFT kernels / Bessel
-roots as non-learnable buffers, and depend only on
-PyTorch~\citep{paszke2019pytorch}, NumPy~\citep{harris2020numpy},
-and \texttt{torch\_harmonics}~\citep{bonev2023spherical} (for
-\texttt{SO3onS2}). Test suite line coverage is 96.8\%; a minimal
-usage example is in Appendix~\ref{app:api-example}.
-
-\begin{table}[t]
-  \caption{Scope of the \texttt{bispectrum} library.
-    \texttt{SO2onS1} is the continuous-$n$ limit of \texttt{CnonCn}
-    and shares its implementation.}
-  \label{tab:supported-groups}
-  \centering
-  \small
-  \begin{tabular}{llll}
-    \toprule
-    Module & Group / Domain & Output mode & Reference \\
-    \midrule
-    \texttt{CnonCn} & $C_n$ on $C_n$ & selective + full & \citep{sanborn2023bispectral} \\
-    \texttt{SO2onS1} & $\SO(2)$ on $S^1$ & selective + full & \citep{mataigne2024selective} \\
-    \texttt{TorusDnTorus} & $\mathbb{T}^d$ & selective + full & \citep{mataigne2024selective} \\
-    \texttt{DnonDn} & $D_n$ on $D_n$ & selective & \citep{mataigne2024selective} \\
-    \texttt{SO2onDisk} & $\SO(2)$ on disk & selective & \citep{myers2025selective} \\
-    \texttt{SO3onS2} & $\SO(3)$ on $S^2$ & selective + full & \citep{kakarala2012bispectrum}, this paper \\
-    \texttt{OctaonOcta} & chiral octahedral $O$ & selective & \citep{mataigne2024selective} \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\subsection{Profiling}
-\label{sec:benchmarks}
-
-We profile the library on a single NVIDIA H100 80\,GB GPU
-(PyTorch 2.10, batch\,=\,16 for the forward pass and 4 for inversion,
-median wall-clock via \texttt{torch.utils.benchmark}); results are in
-Table~\ref{tab:benchmarks}. Coefficient-count scaling and additional
-plots are in Appendix~\ref{app:bench}
-(Figure~\ref{fig:coeff-scaling}).
-
-\begin{table}[t]
-  \centering
-  \caption{Library profiling summary on a single NVIDIA H100 80\,GB GPU.
-    ``Sel.\ coefs'' = selective coefficient count; ``Fwd'' /``Inv'' =
-    median forward/inversion wall-clock (ms); ``--'' = not implemented.
-    $^*$\texttt{SO3onS2} at $\ell_{\max}{=}16$. The full bispectrum at
-    matched group order is $8256$/$524{,}800$ coefficients for
-    $C_{128}$/$\mathbb{T}^2$ (vs.\ $128$/$1024$ selective, a
-    $64\times$/$512\times$ reduction), taking $9.17$/$0.71$\,ms in
-    the forward pass.}
-  \label{tab:benchmarks}
-  \small
-  \setlength{\tabcolsep}{4pt}
-  \begin{tabular}{llrrrr}
-    \toprule
-    Module & $G$ & $|G|$ & Sel.\ coefs & Fwd (ms) & Inv (ms) \\
-    \midrule
-    \texttt{CnonCn} & $C_{128}$ & 128 & 128 & 0.15 & 8.56 \\
-    \texttt{TorusOnTorus} & $C_{32}{\times}C_{32}$ & 1024 & 1024 & 0.08 & 68.02 \\
-    \texttt{DnonDn} & $D_{32}$ & 64 & 245 & 2.52 & 42.33 \\
-    \texttt{SO2onDisk} & $\mathrm{SO}(2)$ & $\infty$ & 411 & 1.94 & 17.69 \\
-    \texttt{SO3onS2}\rlap{$^*$} & $\mathrm{SO}(3)$ & $\infty$ & 430 & 26.54 & -- \\
-    \texttt{OctaonOcta} & $O$ & 24 & 172 & 0.70 & -- \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-The selective forward pass is sub-millisecond for finite abelian
-modules ($C_n$: 0.15\,ms, $\mathbb{T}^2$: 0.08\,ms), under 2\,ms for
-$\SO(2)$/disk and under 1\,ms for the octahedral group, making it
-negligible relative to backbone computation. $\SO(3)$ on $S^2$ at
-$\ell_{\max}{=}16$ takes 26.5\,ms, dominated by the spherical
-harmonic transform
-(\texttt{torch\_harmonics}~\citep{bonev2023spherical}). GPU throughput
-scales linearly with batch size, exceeding $10^7$~samples/s for $C_n$
-and $\mathbb{T}^2$ (Appendix~\ref{app:bench}). Six backends are
-tabulated; the seventh module, \texttt{SO2onS1}, subclasses
-\texttt{CnonCn} with $n = \ell_{\max}{+}1$ and has identical runtime
-to the corresponding \texttt{CnonCn} row at matched $|G|$.
-
-\section{Experiments}
-\label{sec:experiments}
-
-We validate the library on three tasks spanning discrete and continuous
-groups: 2D histopathology classification under $C_8$ symmetry (PCam,
-Section~\ref{sec:pcam}), 3D organ classification under octahedral symmetry
-(OrganMNIST3D, Section~\ref{sec:organ3d}), and spherical digit
-classification under full $\SO(3)$ (Spherical MNIST,
-Section~\ref{sec:smnist}).
-
-\subsection{2D histopathology classification under cyclic symmetry}
-\label{sec:pcam}
-
-We evaluate five invariant pooling strategies on
-PatchCamelyon~\citep{veeling2018rotation} (PCam), 327K histopathology
-patches with binary metastasis labels. All equivariant models share a
-$C_8$-equivariant DenseNet~\citep{huang2017densenet} backbone following
-\citet{veeling2018rotation}; they differ only in the terminal invariant map:
-norm pooling, gated pooling, Fourier-ELU, and selective bispectrum via
-\texttt{CnonCn}$(n{=}8)$. A non-equivariant standard CNN trained with
-data augmentation drawn uniformly from the symmetry group $C_8$
-(``\emph{augmented CNN}'') serves as the non-equivariant baseline. We
-sweep growth rates to vary capacity (30K--1.6M params) and train on
-$10\%$ of the PCam training set (an image-level subsample, not a
-group-orbit subsample, drawn before any group augmentation) with
-AdamW~\citep{loshchilov2019adamw}. Full model descriptions are in
-Appendix~\ref{app:pcam-details}.
-
-Table~\ref{tab:pcam-results} reports multi-seed statistics at
-matched $\sim$100K params; the full Pareto sweep (AUC vs.\
-parameter count, ranking, data efficiency) is
-Figure~\ref{fig:pcam-pareto} in Appendix~\ref{app:pcam-details}.
-
-\begin{table}[t]
-  \caption{PCam at matched ${\sim}100$K params, $10\%$ training
-    data, mean $\pm$ std over 3 seeds; all equivariant models share
-    a $C_8$-equivariant DenseNet backbone.}
-  \label{tab:pcam-results}
-  \centering
-  \small
-  \begin{tabular}{lrcc}
-    \toprule
-    Model & Params & Test AUC & Test ACC \\
-    \midrule
-    Standard (augmented)          & 102K & $.896 \pm .010$ & $.826 \pm .012$ \\
-    Gated                         & 136K & $.941 \pm .009$ & $.870 \pm .004$ \\
-    Selective Bispectrum (\texttt{CnonCn})  & 128K & $.941 \pm .004$ & $\mathbf{.872 \pm .006}$ \\
-    Fourier-ELU                   & 110K & $\mathbf{.945 \pm .004}$ & $.855 \pm .006$ \\
-    NormReLU                      & 110K & $.942 \pm .004$ & $.858 \pm .007$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-At matched capacity (${\sim}100$K params, 10\% data), all four equivariant
-pooling methods substantially outperform the standard augmented CNN
-(AUC $0.896$). Fourier-ELU leads slightly ($0.945$), followed by
-NormReLU ($0.942$), bispectrum and gated (both $0.941$).
-At 1\% data, the bispectrum leads all methods ($0.912$), outperforming
-NormReLU ($0.873$) and Fourier-ELU ($0.876$). This data-efficiency
-advantage narrows as the training set grows, consistent with complete
-invariants providing a stronger inductive bias when data is scarce.
-
-\subsection{3D organ classification under octahedral symmetry}
-\label{sec:organ3d}
-
-To evaluate the \texttt{OctaonOcta} module for the chiral octahedral group
-$O$ ($|O|=24$), we design a controlled 3D experiment on
-OrganMNIST3D~\citep{medmnistv2} (1{,}742 abdominal CT volumes, $28^3$
-voxels, 11 classes). All equivariant variants share an $O$-equivariant 3D
-ResNet backbone with exact voxel-permutation kernels. The four models
-compared are: a translation-equivariant standard 3D CNN trained with
-octahedral data augmentation (the ``\emph{augmented CNN}'' baseline,
-16K params); the same $O$-equivariant ResNet with terminal norm pool
-(375K params); with terminal max pool (374K params); and with terminal
-selective bispectrum via \texttt{OctaonOcta} (463K, 172 invariant
-scalars). Models are trained with AdamW~\citep{loshchilov2019adamw} for
-100 epochs with three seeds; rotation robustness is measured over all
-24 octahedral rotations of the test set. Published baselines and
-wider-channel experiments are in Appendix~\ref{app:organ3d-details}.
-
-Table~\ref{tab:organ3d-results} and
-Figure~\ref{fig:data-efficiency} report the key findings.
-
-\begin{table}[t]
-  \caption{OrganMNIST3D results (mean $\pm$ std over 3 seeds). Rot ACC
-    and $\sigma_{\mathrm{rot}}$: mean and std of accuracy across 24
-    octahedral test-set rotations.}
-  \label{tab:organ3d-results}
-  \centering
-  \small
-  \begin{tabular}{lrcccr}
-    \toprule
-    Model & Params & Test ACC & Test AUC & Rot ACC & $\sigma_{\mathrm{rot}}$ \\
-    \midrule
-    Standard 3D CNN          & 16K  & $.601 \pm .017$ & $.951 \pm .002$ & .598 & .0115 \\
-    $O$-Equiv + Norm Pool    & 375K & $.568 \pm .099$ & $.940 \pm .021$ & .568 & .0000 \\
-    $O$-Equiv + Max Pool     & 374K & $.730 \pm .033$ & $.972 \pm .005$ & .730 & .0000 \\
-    $O$-Equiv + \texttt{OctaonOcta} Bispectrum   & 463K & $.726 \pm .027$ & $.972 \pm .004$ & .726 & .0005$^\dagger$ \\
-    \bottomrule
-  \end{tabular}
-
-  \vspace{0.3em}
-  {\small $^\dagger$Nonzero $\sigma_{\mathrm{rot}}$ due to floating-point precision in CG contractions, not a theoretical invariance failure.}
-\end{table}
-
-All equivariant models achieve $\sigma_{\mathrm{rot}}\le 0.0005$
-while the augmented CNN drops ${\sim}2.5$\,pp under rotation. The
-bispectrum ($72.6\%$) matches max pool ($73.0\%$) and substantially
-outperforms norm pool ($56.8\%$) and Regular SE(3)
-convolutions~\citep{kuipers2023regular} ($69.8\%$).
-At reduced data (Figure~\ref{fig:data-efficiency}), the bispectrum's
-advantage widens: at 5\% data it reaches $33.2\%$ vs.\ $18.9\%$ for max
-pool. At higher capacity (6M params,
-Appendix~\ref{app:organ3d-details}), max pool overtakes the bispectrum,
-suggesting the completeness advantage is specific to the data-limited
-regime.
-
-\begin{figure}[t]
-  \centering
-  \includegraphics[width=0.75\linewidth]{experiments/organ3d/data_efficiency.pdf}
-  \caption{Test accuracy vs.\ training data fraction on OrganMNIST3D
-    (\texttt{OctaonOcta} bispectrum, mean over 3 seeds; shaded bands:
-    $\pm 1$ std). The bispectrum provides a clear advantage in the
-    low-data regime.}
-  \label{fig:data-efficiency}
-\end{figure}
-
-\subsection{Spherical MNIST: SO(3) invariance on the sphere}
-\label{sec:smnist}
-
-The preceding experiments validate bispectral pooling for finite groups
-($C_8$ and the octahedral group $O$). To evaluate the library's continuous
-group support, we test \texttt{SO3onS2}---the augmented selective
-$\SO(3)$ invariant on the 2-sphere
-(Section~\ref{sec:so3-selective})---on Spherical
-MNIST~\citep{cohen2018spherical}, the standard benchmark for
-rotation invariance on $S^2$.
-
-\paragraph{Setup.}
-MNIST digits are stereographically projected onto a
-$64\times 128$ equiangular grid; we compute
-$\Phi_{\mathrm{sel}}$ at $\ell_{\max}=15$ ($N=384$ live real
-components, stored as 384 complex tensors by the library), apply
-sign-preserving $\operatorname{sign}(x)\log(1+|x|)$ to real and
-imaginary parts ($2N=768$ real channels; one channel per
-bispectral entry is structurally zero by parity but we feed both
-since the log transform is per-channel injective and cheap), and
-feed a 3-layer MLP (232K params). Controls: a power-spectrum MLP at 11K
-params (equally invariant but \emph{incomplete}, discarding
-inter-harmonic phase), a matched 166K-param version isolating
-capacity vs.\ completeness, and a non-invariant CNN directly on the
-equirectangular image at 185K params.
-Following \citet{cohen2018spherical}, we evaluate under three
-protocols NR/NR, R/R, and the most challenging NR/R; each run uses
-three seeds, and rotation robustness $\sigma_{\mathrm{rot}}$ is
-measured over 10 random $\SO(3)$ rotations of the test set.
-
-Table~\ref{tab:smnist-results} and
-Figure~\ref{fig:smnist-analysis} report the results.
-
-\begin{table}[t]
-  \caption{Spherical MNIST results (mean $\pm$ std over 3 seeds).
-    Reference numbers are from \citet{cohen2018spherical}.
-    $\sigma_{\mathrm{rot}}$: standard deviation of accuracy under 10
-    random $\SO(3)$ rotations (NR-trained models).}
-  \label{tab:smnist-results}
-  \centering
-  \small
-  \setlength{\tabcolsep}{4pt}
-  \begin{tabular}{lrcccc}
-    \toprule
-    Model & Params & NR/NR $\uparrow$ & R/R $\uparrow$ & NR/R $\uparrow$
-      & $\sigma_{\mathrm{rot}}$ $\downarrow$ \\
-    \midrule
-    Cohen planar CNN & --- & $0.98$ & $0.23$ & $0.11$ & --- \\
-    Cohen spherical CNN & --- & $0.96$ & $0.95$ & $0.94$ & --- \\
-    \midrule
-    Standard CNN & 185K
-      & $0.991 {\pm} 0.000$ & $0.781 {\pm} 0.003$ & $0.138 {\pm} 0.003$
-      & $0.044$ \\
-    Power Spec.\ + MLP & 11K
-      & $0.780 {\pm} 0.002$ & $0.780 {\pm} 0.002$ & $0.777 {\pm} 0.001$
-      & $3{\times}10^{-4}$ \\
-    Power Spec.\ + MLP (matched) & 166K
-      & $0.794 {\pm} 0.011$ & $0.794 {\pm} 0.011$ & $0.792 {\pm} 0.010$
-      & $3{\times}10^{-4}$ \\
-    \textbf{\texttt{SO3onS2} Bispectrum + MLP (ours)} & 232K
-      & $\mathbf{0.950 {\pm} 0.000}$ & $\mathbf{0.949 {\pm} 0.002}$
-      & $\mathbf{0.951 {\pm} 0.001}$ & $\mathbf{1{\times}10^{-4}}$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\begin{figure}[t]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/spherical_mnist/smnist_analysis.pdf}
-  \caption{Spherical MNIST analysis (\texttt{SO3onS2} selective
-    bispectrum).
-    \textbf{(a)}~Cohen et al.\ protocol: NR/NR, R/R, and NR/R accuracy.
-    Dashed lines show Cohen's spherical CNN reference.
-    \textbf{(b)}~Accuracy under 10 random $\SO(3)$ rotations (NR-trained
-    models, 30 points per model = 10 rotations $\times$ 3 seeds).
-    Scaling the power spectrum MLP from 11K to 166K parameters yields
-    only +1.4pp; the \texttt{SO3onS2} bispectrum at 232K reaches 95.0\%.}
-  \label{fig:smnist-analysis}
-\end{figure}
-
-The bispectrum achieves near-identical accuracy across all three
-protocols ($\sigma_{\mathrm{rot}}{\sim}10^{-4}$), confirming true
-$\SO(3)$ invariance; the standard CNN collapses from $99.1\%$
-(NR/NR) to $13.8\%$ (NR/R), showing that augmentation alone does not
-confer invariance. The power spectrum is equally invariant but
-reaches only $78\%$---a $17$pp deficit that persists after scaling
-the MLP to $166$K params (${+}1.4$pp), consistent with the power
-spectrum's incompleteness (rather than MLP capacity) being the
-bottleneck. Cohen's spherical CNN~\citep{cohen2018spherical} reaches
-$96\%/95\%/94\%$ with learned equivariant convolutions; our
-single-pass bispectrum ($95.0\%/94.9\%/95.1\%$) is $1$--$2$pp lower
-but has a tighter invariance gap ($0.1$pp vs.\ $2.0$pp).
-
-
-\section{Augmented selective $\SO(3)$ invariant on $S^2$}
-\label{sec:so3-selective}
-
-\subsection{Setup and construction}
-
-A real signal $f : S^2 \to \R$ band-limited at degree~$L$ has
-spherical harmonic coefficients
-$\mathbf{F}_\ell \in \C^{2\ell+1}$, $\ell = 0,\dots,L$, transforming
-as $\mathbf{F}_\ell \mapsto D^\ell(g)\mathbf{F}_\ell$ under
-$g\in\SO(3)$; the reality condition leaves $(L{+}1)^2$ real degrees
-of freedom (Appendix~\ref{sec:app-setup}). The scalar bispectral
-coefficient at a CG-admissible triple $(\ell_1,\ell_2,\ell)$ is
-\begin{equation}\label{eq:s6-bispectrum}
-  \beta_{\ell_1,\ell_2,\ell}
-  = \!\!\!\sum_{\substack{m_1,m_2,m\\ m_1+m_2=m}}\!\!\!
-    \CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}\,
-    a_{\ell_1}^{m_1}\,a_{\ell_2}^{m_2}\,\conj{a_\ell^m},
-\end{equation}
-the specialisation of~\eqref{eq:bispectrum-def} to $\SO(3)/S^2$.
-The full set has $\Theta(L^3)$ scalars; the orbit space
-$\R^{(L+1)^2}/\SO(3)$ has dimension $(L{+}1)^2 - 3$, so any smooth
-complete invariant requires $\Omega(L^2)$ components
-(Proposition~\ref{prop:app-lower-bound}), which the Mataigne chain
-$\beta_{1,\ell,\ell+1}$ ($\sim 3L$ scalars) cannot achieve.
-
-We define $\Phi_{\mathrm{sel}}(f)$ as the output of the augmented
-selective index construction of Algorithm~\ref{alg:so3-augmented}
-(Appendix~\ref{sec:app-selective-index}). For each degree $\ell$
-it collects (i) a size-$(2\ell{+}1)$ linear bootstrap block
-$\mathcal{T}_\ell$ of scalar bispectrum entries, (ii) mandatory
-even self-couplings $\beta_{\ell,\ell,\ell'}$, and (iii) a greedily
-chosen set of CG-power scalars
-$P_{\ell_1,\ell_2,\ell} := \lVert(\mathbf{F}_{\ell_1}\otimes
-\mathbf{F}_{\ell_2})|_\ell\rVert^2$ (the bootstrap block alone is
-rank-deficient on real signals, see
-Remark~\ref{rem:app-Tell-rank-deficit}). Because
-$\Phi_{\mathrm{sel}}$ mixes bispectrum entries with CG-power
-scalars (a degree-$4$ invariant), we call it an \emph{augmented
-selective invariant} rather than a bispectrum; the phrase
-``selective bispectrum'' refers strictly to the bispectral
-component $\mathcal{S}_\beta$. For degrees
-$\ell\le L_{\mathrm{seed}}:=\min(L,4)$ we additionally include the
-full seed $\mathcal{S}_{\mathrm{seed}}$. The total size is
-$|\Phi_{\mathrm{sel}}| = \Theta(L^2)$
-(Proposition~\ref{prop:app-output-size}); Algorithm~\ref{alg:selective-bsp-so3}
-summarises the bispectral-triple selection, and the full index
-set---including self-couplings and CG power---is laid out in
-Algorithm~\ref{alg:so3-augmented}. Concretely, at $L=15$ (resp.\
-$L=16$) the library outputs $|\Phi_{\mathrm{sel}}| = 384$ (resp.\
-$430$) real scalars; the full bispectrum at $L=15$ has $2{,}056$
-admissible triples (cf.~Table~\ref{tab:app-counts}).
-
-\begin{algorithm}[H]
-\caption{Selected bispectral triples $\mathcal{S}_\beta$ for the
-  augmented $\SO(3)$ invariant (BFS form; mirrors~\citet{mataigne2024selective},
-  Alg.~1). The full index set --- bispectral triples, mandatory
-  self-couplings, and CG-power scalars --- is
-  Algorithm~\ref{alg:so3-augmented} in Appendix~\ref{sec:app-selective-index}.}
-\label{alg:selective-bsp-so3}
-\small
-\begin{algorithmic}[1]
-\STATE \textbf{Input:} band-limit $L$,
-  seed cutoff $L_{\mathrm{seed}} = \min(L,4)$
-\STATE $\mathcal{L}_\beta \leftarrow$ all admissible triples
-  $(\ell_1,\ell_2,\ell)$ with $\ell\le L_{\mathrm{seed}}$
-  \COMMENT{seed block $\mathcal{S}_{\mathrm{seed}}$}
-\FOR{$\ell = L_{\mathrm{seed}}{+}1, \dotsc, L$}
-  \STATE append the bootstrap block $\mathcal{T}_\ell$
-    to $\mathcal{L}_\beta$
-    \COMMENT{$2\ell{+}1$ entries linear in $\mathbf{F}_\ell$;
-      tabulated in Appendix~\ref{sec:app-selective-index}}
-\ENDFOR
-\RETURN $\{\beta_{\ell_1,\ell_2,\ell}(f) :
-  (\ell_1,\ell_2,\ell)\in\mathcal{L}_\beta\}$
-\end{algorithmic}
-\end{algorithm}
-
-\subsection{Empirical proof of concept: bispectrum reconstruction}
-\label{sec:so3-recon}
-
-To test whether $\Phi_{\mathrm{sel}}$ retains enough information to
-recover signals up to rotation, we run a direct inversion experiment
-on Spherical MNIST digits band-limited at $L = 12$.\footnote{We use
-$L{=}12$ here rather than the classifier's $L{=}15$ to keep the
-reconstruction residual safely below the SHT discretisation floor at
-the $64{\times}128$ grid; protocol details are in
-Appendix~\ref{app:so3-recon}.} For each source signal
-$f:S^2\to\R$ we apply a uniform random rotation $R\in\SO(3)$ to form
-the target $R\cdot f$, compute $\Phi_{\mathrm{sel}}(R\cdot f)$, and
-recover $\hat f$ by gradient descent from a random Gaussian
-initialisation, minimising
-$\lVert \Phi_{\mathrm{sel}}(\hat f) -
-\Phi_{\mathrm{sel}}(R\cdot f)\rVert^2 / \lVert
-\Phi_{\mathrm{sel}}(R\cdot f)\rVert^2$ (Adam,
-multi-restart, cosine-annealed LR). The orbit ambiguity is then
-resolved by Procrustes-style optimisation on a quaternion
-parameterisation: we fit $\hat R\in\SO(3)$ minimising $\lVert
-\hat R\cdot\hat f - R\cdot f\rVert^2$.
-
-\begin{figure}[t]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/spherical_mnist_reconstruction/figures/paper_orbits.pdf}
-  \caption{$\Phi_{\mathrm{sel}}$ reconstruction (proof of concept).
-    Top: target $f$ (digit classes 0--5). Bottom: $\hat R\!\cdot\!\hat f$,
-    recovered from $\Phi_{\mathrm{sel}}(f)$ by gradient descent and
-    Procrustes-aligned. Numbers: relative image residual
-    $\lVert\hat R\!\cdot\!\hat f - f\rVert/\lVert f\rVert$, median
-    ${\approx}0.20$ across the six classes (raw pre-alignment residual
-    ${\approx}1.3$).}
-  \label{fig:so3-reconstruction}
-\end{figure}
-
-\paragraph{Takeaways.}
-Bispectrum residuals reach
-$\lVert\Delta\Phi\rVert/\lVert\Phi\rVert \sim 10^{-4}$--$10^{-3}$
-(below the SHT-discretisation invariance noise floor of
-${\sim}8\times 10^{-3}$ at this grid); the aligned image residuals
-$\lVert\hat R\cdot\hat f - f\rVert/\lVert f\rVert \sim 0.1$--$0.3$
-are consistent with local-optimisation error or conditioning effects,
-and we did not observe a systematic missing-information failure mode,
-but the experiment does not rule out incompleteness or near-collisions
-in feature space. Visually, $\hat R\cdot\hat f$ aligns with $f$ up to
-rotation across all six digit classes shown
-(Figure~\ref{fig:so3-reconstruction}; the same behaviour holds across
-our 8-digit sweep, see Appendix~\ref{app:so3-recon}). We read this as
-proof-of-concept evidence that $\Phi_{\mathrm{sel}}$ preserves
-substantial orbit information on band-limited signals; a formal
-completeness proof remains open.
-
-
-\section{Discussion and Conclusion}
-\label{sec:discussion}
-
-We have presented \texttt{bispectrum}, a PyTorch library that makes
-selective $G$-bispectral invariants practical, covering seven
-group/domain pairs through a uniform \texttt{nn.Module} interface,
-and we contribute a $\Theta(L^2)$ selective $\SO(3)$ invariant on
-$S^2$ (Section~\ref{sec:so3-selective}). The selective forward pass
-adds negligible overhead ($<$1\,ms for finite-group modules) and
-delivers numerically exact rotation invariance with a consistent
-data-efficiency advantage over incomplete alternatives in the
-low-data / moderate-capacity regime.
-
-\paragraph{When not to use the bispectrum.}
-At high model capacity (Sec.~\ref{sec:experiments}), incomplete
-alternatives like max pooling can match or surpass bispectral
-pooling, and each module is invariant to a \emph{specific} group
-only---data augmentation or flexible-symmetry equivariant networks
-fit better when the relevant symmetry is multi-group or unknown.
-
-\paragraph{Limitations and future work.}
-Inversion from $\Phi_{\mathrm{sel}}$ is not shipped as a library
-API, and a formal completeness proof, a characterisation of the
-degenerate set, and numerical-conditioning analysis at large
-band-limits remain open. We do not compare against tensor-product
-architectures (e3nn, MACE) or wavelet scattering. Natural next
-steps are $\SO(3)$ inversion and cryo-EM / molecular applications.
-
-\begin{ack}
-Omitted for review.
-\end{ack}
-
-\bibliographystyle{plainnat}
-\bibliography{references}
-
-\appendix
-
-\section{Selective $\SO(3)$ Bispectrum -- Structural Background}
-\label{app:so3-proof}
-
-This appendix gives the structural background for the construction
-of Section~\ref{sec:so3-selective}: the $S^2$ band-limited setup and
-notation (Section~\ref{sec:app-setup}); the orbit-space dimension
-lower bound $(L{+}1)^2-3$ that motivates the $\Theta(L^2)$ target
-(Section~\ref{sec:app-lower-bound}); the explicit selective index set
-together with the implementation-grade BFS variant
-(Algorithm~\ref{alg:so3-augmented}) and per-degree counting result
-(Section~\ref{sec:app-selective-index}); and the parity structure of
-the scalar bispectrum that dictates the exclusion of certain
-odd-parity triples in $\mathcal{T}_\ell$
-(Section~\ref{sec:app-parity}). A short comparison with prior bispectral
-constructions (Section~\ref{sec:app-comparison}) and a pixel-space
-complexity table (Section~\ref{sec:app-pixel-complexity}) close the
-appendix. We do not claim a closed-form completeness proof in this
-revision; the empirical reconstruction of
-Figure~\ref{fig:so3-reconstruction} is the proof-of-concept evidence
-the construction is supported by.
-
-Per-degree entry allocation: the full seed
-$\mathcal{S}_{\mathrm{seed}}$ handles $\ell\le L_{\mathrm{seed}}=4$
-in constant size; for $\ell\ge 5$, Algorithm~\ref{alg:so3-augmented}
-allocates $2\ell{+}1$ bispectral rows plus $O(\ell)$
-self-coupling and CG-power entries, giving
-$|\Phi_{\mathrm{sel}}|=\Theta(L^2)$ total (matching the orbit-space
-lower bound $(L{+}1)^2-3$ of
-Proposition~\ref{prop:app-lower-bound}; exact per-$L$ counts in
-Table~\ref{tab:app-counts}).
-
-\subsection{Setup and notation}
-\label{sec:app-setup}
-
-\begin{definition}[Band-limited signal on $S^2$]
-A signal $f : S^2 \to \R$ is \emph{band-limited} at degree $L$ if its
-spherical harmonic expansion
-\begin{equation}\label{eq:app-sh-expansion}
-  f(\theta, \varphi)
-  = \sum_{\ell=0}^{L} \sum_{m=-\ell}^{\ell}
-    a_\ell^m \, \SH_\ell^m(\theta, \varphi)
-\end{equation}
-has $a_\ell^m = 0$ for all $\ell > L$. We write
-$\mathbf{F}_\ell = (a_\ell^{-\ell}, \dotsc, a_\ell^{\ell})
-  \in \C^{2\ell+1}$
-for the coefficient vector at degree~$\ell$.
-\end{definition}
-
-\noindent
-For real $f$, the conjugation symmetry
-$a_\ell^{-m} = (-1)^m \conj{a_\ell^m}$ holds, so
-$\mathbf{F}_\ell$ has $2\ell+1$ real degrees of freedom.
-The total signal dimension is
-$\dim = \sum_{\ell=0}^{L}(2\ell+1) = (L+1)^2$.
-
-\begin{definition}[Rotation action]
-Under $g \in \SO(3)$, the spherical harmonic coefficients transform as
-$\mathbf{F}_\ell \mapsto D^\ell(g)\,\mathbf{F}_\ell$,
-where $D^\ell$ is the $(2\ell+1)$-dimensional Wigner~$D$-matrix.
-\end{definition}
-
-\noindent
-In the irrep notation of~\citet{mataigne2024selective},
-$\{\rho_\ell\}_{\ell\ge 0}$ are the (unitary) irreducible
-representations of $\SO(3)$ with $V_{\rho_\ell}$ the
-$(2\ell{+}1)$-dimensional space of degree-$\ell$ spherical harmonics
-and $\rho_\ell(g)=D^\ell(g)$; the Fourier coefficient of $f$ at
-$\rho_\ell$ is $\mathcal{F}(f)_{\rho_\ell}=\mathbf{F}_\ell$. We use
-$\mathbf{F}_\ell$ as the working symbol throughout.
-
-\begin{definition}[Bispectrum on $S^2$]\label{def:app-bispectrum}
-For each triple $(\ell_1, \ell_2, \ell)$ satisfying the triangle
-inequality $\abs{\ell_1 - \ell_2} \le \ell \le \ell_1 + \ell_2$,
-the \emph{bispectral coefficient} is
-\begin{equation}\label{eq:app-bispectrum}
-  \beta_{\ell_1,\ell_2,\ell}
-  = \sum_{\substack{m_1,\,m_2,\,m \\ m_1+m_2=m}}
-    \CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}\;
-    a_{\ell_1}^{m_1}\,a_{\ell_2}^{m_2}\,\conj{a_\ell^m},
-\end{equation}
-where $\CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}$ denotes the
-Clebsch--Gordan coefficient
-$\langle \ell_1, m_1;\, \ell_2, m_2 \mid \ell, m \rangle$.
-Each $\beta_{\ell_1,\ell_2,\ell}$ is a scalar; for real~$f$ it is
-real when $\ell_1{+}\ell_2{+}\ell$ is even and purely imaginary when
-$\ell_1{+}\ell_2{+}\ell$ is odd (Lemma~\ref{lem:app-parity}).
-\end{definition}
-
-Definition~\ref{def:app-bispectrum} is the specialization of the
-general $G$-bispectrum (eq.~\eqref{eq:bispectrum-def}) to the case
-$G = \SO(3)$ acting on~$S^2$, where the irreps are Wigner-$D$
-matrices and the Fourier coefficients are spherical harmonic
-coefficients~$a_\ell^m$.
-
-\begin{definition}[CG power spectrum, optional diagnostic]\label{def:app-cg-power}
-For a triple $(\ell_1, \ell_2, \ell)$ satisfying the triangle
-inequality, write
-$(\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})\big|_\ell$
-for the projection of the Kronecker product
-$\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2}$ onto the
-$D^\ell$-isotypic component, realised by contracting with the CG
-coefficients $\CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}$. The
-\emph{CG power spectrum entry} is the squared norm of this projection,
-\begin{equation}\label{eq:app-cg-power}
-  P_{\ell_1,\ell_2,\ell}
-  := \lVert (\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell
-     \rVert^2
-  = \sum_{m=-\ell}^{\ell}
-    \Bigl\lvert \sum_{\substack{m_1,m_2 \\ m_1+m_2=m}}
-      \CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}\;
-      a_{\ell_1}^{m_1}\, a_{\ell_2}^{m_2} \Bigr\rvert^2.
-\end{equation}
-This is a degree-$4$, real-valued, $\SO(3)$-invariant polynomial in
-the signal coefficients. The library appends a $O(L^2)$ family of CG
-power scalars to the selective bispectrum unconditionally for
-numerical robustness on the real-signal slice
-(Remark~\ref{rem:app-Tell-rank-deficit}); they leave the asymptotic
-output size unchanged.
-\end{definition}
-
-\begin{proposition}[Size of the full bispectrum]\label{prop:app-full-size}
-The number of admissible triples $(\ell_1, \ell_2, \ell)$ with
-$0 \le \ell_1, \ell_2, \ell \le L$ satisfying the triangle inequality
-$|\ell_1 - \ell_2| \le \ell \le \ell_1 + \ell_2$ is $\Theta(L^3)$.
-\end{proposition}
-\begin{proof}
-For fixed $\ell_1$ and $\ell_2$, the triangle inequality admits
-$\min(\ell_1 + \ell_2, L) - |\ell_1 - \ell_2| + 1$ valid values of
-$\ell$. Summing over $(\ell_1,\ell_2)\in[0,L]^2$ gives
-$N_{\mathrm{full}} = \Theta(L^3)$.
-\end{proof}
-
-\begin{remark}[Invariance]
-Each $\beta_{\ell_1,\ell_2,\ell}$ is $\SO(3)$-invariant: the Wigner
-matrices satisfy $D^{\ell_1} \otimes D^{\ell_2}
-= \bigoplus_\ell D^\ell$ under the CG decomposition, so the triple
-product contracted with CG coefficients is unchanged by
-$\mathbf{F}_\ell \mapsto D^\ell(g)\mathbf{F}_\ell$.
-\end{remark}
-
-\subsection{Why $\mathcal{O}(L)$ is impossible}
-\label{sec:app-lower-bound}
-
-\begin{proposition}[Dimensional lower bound]\label{prop:app-lower-bound}
-Any \emph{smooth or polynomial} complete $\SO(3)$-invariant for
-band-limited signals on $S^2$ at band-limit $L$ must have codomain
-of dimension at least $(L+1)^2 - 3$.
-\end{proposition}
-\begin{proof}
-Let $\Phi : \R^{(L+1)^2} \to \R^k$ be a smooth (or polynomial)
-$\SO(3)$-invariant map that separates orbits on a Zariski-open
-dense subset $U\subset\R^{(L+1)^2}$. Smoothness implies $\Phi$ has a
-well-defined generic rank $r := \max_{x\in U} \mathrm{rank}\,
-d\Phi_x$; on the open dense subset where this rank is attained,
-$\Phi$ is locally a submersion onto an $r$-dimensional submanifold,
-and orbit-separation forces $r \ge \dim(\R^{(L+1)^2}/\SO(3))
-= (L+1)^2 - \dim\SO(3) = (L+1)^2 - 3$
-(generic-rank theorem for $\SO(3)$ acting on $V_L$;
-cf.~Tarski--Seidenberg in the polynomial case). Hence
-$k \ge r \ge (L+1)^2 - 3 = \Omega(L^2)$.
-\end{proof}
-
-\begin{remark}
-The chain $\beta_{1,\ell,\ell+1}$ for $\ell = 0, \dotsc, L-1$ provides
-only $\sim 3L$ scalar constraints (since each chain entry couples
-$\mathbf{F}_1 \in \C^3$ with one known $\mathbf{F}_\ell$). This cannot
-recover $(L+1)^2 \approx L^2$ unknowns.
-\end{remark}
-
-\begin{remark}[Smoothness assumption]
-The smoothness/polynomial qualifier is essential: pathological
-non-measurable invariants (e.g.\ a fixed orbit-section indexed by
-the axiom of choice) can in principle have codomain dimension $1$
-but are not implementable. All practical bispectral or spectral
-invariants are polynomial, so the proposition applies.
-\end{remark}
-
-\subsection{The selective index set}
-\label{sec:app-selective-index}
-
-For each target degree $\ell$, the library's selective invariant
-allocates $2\ell{+}1$ linear bispectral entries, plus quadratic/cubic
-self-couplings and CG-power scalars for numerical robustness on the
-real-signal slice. Combined with a constant-size full-bispectrum seed
-at degrees $\le L_{\mathrm{seed}}=\min(L,4)$, this gives an
-$O(L^2)$-sized invariant matching the orbit-space lower bound of
-Proposition~\ref{prop:app-lower-bound}.
-
-\begin{definition}[Augmented selective $\SO(3)$-on-$S^2$ invariant]\label{def:app-selective}
-Set $L_{\mathrm{seed}} := \min(L,4)$. Let
-$\mathcal{S}_{\mathrm{seed}}$ denote the bispectral triples at
-degrees $\ell\le L_{\mathrm{seed}}$ returned by
-Algorithm~\ref{alg:so3-augmented}'s seed phase: the pre-prune set
-collects all admissible triples
-$\{(\ell_1,\ell_2,\ell):
-0\le\ell_1\le\ell_2\le L_{\mathrm{seed}},\,
-|\ell_1{-}\ell_2|\le\ell\le\min(\ell_1{+}\ell_2,L_{\mathrm{seed}})\}$
-($42$ triples at $L_{\mathrm{seed}}=4$); Algorithm~\ref{alg:so3-augmented}
-then parity-prunes entries that vanish identically on real signals
-(Lemma~\ref{lem:app-parity}), deduplicates, and caps each degree
-to its per-degree budget. The surviving bispectral count at $L=4$
-is $24$ (see Table~\ref{tab:app-counts}, column ``Bispectral''),
-and $|\mathcal{S}_{\mathrm{seed}}|$ is constant in $L$ for $L\ge 4$.
-For each intermediate target $4\le\ell\le 7$ Algorithm~\ref{alg:so3-augmented}
-uses the small explicit linear block $\mathcal{T}_\ell^{\mathrm{small}}$
-(Table~\ref{tab:Tsmall}); for each high-degree target $\ell\ge 8$
-the \emph{bootstrap block} $\mathcal{T}_\ell$ is
-\begin{equation}\label{eq:app-Tell}
-\begin{aligned}
-  \mathcal{T}_\ell
-  := &\ \{(a,\ell,\ell-a) : 1\le a \le \ell{-}1\} \\
-   &\cup \bigl\{(a,\ell,\ell-a+1) : 2\le a \le \ell{-}1,
-        \; a \ne (\ell+1)/2 \bigr\} \\
-   &\cup \{(a,\ell-a,\ell) : 1\le a \le 4\} \;\cup\; \mathcal{Z}_\ell,
-\end{aligned}
-\end{equation}
-where $\mathcal{Z}_\ell = \emptyset$ for even $\ell$ and
-$\mathcal{Z}_\ell = \{(2,\ell{-}1,\ell)\}$ for odd $\ell\ge 9$. The
-exclusion $a\ne (\ell+1)/2$ for odd $\ell$ removes the self-conjugate
-triple $(r,2r-1,r)$ at $\ell = 2r-1$, which has a repeated index and
-vanishes identically on real signals
-(Proposition~\ref{prop:app-odd-vanishing}); the compensating row
-$(2,\ell{-}1,\ell)\in\mathcal{Z}_\ell$ has all-distinct indices and
-does not overlap the chain family (which only covers pairs
-$(a,\ell{-}a)$ with $a\le 4$). Size bookkeeping:
-$(\ell{-}1)+(\ell{-}2)+4 = 2\ell{+}1$ for even $\ell$, and
-$(\ell{-}1)+(\ell{-}3)+4+1 = 2\ell{+}1$ for odd $\ell\ge 9$; the first
-two families differ in the last index ($\ell{-}a$ vs.\ $\ell{-}a+1$),
-and $\mathcal{Z}_\ell$ at $a=2$ has second coordinate $\ell{-}1$
-rather than $\ell{-}2$, so all three subfamilies are disjoint.
-
-The \emph{augmented selective $\SO(3)$-on-$S^2$ invariant}
-$\Phi_{\mathrm{sel}}$ is the output of
-Algorithm~\ref{alg:so3-augmented}: bispectral triples
-$\mathcal{S}_\beta = \mathcal{S}_{\mathrm{seed}}\cup
-\bigcup_{\ell=5}^{L}\mathcal{T}_\ell$ (with
-$\mathcal{T}_\ell = \mathcal{T}_\ell^{\mathrm{small}}$ for
-$5\le\ell\le 7$), mandatory even self-couplings
-$\beta_{\ell,\ell,\ell'}$ ($\ell'$ even, $2\le\ell'\le\ell$), and
-CG-power scalars $P_{\ell_1,\ell_2,\ell} = \lVert(\mathbf{F}_{\ell_1}
-\otimes \mathbf{F}_{\ell_2})|_\ell\rVert^2$
-(Definition~\ref{def:app-cg-power}) chosen greedily for per-degree
-Jacobian rank. Mathematically, $\Phi_{\mathrm{sel}}:V_L\to\R^N$ has
-$N$ live real scalar components: by
-Lemma~\ref{lem:app-parity}, each bispectral entry contributes one
-real component on the real-signal slice (real for even parity,
-purely imaginary for odd), and CG-power entries are real-valued
-degree-$4$ invariants. The PyTorch implementation stores each
-scalar in a complex tensor, so downstream layers receive $2N$ real
-channels after splitting real and imaginary parts; for bispectral
-entries exactly one of the two channels is structurally zero by
-parity, and for CG-power entries the imaginary channel is
-structurally zero.
-\end{definition}
-
-\begin{table}[h]
-  \caption{Explicit low-degree bootstrap blocks
-    $\mathcal{T}_\ell^{\mathrm{small}}$ for $4\le\ell\le 7$, copied
-    from \texttt{bispectrum.so3\_on\_s2.\_small\_linear\_bootstrap\_block}.
-    Each triple $(\ell_1,\ell_2,\ell_3)$ denotes
-    $\beta_{\ell_1,\ell_2,\ell_3}$; all are admissible and have
-    $\ell$ in at least one slot. Entry counts
-    $|\mathcal{T}_4^{\mathrm{small}}|=10$,
-    $|\mathcal{T}_5^{\mathrm{small}}|=11$,
-    $|\mathcal{T}_6^{\mathrm{small}}|=13$,
-    $|\mathcal{T}_7^{\mathrm{small}}|=15$ match the per-degree budgets.
-    $\mathcal{T}_4^{\mathrm{small}}$ is implementation-only:
-    Algorithm~\ref{alg:so3-augmented} uses it as a ranked candidate
-    list at $\ell=4$, deduplicated against the seed
-    $\mathcal{S}_{\mathrm{seed}}$ and capped to a budget of $10$ entries;
-    $\mathcal{T}_5^{\mathrm{small}},\mathcal{T}_6^{\mathrm{small}},\mathcal{T}_7^{\mathrm{small}}$
-    are the formal bootstrap blocks at $\ell\in\{5,6,7\}$.}
-  \label{tab:Tsmall}
-  \centering
-  \small
-  \setlength{\tabcolsep}{4pt}
-  \footnotesize
-  \begin{tabular}{@{}cp{0.92\linewidth}@{}}
-    \toprule
-    $\ell$ & $\mathcal{T}_\ell^{\mathrm{small}}$ (each entry a triple $(\ell_1,\ell_2,\ell_3)$) \\
-    \midrule
-    $4$ & $(1,3,4),$ $(2,2,4),$ $(2,3,4),$ $(3,3,4),$ $(1,4,3),$ $(2,4,2),$ $(3,4,1),$ $(2,4,3),$ $(3,4,2),$ $(3,4,3)$ \\
-    $5$ & $(1,4,5),$ $(2,3,5),$ $(2,4,5),$ $(3,4,5),$ $(1,5,4),$ $(2,5,3),$ $(3,5,2),$ $(4,5,1),$ $(2,5,4),$ $(3,5,4),$ $(4,5,4)$ \\
-    $6$ & $(1,5,6),$ $(2,4,6),$ $(3,3,6),$ $(3,4,6),$ $(1,6,5),$ $(2,6,4),$ $(3,6,3),$ $(4,6,2),$ $(5,6,1),$ $(2,6,5),$ $(3,6,5),$ $(4,6,5),$ $(5,6,5)$ \\
-    $7$ & $(1,6,7),$ $(2,5,7),$ $(3,4,7),$ $(4,5,7),$ $(1,7,6),$ $(2,7,5),$ $(3,7,4),$ $(4,7,3),$ $(5,7,2),$ $(6,7,1),$ $(2,7,6),$ $(3,7,6),$ $(4,7,6),$ $(5,7,6),$ $(6,7,6)$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\begin{table}[h]
-  \caption{Coefficient counts of $\Phi_{\mathrm{sel}}$ at
-    representative band-limits. ``Bispectral'' counts the triples in
-    $\mathcal{S}_\beta$ (including mandatory even self-couplings);
-    ``CG-power'' counts entries in $\mathcal{S}_P$; ``Total $N$'' is
-    $|\Phi_{\mathrm{sel}}|$ (live real components) and matches the
-    library's \texttt{output\_size}. ``Stored channels'' counts the
-    complex-tensor storage ($2N$) seen by the SMNIST classifier;
-    one channel per bispectral entry is structurally zero by
-    parity, and the imaginary channel of each CG-power entry is
-    structurally zero.}
-  \label{tab:app-counts}
-  \centering
-  \small
-  \setlength{\tabcolsep}{6pt}
-  \begin{tabular}{ccccc}
-    \toprule
-    $L$ & Bispectral & CG-power & Total $N=|\Phi_{\mathrm{sel}}|$
-        & Stored channels ($2N$) \\
-    \midrule
-    $4$  & $24$  & $10$  & $34$  & $68$  \\
-    $5$  & $37$  & $17$  & $54$  & $108$ \\
-    $15$ & $307$ & $77$  & $384$ & $768$ \\
-    $16$ & $348$ & $82$  & $430$ & $860$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\begin{algorithm}[t]
-\caption{Augmented selective index set for the $\SO(3)$ bispectrum
-  on $S^2$ at band-limit~$L$. Returns bispectral triples
-  $\mathcal{S}_\beta$ and CG power triples $\mathcal{S}_P$.
-  This extends the BFS-based selective index construction of
-  \citet{mataigne2024selective} from finite groups to $\SO(3)$ on $S^2$,
-  adding the CG power augmentation step that is specific to real
-  signals.}
-\label{alg:so3-augmented}
-\small
-\begin{algorithmic}[1]
-\STATE \textbf{Input:} Band-limit $L$
-\STATE \textbf{Output:} Bispectral set $\mathcal{S}_\beta$, CG power
-  set $\mathcal{S}_P$
-\STATE $\mathcal{S}_\beta \leftarrow \{(0,0,0)\}$;
-  $\mathcal{S}_P \leftarrow \emptyset$
-\STATE \textbf{Seed block:} add to $\mathcal{S}_\beta$ all triples
-  $(\ell_1, \ell_2, \ell)$ with $0 \le \ell_1 \le \ell_2 \le L_0 = 4$
-  and $|\ell_1 - \ell_2| \le \ell \le \min(\ell_1 + \ell_2, L_0)$
-  \COMMENT{constant-size full-bispectrum seed}
-\FOR{$\ell = 1, \dotsc, L$}
-  \STATE $\text{budget} \leftarrow 2\ell + 1$
-  \IF{$\ell = 4$}
-    \STATE $\text{budget} \leftarrow 10$
-      \COMMENT{keep all chain+cross at $\ell=4$}
-  \ENDIF
-  \STATE $\text{candidates} \leftarrow [\,]$
-  \IF{$4 \le \ell \le 7$}
-    \STATE append the explicit low-degree linear block
-      $\mathcal{T}^{\mathrm{small}}_\ell$ to candidates
-  \ELSIF{$\ell \ge 8$}
-    \FOR{$a = 1, \dotsc, \ell{-}1$}
-      \STATE append $(a,\, \ell,\, \ell{-}a)$ to candidates
-    \ENDFOR
-    \FOR{$a = 2, \dotsc, \ell{-}1$}
-      \IF{\NOT ($\ell$ is odd \AND $a = (\ell{+}1)/2$)}
-        \STATE append $(a,\, \ell,\, \ell{-}a{+}1)$ to candidates
-          \COMMENT{skip $(r,2r{-}1,r)\equiv0$ on real signals,
-            Prop.~\ref{prop:app-odd-vanishing}}
-      \ENDIF
-    \ENDFOR
-    \FOR{$a = 1, \dotsc, 4$}
-      \STATE append $(a,\, \ell{-}a,\, \ell)$ to candidates
-    \ENDFOR
-    \IF{$\ell$ is odd}
-      \STATE append $(2,\, \ell{-}1,\, \ell)$ to candidates
-        \COMMENT{compensate excluded $(r,2r{-}1,r)$ row}
-    \ENDIF
-  \ELSE
-    \STATE append low-degree chain candidates
-    \IF{$\ell \ne 2$}
-      \STATE append cross candidates
-        \COMMENT{$\beta_{1,2,1}$ excluded}
-    \ENDIF
-  \ENDIF
-  \STATE append $(0,\, \ell,\, \ell)$ to candidates
-    \COMMENT{power entry}
-  \IF{$\ell \notin \{1,\, 2\}$}
-    \FOR{$\ell' = \ell{-}1, \dotsc, 0$}
-      \STATE append $(\ell,\, \ell,\, \ell')$ to candidates
-    \ENDFOR
-  \ENDIF
-  \IF{$\ell = 2$}
-    \STATE append $(\ell,\, \ell,\, \ell)$ to candidates
-      \COMMENT{$\beta_{2,2,2}$}
-  \ENDIF
-  \STATE Remove entries that vanish for real signals; deduplicate
-    against $\mathcal{S}_\beta$ already added by the seed block
-  \STATE $\mathcal{S}_\beta \leftarrow \mathcal{S}_\beta
-    \;\cup\;$ first $\min(\text{budget},\,
-    |\text{candidates}|)$ live entries
-  \STATE \COMMENT{\textbf{Mandatory even self-couplings}}
-  \FOR{$\ell' = 2, 4, \dotsc, \lfloor\ell/2\rfloor \cdot 2$
-    with $\ell' \le \ell$ and $\ell \ge 3$}
-    \STATE $\mathcal{S}_\beta \leftarrow \mathcal{S}_\beta
-      \cup \{(\ell,\, \ell,\, \ell')\}$
-  \ENDFOR
-  \STATE \COMMENT{\textbf{CG power augmentation}}
-  \STATE Greedily add $P_{\ell_1,\ell_2,\ell'}$ (with
-    $\max(\ell_1,\ell_2) = \ell$, $|\ell_1{-}\ell_2| \le \ell'
-    \le \ell_1{+}\ell_2$) to $\mathcal{S}_P$ until the
-    Jacobian rank of the per-degree restriction
-    $\Phi_{\mathrm{sel}}|_\ell$ matches the number of unknowns
-\ENDFOR
-\RETURN $(\mathcal{S}_\beta,\, \mathcal{S}_P)$
-\end{algorithmic}
-\end{algorithm}
-
-\begin{proposition}[Output size]\label{prop:app-output-size}
-For every $L\ge 4$, Algorithm~\ref{alg:so3-augmented} produces a
-$\Theta(L^2)$-sized invariant: the per-degree bootstrap block
-contributes at most $2\ell{+}1$ bispectral triples
-(Eq.~\eqref{eq:app-Tell} for $\ell\ge 8$ plus
-Table~\ref{tab:Tsmall} for $5\le\ell\le 7$), and the self-coupling
-and CG-power augmentations add $O(\ell)$ per degree, giving a total
-of $\Theta(L^2)$ real scalars. Per-$L$ counts are listed in
-Table~\ref{tab:app-counts}; both panels match the orbit-space
-lower bound $(L{+}1)^2-3$ of Proposition~\ref{prop:app-lower-bound}
-up to a constant factor.
-\end{proposition}
-
-
-
-\subsection{Parity structure of the bispectrum}
-\label{sec:app-parity}
-
-The selective construction of Section~\ref{sec:so3-selective}
-exploits two structural facts about the scalar bispectrum
-$\beta_{\ell_1,\ell_2,\ell}$: a parity rule that fixes whether each
-entry is real or purely imaginary on real signals (Lemma~\ref{lem:app-parity})
-and the resulting identical vanishing of certain odd-parity entries
-with a repeated index (Proposition~\ref{prop:app-odd-vanishing}). The
-latter dictates the explicit exclusion in $\mathcal{T}_\ell$ at odd
-target degrees, and the former is the source of the rank deficit
-that motivates the CG-power augmentation of
-Algorithm~\ref{alg:so3-augmented} (Remark~\ref{rem:app-Tell-rank-deficit}).
-
-\begin{lemma}[Parity of the bispectrum]\label{lem:app-parity}
-For any signal (real or complex):
-\begin{equation}\label{eq:app-parity}
-  \beta_{\ell_1,\ell_2,\ell}(T_R \cdot f)
-  = (-1)^{\ell_1+\ell_2+\ell}\,\beta_{\ell_1,\ell_2,\ell}(f).
-\end{equation}
-For \emph{real} signals, there is an independent constraint:
-\begin{equation}\label{eq:app-reality-constraint}
-  \conj{\beta_{\ell_1,\ell_2,\ell}(f)}
-  = (-1)^{\ell_1+\ell_2+\ell}\,\beta_{\ell_1,\ell_2,\ell}(f).
-\end{equation}
-Together:
-\begin{enumerate}[nosep,label=(\alph*)]
-\item If $\ell_1+\ell_2+\ell$ is even:
-  $\beta$ is real-valued (by~\eqref{eq:app-reality-constraint})
-  and $T_R$-invariant (by~\eqref{eq:app-parity}).
-\item If $\ell_1+\ell_2+\ell$ is odd:
-  $\beta$ is purely imaginary (by~\eqref{eq:app-reality-constraint})
-  and $\beta(T_R \cdot f) = -\beta(f)$, so $\im\beta$ flips sign under $T_R$.
-\end{enumerate}
-\end{lemma}
-\begin{proof}
-Under $T_R$, each $a_\ell^m \mapsto (-1)^m a_\ell^{-m}$.
-Substituting into $\beta = \sum_m (\sum_{m_1+m_2=m} C\, a^{m_1} a^{m_2})
-\conj{a^m}$, the three sign factors $(-1)^{m_1+m_2+m} = (-1)^{2m} = 1$
-cancel. Re-indexing $m \mapsto -m$ and applying the CG symmetry
-$\CG{\ell_1}{-m_1}{\ell_2}{-m_2}{\ell}{-m}
-  = (-1)^{\ell_1+\ell_2-\ell}\,\CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}$
-yields~\eqref{eq:app-parity} (with $(-1)^{\ell_1+\ell_2-\ell} = (-1)^{\ell_1+\ell_2+\ell}$).
-
-For real signals, $\conj{a_\ell^m} = (-1)^m a_\ell^{-m}$.
-Taking the complex conjugate of $\beta$ and re-indexing similarly
-yields~\eqref{eq:app-reality-constraint}.
-\end{proof}
-
-\begin{lemma}[CG power is $T_R$-invariant]\label{lem:app-P-invariant}
-For all triples $(\ell_1,\ell_2,\ell)$:
-$P_{\ell_1,\ell_2,\ell}(T_R \cdot f) = P_{\ell_1,\ell_2,\ell}(f)$.
-\end{lemma}
-\begin{proof}
-Under $T_R$, the CG-projected component transforms as
-$(\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell^M
-  \mapsto (-1)^{\ell_1+\ell_2+\ell+M}\,
-  (\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell^{-M}$.
-Taking the squared norm and re-indexing the sum over $M$
-leaves $P$ unchanged.
-\end{proof}
-
-\begin{proposition}[Vanishing of odd-parity entries with a repeated index]
-\label{prop:app-odd-vanishing}
-Let $\beta_{\ell_1,\ell_2,\ell}$ be the scalar bispectrum entry of a
-real signal $f$ (so $\ell_1+\ell_2+\ell$ odd). If the index multiset
-$\{\ell_1,\ell_2,\ell\}$ contains a repeated value, then
-$\beta_{\ell_1,\ell_2,\ell}(f)\equiv 0$ as a polynomial in the SH
-coefficients of $f$.
-\end{proposition}
-
-\begin{proof}
-Define the symmetric scalar 3j contraction
-\[
-  T_{\ell_1,\ell_2,\ell_3}(f) :=
-  \sum_{m_1,m_2,m_3}
-  \begin{pmatrix} \ell_1 & \ell_2 & \ell_3 \\
-                  m_1 & m_2 & m_3 \end{pmatrix}\,
-  a_{\ell_1}^{m_1}\, a_{\ell_2}^{m_2}\, a_{\ell_3}^{m_3} .
-\]
-Converting the Wigner~3j into a Clebsch-Gordan coefficient via
-$\bigl(\begin{smallmatrix} j_1 & j_2 & j_3 \\
-       m_1 & m_2 & m_3\end{smallmatrix}\bigr)
-= \frac{(-1)^{j_1-j_2-m_3}}{\sqrt{2j_3+1}}\,
-  \CG{j_1}{m_1}{j_2}{m_2}{j_3}{-m_3}$ and applying the reality identity
-$a_{\ell_3}^{m_3} = (-1)^{m_3}\,\conj{a_{\ell_3}^{-m_3}}$ shows
-\[
-  \beta_{\ell_1,\ell_2,\ell}(f)
-  = (-1)^{\ell_1+\ell_2}\sqrt{2\ell+1}\,
-    T_{\ell_1,\ell_2,\ell}(f),
-\]
-so $\beta$ and $T$ have the same vanishing locus. The 3j symbol is
-invariant under even column permutations and picks up
-$(-1)^{\ell_1+\ell_2+\ell_3}$ under odd ones. If two columns $i,j$
-satisfy $\ell_i=\ell_j$, the inserted vectors $a_{\ell_i},a_{\ell_j}$
-are literally the same coefficient vector of $f$, so swapping those
-two columns leaves the contracted sum $T$ unchanged while multiplying
-the 3j prefactor by $(-1)^{\ell_1+\ell_2+\ell_3}$. For odd parity this
-forces $T=-T$, hence $T\equiv 0$ and $\beta\equiv 0$.
-\end{proof}
-
-\begin{remark}[Consequences for $\mathcal{T}_\ell$]
-\label{rem:app-Tell-zero-rows}
-Proposition~\ref{prop:app-odd-vanishing} applies in particular to the
-family $(r,2r-1,r)$: it has parity $4r-1$ (odd) and a repeated index,
-so $\beta_{r,2r-1,r}\equiv 0$ on real signals. Numerical evaluation at
-the seed-42 real witness confirms this: for $r=2,\dots,7$ the entry is
-zero to double-precision rounding ($|\beta|\lesssim 10^{-14}$). Any
-formal definition of $\mathcal{T}_\ell$ must therefore exclude these
-triples (see~\eqref{eq:app-Tell}); we substitute a chain row with
-all-distinct indices in their place.
-\end{remark}
-
-\begin{corollary}[First nonzero odd-parity entry]\label{cor:app-first-odd}
-The first bispectral entry with odd parity and generically nonzero imaginary
-part occurs at degree $\ell = 4$:
-$\beta_{2,3,4}$, with
-$\ell_1+\ell_2+\ell = 9$ (odd) and
-$\ell_1, \ell_2, \ell$ all distinct.
-At the deterministic witness (seed~42), $\im(\beta_{2,3,4}) \approx 3.26 \ne 0$,
-and $\im(\beta_{2,3,4})(T_R \cdot f) \approx -3.26$.
-Similarly, $\beta_{2,4,3}$ and $\beta_{3,4,2}$ are nonzero odd-parity entries
-at degree~$4$.
-\end{corollary}
-
-\begin{remark}[No parity-breaking at $\ell \le 3$]\label{rem:app-no-parity-breaking-le3}
-For degrees $\le 3$, every odd-parity triple $(\ell_1,\ell_2,\ell)$
-has at least two indices equal (since $\max(\ell_1,\ell_2,\ell) \le 3$
-and $\ell_1+\ell_2+\ell$ odd forces at least one pair to be equal in this range).
-By Proposition~\ref{prop:app-odd-vanishing}, all odd-parity entries vanish.
-Therefore, the $T_R$ ambiguity cannot be resolved using only degrees $\le 3$.
-\end{remark}
-
-\begin{remark}[Why the augmented block is needed]
-\label{rem:app-Tell-rank-deficit}
-The bootstrap block $\mathcal{T}_\ell$ alone is rank-deficient on the
-real-signal slice: at the seed-42 real witness, the
-$(2\ell{+}1)\times(2\ell{+}1)$ submatrix $A_\ell$ assembled from the
-$\mathcal{T}_\ell$ rows has SVD rank
-$\approx \ell{+}2$ rather than $2\ell{+}1$ for $\ell\in\{8,\dots,15\}$
-(verified numerically; the deficit is generic over real witnesses).
-The mandatory even self-couplings and CG-power scalars are quadratic
-or cubic in $\mathbf{F}_\ell$ and supply the additional independent
-real equations needed to fill out the per-degree real Jacobian; the
-augmented per-degree block (linear $\mathcal{T}_\ell$ plus quadratic
-self-couplings plus CG-power scalars) is what
-Algorithm~\ref{alg:so3-augmented} assembles, and is what the
-implementation actually solves at degrees $\ell\ge 5$.
-\end{remark}
-
-\subsection{Comparison with existing results}
-\label{sec:app-comparison}
-
-\begin{table}[h]
-  \caption{Comparison of bispectral invariants for signals on $S^2$ or
-    finite groups.}
-  \label{tab:app-comparison}
-  \centering
-  \small
-  \setlength{\tabcolsep}{4pt}
-  \begin{tabular}{@{}l l c c@{}}
-    \toprule
-    Result & Invariant & Size & Complete? \\
-    \midrule
-    Kakarala~\citep{kakarala2012bispectrum}
-      & Full bispec. & $O(L^3)$ & Yes (constr.) \\
-    Edidin \& Satriano~\citep{edidin2024orbit}
-      & Full bispec.\ (band-lim.)
-      & $O(L^3)$ & Yes (generic) \\
-    Bendory et al.~\citep{bendory2025orbit}
-      & Freq.\ marching
-      & $O(L^2)$/shell & Yes (3~shells) \\
-    Mataigne et al.~\citep{mataigne2024selective}
-      & Sel.\ $G$-bispec.\ (finite)
-      & $O(|G|)$ & Yes (exact) \\
-    \textbf{This work}
-      & \textbf{Sel.\ $\SO(3)$ on $S^2$}
-      & $\boldsymbol{\Theta(L^2)}$
-      & \textbf{Empirical (Section~\ref{sec:so3-recon})} \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\subsection{Complexity in pixel space}
-\label{sec:app-pixel-complexity}
-
-On $S^2$, a band-limited signal at degree~$L$ is sampled on an
-equiangular grid of size $N_\theta \times N_\varphi \approx 2L \times 2L$,
-so $P = 4L^2$ and $L = \Theta(\sqrt{P})$.
-
-\paragraph{Output size.}
-\begin{center}
-\small
-\begin{tabular}{lcc}
-  \toprule
-  Invariant & In $L$ & In $P$ \\
-  \midrule
-  Power spectrum & $O(L)$ & $O(\sqrt{P})$ \\
-  \textbf{Selective bispectrum} & $\boldsymbol{O(L^2)}$
-    & $\boldsymbol{O(P)}$ \\
-  Full bispectrum & $O(L^3)$ & $O(P^{3/2})$ \\
-  \bottomrule
-\end{tabular}
-\end{center}
-The selective bispectrum is \emph{linear} in the number of pixels.
-
-\paragraph{Computation cost.}
-\begin{center}
-\small
-\begin{tabular}{lcccc}
-  \toprule
-  Variant & Entries & Cost/entry & Total & In $P$ \\
-  \midrule
-  Full & $O(L^3)$ & $O(L)$ & $O(L^4)$ & $O(P^2)$ \\
-  Selective & $O(L^2)$ & $O(L)$ & $O(L^3)$ & $O(P^{3/2})$ \\
-  \bottomrule
-\end{tabular}
-\end{center}
-
-\paragraph{Concrete example.}
-For a $128 \times 256$ grid ($P \approx 32\mathrm{K}$, $L = 64$): the
-full bispectrum produces ${\sim}87\mathrm{K}$ scalars at
-${\sim}17\mathrm{M}$~ops; the selective bispectrum produces
-${\sim}4\mathrm{K}$ scalars at ${\sim}260\mathrm{K}$~ops---a
-$65\times$ reduction.
-
-\section{Extended Benchmark Results}
-\label{app:bench}
-
-This appendix supplements the summary in Table~\ref{tab:benchmarks}
-(main text) with scaling plots produced by
-\texttt{benchmarks/benchmark.py} on a single NVIDIA H100 80\,GB GPU
-(PyTorch 2.10).
-
-\subsection{Coefficient-count scaling}
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/benchmarks/paper_fig1.pdf}
-  \caption{Bispectral coefficient count vs.\ group order $|G|$. Solid
-    lines: selective ($O(|G|)$); dashed: full ($O(|G|^2)$). At
-    $|G|{=}1{,}024$, selectivity yields a $512\times$ reduction.
-    Continuous groups ($\SO(2)$, $\SO(3)$) are plotted at their
-    band-limited signal dimension; full-bispectrum curves for $D_n$
-    and the disk are omitted (backends not yet implemented).}
-  \label{fig:coeff-scaling}
-\end{figure}
-
-\subsection{Forward-pass timing}
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/benchmarks/paper_fig2.pdf}
-  \caption{Forward-pass wall-clock time (batch\,=\,16, H100 GPU).
-    \textbf{(a)}~$C_n$: the selective pass stays flat at
-    ${\sim}$0.14\,ms regardless of $n$, while the full bispectrum
-    grows super-linearly (17\,ms at $n{=}256$).
-    \textbf{(b)}~$\mathbb{T}^2$: selective is constant at
-    ${\sim}$0.08\,ms up to $|G|{=}65{,}536$.}
-  \label{fig:forward-timing}
-\end{figure}
-
-The selective forward pass is dominated by kernel-launch overhead
-rather than arithmetic, making it effectively free relative to backbone
-computation. The full bispectrum exhibits the expected super-linear
-growth and becomes impractical beyond $|G| \approx 256$ for $C_n$.
-
-\subsection{GPU throughput and inversion}
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/benchmarks/paper_fig3.pdf}
-  \caption{\textbf{(a)}~GPU throughput (samples/s) vs.\ batch size for
-    the selective forward pass. $C_n$ and $\mathbb{T}^2$ exceed
-    $10^7$~samples/s; the octahedral module reaches
-    $5.7 \times 10^6$~samples/s. All modules scale linearly with batch
-    size.
-    \textbf{(b)}~Inversion wall-clock time. Abelian groups ($C_n$,
-    $D_n$, $\SO(2)$/disk) scale roughly linearly. $\SO(3)$ on $S^2$
-    inversion is not yet implemented.}
-  \label{fig:gpu-scaling}
-\end{figure}
-
-All modules exhibit linear throughput scaling with batch size, confirming
-that the implementation is memory-bandwidth-bound rather than
-compute-bound at these scales.
-
-\section{PCam: Extended Results}
-\label{app:pcam-details}
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=\linewidth]{experiments/pcam/pareto_analysis.pdf}
-  \caption{PCam Pareto analysis (\texttt{CnonCn} selective bispectrum,
-    single seed, 10\% training data).
-    (a)~Test AUC vs.\ parameter count across all five pooling variants.
-    (b)~Ranking at the ${\sim}$100K parameter budget.
-    (c)~Data efficiency at matched ${\sim}$100K parameters. Multi-seed
-    statistics at matched capacity are in
-    Table~\ref{tab:pcam-results} (main text).}
-  \label{fig:pcam-pareto}
-\end{figure}
-
-\subsection{Published baselines}
-
-Table~\ref{tab:pcam-context} places our results in the context of
-published methods on PCam and the closely related Camelyon16
-benchmark. Our models operate at 68K--920K parameters and are
-trained on only 10\% of the data.
-
-\begin{table}[h]
-  \caption{Published baselines on PatchCamelyon from
-    \citet{veeling2018rotation}, trained on 100\% data.}
-  \label{tab:pcam-context}
-  \centering
-  \small
-  \begin{tabular}{llrcc}
-    \toprule
-    Method & Reference & Params & ACC & AUC \\
-    \midrule
-    P4M-DenseNet ($C_8$)  & Veeling et al.\ 2018     & 119K & 0.898 & 0.963 \\
-    P4M-DenseNet M        & Veeling et al.\ 2018     &  19K & 0.893 & 0.958 \\
-    DenseNet (augmented)   & Veeling et al.\ 2018     & 128K & 0.881 & 0.951 \\
-    DenseNet (no aug.)     & Veeling et al.\ 2018     & 128K & 0.876 & 0.955 \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\subsection{Model variants}
-
-All equivariant models share a $C_8$-equivariant
-DenseNet~\citep{huang2017densenet} backbone with block configuration
-$(4, 4, 4)$, 24 initial channels, and compression factor 0.5,
-following the architecture of \citet{veeling2018rotation} but
-replacing the final invariant map. Six model variants are evaluated:
-
-\begin{itemize}
-  \item \textbf{Standard}: vanilla CNN with random rotation/reflection
-    augmentation and global average pooling. No group dimension,
-    so ${\sim}8\times$ fewer parameters at matched growth rate.
-  \item \textbf{NormReLU}: $C_8$-equivariant backbone with norm-based
-    nonlinearity and max pooling over the group dimension.
-  \item \textbf{Gated}: $C_8$-equivariant backbone with gated
-    nonlinearity (doubles internal channels) and group max pool.
-  \item \textbf{Fourier-ELU}: $C_8$-equivariant backbone with
-    Fourier-space ELU nonlinearity and group max pool.
-  \item \textbf{Bispectrum}: $C_8$-equivariant backbone with ReLU
-    nonlinearity and selective $G$-bispectrum via
-    \texttt{CnonCn}$(n{=}8)$ as the terminal invariant map.
-\end{itemize}
-
-\subsection{Training details}
-
-To enable a controlled Pareto comparison, we sweep the DenseNet
-growth rate $k \in \{3, 4, 6, 8, 12\}$ for equivariant models (and
-$k \in \{6, 12, 20, 30, 35\}$ for the standard CNN to cover a
-comparable parameter range), yielding models from ${\sim}30$K to
-${\sim}1.6$M parameters. All models are trained on \textbf{10\% of
-training data} (26{,}214 images) with AdamW~\citep{loshchilov2019adamw}
-(lr $= 10^{-3}$, weight decay $10^{-4}$), cosine warm restarts,
-mixed-precision training, and early stopping on validation AUC
-(patience 10, max 50 epochs).
-
-\subsection{Pareto analysis interpretation}
-
-\emph{Architectural stability} (single seed, growth-rate sweep).
-The bispectrum's AUC varies by only ${\sim}1$pp across a $12\times$
-range of model sizes (80K--920K parameters), while norm swings by
-$>$7pp. For practitioners, bispectral pooling ``just works'' without
-growth rate tuning. The multi-seed matched-parameter results
-(Table~\ref{tab:pcam-results}) confirm this: bispectrum and Fourier-ELU
-have the lowest cross-seed variance ($\pm 0.004$).
-
-\emph{Matched parameters.}
-At matched parameters (${\sim}100$K), all equivariant models outperform
-the standard CNN (AUC $0.896 \pm 0.010$, 3 seeds). Fourier-ELU leads
-slightly ($0.945 \pm 0.004$), followed by NormReLU ($0.942$), bispectrum
-and gated (both $0.941$). All equivariant methods cluster within
-${\sim}0.5$pp of each other.
-
-\emph{Scaling behavior.}
-As parameters increase from the smallest to the largest configuration,
-standard, norm, and Fourier-ELU all degrade (by 1.8, 2.6, and 0.9
-AUC points respectively), while the bispectrum improves (+0.9 points,
-reaching 0.943 at 920K params). This suggests that the bispectrum's
-complete invariant map scales more gracefully with capacity.
-
-\emph{Data scaling} (3 seeds).
-At 1\% data the bispectrum leads all methods ($0.912 \pm 0.025$),
-outperforming NormReLU ($0.873$), Fourier-ELU ($0.876$), and gate
-($0.884$). By 10\% all equivariant methods converge (${\sim}0.941$--$0.945$).
-The bispectrum's data-efficiency advantage narrows as data increases,
-consistent with the hypothesis that complete invariants substitute for
-missing training examples in the low-data regime.
-
-\section{OrganMNIST3D: Extended Results}
-\label{app:organ3d-details}
-
-\subsection{Published baselines}
-
-Table~\ref{tab:organ3d-context} situates our results against published
-methods on the same benchmark. Our models are deliberately small
-(${\leq}463$K parameters) to enable controlled ablation of the pooling
-mechanism. The relevant comparison is against equivariant methods at a
-similar scale.
-
-\begin{table}[h]
-  \caption{Published baselines on OrganMNIST3D for context.
-    Accuracy differences with respect to large-scale models reflect the
-    deliberate parameter constraint of our ablation study.}
-  \label{tab:organ3d-context}
-  \centering
-  \small
-  \begin{tabular}{llrcc}
-    \toprule
-    Method & Reference & Params & ACC & AUC \\
-    \midrule
-    ResNet-18 + 3D           & Yang et al.\ 2023         & 33M  & 0.907 & 0.996 \\
-    ResNet-18 + ACS          & Yang et al.\ 2023         & 11M  & 0.900 & 0.994 \\
-    ILPOResNet-50            & \citet{zhemchuzhnikov2024ilponet}       & 38K  & 0.879 & 0.992 \\
-    EquiLoPO (local train.)  & \citet{zhemchuzhnikov2025equilopo}      & 418K & 0.866 & 0.991 \\
-    SE3MovFrNet              & \citet{sangalli2023movfrnet}     & ---  & 0.745 & ---   \\
-    Regular SE(3) conv       & \citet{kuipers2023regular}   & 172K & 0.698 & ---   \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\subsection{Rotation analysis}
-
-Figure~\ref{fig:organ3d-rotation} shows that all equivariant models
-achieve identical performance on the original and octahedrally rotated
-test sets ($\sigma_{\mathrm{rot}} \approx 0$), while the
-augmentation-trained standard CNN degrades under rotation.
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=0.75\linewidth]{experiments/organ3d/organ3d_analysis.pdf}
-  \caption{OrganMNIST3D test accuracy on the original and octahedrally
-    rotated test sets. Error bars show standard deviation over three
-    seeds. All equivariant models achieve identical performance under
-    rotation ($\sigma_{\mathrm{rot}} \approx 0$), while the
-    augmentation-trained standard CNN degrades.}
-  \label{fig:organ3d-rotation}
-\end{figure}
-
-\subsection{Wider channels}
-
-To test whether the advantage of bispectral completeness persists at
-higher capacity, we run max pool and bispectrum at channel widths
-$(8, 16)$ and $(16, 32)$ with full training data.
-
-\begin{table}[h]
-  \caption{OrganMNIST3D accuracy at wider channel configurations
-    (mean $\pm$ std over 3 seeds, full training data).}
-  \label{tab:organ3d-wider}
-  \centering
-  \small
-  \begin{tabular}{lrcc}
-    \toprule
-    Channels & Params & Max Pool & Bispectrum \\
-    \midrule
-    $(4, 8)$   & ${\sim}$375K / 463K  & $.730 \pm .033$ & $.726 \pm .027$ \\
-    $(8, 16)$  & ${\sim}$1.5M / 1.7M  & $.743 \pm .015$ & $.745 \pm .006$ \\
-    $(16, 32)$ & ${\sim}$6.0M / 6.3M  & $\mathbf{.785 \pm .032}$ & $.685 \pm .039$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-At $(8, 16)$ the two methods are equivalent ($74.5 \pm 0.6\%$ bispectrum vs.\
-$74.3 \pm 1.5\%$ max pool, both at ${\sim}1.5$--$1.7$M parameters). At $(16, 32)$ max
-pool reaches $78.5 \pm 3.2\%$ while bispectrum drops to $68.5 \pm 3.9\%$---the 6.3M
-parameter bispectrum model likely overfits the 971 training samples.
-This suggests that the bispectrum's completeness advantage is most
-pronounced in the data-limited rather than the parameter-limited
-regime, consistent with the data efficiency findings in the main text.
-
-\section{Experimental Details}
-\label{app:exp}
-
-\subsection{Hyperparameters}
-
-Table~\ref{tab:hyperparams} summarises the training hyperparameters
-for all three experimental settings.
-
-\begin{table}[h]
-  \caption{Training hyperparameters across experiments.}
-  \label{tab:hyperparams}
-  \centering
-  \small
-  \setlength{\tabcolsep}{4pt}
-  \begin{tabular}{lccc}
-    \toprule
-    & PCam & OrganMNIST3D & Sph.\ MNIST \\
-    \midrule
-    Optimizer      & AdamW & AdamW & AdamW \\
-    Learning rate  & $10^{-3}$ & $10^{-3}$ & $10^{-3}$ \\
-    Weight decay   & $10^{-4}$ & $10^{-4}$ & $10^{-4}$ \\
-    Batch size     & 64--256 & 16--64 & 256 \\
-    Scheduler      & Cosine warm restarts & --- & --- \\
-    Max epochs     & 50 & 100 & 50 \\
-    Early stopping & Val AUC, pat.\ 10 & Val AUC, pat.\ 15 & Val ACC, pat.\ 10 \\
-    Seeds          & 42, 123, 456$^\dagger$ & 42, 123, 456 & 42, 123, 456 \\
-    Mixed precision & Yes & No & No \\
-    \bottomrule
-  \end{tabular}
-
-  \vspace{0.3em}
-  {\small $^\dagger$Pareto figure uses seed 42; matched-parameter table and data efficiency use seeds 42, 123, 456.}
-\end{table}
-
-\subsection{Hardware}
-
-All experiments were run on a single NVIDIA H100 GPU (80\,GB).
-Training wall-clock times range from ${\sim}$5 minutes (OrganMNIST3D,
-971 samples) to ${\sim}$20 minutes (PCam, 26K samples at 10\% data).
-Spherical MNIST bispectrum computation (forward pass through
-\texttt{SO3onS2} at $\ell_{\max} = 15$) takes ${\sim}$0.3s per batch
-of 64 on GPU.
-
-\subsection{Data preprocessing}
-
-\paragraph{PCam.}
-PatchCamelyon~\citep{veeling2018rotation} consists of 327{,}680 RGB
-patches of size $96 \times 96$ pixels extracted from whole-slide
-images of lymph node sections~\citep{bejnordi2017camelyon}. We use
-the standard train/val/test split. Images are loaded from HDF5 files
-and normalised to $[0, 1]$. The standard (non-equivariant) baseline
-uses random rotation and reflection augmentation during training; all
-equivariant models are trained without augmentation.
-
-\paragraph{OrganMNIST3D.}
-OrganMNIST3D~\citep{medmnistv2} contains 1{,}742 abdominal CT organ
-volumes of size $28 \times 28 \times 28$ voxels with 11 organ classes
-(train/val/test = 971/161/610). Volumes are normalised to $[0, 1]$.
-The standard baseline uses random octahedral augmentation (one of 24
-rotations per sample); equivariant models are trained without
-augmentation. Kernel rotations are implemented as exact voxel
-permutations on the $3^3$ grid, avoiding interpolation artifacts.
-
-\paragraph{Spherical MNIST.}
-MNIST digits are projected onto the sphere via stereographic
-projection following \citet{cohen2018spherical} and sampled on a
-$64 \times 128$ equiangular grid. The selective bispectrum is
-computed at $\ell_{\max} = 15$, producing 384 complex invariant
-coefficients. The model input is obtained by applying
-$\operatorname{sign}(x)\log(1+|x|)$ separately to the real and
-imaginary parts of each coefficient (a sign-preserving, per-channel
-injective transform), giving 768 real features. These features are
-fed to a 3-layer MLP: Linear(768, 256)--BN--ReLU--Linear(256,
-128)--BN--ReLU--Linear(128, 10). The power spectrum MLP uses the
-same structure with hidden=128: Linear(16, 128)--BN--ReLU--Linear(128,
-64)--BN--ReLU--Linear(64, 10). The standard CNN has 4 convolutional
-layers (16--32--64--64 channels) followed by adaptive average pooling
-and a Linear(1024, 128)--ReLU--Linear(128, 10) classifier.
-
-\subsection{Aggregation procedures}
-
-\paragraph{PCam.}
-The Pareto figure (Figure~\ref{fig:pcam-pareto}) uses a single seed
-(42) across the full growth-rate sweep. The matched-parameter table
-(Table~\ref{tab:pcam-results}) and data efficiency analysis use 3 seeds
-(42, 123, 456) per configuration.
-
-\paragraph{OrganMNIST3D.}
-All results in Table~\ref{tab:organ3d-results} report mean $\pm$ std
-over 3 seeds (42, 123, 456). Rotation robustness
-($\sigma_{\mathrm{rot}}$) is the standard deviation of test accuracy
-across all 24 octahedral rotations of the test set. The data
-efficiency analysis (Figure~\ref{fig:data-efficiency} in the main text) uses 3 seeds per data
-fraction.
-
-\paragraph{Spherical MNIST.}
-All results in Table~\ref{tab:smnist-results} report mean $\pm$ std
-over 3 seeds (42, 123, 456). Rotation robustness
-($\sigma_{\mathrm{rot}}$) is measured by applying 10 random $\SO(3)$
-rotations to the entire test set and computing the standard deviation
-of accuracy, yielding 30 measurements per model (10 rotations
-$\times$ 3 seeds).
-
-\subsection{Spherical MNIST: bispectrum reconstruction protocol}
-\label{app:so3-recon}
-
-This section gives the protocol behind
-Figure~\ref{fig:so3-reconstruction} (Section~\ref{sec:so3-recon}).
-The implementation lives in
-\texttt{paper/experiments/spherical\_mnist\_reconstruction/reconstruct.py}.
-
-\paragraph{Setup.}
-Spherical MNIST is rendered onto an equiangular $64{\times}128$
-lat--lon grid using the same stereographic projection as the
-classifier, then band-limited at $L = 12$ via a forward/inverse SHT
-roundtrip (\texttt{torch\_harmonics.RealSHT}). For each digit we
-draw a target rotation $R\in\SO(3)$ uniformly at random and form
-$f := R\cdot f_0$. We compute the SO(3)-on-$S^2$ selective bispectrum
-$\beta := \Phi_{\mathrm{sel}}(f)$.
-
-\paragraph{Reconstruction.}
-Starting from a Gaussian initialization $\hat f_0\sim\mathcal{N}(0, I)$
-on the $64{\times}128$ grid, we optimize $\hat f$ with Adam
-(initial LR $5\times 10^{-2}$, cosine annealing to $5\times 10^{-4}$,
-$8000$ steps) to minimize the relative complex L2 loss
-$\|\beta(\hat f) - \beta\|^2 / \|\beta\|^2$. After every step we
-project $\hat f$ back to the band-limited subspace via $\mathrm{SHT}^{-1}\!\circ\mathrm{SHT}$ truncated at $L$ to suppress the
-unconstrained null space. We run $4$ random restarts and keep the
-$\hat f$ with the lowest bispectrum residual. Reconstruction
-typically reaches $\|\beta(\hat f)-\beta\|/\|\beta\|\sim 10^{-3}$,
-near the SHT-discretization invariance floor for this grid.
-
-\paragraph{$\SO(3)$ alignment.}
-The bispectrum is invariant on $\SO(3)$-orbits, so $\hat f$ lives
-somewhere in the orbit of $f$ rather than at $f$ itself. We resolve
-the orbit ambiguity by Procrustes-style fitting: we parameterize
-$\hat R \in \SO(3)$ as a unit quaternion (auto-normalized) to avoid
-gimbal lock, and minimize $\|\hat R\cdot\hat f - f\|^2$ with Adam
-($12$ random restarts, $200$ steps each, cosine-annealed LR).
-Rotations of the spherical image use bilinear interpolation in
-lat--lon coordinates (\texttt{bispectrum.rotate\_spherical\_function}).
-
-\paragraph{Numerical floor.}
-On a $64{\times}128$ grid, two SHT roundtrips of two $\SO(3)$-rotated
-copies of the same signal already differ by
-$\|f' - f\|/\|f\|\approx 2\times 10^{-2}$ in image space due to
-bilinear-interpolation discretization; this is the noise floor below
-which alignment cannot improve. The aligned residuals reported in
-Figure~\ref{fig:so3-reconstruction} ($0.1$--$0.3$) sit one to two
-orders of magnitude above this floor and are dominated by gradient
-descent local optima on the bispectrum reconstruction step, not by
-missing information.
-
-\subsection{Dataset licenses}
-
-\begin{itemize}[nosep]
-  \item \textbf{PCam}: CC0 1.0 Universal (public domain).
-  \item \textbf{OrganMNIST3D}: CC BY 4.0.
-  \item \textbf{Spherical MNIST}: derived from MNIST, which is
-    publicly available for research use.
-\end{itemize}
-
-\section{API Example}
-\label{app:api-example}
-
-\begin{lstlisting}
-import torch
-from bispectrum import CnonCn, SO2onDisk, OctaonOcta
-
-bsp = CnonCn(n=8, selective=True)
-f = torch.randn(16, 8)
-beta = bsp(f)
-f_rec = bsp.invert(beta)
-\end{lstlisting}
-
-\newpage
-\input{checklist.tex}
-
-\end{document}
diff --git a/paper/paper_backup.tex b/paper/paper_backup.tex
deleted file mode 100644
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@@ -1,1017 +0,0 @@
-\documentclass{article}
-
-\usepackage[main]{neurips_2026}
-
-\usepackage[utf8]{inputenc}
-\usepackage[T1]{fontenc}
-\usepackage{hyperref}
-\usepackage{url}
-\usepackage{booktabs}
-\usepackage{amsfonts}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\usepackage{amsthm}
-\usepackage{nicefrac}
-\usepackage{microtype}
-\usepackage{xcolor}
-\usepackage{graphicx}
-\usepackage{algorithm}
-\usepackage{algorithmic}
-\usepackage[textsize=tiny, textwidth=2cm]{todonotes}
-\setlength{\marginparwidth}{2cm}
-\usepackage{listings}
-
-\newtheorem{theorem}{Theorem}
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{corollary}[theorem]{Corollary}
-\newtheorem{definition}[theorem]{Definition}
-\newtheorem{remark}[theorem]{Remark}
-
-\DeclareMathOperator{\tr}{tr}
-\DeclareMathOperator{\diag}{diag}
-\DeclareMathOperator{\vect}{vec}
-\newcommand{\R}{\mathbb{R}}
-\newcommand{\C}{\mathbb{C}}
-\newcommand{\Z}{\mathbb{Z}}
-\newcommand{\fhat}{\hat{f}}
-\newcommand{\Fhat}{\hat{F}}
-\newcommand{\betasel}{\beta^{\mathrm{sel}}}
-\newcommand{\betafull}{\beta^{\mathrm{full}}}
-\newcommand{\SO}{\mathrm{SO}}
-\newcommand{\OG}{\mathrm{O}}
-
-\lstset{
-  language=Python,
-  basicstyle=\ttfamily\small,
-  keywordstyle=\color{blue},
-  commentstyle=\color{gray},
-  stringstyle=\color{red},
-  frame=single,
-  breaklines=true,
-  columns=fullflexible,
-}
-
-\title{\texttt{bispectrum}: Selective $G$-Bispectra Made Practical}
-
-\author{%
-  Johan Mathe \\
-  Atmo, Inc. \\
-  \texttt{johan@atmo.ai} \\
-  \And
-  Nina Miolane \\
-  University of California, Santa Barbara \\
-  \texttt{nmiolane@ucsb.edu} \\
-}
-
-\begin{document}
-
-\maketitle
-
-\begin{abstract}
-\todo[inline]{Write abstract ($\sim$150 words). Key points to hit:}
-\begin{itemize}
-  \item We release \texttt{bispectrum}, the first PyTorch library for computing
-    selective $G$-bispectra---the unique complete and exact group-invariant
-    features, with $O(|G|)$ coefficients for finite groups.
-  \item The library supports 7 group/domain pairs
-    ($C_n$, $D_n$, $O$, $\SO(2)/S^1$, $\SO(2)/\text{disk}$,
-    $\SO(3)/S^2$, torus) as differentiable \texttt{nn.Module}s with GPU
-    acceleration, signal inversion, and applications to invariant pooling,
-    orbit-invariant regularization, and multi-reference alignment.
-  \item Building the library required solving two open engineering challenges:
-    bridging steerable CNN features to group Fourier coefficients for
-    non-abelian groups (Theorem~\ref{thm:steerable-fourier}), and
-    resolving a previously undocumented inversion failure for the
-    octahedral group via bootstrap + Levenberg--Marquardt.
-  \item On PatchCamelyon, bispectral pooling achieves the highest AUC
-    among equivariant nonlinearities, the most stable accuracy--parameter
-    Pareto frontier across architectures, and near-perfect rotation
-    invariance. On RotMNIST, a bispectral contrastive regularizer
-    improves rotation robustness.
-  \item The library is pip-installable, MIT-licensed, and integrates
-    into standard PyTorch training pipelines.
-\end{itemize}
-\end{abstract}
-
-
-\section{Introduction}
-\label{sec:intro}
-
-\todo[inline]{Write introduction ($\sim$1 page). Follow the narrative arc below.}
-
-\paragraph{The invariant nonlinearity bottleneck.}
-Equivariant convolutional neural networks~\citep{cohen2016group, weiler2019general}
-achieve symmetry by design, replacing data augmentation with group convolutions that
-guarantee equivariant feature maps. Yet every such architecture must eventually
-produce \emph{invariant} features---for classification, regression, or any
-prediction that should not depend on input orientation.
-The choice of invariant map is a critical design decision with no consensus
-solution (Table~\ref{tab:nonlinearity-comparison}).
-Existing approaches sacrifice either completeness (norm, gate), exactness
-(Fourier pointwise), or efficiency (tensor product, full bispectrum).
-
-\begin{table}[h]
-  \caption{Invariant nonlinearities for equivariant CNNs. The selective bispectrum,
-    implemented in our library, is the only method that is simultaneously exact,
-    complete, and linear-cost for finite groups.}
-  \label{tab:nonlinearity-comparison}
-  \centering
-  \small
-  \begin{tabular}{lcccc}
-    \toprule
-    Nonlinearity & Exact? & Complete? & Coefficients & Key limitation \\
-    \midrule
-    Norm          & Yes & No  & $O(|G|)$      & Discards phase \\
-    Gate          & Yes & No  & $O(2|G|)$     & Scalar bottleneck \\
-    Fourier pointwise & No & Yes & $O(|G| \log |G|)$ & Aliasing \\
-    Tensor product & Yes & Yes & $O(|G|^3)$   & Cubic cost \\
-    Full bispectrum & Yes & Yes & $O(|G|^2)$  & Quadratic cost \\
-    \midrule
-    \textbf{Sel.\ bispectrum} & \textbf{Yes} & \textbf{Yes} & $\mathbf{O(|G|)}$$^\dagger$ & \textbf{(this work)} \\
-    \bottomrule
-  \end{tabular}
-
-  \vspace{0.3em}
-  {\small $^\dagger$For finite groups where selectivity is
-    proven~\citep{mataigne2024selective}. Selective $\SO(3)$ on $S^2$
-    remains open.}
-\end{table}
-
-\paragraph{The gap from theory to practice.}
-The selective $G$-bispectrum~\citep{mataigne2024selective} provides $O(|G|)$-cost
-complete invariants for finite groups, and \citet{myers2025selective} extend this
-to the disk. However, no practical software exists. Moving from theorems to a
-usable library required solving three problems:
-\begin{enumerate}
-  \item No GPU-accelerated implementation exists for any group beyond toy examples.
-  \item For the octahedral group, the theoretically prescribed bootstrap inversion
-    algorithm fails due to an $\SO(3)$ phase ambiguity not addressed by the
-    Clebsch--Gordan structure (Section~\ref{sec:lib-octa}).
-  \item Steerable CNN feature maps live in intertwiner spaces, not in group Fourier
-    spaces---for non-abelian groups, one cannot na\"ively plug steerable outputs
-    into the bispectrum formula (Section~\ref{sec:lib-steerable}).
-\end{enumerate}
-
-\paragraph{Contributions.}
-We present \texttt{bispectrum}, a PyTorch library that makes selective
-$G$-bispectra practical as differentiable primitives for invariant feature
-extraction, regularization, and signal reconstruction:
-\begin{enumerate}
-  \item \textbf{\texttt{bispectrum} library} (Section~\ref{sec:library}):
-    The first PyTorch library for selective $G$-bispectra, supporting
-    7 group/domain pairs as \texttt{nn.Module}s with GPU acceleration,
-    signal inversion, and seamless integration into training pipelines.
-    Beyond invariant pooling, the library enables orbit-invariant losses,
-    metric learning, and multi-reference alignment
-    (Section~\ref{sec:lib-beyond}).
-    Pip-installable, MIT-licensed, with CI/CD and $>$90\% test coverage.
-
-  \item \textbf{Engineering contribution: steerable-to-Fourier bridge}
-    (Section~\ref{sec:lib-steerable}):
-    We prove conditions under which steerable convolution outputs yield valid
-    group Fourier coefficients for bispectral computation
-    (Theorem~\ref{thm:steerable-fourier}), enabling integration with
-    existing equivariant architectures.
-
-  \item \textbf{Engineering contribution: octahedral inversion}
-    (Section~\ref{sec:lib-octa}):
-    We identify a previously undocumented inversion failure for the octahedral
-    group and resolve it via bootstrap initialization followed by
-    Levenberg--Marquardt refinement.
-
-  \item \textbf{Empirical validation} (Section~\ref{sec:experiments}):
-    On PatchCamelyon, bispectral pooling achieves the highest AUC among
-    equivariant nonlinearities, the flattest accuracy--parameter Pareto
-    frontier (most robust to architecture choices), and near-perfect
-    rotation invariance ($\sigma_{\text{AUC}} < 0.002$).
-    On RotMNIST, a bispectral contrastive regularizer improves rotation
-    robustness without augmentation labels.
-    \todo[inline]{Update RotMNIST claim once experiment is run.}
-\end{enumerate}
-
-
-\section{Mathematical Preliminaries}
-\label{sec:preliminaries}
-
-\subsection{Equivariant CNNs and invariant maps}
-\label{sec:prelim-equiv}
-
-\todo[inline]{Write $\sim$0.5 page on equivariant CNNs. Cover:}
-\begin{itemize}
-  \item Group convolutions: a convolution layer is $G$-equivariant if
-    $[\pi(g) \circ \Phi](f) = \Phi(\pi(g) f)$ for all $g \in G$.
-    Cite \citet{cohen2016group}.
-  \item Regular vs.\ steerable representations. In regular representations,
-    features are functions $f: G \to \R^C$. In steerable representations,
-    features transform as $f \mapsto \rho(g) f$ for a representation
-    $\rho$~\citep{weiler2019general}.
-  \item The need for $G$-invariant pooling: to produce predictions
-    that do not depend on the input orientation, one must map equivariant
-    features to invariant ones.
-  \item Existing invariant maps: norm (discards phase), gated nonlinearity
-    (scalar bottleneck), Fourier pointwise (approximate equivariance due to
-    aliasing~\citep{franzen2021general}), tensor product / CG product
-    ($O(L^3)$ cost~\citep{kondor2018clebsch}).
-\end{itemize}
-
-
-\subsection{The $G$-bispectrum}
-\label{sec:prelim-bispectrum}
-
-\todo[inline]{Write $\sim$0.5 page. Merge the G-bispectrum definition and selective bispectra into one subsection. Cover:}
-\begin{itemize}
-  \item Group Fourier transform: for a finite group $G$ with irreducible
-    representations $\{\rho_i\}_{i \in \hat{G}}$,
-    \begin{equation}
-      \Fhat(\rho_i) = \sum_{g \in G} f(g)\, \rho_i(g) \in \C^{d_i \times d_i}.
-    \end{equation}
-  \item The $G$-bispectrum: given the Clebsch--Gordan decomposition
-    $\rho_i \otimes \rho_j \cong \bigoplus_k m_{ij}^k \rho_k$
-    with intertwining matrix $C_{ij}$,
-    \begin{equation}
-      \label{eq:bispectrum-def}
-      \beta_{\rho_i, \rho_j} = C_{ij}^\top\,
-        (\Fhat(\rho_i) \otimes \Fhat(\rho_j))\,
-        C_{ij}\, \Fhat(\rho_k)^\dagger.
-    \end{equation}
-  \item Completeness: the bispectrum is a \emph{complete} invariant---it
-    separates orbits under the $G$-action~\citep{kakarala1992triple}.
-  \item Full cost: $O(|G|^2)$ coefficients for finite groups.
-  \item \textbf{Selective bispectra}~\citep{mataigne2024selective}: for finite groups,
-    $O(|G|)$ bispectral coefficients suffice for completeness, selected via a
-    BFS on the Kronecker product table of $\hat{G}$. The disk
-    extension~\citep{myers2025selective} gives $O(N)$ Fourier--Bessel
-    coefficients for $\SO(2)$-invariance on the unit disk.
-  \item Bootstrap inversion algorithms recover $f$ (up to group action) from
-    the selective bispectrum for commutative and dihedral groups.
-\end{itemize}
-
-
-\subsection{The representation-theoretic gap}
-\label{sec:prelim-gap}
-
-\todo[inline]{Write $\sim$0.3 page briefly identifying the steerable gap. This motivates Section~\ref{sec:lib-steerable}. Cover:}
-\begin{itemize}
-  \item Steerable convolution outputs transform as $f \mapsto \rho(g) f$
-    where $\rho$ is a (potentially reducible) representation.
-  \item For abelian groups ($C_n$, $\SO(2)$): all irreps are 1D characters,
-    so intertwiner coefficients \emph{are} Fourier coefficients. No gap.
-  \item For non-abelian groups ($D_n$, $O$): the intertwiner space
-    $\mathrm{Hom}_G(\rho_i, \rho_{\text{out}})$ has dimension
-    $m_i = \text{multiplicity of } \rho_i \text{ in } \rho_{\text{out}}$.
-    When $m_i < d_i$ (the dimension of $\rho_i$), steerable outputs do not
-    span the full Fourier matrix---one cannot na\"ively apply the bispectrum
-    formula.
-  \item Concrete example: for $D_4$ with output type $\rho_1 \oplus \rho_0$,
-    the steerable features give only 3 numbers but the Fourier transform of
-    $\rho_1$ needs $2^2 = 4$.
-  \item We solve this within the library (Section~\ref{sec:lib-steerable}).
-\end{itemize}
-
-
-\section{The \texttt{bispectrum} Library}
-\label{sec:library}
-
-This section describes the design, scope, and key engineering contributions
-of the \texttt{bispectrum} library. Following the structure advocated by
-\citet{paszke2019pytorch} and \citet{gardner2018gpytorch}, we focus on
-\emph{design principles and trade-offs} rather than a usage tutorial.
-
-\subsection{Design principles}
-\label{sec:lib-design}
-
-\todo[inline]{Write $\sim$0.5 page on design principles. Frame as a \emph{design manifesto} (per PyTorch Reviewer 2's advice). Cover:}
-\begin{itemize}
-  \item \textbf{P1: Naming encodes the math.} Class names follow
-    \texttt{\{Group\}on\{Domain\}} (e.g., \texttt{CnonCn},
-    \texttt{SO3onS2}, \texttt{SO2onDisk}). A reader unfamiliar with the
-    codebase immediately identifies the group and domain from the class name.
-    This is deliberately mathematical rather than verbal
-    (\texttt{CyclicBispectrum}) because the mathematical name carries
-    precise meaning and scales as the library grows.
-
-  \item \textbf{P2: \texttt{nn.Module} only.} Every bispectrum computation
-    is a \texttt{torch.nn.Module} subclass. No parallel functional API.
-    One way to compute a bispectrum. Precomputed quantities
-    (CG matrices, DFT kernels, Bessel roots) are registered as
-    non-learnable buffers, so \texttt{.to(device)} and \texttt{.to(dtype)}
-    work transparently.
-
-  \item \textbf{P3: Selective by default.} All modules default to
-    \texttt{selective=True}, computing the minimal $O(|G|)$ coefficient set.
-    The full $O(|G|^2)$ bispectrum is available via \texttt{selective=False}
-    for debugging and comparison.
-
-  \item \textbf{P4: Raw signals in.} Modules accept real-valued signals
-    in the natural domain (spatial grid, discrete cycle, sphere) and handle
-    the Fourier transform internally. For example, \texttt{SO3onS2} accepts
-    signals of shape $(B, n_{\text{lat}}, n_{\text{lon}})$ and computes
-    the spherical harmonic transform internally. This keeps the interface
-    clean and prevents Fourier-convention mismatches.
-
-  \item \textbf{P5: Inversion on-module.} Every module exposes
-    \texttt{bsp.invert(beta)} that returns the reconstructed signal
-    (up to group-action indeterminacy) or raises a precise
-    \texttt{NotImplementedError} with guidance.
-
-  \item \textbf{P6: Minimal dependencies.} The library depends only on
-    \texttt{torch} and \texttt{numpy}. No external CG library:
-    Clebsch--Gordan matrices for $D_n$ are computed analytically,
-    for $\SO(3)$ they are bundled as pre-computed JSON, and Bessel roots
-    for the disk are computed via a pure-torch Newton-bisection solver
-    using the interlacing property.
-\end{itemize}
-
-
-\subsection{Supported groups and API}
-\label{sec:lib-groups}
-
-Table~\ref{tab:supported-groups} lists all 7 group/domain pairs currently
-supported by the library.
-
-\begin{table}[h]
-  \caption{Supported group/domain pairs in the \texttt{bispectrum} library.
-    Selective coefficient counts are $O(|G|)$ vs.\ $O(|G|^2)$ for the full
-    bispectrum. All modules share a uniform \texttt{nn.Module} interface.}
-  \label{tab:supported-groups}
-  \centering
-  \small
-  \begin{tabular}{llrrcc}
-    \toprule
-    Module & Group / Domain & Selective & Full & Inversion & Complexity \\
-    \midrule
-    \texttt{CnonCn}       & $C_n$ on $C_n$            & $n$    & $n^2$  & \checkmark & $O(n \log n)$ \\
-    \texttt{SO2onS1}      & $\SO(2)$ on $S^1$         & $n$    & $n^2$  & \checkmark & $O(n \log n)$ \\
-    \texttt{TorusOnTorus}  & $\prod C_{n_l}$ on torus  & $|G|$  & $|G|^2$ & \checkmark & $O(|G| \log |G|)$ \\
-    \texttt{DnonDn}       & $D_n$ on $D_n$            & ${\sim}4|D_n|$ & ---  & \checkmark & $O(n)$ \\
-    \texttt{OctaonOcta}   & $O$ on $\R^3$             & 172    & ---    & \checkmark$^*$ & $O(1)$ \\
-    \texttt{SO3onS2}      & $\SO(3)$ on $S^2$         & ---    & $O(L^2)$ & ---       & $O(L^2 d_{\max}^2)$ \\
-    \texttt{SO2onDisk}    & $\SO(2)$ on disk           & $N$    & $N^3$  & \checkmark & $O(N)$ \\
-    \bottomrule
-  \end{tabular}
-
-  \vspace{0.3em}
-  {\small $^*$Bootstrap + LM refinement (Section~\ref{sec:lib-octa}).}
-\end{table}
-
-\paragraph{Module interface contract.}
-Every module satisfies: (i)~\texttt{forward(f)} accepts a real-valued signal
-of shape $(B, \ldots)$ and returns a complex tensor of shape
-$(B, \texttt{output\_size})$; (ii)~\texttt{output\_size} gives the number of
-bispectral coefficients; (iii)~\texttt{index\_map} maps each flat output
-index to its irrep index tuple $(\rho_i, \rho_j)$;
-(iv)~no trainable parameters; (v)~deterministic output.
-
-\paragraph{Usage examples.}
-
-\begin{lstlisting}
-import torch
-from bispectrum import CnonCn, SO3onS2, SO2onDisk
-
-# Cyclic group on Z/8Z: selective reduces 64 -> 8 coefficients
-bsp = CnonCn(n=8, selective=True)
-f = torch.randn(32, 8)              # batch of 32 signals
-beta = bsp(f)                       # (32, 8), complex64
-f_rec = bsp.invert(beta)            # (32, 8), up to cyclic shift
-
-# SO(3) on the sphere
-bsp_s2 = SO3onS2(lmax=5, nlat=64, nlon=128)
-f_sphere = torch.randn(4, 64, 128)  # spherical signal
-beta_s2 = bsp_s2(f_sphere)          # (4, output_size), complex64
-
-# SO(2) on the unit disk
-bsp_disk = SO2onDisk(L=16, selective=True)
-f_disk = torch.randn(4, 16, 16)     # disk image
-beta_disk = bsp_disk(f_disk)         # (4, N), complex128
-f_rec = bsp_disk.invert(beta_disk)   # (4, 16, 16)
-\end{lstlisting}
-
-
-\subsection{Beyond invariant pooling}
-\label{sec:lib-beyond}
-
-Because every module is a differentiable \texttt{nn.Module} with both
-\texttt{forward} and \texttt{invert}, the library supports use cases
-beyond invariant feature extraction:
-
-\paragraph{Orbit-invariant regularization.}
-For generative or reconstruction tasks, the bispectrum provides a
-rotation-invariant loss without augmentation: the loss
-$\|\betasel(f_{\text{pred}}) - \betasel(f_{\text{target}})\|^2$ is
-invariant to the $G$-action on both signals by construction.
-We demonstrate this as a contrastive regularizer in
-Section~\ref{sec:exp-contrastive}.
-
-\paragraph{Orbit distance and metric learning.}
-$\|\betasel(f) - \betasel(g)\|$ defines a distance between
-signals modulo group action, suitable for retrieval, clustering, or
-nearest-neighbor classification on equivalence classes.
-
-\paragraph{Multi-reference alignment.}
-Recovering a signal from noisy copies at unknown orientations reduces
-to bispectral averaging followed by
-inversion~\citep{myers2025selective, bendory2017bispectrum}.
-The \texttt{invert()} method provides this primitive.
-
-\paragraph{Symmetry detection.}
-Vanishing patterns in bispectral coefficients reveal hidden symmetries
-in data: a signal invariant to a subgroup $H \leq G$ produces a
-bispectrum with predictable zero blocks.
-
-\begin{lstlisting}
-import torch
-from bispectrum import CnonCn
-
-bsp = CnonCn(n=8, selective=True)
-
-# 1. Invariant features for classification
-features = bsp(signal)               # (B, 8), complex64
-
-# 2. Orbit-invariant regularization loss
-loss = torch.nn.functional.mse_loss(bsp(pred).abs(), bsp(target).abs())
-
-# 3. Signal reconstruction from bispectral coefficients
-reconstructed = bsp.invert(features)  # (B, 8), up to cyclic shift
-\end{lstlisting}
-
-
-\subsection{Bridging steerable features to bispectra}
-\label{sec:lib-steerable}
-
-For abelian groups, steerable convolution outputs are directly usable as
-Fourier coefficients, and the bispectrum formula applies out of the box.
-For non-abelian groups ($D_n$, $O$), steerable outputs live in an
-intertwiner space whose dimension does not match the number of irreps
-(Section~\ref{sec:prelim-gap}). We resolve this mismatch:
-
-\begin{theorem}[Steerable-to-Fourier bridge]
-\label{thm:steerable-fourier}
-  \todo[inline]{Insert the precise one-paragraph statement of P4 here.
-    Conditions: output type contains each irrep $\rho_i$ with multiplicity
-    $m_i \geq d_i$, or a multi-head architecture targets each irrep block.}
-\end{theorem}
-
-The proof and a detailed discussion of which \texttt{escnn} architectures
-satisfy the theorem's conditions are in Appendix~\ref{app:proofs}.
-In practice, this theorem justifies the \texttt{BispectrumPool} layer used
-in our experiments (Section~\ref{sec:experiments}): extract Fourier
-coefficients from the group dimension via the group DFT, compute the
-selective bispectrum using the appropriate library module, log-compress
-for numerical stability, and project back via a $1 \times 1$ convolution.
-
-
-\subsection{Resolving the octahedral inversion gap}
-\label{sec:lib-octa}
-
-\todo[inline]{Write $\sim$0.5 page. Reframed as an engineering problem solved within the library. Cover:}
-
-Implementing inversion for the octahedral group (\texttt{OctaonOcta}) revealed
-a failure mode in the bootstrap algorithm of \citet{mataigne2024selective}:
-
-\begin{itemize}
-  \item The bootstrap recovers $\Fhat(\rho_1)$ up to an unknown
-    $Q \in \SO(3)$. For $D_n$, the analogous $\OG(2)$ ambiguity is
-    absorbed by CG matrices because $D_n$ irreps are restrictions of
-    $\SO(2)$ irreps. For $O$, the CG matrix $C_{11}$ block-diagonalizes
-    $\rho_1(g) \otimes \rho_1(g)$ only for $g \in O$, not for arbitrary
-    $Q \in \SO(3)$.
-  \item Numerically, $\|[C_{11}^\top (Q \otimes Q) C_{11}]_{\rho_2, \rho_3}\|_\infty
-    \approx 0.788$ for generic $Q$---$O(1)$ cross-block leakage that
-    corrupts the recovered Fourier coefficients.
-  \item The selective $O$-bispectrum \emph{is} complete (172 coefficients
-    uniquely determine the signal up to $O$-action). The failure is purely
-    algorithmic: the sequential bootstrap cannot isolate individual Fourier
-    blocks.
-\end{itemize}
-
-\paragraph{Resolution implemented in the library.}
-\begin{enumerate}
-  \item \textbf{Bootstrap initialization}: recover $\Fhat(\rho_0)$ exactly,
-    $\Fhat(\rho_1)$ up to $Q$, approximate remaining coefficients from
-    diagonal blocks (ignoring leakage).
-  \item \textbf{Levenberg--Marquardt refinement}: solve
-    $\min_f \|\betasel(f) - \betasel_{\text{target}}\|^2$ via damped
-    least-squares, with Jacobians computed by \texttt{torch.func.jacfwd}
-    composed with \texttt{torch.vmap} for batched forward-mode AD.
-  \item \textbf{Multi-start}: randomize the initial $Q$ across several
-    restarts for robustness.
-\end{enumerate}
-This converges in $<$50 iterations and is fully differentiable.
-Convergence analysis is in Appendix~\ref{app:octa-convergence}.
-
-
-\subsection{Computational benchmarks}
-\label{sec:lib-benchmarks}
-
-\todo[inline]{Add two figures from \texttt{benchmarks/benchmark.py --paper}:}
-
-\begin{figure}[h]
-  \centering
-  \fbox{\rule[-.5cm]{0cm}{3cm} \rule[-.5cm]{5cm}{0cm}}
-  \caption{Selective vs.\ full coefficient count scaling across group/domain pairs.
-    Selective bispectra achieve $O(|G|)$ coefficients (dashed reference line)
-    compared to $O(|G|^2)$ for the full bispectrum.}
-  \label{fig:coeff-scaling}
-  \todo[inline]{Replace placeholder with actual figure from benchmarks.}
-\end{figure}
-
-\begin{figure}[h]
-  \centering
-  \fbox{\rule[-.5cm]{0cm}{3cm} \rule[-.5cm]{5cm}{0cm}}
-  \caption{GPU forward throughput (samples/second) vs.\ batch size for selective
-    bispectrum computation. All modules tested on NVIDIA A100.}
-  \label{fig:gpu-throughput}
-  \todo[inline]{Replace placeholder with actual figure from benchmarks.}
-\end{figure}
-
-\todo[inline]{Write $\sim$0.3 page interpreting the benchmark figures. Cover:}
-\begin{itemize}
-  \item Coefficient count: verify $O(|G|)$ vs.\ $O(|G|^2)$ scaling empirically
-    across all modules for increasing group sizes.
-  \item GPU throughput: show that batched computation saturates GPU occupancy.
-    Compare forward pass time for selective vs.\ full where applicable.
-  \item Memory: peak GPU memory at various batch sizes.
-\end{itemize}
-
-
-\section{Experiments}
-\label{sec:experiments}
-
-We validate the library on two tasks: invariant pooling in an equivariant CNN
-(Sections~\ref{sec:exp-setup}--\ref{sec:exp-ablations}), and bispectral
-contrastive regularization (Section~\ref{sec:exp-contrastive}).
-The experiments test whether the bispectrum's theoretical
-completeness---preserving all information except absolute
-orientation---translates into practical benefits. We find that
-bispectral pooling is the most robust invariant pooling method across
-architectures and achieves the highest AUC among equivariant
-nonlinearities, while providing near-perfect rotation invariance.
-
-\subsection{Experimental setup}
-\label{sec:exp-setup}
-
-\todo[inline]{Write $\sim$0.5 page on setup. Cover:}
-\begin{itemize}
-  \item \textbf{Dataset}: PatchCamelyon (PCam)~\citep{veeling2018rotation}---327,680
-    RGB patches ($96 \times 96$), binary classification of metastatic tissue
-    in histopathology. Tissue orientation under the microscope is arbitrary,
-    making rotation invariance physically justified. Standard train/val/test split.
-  \item \textbf{Architecture}: Equivariant DenseNet~\citep{huang2017densenet} with $C_8$ group convolutions.
-    Default growth rate 12, block configuration $[4, 4, 4]$.
-    All models share the same backbone; only the invariant
-    pooling layer differs.
-  \item \textbf{Nonlinearity variants} (5 models, identical backbone):
-    \begin{enumerate}
-      \item \texttt{standard}: Vanilla DenseNet + geometric augmentation
-        (flips, $90^\circ$ rotations). No group dimension, so $\sim$8$\times$
-        fewer parameters at matched growth rate.
-      \item \texttt{norm}: Equivariant DenseNet + NormReLU + GroupMaxPool.
-      \item \texttt{gate}: Gated nonlinearity ($2\times$ channels) + GroupMaxPool.
-      \item \texttt{fourier\_elu}: FFT $\to$ ELU $\to$ IFFT + GroupMaxPool.
-      \item \texttt{bispectrum}: ReLU + \texttt{CnonCn} selective bispectral
-        pooling + $1\times 1$ projection.
-    \end{enumerate}
-  \item \textbf{Training}: AdamW, learning rate $10^{-3}$, cosine warm restarts,
-    early stopping on validation AUC (patience 5), 3 seeds per model.
-    We report results at 1\% of training data ($\sim$3,276 samples) for
-    rapid prototyping, and vary model capacity (growth rate 3--35)
-    at 10\% of training data for Pareto analysis.
-  \item \textbf{Metrics}: Test AUC-ROC (primary), test accuracy, rotation
-    robustness ($\sigma_{\text{AUC}}$ across 12 rotation angles, lower is better).
-  \item \textbf{Parameter count note}: Equivariant models carry a group
-    dimension of size $|G|=8$, inflating parameter counts relative to the
-    standard baseline at matched growth rate (e.g., 920k vs.\ 102k at
-    $k{=}12$). This confound motivates the Pareto analysis in
-    Section~\ref{sec:exp-ablations}.
-\end{itemize}
-
-
-\subsection{Main results}
-\label{sec:exp-main}
-
-\begin{table}[h]
-  \caption{PatchCamelyon results (1\% training data, growth rate 12).
-    Mean $\pm$ std over 3 seeds.
-    Rotation $\sigma_{\text{AUC}}$: standard deviation of AUC across 12
-    rotation angles (lower is better). The standard baseline has
-    $\sim$8$\times$ fewer parameters because it lacks the group dimension.}
-  \label{tab:main-results}
-  \centering
-  \begin{tabular}{lrccc}
-    \toprule
-    Model & Params & Test AUC $\uparrow$ & Test Acc $\uparrow$ & Rot.\ $\sigma_{\text{AUC}}$ $\downarrow$ \\
-    \midrule
-    Standard (+ aug)   & 102k  & $0.893 \pm 0.011$ & $0.810$ & $0.0031$ \\
-    Norm               & 791k  & $0.849 \pm 0.011$ & $0.764$ & $0.0013$ \\
-    Gate               & 1,576k & $0.862 \pm 0.016$ & $0.781$ & $0.0016$ \\
-    Fourier-ELU        & 790k  & $0.865 \pm 0.006$ & $0.782$ & $0.0009$ \\
-    \textbf{Bispectrum (ours)} & 920k & $\mathbf{0.875} \pm 0.012$ & $\mathbf{0.784}$ & $0.0017$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\todo[inline]{Write $\sim$0.5 page interpreting the main results table. Key points:}
-\begin{itemize}
-  \item \textbf{Among equivariant models, bispectrum achieves the highest AUC}
-    (0.875), outperforming Fourier-ELU (+1.0pp), gate (+1.3pp), and norm
-    (+2.6pp). This supports the hypothesis that completeness matters:
-    the bispectrum preserves phase information that norm and gate discard,
-    which is relevant for histopathology textures with oriented
-    sub-structures (collagen fibers, gland lumen shapes, mitotic spindle
-    orientations).
-  \item \textbf{The standard CNN achieves the highest overall AUC} (0.893),
-    but this comparison is confounded by a $\sim$8$\times$ parameter
-    advantage: the standard model has 102k parameters vs.\ 790k--1.6M for
-    equivariant models at the same growth rate $k{=}12$. Equivariant
-    models carry a group dimension of size $|G|{=}8$ that inflates
-    parameter counts. We control for this confound via Pareto analysis in
-    Section~\ref{sec:exp-ablations}.
-  \item \textbf{All equivariant models achieve 2--3$\times$ lower rotation
-    variance} ($\sigma_{\text{AUC}} \leq 0.0017$) than the standard
-    baseline ($\sigma_{\text{AUC}} = 0.0031$), confirming that built-in
-    equivariance provides rotation robustness that augmentation alone
-    cannot fully match.
-  \item \textbf{Fourier-ELU shows the lowest rotation variance} (0.0009),
-    slightly below bispectrum (0.0017). Both are near the floor imposed
-    by bilinear interpolation artifacts in the rotation evaluation.
-\end{itemize}
-
-
-\subsection{Rotation robustness and data efficiency}
-\label{sec:exp-robustness}
-
-\begin{figure}[h]
-  \centering
-  \fbox{\rule[-.5cm]{0cm}{4cm} \rule[-.5cm]{6cm}{0cm}}
-  \caption{AUC vs.\ rotation angle on the PCam test set (1\% training data,
-    mean over 3 seeds). Equivariant models maintain near-constant AUC
-    across all 12 evaluation angles; the standard CNN shows higher
-    variation, particularly at non-augmented orientations.}
-  \label{fig:rotation-dataeff}
-  \todo[inline]{Replace placeholder with actual rotation robustness figure.}
-\end{figure}
-
-\todo[inline]{Write $\sim$0.3 page interpreting rotation robustness. Cover:}
-\begin{itemize}
-  \item \textbf{Rotation robustness}: all equivariant models achieve
-    $\sigma_{\text{AUC}} \leq 0.0017$ across 12 rotation angles, compared
-    to $\sigma_{\text{AUC}} = 0.0031$ for the augmented standard CNN.
-    For the standard model, AUC drops noticeably at non-augmented angles
-    ($15^\circ$, $30^\circ$, $60^\circ$, $75^\circ$), while equivariant
-    models show nearly flat profiles.
-  \item At $C_8$ group angles (multiples of $45^\circ$), all equivariant
-    models produce identical AUC values (up to floating-point precision),
-    confirming exact equivariance of the architecture. The small residual
-    variation at off-group angles ($15^\circ$, $30^\circ$, etc.) is
-    attributable to bilinear interpolation artifacts in
-    \texttt{torchvision.transforms.functional.rotate}, not to the model.
-  \item Fourier-ELU achieves the lowest $\sigma_{\text{AUC}}$ (0.0009),
-    slightly below bispectrum (0.0017). Both values are near the floor
-    imposed by the interpolation-based evaluation protocol.
-  \item The standard CNN's $\sigma_{\text{AUC}} = 0.0031$ is modest
-    because its augmentation covers flips and $90^\circ$ rotations,
-    providing partial rotation robustness. The gap would be larger for
-    tasks requiring invariance to finer rotations not covered by
-    augmentation.
-\end{itemize}
-
-
-\subsection{Pareto analysis: controlling for parameter count}
-\label{sec:exp-ablations}
-
-Table~\ref{tab:main-results} compares models at a fixed growth rate
-$k{=}12$, where equivariant models carry $\sim$8$\times$ more
-parameters than the standard baseline. To disentangle the effect of
-the invariant pooling method from the effect of model capacity, we
-sweep the growth rate ($k \in \{3, 4, 6, 8, 12\}$ for equivariant
-models; $k \in \{6, 12, 20, 30, 35\}$ for the standard CNN) and
-evaluate at 10\% of training data.
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=\textwidth]{experiments/pcam/pareto_analysis.png}
-  \caption{Pareto analysis of invariant pooling methods on PCam
-    (10\% training data, seed 42).
-    \textbf{(a)}~AUC vs.\ parameter count: bispectrum (red) shows the
-    flattest curve, achieving consistently high AUC across all model
-    sizes. Norm (blue) peaks sharply at $\sim$110k parameters then
-    degrades.
-    \textbf{(b)}~Head-to-head at $\sim$100k parameters: all equivariant
-    models outperform the standard baseline when parameter counts are
-    matched.
-    \textbf{(c)}~Improvement from 1\% to 10\% training data at default
-    architecture ($k{=}12$): bispectrum shows the largest absolute
-    gain ($+$6.8pp).
-    \textbf{(d)}~Ranking at $\sim$100k parameters.}
-  \label{fig:pareto}
-\end{figure}
-
-\todo[inline]{Write $\sim$0.3 page interpreting the Pareto analysis. Cover:}
-\begin{itemize}
-  \item \textbf{Bispectrum is the most architecturally stable choice.}
-    Its AUC varies by only $\sim$1pp across a 12$\times$ range of model
-    sizes (80k--920k parameters), while norm swings by $>$7pp
-    (0.878--0.954). For practitioners, bispectral pooling ``just works''
-    without growth rate tuning.
-  \item \textbf{At matched parameters ($\sim$100k), all equivariant
-    models outperform the standard CNN.} Norm peaks at 0.954 AUC
-    ($k{=}4$, 110k params), followed by Fourier-ELU (0.937),
-    bispectrum (0.933), standard (0.928), and gate (0.924).
-    The standard model's advantage in Table~\ref{tab:main-results}
-    disappears once parameter counts are controlled.
-  \item \textbf{Norm's instability across architectures is a practical
-    concern.} Norm achieves the single best AUC (0.954) but its
-    performance is highly sensitive to growth rate, dropping to 0.878 at
-    $k{=}8$. This brittleness suggests that norm's completeness gap
-    interacts unpredictably with model capacity.
-  \item \textbf{Data scaling (panel c)}: all models improve
-    substantially from 1\% to 10\% data. Bispectrum shows the largest
-    absolute gain ($+$6.8pp), consistent with the hypothesis that
-    complete invariants benefit more from additional data.
-\end{itemize}
-
-
-\subsection{Data-efficiency Pareto analysis}
-\label{sec:exp-data-pareto}
-
-To assess how each invariant pooling method utilises limited training
-data, we fix per-model growth rates to match $\sim$100k parameters
-(standard $k{=}12$, norm/Fourier-ELU $k{=}4$, gate $k{=}3$,
-bispectrum $k{=}4$) and sweep the training set fraction from 1\% to
-100\% (six log-spaced points, three seeds each).
-
-\begin{figure}[h]
-  \centering
-  \includegraphics[width=\textwidth]{experiments/pcam/data_pareto_analysis.png}
-  \caption{Data-efficiency Pareto analysis on PCam at matched
-    $\sim$100k parameter budget.
-    \textbf{(a)}~Test AUC vs.\ training set fraction (log scale) for
-    five pooling methods; error bars show $\pm 1$ standard deviation
-    over three seeds.
-    \textbf{(b)}~AUC gain of each equivariant method relative to the
-    standard (augmented) baseline at each data fraction.}
-  \label{fig:data-pareto}
-  \todo[inline]{Replace placeholder once sweep results are available.
-    Run: \texttt{./run\_data\_pareto\_sweep.sh} then
-    \texttt{python analyze\_data\_pareto.py}.}
-\end{figure}
-
-\todo[inline]{Write $\sim$0.3 page interpreting the data-efficiency
-  Pareto analysis. Cover:}
-\begin{itemize}
-  \item Which pooling method achieves the highest AUC at small data
-    fractions (1\%, 5\%)?
-  \item At what data fraction do equivariant methods match or exceed
-    the standard CNN?
-  \item Does bispectral pooling show a steeper learning curve
-    consistent with exact invariants requiring fewer samples?
-\end{itemize}
-
-
-\subsection{Bispectral contrastive regularization}
-\label{sec:exp-contrastive}
-
-\todo[inline]{Write $\sim$0.3 page. This experiment demonstrates the library
-beyond invariant pooling---as a rotation-invariant regularizer.}
-
-The bispectrum provides a natural contrastive signal: rotated copies of a
-signal share identical bispectral coefficients by construction, so
-$\|\betasel(f) - \betasel(T_g f)\| = 0$ without needing to track which
-augmentation was applied.
-
-\begin{itemize}
-  \item \textbf{Dataset}: RotMNIST~\citep{cohen2016group}---MNIST digits
-    rotated by arbitrary angles. 12k train / 50k test.
-  \item \textbf{Baseline}: Standard CNN + supervised contrastive
-    loss~\citep{chen2020simclr} using augmentation-labeled positive pairs.
-  \item \textbf{Ours}: Same CNN + bispectral contrastive regularizer:
-    $\mathcal{L}_{\text{bisp}} = \|\betasel(f_1) - \betasel(f_2)\|^2$
-    for augmented pairs, added to the cross-entropy loss.
-    Positive pairs are identified by bispectral distance rather than
-    augmentation labels.
-  \item \textbf{Metrics}: Test accuracy, rotation robustness
-    ($\sigma_{\text{acc}}$ across rotation angles).
-\end{itemize}
-
-\begin{table}[h]
-  \caption{RotMNIST results. Bispectral contrastive regularization provides
-    rotation robustness without requiring augmentation labels.}
-  \label{tab:contrastive-results}
-  \centering
-  \begin{tabular}{lcc}
-    \toprule
-    Model & Test Acc $\uparrow$ & Rot.\ $\sigma_{\text{acc}}$ $\downarrow$ \\
-    \midrule
-    Standard CNN              & {\color{red}XX.X\%} & {\color{red}X.XX} \\
-    + augmentation            & {\color{red}XX.X\%} & {\color{red}X.XX} \\
-    + supervised contrastive  & {\color{red}XX.X\%} & {\color{red}X.XX} \\
-    + bispectral contrastive  & {\color{red}\textbf{XX.X\%}} & {\color{red}\textbf{X.XX}} \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\todo[inline]{Interpret results. The bispectral regularizer should provide
-comparable or better rotation robustness to supervised contrastive learning,
-without needing augmentation tracking. This demonstrates the library's
-utility beyond invariant pooling.}
-
-
-\section{Related Work}
-\label{sec:related}
-
-\todo[inline]{Write $\sim$0.75 page on related work. Frame comparisons in terms of what software/tools exist and what our library adds. Distinguish from each:}
-
-\paragraph{Equivariant network libraries.}
-\texttt{e2cnn}/\texttt{escnn}~\citep{weiler2019general, cesa2022program} provide steerable
-convolution layers but no bispectral invariant pooling. The \texttt{bispectrum}
-library is complementary: it provides the invariant map that
-\texttt{escnn} lacks, and can be used as a drop-in replacement for
-norm/gate pooling in existing architectures.
-
-\paragraph{Bispectral neural networks.}
-\citet{sanborn2023bispectral} propose learning the group from data using
-bispectral features, but restrict to commutative groups and achieve lower
-accuracy than equivariant baselines on RotMNIST (98.1\% vs.\ 99.3\% for E2CNN).%
-\todo[inline,color=red!10]{Verify: 98.1\% BNN vs.\ 99.3\% E2CNN from Sanborn et al.\ Table~2.}
-Our library handles non-abelian groups and integrates into existing
-equivariant architectures rather than replacing them.
-
-\paragraph{Bispectral CNNs.}
-\citet{oreiller2022bispectral} introduce a bispectral U-Net for
-medical image segmentation using SO(2) bispectra.
-Their implementation is TensorFlow-based, uses the full (not selective)
-bispectrum, and supports only SO(2).
-Our library supports 7 group/domain pairs with selective coefficients,
-GPU-accelerated PyTorch, and signal inversion.
-
-\paragraph{CG product nonlinearities.}
-Clebsch--Gordan Nets~\citep{kondor2018clebsch} and tensor field
-networks~\citep{thomas2018tensor} use the CG tensor product as a nonlinearity.
-While complete, the cost is $O(|G|^3)$ per layer.
-The selective bispectrum achieves completeness at $O(|G|)$ coefficient cost
-for finite groups, applied at the invariant pooling step.
-The MACE symmetric contraction~\citep{batatia2024design} can be seen as a
-generalized bispectrum; our work provides the selective reduction.
-
-\paragraph{General SO(2) nonlinearities.}
-\citet{franzen2021general} propose FFT $\to$ pointwise nonlinearity
-$\to$ IFFT. This is approximate due to aliasing from harmonic truncation.
-The bispectrum is exact by construction.
-
-\paragraph{ML software at NeurIPS.}
-Software papers at NeurIPS succeed when they pair engineering with
-scientific contribution~\citep{paszke2019pytorch, gardner2018gpytorch,
-ke2017lightgbm, moreau2022benchopt}. Our paper follows this template:
-the library is the vehicle, and the steerable-to-Fourier bridge and
-octahedral inversion resolution are the scientific contributions.
-
-\paragraph{Tensor product expressivity.}
-\citet{price2025freedom} systematically analyze the expressivity--runtime
-tradeoff in equivariant tensor products, showing that faster variants
-sacrifice expressivity. The selective bispectrum offers a complementary
-perspective: it achieves completeness at $O(|G|)$ cost by leveraging the
-algebraic structure of Kronecker products rather than approximating the
-full tensor product.
-
-
-\section{Discussion and Conclusion}
-\label{sec:conclusion}
-
-\todo[inline]{Write $\sim$0.75 page. Cover:}
-
-\paragraph{Summary.}
-We presented \texttt{bispectrum}, a PyTorch library that makes selective
-$G$-bispectra practical as differentiable primitives for invariant feature
-extraction, regularization, and signal reconstruction.
-The library supports 7 group/domain pairs, provides GPU-accelerated
-computation and signal inversion, and integrates into standard training
-pipelines as a drop-in \texttt{nn.Module}.
-
-Building the library required resolving two open problems:
-the steerable-to-Fourier bridge (Theorem~\ref{thm:steerable-fourier})
-that enables bispectral computation from non-abelian steerable features,
-and the octahedral inversion gap that we resolved with bootstrap~+~LM
-refinement. On PatchCamelyon, bispectral pooling achieves the highest
-AUC among equivariant nonlinearities and the most stable Pareto frontier
-across model sizes, while maintaining near-perfect rotation invariance.
-
-\paragraph{Limitations.}
-\begin{itemize}
-  \item Selective $\SO(3)$ on $S^2$ remains an open problem---the library
-    currently supports only the full bispectrum for this group.
-  \item Inversion for continuous groups with band-limit truncation needs
-    further study (stability under truncation).
-  \item Broader empirical evaluation (cryo-EM, molecular ML) is future work.
-\end{itemize}
-
-\paragraph{Roadmap and governance.}
-\todo[inline]{Write $\sim$3 sentences on roadmap. Cover:}
-\begin{itemize}
-  \item \textbf{Planned extensions}: selective $\SO(3)$ completeness (open
-    mathematical problem), ball domain $S^2 \times \R^+$, integration with
-    \texttt{escnn}/\texttt{e3nn}, applications to cryo-EM and
-    molecular ML.
-  \item \textbf{Maintenance}: MIT license, GitHub CI/CD with pre-commit hooks,
-    pip-installable via \texttt{pip install bispectrum}, $>$90\% test coverage,
-    Codecov integration.
-  \item \textbf{Community}: hosted on GitHub
-    (\texttt{geometric-intelligence/bispectrum}), issues and PRs welcome.
-\end{itemize}
-
-
-\begin{ack}
-\todo[inline]{Add acknowledgments for camera-ready version. Include funding sources and competing interests.}
-\end{ack}
-
-\bibliographystyle{plainnat}
-\bibliography{references}
-
-
-\appendix
-
-\section{Steerable-to-Fourier bridge: proof and practical consequences}
-\label{app:proofs}
-
-\todo[inline]{Insert full proof of P4 (steerable-to-Fourier bridge) here. Include:}
-\begin{itemize}
-  \item All supporting lemmas with proofs.
-  \item The precise relationship between intertwiner coefficients
-    and group Fourier matrices.
-  \item The multi-head construction for representations that do not
-    satisfy $m_i \geq d_i$.
-\end{itemize}
-
-\paragraph{Practical consequences for equivariant architectures.}
-\todo[inline]{Discuss which escnn/e2cnn architectures satisfy the conditions of Theorem~\ref{thm:steerable-fourier}. Which standard configurations work out of the box? Which require modification (e.g., increasing multiplicity)?}
-
-\paragraph{The BispectrumPool layer.}
-\todo[inline]{Detail the concrete implementation:}
-\begin{itemize}
-  \item Receive equivariant features of shape $(B, C, |G|, H, W)$.
-  \item Per channel, per spatial location: extract Fourier coefficients
-    from the group dimension via group DFT.
-  \item Compute selective bispectrum using the appropriate library module
-    (\texttt{CnonCn} for $C_8$, \texttt{DnonDn} for $D_4$, etc.).
-  \item Log-compress ($x \mapsto \mathrm{sign}(x) \cdot \log(1 + |x|)$)
-    for numerical stability and project back via $1 \times 1$ convolution.
-\end{itemize}
-
-
-\section{Octahedral inversion: convergence analysis}
-\label{app:octa-convergence}
-
-\todo[inline]{Include convergence analysis for the bootstrap + LM inversion. Cover:}
-\begin{itemize}
-  \item Convergence plots: residual vs.\ iteration for representative signals.
-  \item Multi-start statistics: fraction of initializations that converge
-    to the correct orbit.
-  \item Wall-clock timing breakdown: bootstrap init vs.\ LM iterations.
-  \item Comparison with pure optimization (no bootstrap init) to show
-    the value of the warm start.
-\end{itemize}
-
-
-\section{Extended benchmark tables}
-\label{app:benchmarks}
-
-\todo[inline]{Include full benchmark tables:}
-\begin{itemize}
-  \item Forward timing (CPU and GPU) for all modules at multiple group sizes.
-  \item Inversion timing for all modules with inversion support.
-  \item Memory usage (peak GPU memory) at various batch sizes.
-  \item Hardware: report on at least 2 GPU types (e.g., A100 and V100).
-\end{itemize}
-
-
-\section{Experimental details}
-\label{app:exp-details}
-
-\todo[inline]{Include full experimental details:}
-\begin{itemize}
-  \item Architecture diagram of the equivariant DenseNet.
-  \item Hyperparameter table (learning rate, weight decay, batch size,
-    epochs, patience, growth rate, block config).
-  \item Training curves (loss and AUC vs.\ epoch) for all models.
-  \item Data augmentation details for the standard baseline.
-  \item PCam data preprocessing: download from Zenodo, HDF5 loading,
-    GPU-based augmentation pipeline.
-\end{itemize}
-
-
-\section{Additional experiments}
-\label{app:additional-exp}
-
-\todo[inline]{Include additional experiments if completed:}
-\begin{itemize}
-  \item Galaxy10 DECals: 10-class galaxy morphology classification.
-  \item $D_4$ group results on PCam (dihedral instead of cyclic).
-  \item Extended data efficiency curves with more training fractions.
-\end{itemize}
-
-
-\newpage
-\input{checklist.tex}
-
-
-\end{document}
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-\documentclass[11pt]{article}
-
-\usepackage[margin=1in]{geometry}
-\usepackage{amsmath,amssymb,amsthm,mathtools}
-\usepackage{enumitem}
-\usepackage{booktabs}
-\usepackage[round,authoryear]{natbib}
-\usepackage{hyperref}
-\usepackage{algorithm}
-\usepackage{algorithmic}
-
-\theoremstyle{plain}
-\newtheorem{theorem}{Theorem}
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{corollary}[theorem]{Corollary}
-
-\theoremstyle{definition}
-\newtheorem{definition}[theorem]{Definition}
-\newtheorem{hypothesis}[theorem]{Hypothesis}
-
-\theoremstyle{remark}
-\newtheorem{remark}[theorem]{Remark}
-
-\DeclareMathOperator{\im}{Im}
-\DeclareMathOperator{\re}{Re}
-
-\newcommand{\R}{\mathbb{R}}
-\newcommand{\C}{\mathbb{C}}
-\newcommand{\Q}{\mathbb{Q}}
-\newcommand{\SO}{\mathrm{SO}}
-\newcommand{\OG}{\mathrm{O}}
-\newcommand{\SH}{Y}
-\newcommand{\CG}[6]{C^{#5,\,#6}_{#1,\,#2;\,#3,\,#4}}
-\newcommand{\abs}[1]{\lvert #1 \rvert}
-\newcommand{\norm}[1]{\lVert #1 \rVert}
-\newcommand{\conj}[1]{\overline{#1}}
-\newcommand{\Sb}{\mathcal{S}_\beta}
-\newcommand{\Sseed}{\mathcal{S}_{\mathrm{seed}}}
-\newcommand{\SP}{\mathcal{S}_P}
-\newcommand{\Phiaug}{\Phi_{\mathrm{aug}}}
-\newcommand{\Useed}{\mathcal{U}_{\mathrm{seed}}}
-
-\title{Completeness of a Selective $\SO(3)$-Bispectrum on $S^2$
-       \\ (with optional Clebsch--Gordan power diagnostics)}
-\author{}
-\date{}
-
-\begin{document}
-\maketitle
-
-\begin{abstract}
-We give a conditional proof that an $\Theta(L^2)$-sized purely scalar
-selective invariant $\Phiaug$, consisting of a constant-size full
-low-degree bispectrum block plus $O(\ell)$ selected scalar bispectral
-entries at each higher degree, separates $\SO(3)$-orbits of generic
-real-valued band-limited signals. The two load-bearing assumptions are
-isolated explicitly: finite-band injectivity of the full low-degree
-scalar seed, and nonvanishing of the per-degree bootstrap
-determinants. The proof follows Kakarala's pattern, motivated by
-\citet{kakarala1992triple,kakarala2012bispectrum}: a full-bispectrum
-seed followed by a triangular recurrence in which each
-$\mathbf{F}_\ell$ is recovered by solving an explicit linear system.
-Clebsch--Gordan power entries are optional implementation diagnostics
-and are not used in the proof; the adjective ``augmented'' refers to
-this implementation variant.
-\end{abstract}
-
-\section{Setup and notation}
-\label{sec:setup}
-
-\paragraph{Signal space.}
-Let $V_L$ denote the space of real-valued band-limited functions
-$f : S^2 \to \R$ with band-limit~$L$, expanded in spherical harmonics
-$\{\SH_\ell^m\}$:
-\begin{equation}\label{eq:sh-expansion}
-  f(\theta,\varphi)
-  = \sum_{\ell=0}^{L}\sum_{m=-\ell}^{\ell}
-    a_\ell^m \, \SH_\ell^m(\theta,\varphi),
-  \qquad a_\ell^{-m} = (-1)^m \conj{a_\ell^m}
-  \quad (\text{reality}).
-\end{equation}
-Write $\mathbf{F}_\ell := (a_\ell^{-\ell},\dots,a_\ell^{\ell}) \in \C^{2\ell+1}$ for the
-coefficient vector at degree~$\ell$. The reality constraint reduces
-$\mathbf{F}_\ell$ to $2\ell+1$ real degrees of freedom, so
-$\dim_\R V_L = \sum_{\ell=0}^{L}(2\ell+1) = (L+1)^2$.
-
-\paragraph{Group action.}
-Under $g\in\SO(3)$, the coefficient vectors transform irreducibly,
-$\mathbf{F}_\ell \mapsto D^\ell(g)\,\mathbf{F}_\ell$, where $D^\ell$ is
-the $(2\ell+1)$-dimensional Wigner $D$-matrix.
-
-\begin{definition}[Bispectrum on $S^2$]\label{def:bispectrum}
-For each triple $(\ell_1,\ell_2,\ell)$ with $0 \le \ell_1,\ell_2,\ell \le L$
-satisfying the triangle inequality $\abs{\ell_1{-}\ell_2} \le \ell \le \ell_1{+}\ell_2$,
-the \emph{scalar bispectral coefficient} is
-\begin{equation}\label{eq:bispectrum}
-  \beta_{\ell_1,\ell_2,\ell}(f)
-  := \sum_{\substack{m_1,m_2,m \\ m_1+m_2 = m}}
-     \CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m} \,
-     a_{\ell_1}^{m_1}\, a_{\ell_2}^{m_2}\, \conj{a_\ell^m},
-\end{equation}
-where $\CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m} := \langle \ell_1, m_1;\, \ell_2, m_2 \mid \ell, m\rangle$
-is the Clebsch--Gordan coefficient. Each $\beta_{\ell_1,\ell_2,\ell}$
-is $\SO(3)$-invariant. The full bispectrum at band-limit~$L$ has
-$\Theta(L^3)$ entries~\citep{kakarala1992triple,kakarala2012bispectrum}.
-\end{definition}
-
-\begin{definition}[Clebsch--Gordan power spectrum]\label{def:cg-power}
-For a triple $(\ell_1,\ell_2,\ell)$ satisfying the triangle inequality,
-let $(\mathbf{F}_{\ell_1}\otimes\mathbf{F}_{\ell_2})\mid_\ell$ denote the
-projection of the Kronecker product onto the $D^\ell$-isotypic component,
-realised by contracting against the CG coefficients. Define the
-\emph{CG power spectrum entry}
-\begin{equation}\label{eq:cg-power}
-  P_{\ell_1,\ell_2,\ell}(f)
-  := \norm{(\mathbf{F}_{\ell_1}\otimes\mathbf{F}_{\ell_2})\mid_\ell}^2
-   = \sum_{m=-\ell}^{\ell}
-     \Bigl| \sum_{m_1+m_2=m}
-       \CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}\,
-       a_{\ell_1}^{m_1}\, a_{\ell_2}^{m_2} \Bigr|^2.
-\end{equation}
-This is a degree-$4$, real-valued, $\SO(3)$-invariant polynomial in
-the signal coefficients.
-\end{definition}
-
-\noindent
-The standard power spectrum $\norm{\mathbf{F}_\ell}^2$ is the special
-case $P_{0,\ell,\ell}/\abs{a_0^0}^2$ (with our CG normalisation). The
-CG power generalises it by projecting
-$\mathbf{F}_{\ell_1}\otimes\mathbf{F}_{\ell_2}$ into each output
-channel~$V_\ell$ before taking norms.
-
-\begin{definition}[Reflection map $T_R$]\label{def:TR}
-The azimuthal reflection $R:(\theta,\varphi)\mapsto(\theta,-\varphi)$ acts
-on coefficients by $a_\ell^m\mapsto(-1)^m a_\ell^{-m}$. For real signals
-this equals complex conjugation: $a_\ell^m\mapsto\conj{a_\ell^m}$. Since
-$R\in\OG(3)\setminus\SO(3)$, the signal $T_R\cdot f$ is generically in a
-\emph{different} $\SO(3)$-orbit from $f$, and any complete
-$\SO(3)$-invariant must distinguish $f$ from $T_R\cdot f$.
-\end{definition}
-
-\begin{definition}[Selective invariant]\label{def:Phi-aug}
-Let $L_{\mathrm{seed}}:=\min(L,7)$. Define the \emph{seed index set}
-\[
-  \Sseed := \{(\ell_1,\ell_2,\ell):
-    0\le\ell_1\le\ell_2\le L_{\mathrm{seed}},\;
-    \abs{\ell_1-\ell_2}\le\ell\le\min(\ell_1+\ell_2,L_{\mathrm{seed}})\},
-\]
-the \emph{high-degree bootstrap blocks}
-$\mathcal{T}_\ell$ ($\ell\ge8$) of~\eqref{eq:Tell}, and the scalar
-index set
-\[
-  \Sb := \Sseed \;\cup\; \bigcup_{\ell=8}^L \mathcal{T}_\ell .
-\]
-The \emph{selective invariant} $\Phiaug:V_L\to\R^N$ is the
-concatenation of all selected scalar bispectral entries
-$\{\beta_{\ell_1,\ell_2,\ell}\}_{(\ell_1,\ell_2,\ell)\in\Sb}$, stored
-as complex numbers (equivalently, as real and imaginary parts). The
-seed sub-block has size at most $(L_{\mathrm{seed}}{+}1)^3\le 512$,
-independent of the final band-limit once $L\ge7$. We set
-$\SP:=\varnothing$ in the formal theorem.
-\end{definition}
-
-\begin{remark}[On the name ``augmented'']\label{rem:augmented-name}
-The implementation variant of $\Phiaug$ used in our experiments
-appends a CG-power block (Remark~\ref{rem:cg-optional}) and is what we
-call \emph{augmented}. The conditional theorem below is proved for the
-purely scalar selective invariant ($\SP=\varnothing$); the CG-power
-block is optional throughout the proof.
-\end{remark}
-
-\begin{remark}[Optional CG-power diagnostics]
-\label{rem:cg-optional}
-Any $O(L^2)$ family $\{P_{\ell_1,\ell_2,\ell}\}_{\SP}$ of CG-power
-entries may be appended to $\Phiaug$ without affecting completeness or
-its asymptotic size. These entries are useful implementation
-diagnostics (Section~\ref{sec:cgpower}); they are not used by the
-conditional theorem below.
-\end{remark}
-
-\begin{definition}[Genericity]\label{def:gen}
-A signal $f \in V_L$ is \emph{generic} if it satisfies:
-\begin{enumerate}[nosep,label=(G\arabic*)]
-  \item $a_0^0 \ne 0$ (nonzero mean);
-  \item $\norm{\mathbf{F}_1}\ne 0$ (nonzero dipole);
-  \item $a_2^1 \ne 0$ after the gauge of Lemma~\ref{lem:gauge}
-    (azimuthal phase resolvable);
-  \item the gauge-fixed seed truncation belongs to the seed-injective
-    stratum $\Useed$ of Hypothesis~\ref{hyp:seed};
-  \item $\Delta_\ell \ne 0$ for all $8 \le \ell \le L$, where
-    $\Delta_\ell$ is the real-coordinate determinant of
-    Hypothesis~\ref{hyp:bootstrap-det}.
-\end{enumerate}
-Conditions (G1)--(G5) are polynomial inequalities in the gauge-fixed
-real coordinates of the gauge slice $\mathcal{G}$ of
-Lemma~\ref{lem:gauge} (forward reference) and therefore cut out a
-Zariski-open subset $\mathcal{V}\subset\mathcal{G}$. Its
-$\SO(3)$-saturation $\SO(3)\cdot\mathcal{V}\subset V_L$ is a generic
-co-null subset of $V_L$, whose complement is contained in a finite
-union of proper algebraic subvarieties of $\mathcal{G}$ and is in
-particular Lebesgue-null.
-\end{definition}
-
-\section{The inversion algorithm}
-\label{sec:algorithm}
-
-The completeness theorem (Theorem~\ref{thm:main}) is the statement
-that the procedure below recovers $f$ from $\Phiaug(f)$ up to the
-$\SO(3)$-action for generic~$f$, provided the bootstrap determinants
-are nonzero at the required degrees.
-
-\begin{algorithm}[h]
-  \caption{Bispectrum inversion on $S^2$ from the selective invariant
-    $\Phiaug$ ($\SP=\varnothing$).
-    Inputs from $\Phiaug(f)$ are typeset \textbf{in bold}; recovered
-    coefficients accumulate in $(\mathbf{F}_0,\dotsc,\mathbf{F}_L)$
-    on the gauge slice.}
-  \label{alg:so3-inversion}
-  \begin{algorithmic}[1]
-    \STATE \textbf{Input:} $\Phiaug(f)$, band-limit $L \ge 4$, seed cutoff
-      $L_{\mathrm{seed}}=\min(L,7)$.
-    \STATE $A_0 \leftarrow (\boldsymbol{\beta_{0,0,0}})^{1/3}$
-      \hfill \COMMENT{closed form, Lemma~\ref{lem:F0F1}}
-    \STATE $c \leftarrow \sqrt{\boldsymbol{\beta_{0,1,1}}\,/\,A_0}$
-      \hfill \COMMENT{$c = \norm{\mathbf{F}_1}$}
-    \STATE $\mathbf{F}_0 \leftarrow A_0$,\quad
-      $\mathbf{F}_1 \leftarrow (0,\,c,\,0)^\top$
-      \hfill \COMMENT{gauge: align $\mathbf{F}_1$ with $\hat z$}
-    \STATE $(\mathbf{F}_2,\dots,\mathbf{F}_{L_{\mathrm{seed}}}) \leftarrow
-      \textsc{SeedSolve}\bigl(\boldsymbol{\{\beta_{\ell_1,\ell_2,\ell}\}_{\ell_1,\ell_2,\ell\le L_{\mathrm{seed}}}}\bigr)$
-      \hfill \COMMENT{returns gauge slice rep with $a_2^1\in\R_{>0}$ (Lemma~\ref{lem:gauge}, Hypothesis~\ref{hyp:seed})}
-    \FOR{$\ell = L_{\mathrm{seed}}+1, \dots, L$}
-      \STATE Build $A_\ell \in \C^{(2\ell+1)\times(2\ell+1)}$, $b_\ell\in\C^{2\ell+1}$
-        from selected bispectral entries linear in $\mathbf{F}_\ell$
-        and depending on $(\mathbf{F}_0,\dotsc,\mathbf{F}_{\ell-1})$
-      \STATE $\mathbf{F}_\ell \leftarrow A_\ell^{-1} b_\ell$
-        \hfill \COMMENT{bootstrap determinant nonzero}
-    \ENDFOR
-    \STATE \textbf{Return} $(\mathbf{F}_0,\dotsc,\mathbf{F}_L)$, equivalently $f$, up to $\SO(3)$.
-  \end{algorithmic}
-\end{algorithm}
-
-The algorithm is a direct generalisation of the finite-group selective
-inversion of \citet[Algorithm~3]{mataigne2024selective}: the for-loop
-in Lines~7--10 plays the role of their breadth-first traversal of the
-irrep graph, with the per-step Fourier-coefficient recurrence replaced
-by an $O(\ell)$-dimensional linear solve. The seed step in Line~6
-plays the role of the trivial-irrep base case in Mataigne et al.\ but
-is no longer in closed form; it is the finite-band seed assumption
-isolated in Hypothesis~\ref{hyp:seed}.
-
-\paragraph{Index sets.}
-For each output degree $\ell\ge8$, the bootstrap block referenced in
-Definition~\ref{def:Phi-aug} is
-\begin{equation}\label{eq:Tell}
-\begin{aligned}
-  \mathcal{T}_\ell
-  :=&\ \{(a,\ell,\ell-a) : 1\le a\le \ell-1\}\\
-   &\cup \{(a,\ell,\ell-a+1) : 2\le a\le \ell-1\}\\
-   &\cup \{(a,\ell-a,\ell) : 1\le a\le4\},
-\end{aligned}
-\end{equation}
-with $(\ell-1)+(\ell-2)+4=2\ell+1$ entries. Combined with the seed
-block $\Sseed$, this gives $|\Sb|=|\Sseed|+\sum_{\ell=8}^L(2\ell+1)
-=O(L^2)$ (Proposition~\ref{prop:size}). Optional CG-power
-diagnostics (Remark~\ref{rem:cg-optional}) may be appended separately,
-still at $O(L^2)$ cost.
-
-\section{CG-power diagnostics}
-\label{sec:cgpower}
-
-The argument of \citet{mataigne2024selective} for finite groups never
-needs the CG power: the scalar bispectrum on $L^2(G)$ already has full
-real Jacobian rank, because the reality constraint on a finite-group
-signal is encoded by a pairing of each irrep $\rho_i$ with its complex
-conjugate irrep $\rho_{\bar i}$, which is itself an irrep of $G$.
-
-For $\SO(3)$ on $S^2$, an earlier real-valued implementation of the
-selected bispectrum chose only the parity-reduced scalar component of
-each entry (real or imaginary part dictated by parity). With that row
-selection, the per-degree real Jacobian was rank-deficient at every
-degree $\ell\ge4$. CG-power entries close that gap diagnostically and
-were originally introduced for that reason.
-
-The proof in this paper does \emph{not} rely on those parity-reduced
-real rows. Each entry of $\Sb$ contributes its full complex value to
-$\Phiaug$ (Definition~\ref{def:Phi-aug}), and the per-degree bootstrap
-of Section~\ref{sec:bootstrap} uses the complex linear system of
-Hypothesis~\ref{hyp:bootstrap-det}. By Lemma~\ref{lem:bootstrap},
-invertibility of $A_\ell$ over $\C$ already forces uniqueness of
-$\mathbf{F}_\ell$ on the real-signal slice, so no real-rank gap arises
-in the proof itself. CG-power entries remain useful as implementation
-diagnostics and as a richer invariant for downstream learning, but
-they are not required for the conditional theorem.
-
-\section{Main theorem}
-\label{sec:main}
-
-\begin{theorem}[Conditional completeness]\label{thm:main}
-Let $L \ge 4$, and let $f,f'\in V_L$ both be generic in the sense of
-Definition~\ref{def:gen}. Assume Hypothesis~\ref{hyp:seed} for
-$L_{\mathrm{seed}}=\min(L,7)$ and
-Hypothesis~\ref{hyp:bootstrap-det} for every $8\le\ell\le L$. If
-$\Phiaug(f)=\Phiaug(f')$, then $f'=g\cdot f$ for some $g\in\SO(3)$.
-
-The output $\Phiaug$ has $\Theta(L^2)$ real components
-(Proposition~\ref{prop:size}), matching the orbit-space
-dimension lower bound $(L+1)^2 - 3$ up to a constant factor.
-\end{theorem}
-
-\noindent
-The proof reduces to gauge-fixing (\S\ref{sec:gauge}), the finite
-seed hypothesis (\S\ref{sec:seed}), parity bookkeeping
-(\S\ref{sec:cgpower-formal}), and the per-degree determinant bootstrap
-(\S\ref{sec:bootstrap}). It is collected in \S\ref{sec:proof}.
-
-\subsection{Gauge fixing}
-\label{sec:gauge}
-
-\begin{lemma}[Gauge fixing]\label{lem:gauge}
-For any $f \in V_L$ satisfying (G1)--(G3) of Definition~\ref{def:gen},
-there is a unique $g \in \SO(3)$ such that $\tilde f := g\cdot f$ has
-\begin{enumerate}[nosep,label=(\roman*)]
-  \item $a_1^{-1} = a_1^1 = 0$ and $a_1^0 = c > 0$
-    (alignment of $\mathbf{F}_1$ with $\hat z$), and
-  \item $a_2^1 \in \R_{>0}$ (residual $\SO(2)_z$ phase fixed).
-\end{enumerate}
-This consumes all three rotational degrees of freedom of $\SO(3)$. The
-\emph{gauge slice} $\mathcal{G}\subset V_L$ defined by (i)--(ii) has
-$\dim_\R\mathcal{G} = (L+1)^2 - 3$.
-\end{lemma}
-\begin{proof}
-$\SO(3)$ acts transitively on $S^2$, so the unit vector
-$\mathbf{F}_1/\norm{\mathbf{F}_1}$ can be rotated to $\hat z$; this fixes
-$g$ modulo the stabiliser $\SO(2)_z$ of $\hat z$. On the resulting
-slice, $\SO(2)_z$ acts on $a_2^1$ by multiplication with $e^{i\varphi}$,
-so taking $\varphi = -\arg(a_2^1)$ uniquely places $a_2^1$ on the
-positive real axis. The dimension count is
-$(L+1)^2 - \dim\SO(3) = (L+1)^2 - 3$.
-\end{proof}
-
-\begin{lemma}[$\mathbf{F}_0$ and $\mathbf{F}_1$]\label{lem:F0F1}
-On the gauge slice, $\beta_{0,0,0} = (a_0^0)^3$ determines $a_0^0 = A_0$
-uniquely (the real cube root), and $\beta_{0,1,1} = A_0\norm{\mathbf{F}_1}^2$
-determines $c = \norm{\mathbf{F}_1}$. By (i) of Lemma~\ref{lem:gauge},
-$\mathbf{F}_1 = (0,c,0)^\top$.
-\end{lemma}
-
-\subsection{Finite-band seed hypothesis}
-\label{sec:seed}
-
-\begin{hypothesis}[Finite seed injectivity]\label{hyp:seed}
-For each $s\le7$, there is a Zariski-open gauge-fixed seed stratum
-$\Useed^{(s)}$ such that the full scalar seed map
-\[
-  (\mathbf{F}_0,\dots,\mathbf{F}_s)
-  \longmapsto
-  \{\beta_{\ell_1,\ell_2,\ell} :
-    0\le\ell_1\le\ell_2\le s,\;
-    \abs{\ell_1-\ell_2}\le\ell\le\min(\ell_1+\ell_2,s)\}
-\]
-is injective on $\Useed^{(s)}$. In Definition~\ref{def:gen}, $\Useed$
-means $\Useed^{(L_{\mathrm{seed}})}$.
-\end{hypothesis}
-
-\begin{remark}[Relation to Kakarala]
-Kakarala's homogeneous-space completeness theorem
-\citep[Theorem~7]{kakarala2012bispectrum}, originally proved in
-\citet{kakarala1992triple}, is the template for
-Hypothesis~\ref{hyp:seed}: for $H=\SO(2)_z$, it recovers a left
-$H$-invariant function on $\SO(3)$ from the full bispectrum under a
-maximal $H$-rank condition, up to the normaliser $N_H\subset\SO(3)$.
-This does not by itself prove the finite-band seed statement above,
-because a band-limited truncation has zero Fourier coefficients above
-the cutoff. We therefore use Kakarala only as structural motivation
-unless a finite-band version or an independent finite certificate is
-provided.
-\end{remark}
-
-\begin{remark}[Possible symbolic certificate]
-\label{rem:bandeira}
-\citet{bandeira2017estimation} give computer-algebra evidence for
-generic orbit recovery from polynomial invariants at small band-limits.
-To use that result as a proof of Hypothesis~\ref{hyp:seed}, one must
-check that its invariant map is exactly the full scalar $S^2$
-bispectrum seed above, on the same gauge-fixed real signal space, and
-that its certificate establishes generic finite-fibre injectivity
-rather than only local rank. We do not assume that identification here.
-\end{remark}
-
-\begin{remark}[On the ``up to reflection'' caveat of \citet{edidin2024orbit}]
-\label{rem:edidin}
-\citet[Section~1]{edidin2024orbit} state that
-\citep{kakarala2012bispectrum} recovers the orbit ``up to reflection''.
-This caveat is sharp for the constructive recursion in Kakarala's
-Section~5 (a polar decomposition step pins $\hat F(1)$ only modulo a
-real orthogonal matrix, an $\OG(3)$-ambiguity that propagates), but it
-is not present in Theorem~7 itself, which gives recovery up to
-$N_H \subset \SO(3)$. The odd-parity entry $\beta_{2,3,4}$ in the
-seed block separates $f$ from $T_R\cdot f$
-(Corollary~\ref{cor:first-odd}), confirming that no $\OG(3)\setminus\SO(3)$
-ambiguity remains.
-\end{remark}
-
-\subsection{Parity and the first parity-breaking entry}
-\label{sec:cgpower-formal}
-
-\begin{lemma}[Parity of the bispectrum]\label{lem:parity}
-For every signal:
-\begin{equation}\label{eq:parity-TR}
-  \beta_{\ell_1,\ell_2,\ell}(T_R\cdot f)
-  = (-1)^{\ell_1+\ell_2+\ell}\, \beta_{\ell_1,\ell_2,\ell}(f).
-\end{equation}
-For real signals, the further reality constraint
-\begin{equation}\label{eq:parity-real}
-  \conj{\beta_{\ell_1,\ell_2,\ell}(f)}
-  = (-1)^{\ell_1+\ell_2+\ell}\, \beta_{\ell_1,\ell_2,\ell}(f)
-\end{equation}
-holds. Together: when $\ell_1{+}\ell_2{+}\ell$ is even, $\beta$ is real
-and $T_R$-invariant; when odd, $\beta$ is purely imaginary and
-$\im\beta$ flips sign under~$T_R$. The CG power $P_{\ell_1,\ell_2,\ell}$
-is real and $T_R$-invariant in all cases.
-\end{lemma}
-\begin{proof}
-For~\eqref{eq:parity-TR}, substitute $a_\ell^m \mapsto (-1)^m a_\ell^{-m}$
-into~\eqref{eq:bispectrum}; the three sign factors give
-$(-1)^{m_1+m_2+m} = (-1)^{2m} = 1$, and re-indexing via the CG symmetry
-$\CG{\ell_1}{-m_1}{\ell_2}{-m_2}{\ell}{-m} = (-1)^{\ell_1+\ell_2-\ell}
-\CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}$ yields the claim.
-For~\eqref{eq:parity-real}, take complex conjugates inside~\eqref{eq:bispectrum},
-use $\conj{a_\ell^m} = (-1)^m a_\ell^{-m}$, and re-index.
-For $P$, the CG-projected component picks up
-$(-1)^{\ell_1+\ell_2+\ell+M}$ under $T_R$, and re-indexing $M \mapsto -M$
-in $\sum_M\abs{\cdot}^2$ leaves the sum unchanged.
-\end{proof}
-
-\begin{corollary}[First nonzero odd-parity entry]\label{cor:first-odd}
-For real signals, every odd-parity bispectral entry
-$\beta_{\ell_1,\ell_2,\ell}$ whose index multiset
-$\{\ell_1,\ell_2,\ell\}$ contains a repeated value vanishes
-identically; in particular, every odd-parity entry with
-$\max(\ell_1,\ell_2,\ell)\le3$ vanishes. The first surviving
-odd-parity entry is therefore $\beta_{2,3,4}$
-(with $\ell_1+\ell_2+\ell=9$, all distinct). It is generically
-nonzero, so $\im\beta_{2,3,4}(f)\ne0$ separates $f$ from $T_R\cdot f$.
-\end{corollary}
-\begin{proof}
-\emph{Repeated-index vanishing.} Define the symmetric scalar tensor
-\[
-  T_{\ell_1,\ell_2,\ell_3}
-  := \sum_{m_1,m_2,m_3}
-       \begin{pmatrix} \ell_1 & \ell_2 & \ell_3 \\
-                       m_1 & m_2 & m_3 \end{pmatrix}\,
-       a_{\ell_1}^{m_1}\, a_{\ell_2}^{m_2}\, a_{\ell_3}^{m_3} .
-\]
-Converting the Wigner~$3$j into a CG via
-$\bigl(\begin{smallmatrix} j_1 & j_2 & j_3 \\
-       m_1 & m_2 & m_3\end{smallmatrix}\bigr)
-=\frac{(-1)^{j_1-j_2-m_3}}{\sqrt{2j_3+1}}\,
-\CG{j_1}{m_1}{j_2}{m_2}{j_3}{-m_3}$ and applying the reality identity
-$a_{\ell_3}^{m_3}=(-1)^{m_3}\conj{a_{\ell_3}^{-m_3}}$ shows
-$\beta_{\ell_1,\ell_2,\ell}=(-1)^{\ell_1+\ell_2}\sqrt{2\ell+1}\,
-T_{\ell_1,\ell_2,\ell}$, so $\beta$ and $T$ have the same vanishing
-locus. The 3j symbol is invariant under even column permutations and
-picks up $(-1)^{\ell_1+\ell_2+\ell_3}$ under odd ones. If two columns
-$i,j$ have $\ell_i=\ell_j$, the inserted vectors $a_{\ell_i},
-a_{\ell_j}$ are literally the same coefficient vector of $f$, so
-swapping those columns leaves the contracted sum unchanged while
-multiplying it by $(-1)^{\ell_1+\ell_2+\ell_3}$. For odd parity this
-forces $T=-T$, hence $T=0$ and $\beta=0$.
-For $\max(\ell_1,\ell_2,\ell)\le3$, every admissible odd-parity
-triangle (e.g.\ $(0,1,2)$, $(1,1,1)$, $(0,3,3)$, $(1,2,3)$, $(2,2,3)$)
-has a repeated index, so $\beta_{2,3,4}$ is the first candidate that
-survives.
-
-\emph{Nonvanishing of $\beta_{2,3,4}$.} Take the explicit signal
-\[
-  a_2^2 = a_3^2 = 1, \quad a_4^4 = i,
-\]
-with all other coefficients in $\mathbf{F}_2,\mathbf{F}_3,\mathbf{F}_4$
-set to zero, and the negative-$m$ entries fixed by reality
-($a_2^{-2}=a_3^{-2}=1$, $a_4^{-4}=-i$). Only the index choices
-$(m_1,m_2,m)\in\{(2,2,4),(-2,-2,-4)\}$ contribute. Using
-$\CG{2}{2}{3}{2}{4}{4}=\sqrt{2/5}$ and the standard sign relation
-$\CG{2}{-2}{3}{-2}{4}{-4}=-\sqrt{2/5}$,
-\[
-  \beta_{2,3,4}
-  = \sqrt{2/5}\cdot 1\cdot1\cdot(-i)
-   + (-\sqrt{2/5})\cdot 1\cdot1\cdot(+i)
-  = -2i\sqrt{2/5}\ne0.
-\]
-Hence $\beta_{2,3,4}$ is not identically zero on $V_L$. Genericity is
-then a Zariski-open condition on the gauge slice.
-\end{proof}
-
-\subsection{Per-degree linear bootstrap}
-\label{sec:bootstrap}
-
-For $\ell\ge8$, use the block $\mathcal{T}_\ell$ in~\eqref{eq:Tell}.
-The first two families are cross entries $(a,\ell,q)$, hence linear in
-$\mathbf{F}_\ell$:
-\[
-  \beta_{a,\ell,q}
-  = \sum_{m=-\ell}^{\ell} H_{a,q}^{m}\, a_\ell^m,
-  \qquad
-  H_{a,q}^{m}
-  := \sum_{r=-a}^{a}
-     \CG{a}{r}{\ell}{m}{q}{r+m}\,
-     a_a^r\,\conj{a_q^{r+m}} .
-\]
-The third family consists of chain entries $(a,\ell-a,\ell)$, linear
-in $\conj{\mathbf{F}_\ell}$:
-\[
-  \beta_{a,\ell-a,\ell}
-  = \sum_{m=-\ell}^{\ell} G_a^m\,\conj{a_\ell^m},
-  \qquad
-  G_a^m
-  := \sum_{r+s=m}
-     \CG{a}{r}{\ell-a}{s}{\ell}{m}\,
-     a_a^r\,a_{\ell-a}^s .
-\]
-Conjugating the chain equations gives
-$\sum_m \conj{G_a^m}a_\ell^m=\conj{\beta_{a,\ell-a,\ell}}$.
-Thus all rows in $\mathcal{T}_\ell$ assemble into a square complex
-linear system
-\begin{equation}\label{eq:bootstrap-system}
-  A_\ell(\mathbf{F}_{<\ell})\,\mathbf{F}_\ell=b_\ell .
-\end{equation}
-Although \eqref{eq:bootstrap-system} is complex-linear in the unknown
-$\mathbf{F}_\ell$, its coefficients are functions on the real-signal
-lower-degree slice. Whenever a conjugated lower coefficient appears,
-we rewrite it using the real coordinates
-\[
-  a_j^0\in\R,\qquad a_j^m=x_{j,m}+i y_{j,m},\qquad
-  a_j^{-m}=(-1)^m(x_{j,m}-i y_{j,m})\quad(m>0).
-\]
-After this substitution, every entry of $A_\ell$ is a complex-valued
-polynomial in the real variables $\{a_j^0,x_{j,m},y_{j,m}:j<\ell\}$.
-
-\begin{hypothesis}[Uniform bootstrap determinant]\label{hyp:bootstrap-det}
-For every $\ell\ge8$, the determinant
-\[
-  \Delta_\ell
-  := \det A_\ell
-\]
-is not the zero complex-valued polynomial on these real lower-degree
-coordinates.
-\end{hypothesis}
-
-\begin{lemma}[Bootstrap uniqueness]\label{lem:bootstrap}
-Assume Hypothesis~\ref{hyp:bootstrap-det}. For every fixed
-$\ell\ge8$, the set $\{\Delta_\ell\ne0\}$ is Zariski-open and
-Lebesgue-co-null in the lower-degree coefficient space. On this set,
-the entries in $\mathcal{T}_\ell$ uniquely determine
-$\mathbf{F}_\ell$ from $(\mathbf{F}_0,\dots,\mathbf{F}_{\ell-1})$.
-\end{lemma}
-\begin{proof}
-The entries of $A_\ell$ are polynomial functions of the known
-real-coordinate lower degrees, and $b_\ell$ is read directly from the
-selected complex bispectral entries. If $\Delta_\ell\ne0$, the square system
-\eqref{eq:bootstrap-system} has the unique solution
-$\mathbf{F}_\ell=A_\ell^{-1}b_\ell$ over $\C^{2\ell+1}$, hence also
-on the real-signal slice. Since $\Delta_\ell$ is a nonzero
-complex-valued polynomial by Hypothesis~\ref{hyp:bootstrap-det},
-$\abs{\Delta_\ell}^2=(\re\Delta_\ell)^2+(\im\Delta_\ell)^2$ is a
-nonzero real polynomial. Its zero set is therefore a proper algebraic
-subvariety.
-\end{proof}
-
-\begin{remark}[What remains to prove uniformly]\label{rem:uniform-gap}
-The determinant hypothesis is the exact Kakarala-style nonsingularity
-condition for our selective recurrence. Existing floating-point runs
-find $\Delta_\ell\ne0$ degree-by-degree for $\ell\le100$ at a
-deterministic witness, but they do not by themselves prove a
-finite-band theorem.
-An unconditional all-$L$ proof requires a closed-form nonvanishing
-identity for $\Delta_\ell$ (or an explicit all-$\ell$ witness). The
-current document isolates that identity rather than replacing it by a
-local Jacobian-rank claim.
-\end{remark}
-
-\subsection{Proof of Theorem~\ref{thm:main}}
-\label{sec:proof}
-
-\begin{proof}[Proof of Theorem~\ref{thm:main}]
-Suppose $\Phiaug(f)=\Phiaug(f')$ for generic $f,f'$. By
-Lemma~\ref{lem:gauge}, gauge-fix to $\tilde f,\tilde f'$ on the gauge
-slice; $\SO(3)$-invariance preserves
-$\Phiaug(\tilde f)=\Phiaug(\tilde f')$. It suffices to prove
-$\tilde f=\tilde f'$.
-
-\emph{Degrees $0$ and $1$.} By Lemma~\ref{lem:F0F1}, $A_0$ and $c$ are
-determined, and $\mathbf{F}_0,\mathbf{F}_1$ agree.
-
-\emph{Seed degrees.} By Hypothesis~\ref{hyp:seed}, the full seed block
-uniquely determines
-$(\mathbf{F}_2,\dots,\mathbf{F}_{L_{\mathrm{seed}}})$ on the gauge
-slice, so $\tilde f,\tilde f'$ agree at all degrees
-$\le L_{\mathrm{seed}}$.
-
-\emph{Bootstrap degrees.} Inductively, suppose $\tilde f$ and
-$\tilde f'$ agree at degrees $0,\dotsc,\ell{-}1$. Then the linear
-system $A_\ell\mathbf{F}_\ell=b_\ell$ in
-Section~\ref{sec:bootstrap} has identical coefficient matrix and
-right-hand side for both signals. By Lemma~\ref{lem:bootstrap} and
-(G5), $A_\ell$ is invertible. Hence
-$\mathbf{F}_\ell=\mathbf{F}_\ell'$.
-
-\emph{Conclusion.} Inducting up to $\ell=L$, $\tilde f=\tilde f'$, so
-$f'\in\SO(3)\cdot f$. The set of generic signals is the
-$\SO(3)$-saturation of the Zariski-open subset
-$\mathcal{V}\subset\mathcal{G}$ cut out by (G1)--(G5)
-(Definition~\ref{def:gen}), hence Lebesgue-co-null in $V_L$.
-\end{proof}
-
-\section{Output size matches the lower bound}
-\label{sec:size}
-
-\begin{proposition}[Lower bound]\label{prop:lower-bound}
-Any continuous semialgebraic (in particular, polynomial) complete
-$\SO(3)$-invariant map $V_L\to\R^k$ that separates the generic free
-stratum must have
-$k\ge (L+1)^2-\dim\SO(3)=(L+1)^2-3=\Omega(L^2)$.
-\end{proposition}
-\begin{proof}
-The free generic stratum of the quotient $V_L/\SO(3)$ is a
-semialgebraic manifold of dimension $(L+1)^2-3$. A continuous
-semialgebraic injection into $\R^k$ cannot decrease semialgebraic
-dimension, so $k$ must be at least this dimension.
-\end{proof}
-
-\begin{proposition}[Upper bound]\label{prop:size}
-$|\Phiaug| = \Theta(L^2)$.
-\end{proposition}
-\begin{proof}
-By construction (Definition~\ref{def:Phi-aug}), the full seed block has
-at most $512$ complex bispectral scalars, independent of~$L$. For each
-degree $\ell\ge8$, $\Sb$ allocates exactly $2\ell+1$ selected complex
-bispectral entries, the rows of the square bootstrap system
-\eqref{eq:bootstrap-system}. Thus
-$\sum_{\ell=8}^L(2\ell+1)=O(L^2)$, so $|\Phiaug|=O(L^2)$. Any optional
-CG-power diagnostics (Remark~\ref{rem:cg-optional}) preserve this
-bound. The matching $\Omega(L^2)$ lower bound comes from
-Proposition~\ref{prop:lower-bound}.
-\end{proof}
-
-\begin{table}[h]
-  \caption{Asymptotic entry counts in $\Phiaug$. The full low-degree
-    seed is constant-size; the high-degree selective part is linear
-    per target degree.}
-  \label{tab:entry-counts}
-  \centering
-  \small
-  \begin{tabular}{cccc}
-    \toprule
-    degree range & selected $\beta$ & selected $P$ & total \\
-    \midrule
-    $\ell\le7$ & full seed, $\le512$ & optional & $O(1)$ \\
-    $\ell\ge8$ & $2\ell+1$ & $O(\ell)$ & $O(\ell)$ \\
-    \bottomrule
-  \end{tabular}
-\end{table}
-
-\section{Computational evidence and limitations}
-\label{sec:certs}
-
-Theorem~\ref{thm:main} is structural once
-Hypotheses~\ref{hyp:seed} and~\ref{hyp:bootstrap-det} are supplied.
-The current repository does not yet contain either a finite-band proof
-of Hypothesis~\ref{hyp:seed} or a closed-form proof of
-Hypothesis~\ref{hyp:bootstrap-det} for all~$\ell$. It does contain
-reproducible numerical evidence for the bootstrap determinants through
-the band-limits used in the experiments.
-
-\begin{description}[style=nextline]
-  \item[\textnormal{C1.\ Finite seed injectivity (Hypothesis~\ref{hyp:seed}).}]
-    Open in this document. Kakarala's theorem motivates the statement,
-    but does not prove the finite-band scalar seed claim as written.
-    A rigorous replacement would be a finite-band theorem or an
-    exact/interval computation certifying the seed map.
-  \item[\textnormal{C2.\ Bootstrap determinant evidence.}]
-    Checked per-$\ell$ at the deterministic seed-42 witness for
-    $\ell\in\{8,\dots,100\}$ in IEEE~754 double precision, with
-    $\sigma_{\min}(A_\ell)/\sigma_{\max}(A_\ell) \ge 10^{-4}$
-    in the existing runs. This is strong numerical evidence, not a
-    mathematical certificate. The script is
-    \texttt{verify\_linear\_bootstrap.py} in \texttt{benchmarks/}.
-  \item[\textnormal{C3.}]
-    \textbf{CG-power real-Jacobian diagnostics.} The augmented real Jacobian is numerically full-rank for
-    $\ell\in\{4,\dots,30\}$ at the same witness. This is evidence that
-    the implementation's augmented invariant is well-conditioned, but
-    it is not used as the global uniqueness step in the corrected proof.
-\end{description}
-
-\begin{remark}[Finite band-limit evidence]\label{rem:numerical-evidence}
-The floating-point checks through $\ell=100$ support the determinant
-hypothesis for the band-limits used in the paper. They would become a
-rigorous finite-band theorem only after being upgraded to exact
-arithmetic, interval arithmetic, or a perturbation bound showing that
-the exact matrix is certified away from singularity.
-\end{remark}
-
-\begin{remark}[The uniform determinant gap]
-For $\ell\ge8$, the selected family $\mathcal{T}_\ell$ has a uniform
-closed form, but we do not yet have a closed-form identity proving
-$\Delta_\ell\not\equiv0$ for every $\ell$. Kakarala's full-bispectrum
-recurrence avoids this issue by using full matrix Fourier blocks and
-ordinary matrix inverses. Our selective scalar recurrence needs an
-analogous nonvanishing identity for the determinant of the
-$O(\ell)$-row matrix in~\eqref{eq:bootstrap-system}. Establishing that
-identity is the remaining step needed to turn the conditional theorem
-into an unconditional all-$L$ statement.
-\end{remark}
-
-\begin{remark}[Constructive seed alternative]
-Earlier drafts used an explicit closed-form / finite-fibre /
-degree-$4$ filter as the seed solver. That route is useful
-algorithmically but depends on fibre enumeration and nonlinear
-least-squares residual checks; it is deliberately not load-bearing in
-the proof above. The theorem instead assumes finite seed injectivity
-directly in Hypothesis~\ref{hyp:seed}.
-\end{remark}
-
-\section{Relationship to prior work}
-\label{sec:related}
-
-\begin{table}[h]
-  \caption{Comparison of bispectral invariants for signals on $S^2$ or
-    finite groups~$G$.}
-  \label{tab:comparison}
-  \centering
-  \small
-  \setlength{\tabcolsep}{4pt}
-  \begin{tabular}{@{}lllc@{}}
-    \toprule
-    Result & Domain & Output size & Complete? \\
-    \midrule
-    \citet{kakarala2012bispectrum}
-      & $S^2$, full bispec. & $O(L^3)$ & Yes \\
-    \citet{edidin2024orbit}
-      & $S^2$, full bispec.\ (band-lim.) & $O(L^3)$ & Yes (generic) \\
-    \citet{bendory2025orbit}
-      & $S^2$, frequency marching & $O(L^2)$/shell & Yes (3 shells) \\
-    \citet{mataigne2024selective}
-      & finite $G$, sel.\ bispec. & $O(\abs{G})$ & Yes (exact) \\
-    \textbf{This work}
-      & \boldmath$S^2$, sel.\ scalar bispec.\textsuperscript{$\dagger$}
-      & $\boldsymbol{\Theta(L^2)}$ & Conditional; num.\ evidence \\
-    \bottomrule
-  \end{tabular}
-  \par\smallskip
-  {\footnotesize $\dagger$ Optional CG-power diagnostics may be
-  appended (Remark~\ref{rem:cg-optional}); $\SP=\varnothing$ in the
-  proof.}
-\end{table}
-
-\paragraph{Mataigne et al.\ (finite groups).}
-\citet{mataigne2024selective} prove that for finite groups, an
-$O(\abs{G})$-sized subset of the scalar bispectrum suffices for
-completeness, by a breadth-first traversal of the irrep graph. Our
-algorithm is the analog for $\SO(3)$ on $S^2$: selected bootstrap entries play the
-role of edges, and the for-loop in
-Algorithm~\ref{alg:so3-inversion} is a continuum BFS. Applied to a
-finite group, our argument collapses to theirs --- the CG power
-machinery becomes redundant because the finite-group reality
-constraint is algebraic, and Lemma~\ref{lem:bootstrap} is replaced by
-their closed-form Fourier-coefficient recurrence.
-
-\paragraph{Kakarala (full bispectrum).}
-\citet{kakarala1992triple,kakarala2012bispectrum} prove completeness
-of the full $O(L^3)$-sized bispectrum on homogeneous spaces.
-Our finite seed hypothesis is modeled on this full-bispectrum result,
-but the theorem here does not derive finite-band seed injectivity from
-Kakarala alone. The per-degree determinant bootstrap then amplifies
-the assumed seed to higher degrees at $\Theta(L^2)$ output.
-
-\paragraph{Edidin and Satriano (band-limited bispectrum).}
-\citet{edidin2024orbit} prove generic separation by the full
-band-limited bispectrum, with an ``up to reflection'' caveat
-(Remark~\ref{rem:edidin}). The parity argument of
-Corollary~\ref{cor:first-odd} explains why the apparent
-orientation-reversing $T_R$ ambiguity is visible to the scalar
-bispectrum once the seed includes degree~$4$.
-
-\paragraph{Bendory et al.\ (frequency marching).}
-\citet{bendory2025orbit} reconstruct band-limited spherical signals
-from a frequency-marching scheme using $O(L^2)$ third-order moments
-per shell. Our invariant is closer to the bispectral-coefficient
-formulation of \citet{kakarala2012bispectrum} but achieves the same
-asymptotic $\Theta(L^2)$ size.
-
-\paragraph{Bispectral neural networks.}
-\citet{sanborn2023bispectral} introduced bispectrum-based learnable
-invariant pooling for finite groups; \citet{myers2025selective}
-extend the selective-bispectrum construction to the disk. Our $\SO(3)$
-conditional result is aimed at enabling the analogous architecture for
-spherical signals.
-
-\bibliographystyle{plainnat}
-\bibliography{references}
-
-\end{document}
diff --git a/paper/proof_completeness.tex b/paper/proof_completeness.tex
deleted file mode 100644
index ac90c60..0000000
--- a/paper/proof_completeness.tex
+++ /dev/null
@@ -1,635 +0,0 @@
-\documentclass[11pt]{article}
-\usepackage{amsmath,amssymb,amsthm,enumitem,hyperref}
-\usepackage[margin=1in]{geometry}
-
-\newtheorem{theorem}{Theorem}
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{corollary}[theorem]{Corollary}
-\newtheorem{definition}[theorem]{Definition}
-\newtheorem{remark}[theorem]{Remark}
-
-\DeclareMathOperator{\tr}{tr}
-\DeclareMathOperator{\sgn}{sgn}
-\DeclareMathOperator{\im}{Im}
-\newcommand{\R}{\mathbb{R}}
-\newcommand{\C}{\mathbb{C}}
-\newcommand{\Z}{\mathbb{Z}}
-\newcommand{\SO}{\mathrm{SO}}
-\newcommand{\OG}{\mathrm{O}}
-\newcommand{\SH}{Y}
-\newcommand{\CG}[6]{C^{#5,\,#6}_{#1,\,#2;\,#3,\,#4}}
-\newcommand{\abs}[1]{\lvert #1 \rvert}
-\newcommand{\norm}[1]{\lVert #1 \rVert}
-\newcommand{\conj}[1]{\overline{#1}}
-\newcommand{\ip}[2]{\langle #1,\, #2 \rangle}
-
-\title{Completeness of the Augmented Selective $\SO(3)$-Bispectrum on $S^2$\\
-  {\large Full Proof for All Band-Limits $L \ge 4$}}
-\author{}
-\date{}
-
-\begin{document}
-\maketitle
-
-\section{Setup}
-
-\begin{definition}[Signal space]
-Let $V_L$ denote the space of real-valued band-limited functions
-$f: S^2 \to \R$ with band-limit $L$:
-\[
-  f(\theta,\varphi) = \sum_{\ell=0}^{L}\sum_{m=-\ell}^{\ell}
-    a_\ell^m \, \SH_\ell^m(\theta,\varphi),
-  \qquad
-  a_\ell^{-m} = (-1)^m \conj{a_\ell^m}.
-\]
-The coefficient vector at degree $\ell$ is
-$\mathbf{F}_\ell = (a_\ell^{-\ell}, \ldots, a_\ell^{\ell}) \in \C^{2\ell+1}$,
-with $2\ell+1$ real degrees of freedom.
-The total real dimension is $\dim V_L = (L+1)^2$.
-\end{definition}
-
-\begin{definition}[Augmented bispectrum]
-The invariant map $\Phi_{\mathrm{aug}}: V_L \to \C^{N_\beta} \times \R^{N_P}$ consists of:
-\begin{enumerate}[nosep]
-\item \textbf{Bispectral entries} (complex-valued):
-  $\beta_{\ell_1,\ell_2,\ell}
-  = \sum_{m_1+m_2=m} \CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}\,
-    a_{\ell_1}^{m_1}\,a_{\ell_2}^{m_2}\,\conj{a_\ell^m}$,
-  for triples $(\ell_1,\ell_2,\ell)$ selected by
-  Algorithm~1 of the main paper.
-\item \textbf{CG power entries} (real-valued):
-  $P_{\ell_1,\ell_2,\ell}
-  = \norm{(\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell}^2$.
-\end{enumerate}
-Both types are $\SO(3)$-invariant.
-Each complex $\beta$ contributes two real outputs $(\operatorname{Re}\beta, \im\beta)$.
-\end{definition}
-
-\begin{definition}[Reflection map $T_R$]\label{def:TR}
-The azimuthal reflection $R: (\theta,\varphi) \mapsto (\theta,-\varphi)$
-acts on spherical harmonic coefficients as
-$a_\ell^m \mapsto (-1)^m a_\ell^{-m}$.
-For real signals, this equals complex conjugation: $a_\ell^m \mapsto \conj{a_\ell^m}$.
-Since $R \in \OG(3) \setminus \SO(3)$, the signal $R \cdot f$ is generically
-in a \emph{different} $\SO(3)$-orbit from $f$.
-\end{definition}
-
-\section{Gauge Fixing}
-
-\begin{lemma}[Gauge fixing]\label{lem:gauge}
-For generic $f \in V_L$ (with $\norm{\mathbf{F}_1} \ne 0$ and $a_2^1 \ne 0$
-after aligning $\mathbf{F}_1$), there exists a unique $g \in \SO(3)$ such that
-the gauge-fixed signal $\tilde{f} = g \cdot f$ satisfies:
-\begin{enumerate}[nosep,label=(\roman*)]
-\item $a_1^{-1} = a_1^1 = 0$, \quad $a_1^0 = c > 0$ \quad
-  (alignment of $\mathbf{F}_1$ to the $z$-axis),
-\item $\im(a_2^1) = 0$, \quad $a_2^1 = x > 0$ \quad
-  (fixing the residual $\SO(2)_z$ symmetry).
-\end{enumerate}
-This consumes all $3$ degrees of freedom of $\SO(3)$.
-The gauge slice $\mathcal{G} \subset V_L$ is the set of signals
-satisfying (i)--(ii); it has $\dim\mathcal{G} = (L+1)^2 - 3$.
-\end{lemma}
-
-\begin{remark}[Gauge-fixed parameterisation]\label{rem:gauge-param}
-On the gauge slice, the real parameters are:
-\[
-  A_0 = a_0^0, \quad c = a_1^0, \quad
-  (y, x, u, v) = (a_2^0, \operatorname{Re}a_2^1, \operatorname{Re}a_2^2, \im a_2^2),
-\]
-and for $\ell \ge 3$: $t_\ell = a_\ell^0$, and
-$(p_{\ell,m}, q_{\ell,m}) = (\operatorname{Re}a_\ell^m, \im a_\ell^m)$ for $m = 1,\ldots,\ell$.
-The reflection $T_R$ acts on the gauge slice as:
-\begin{equation}\label{eq:TR-action}
-  T_R: \quad v \mapsto -v, \qquad
-  q_{\ell,m} \mapsto -q_{\ell,m} \quad \forall\, \ell \ge 3, \; m \ge 1.
-\end{equation}
-All other gauge-fixed parameters are unchanged.
-Crucially, $T_R$ preserves the gauge conditions (i)--(ii).
-\end{remark}
-
-\section{Recovery of $\mathbf{F}_0$ and $\mathbf{F}_1$}
-
-\begin{lemma}\label{lem:F0F1}
-For $a_0^0 \ne 0$:
-$\beta_{0,0,0} = (a_0^0)^3$ determines $A_0 = a_0^0$ uniquely.
-Then $\beta_{0,1,1} = A_0 \norm{\mathbf{F}_1}^2$ determines
-$c = \norm{\mathbf{F}_1}$, and $\mathbf{F}_1 = (0, c, 0)^\top$
-is fixed by the gauge.
-\end{lemma}
-
-\section{Parity Structure of the Bispectrum}
-
-This section contains the key new ingredient: a complete classification of
-which bispectral entries carry parity-breaking information.
-
-\begin{lemma}[Parity of the bispectrum]\label{lem:parity}
-For any signal (real or complex):
-\begin{equation}\label{eq:parity}
-  \beta_{\ell_1,\ell_2,\ell}(T_R \cdot f)
-  = (-1)^{\ell_1+\ell_2+\ell}\,\beta_{\ell_1,\ell_2,\ell}(f).
-\end{equation}
-For \emph{real} signals, there is an independent constraint:
-\begin{equation}\label{eq:reality-constraint}
-  \conj{\beta_{\ell_1,\ell_2,\ell}(f)}
-  = (-1)^{\ell_1+\ell_2+\ell}\,\beta_{\ell_1,\ell_2,\ell}(f).
-\end{equation}
-Together:
-\begin{enumerate}[nosep,label=(\alph*)]
-\item If $\ell_1+\ell_2+\ell$ is even:
-  $\beta$ is real-valued (by~\eqref{eq:reality-constraint})
-  and $T_R$-invariant (by~\eqref{eq:parity}).
-\item If $\ell_1+\ell_2+\ell$ is odd:
-  $\beta$ is purely imaginary (by~\eqref{eq:reality-constraint})
-  and $\beta(T_R \cdot f) = -\beta(f)$, so $\im\beta$ flips sign under $T_R$.
-\end{enumerate}
-\end{lemma}
-\begin{proof}
-Under $T_R$, each $a_\ell^m \mapsto (-1)^m a_\ell^{-m}$.
-Substituting into $\beta = \sum_m (\sum_{m_1+m_2=m} C\, a^{m_1} a^{m_2})
-\conj{a^m}$, the three sign factors $(-1)^{m_1+m_2+m} = (-1)^{2m} = 1$
-cancel. Re-indexing $m \mapsto -m$ and applying the CG symmetry
-$\CG{\ell_1}{-m_1}{\ell_2}{-m_2}{\ell}{-m}
-  = (-1)^{\ell_1+\ell_2-\ell}\,\CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}$
-yields~\eqref{eq:parity} (with $(-1)^{\ell_1+\ell_2-\ell} = (-1)^{\ell_1+\ell_2+\ell}$).
-
-For real signals, $\conj{a_\ell^m} = (-1)^m a_\ell^{-m}$.
-Taking the complex conjugate of $\beta$ and re-indexing similarly
-yields~\eqref{eq:reality-constraint}.
-\end{proof}
-
-\begin{lemma}[CG power is $T_R$-invariant]\label{lem:P-invariant}
-For all triples $(\ell_1,\ell_2,\ell)$:
-$P_{\ell_1,\ell_2,\ell}(T_R \cdot f) = P_{\ell_1,\ell_2,\ell}(f)$.
-\end{lemma}
-\begin{proof}
-Under $T_R$, the CG-projected component transforms as
-$(\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell^M
-  \mapsto (-1)^{\ell_1+\ell_2+\ell+M}\,
-  (\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell^{-M}$.
-Taking the squared norm and re-indexing the sum over $M$
-leaves $P$ unchanged.
-\end{proof}
-
-\begin{proposition}[Vanishing of odd-parity entries at low degrees]
-\label{prop:odd-vanishing}
-For real signals, the odd-parity bispectral entry
-$\beta_{\ell_1,\ell_2,\ell}$ (with $\ell_1+\ell_2+\ell$ odd)
-vanishes identically in each of the following cases:
-\begin{enumerate}[nosep,label=(\roman*)]
-\item $\ell_1 = \ell_2$: the CG coefficients are antisymmetric
-  under swapping $m_1 \leftrightarrow m_2$, while
-  $a_{\ell_1}^{m_1} a_{\ell_1}^{m_2}$ is symmetric, giving
-  exact cancellation.
-\item Two of $\{\ell_1, \ell_2, \ell\}$ are equal and the entry
-  involves gauge-fixed $\mathbf{F}_1$: the gauge $a_1^{\pm 1} = 0$
-  forces only the $m_1 = 0$ component to contribute, and the
-  resulting sum over $m$ cancels by the CG antisymmetry
-  $\CG{1}{0}{\ell'}{-m}{\ell'}{-m}
-    = -\CG{1}{0}{\ell'}{m}{\ell'}{m}$
-  paired with $|a_{\ell'}^{-m}|^2 = |a_{\ell'}^m|^2$.
-\item More generally, whenever two of the three representation indices
-  coincide, a symmetry argument yields $\beta = -\conj\beta$ \textbf{and}
-  $\beta = 0$, as verified by symbolic computation with sympy for all
-  triples with indices $\le 5$.
-\end{enumerate}
-\end{proposition}
-
-\begin{corollary}[First nonzero odd-parity entry]\label{cor:first-odd}
-The first bispectral entry with odd parity and generically nonzero imaginary
-part occurs at degree $\ell = 4$:
-\[
-  \beta_{2,3,4}, \qquad
-  \ell_1+\ell_2+\ell = 9 \;\text{(odd)},
-  \qquad
-  \ell_1, \ell_2, \ell \;\text{all distinct}.
-\]
-At the deterministic witness (seed~42), $\im(\beta_{2,3,4}) \approx 3.26 \ne 0$,
-and $\im(\beta_{2,3,4})(T_R \cdot f) \approx -3.26 = -\im(\beta_{2,3,4})(f)$.
-Similarly, $\beta_{2,4,3}$ and $\beta_{3,4,2}$ are nonzero odd-parity entries
-at degree~$4$.
-\end{corollary}
-
-\begin{remark}[No parity-breaking at $\ell \le 3$]
-For degrees $\le 3$, every odd-parity triple $(\ell_1,\ell_2,\ell)$
-has at least two indices equal (since $\max(\ell_1,\ell_2,\ell) \le 3$
-and $\ell_1+\ell_2+\ell$ odd forces at least one pair to be equal in this range).
-By Proposition~\ref{prop:odd-vanishing}, all odd-parity entries vanish.
-Therefore, the $T_R$ ambiguity cannot be resolved using only degrees $\le 3$.
-\end{remark}
-
-
-\section{Seed Recovery ($\ell \le 3$)}
-
-\begin{lemma}[Degree-2 recovery]\label{lem:ell2}
-Given $(A_0, c)$ from Lemma~\ref{lem:F0F1}, the gauge-fixed degree-2
-parameters $(y, x, u, v)$ satisfy:
-\begin{align}
-  \beta_{1,1,2} &= \tfrac{\sqrt{6}}{3}\, c^2\, y
-    &\Longrightarrow\quad& y \;\text{determined}, \label{eq:y}\\
-  P_{1,2,1} &= \tfrac{c^2}{5}(3x^2 + 2y^2)
-    &\Longrightarrow\quad& x^2 \;\text{determined}, \label{eq:x2}\\
-  \beta_{0,2,2} &= A_0(2u^2 + 2v^2 + 2x^2 + y^2)
-    &\Longrightarrow\quad& S := u^2+v^2 \;\text{determined}, \label{eq:S}\\
-  \beta_{2,2,2} &= \tfrac{6\sqrt{14}\,S\,y - 6\sqrt{21}\,u\,x^2
-    - 3\sqrt{14}\,x^2 y - \sqrt{14}\,y^3}{7}
-    &\Longrightarrow\quad& u \;\text{determined}
-    \;\text{(linear in $u$)}. \label{eq:u}
-\end{align}
-Here $x = \sqrt{x^2} > 0$ by the gauge condition.
-Then $v^2 = S - u^2$, giving $|v|$ but not $\sgn(v)$.
-The fibre at degree~$2$ is generically $\{(y,x,u,v),\; (y,x,u,-v)\}$.
-\end{lemma}
-\begin{proof}
-Equations~\eqref{eq:y}--\eqref{eq:u} are derived by direct symbolic computation
-using the gauge-fixed parameterisation (Remark~\ref{rem:gauge-param}) with
-$a_1^{\pm 1} = 0$, $a_1^0 = c$.
-The expression for $\beta_{2,2,2}$ involves $v$ only through $v^2$
-(which combines with $u^2$ into $S$), and is \emph{linear} in $u$
-(the coefficient of $u$ is $-6\sqrt{21}\,x^2/7 \ne 0$ for $x > 0$).
-\end{proof}
-
-\begin{lemma}[Degree-3 recovery: finite fibre]\label{lem:ell3}
-Given $\mathbf{F}_2 = (y,x,u,\pm v)$, the degree-3 parameters
-$\mathbf{F}_3 = (t, p_1, q_1, p_2, q_2, p_3, q_3)$ are constrained by the
-augmented entries:
-\begin{itemize}[nosep]
-\item $\beta_{1,2,3}$, $\beta_{1,3,2}$, $\beta_{2,3,1}$: bilinear in
-  $\mathbf{F}_2 \otimes \mathbf{F}_3$.
-  The entries $\beta_{1,3,2}$ and $\beta_{2,3,1}$ are in the Jacobian
-  row span of $\beta_{1,2,3}$ and contribute no additional rank.
-\item $\beta_{0,3,3} = A_0\norm{\mathbf{F}_3}^2$: gives the norm.
-\item $\beta_{3,3,2}$: quadratic in $\mathbf{F}_3$, linear in $\mathbf{F}_2$.
-\item $\beta_{2,3,2}$, $\beta_{2,2,3}$: odd parity with repeated indices,
-  vanish identically for real signals (Proposition~\ref{prop:odd-vanishing}).
-\item $P_{1,2,1}$: constrains $3x^2+2y^2$, resolving $x^2$ once $y$ is known.
-\item $P_{1,3,2}$, $P_{2,3,1}$, $P_{2,3,2}$, $P_{3,3,2}$: CG power entries
-  quadratic in $\mathbf{F}_3$, providing the missing rank directions that the
-  scalar bispectrum alone cannot achieve.
-\end{itemize}
-The full augmented seed system uses $13$ nonzero entries for $11$ unknowns.
-Two additional entries ($\beta_{2,3,2}$ and $\beta_{2,2,3}$, both odd parity with
-repeated indices) vanish identically for real signals by
-Proposition~\ref{prop:odd-vanishing}.
-The incremental rank structure at a rational witness is:
-\begin{center}\small
-\begin{tabular}{llc}
-  Entries added & Rank \\
-  \hline
-  $\mathrm{Re}\,\beta_{1,1,2}$ & 1 \\
-  $+\;\mathrm{Re}\,\beta_{0,2,2}$ & 2 \\
-  $+\;\mathrm{Re}\,\beta_{2,2,2}$ & 3 \\
-  $+\;\mathrm{Re}\,\beta_{1,2,3}$ & 4 \\
-  $+\;\mathrm{Re}\,\beta_{0,3,3}$ & 5 \\
-  $+\;\mathrm{Re}\,\beta_{3,3,2}$ & 6 \\
-  $+\;P_{1,2,1}$ & 7 \\
-  $+\;P_{1,3,2}$ & 8 \\
-  $+\;P_{2,3,1}$ & 9 \\
-  $+\;P_{2,3,2}$ & 10 \\
-  $+\;P_{3,3,2}$ & \textbf{11/11} \\
-\end{tabular}
-\end{center}
-The bispectral entries $\mathrm{Re}\,\beta_{1,3,2}$ and
-$\mathrm{Re}\,\beta_{2,3,1}$ are in the row span of the first four
-bispectral rows and contribute no additional rank.
-Thus the scalar bispectrum achieves rank~$6$, and \emph{all five}
-CG power entries are essential: each contributes one additional rank,
-reaching the full rank~$11$.
-The generic fibre is $0$-dimensional (finitely many points)
-and contains the $T_R$ pair
-$\{(\mathbf{F}_2, \mathbf{F}_3),\; T_R(\mathbf{F}_2, \mathbf{F}_3)\}$.
-\end{lemma}
-\begin{proof}
-The Jacobian rank is verified by exact symbolic computation (sympy) at the
-rational witness $(A_0, c, y, x, u, v, t, p_1, q_1, p_2, q_2, p_3, q_3)
-= (1, 1, \tfrac{3}{7}, \tfrac{2}{5}, \tfrac{1}{3}, \tfrac{4}{9},
-   \tfrac{1}{2}, \tfrac{3}{8}, \tfrac{2}{7}, \tfrac{5}{11}, \tfrac{1}{4},
-   \tfrac{3}{13}, \tfrac{7}{17})$.
-The $13 \times 11$ Jacobian has rank~$11$ (full).
-The six contributing bispectral entries achieve rank~$6$; each of the five
-CG power entries $P_{1,2,1}$, $P_{1,3,2}$, $P_{2,3,1}$, $P_{2,3,2}$,
-$P_{3,3,2}$ contributes one additional rank.
-
-The $T_R$ invariance follows from Lemmas~\ref{lem:parity}--\ref{lem:P-invariant}
-and Proposition~\ref{prop:odd-vanishing}: every entry in the degree~$\le 3$
-augmented system is either real-valued and $T_R$-invariant (even parity $\beta$
-and all~$P$) or identically zero (odd parity $\beta$).
-Since $T_R$ preserves all augmented invariants and maps
-$(y,x,u,v,\ldots) \mapsto (y,x,u,-v,\ldots)$ (a different point on the gauge
-slice when $v \ne 0$), the fibre contains the $T_R$ pair.
-
-Full Jacobian rank gives $0$-dimensional (finite) fibres, but does
-\emph{not} by itself bound the fibre to two points:
-the degree-3 subsystem (7~real equations of degrees $1$, $2$, and $4$
-in 7~unknowns) can have additional isolated real solutions.
-Exhaustive numerical search (multi-start least-squares with 5000 random
-initial points per sign-of-$v$ branch) finds $10$ solutions per branch
-($20$ total, forming $10$ $T_R$-pairs) at the rational witness.
-These spurious solutions are eliminated at degree~$4$
-(Lemma~\ref{lem:degree4-filter}).
-\end{proof}
-
-\begin{remark}[The seed fibre is finite but not two-point]
-\label{rem:seed-fibre-size}
-The seed polynomial system (degrees $0$--$3$) is a system of
-$7$ independent real equations of mixed degree ($1$ linear, $5$ quadratic,
-$1$ quartic) in $7$ real unknowns (for each branch of $\sgn(v)$).
-By B\'ezout's theorem the number of isolated complex solutions is at most
-$1 \times 2^5 \times 4 = 128$.
-Numerically, $10$ real solutions are found per branch.
-The proof does not require the seed fibre to be a two-point set;
-fibre reduction is deferred to degree~$4$.
-\end{remark}
-
-
-\section{Per-Degree Recovery ($\ell \ge 4$)}
-
-\begin{lemma}[Complex bootstrap rank]\label{lem:bootstrap-rank}
-For each $\ell \ge 4$, the selective bispectral entries at degree $\ell$
-that are linear in $\mathbf{F}_\ell$ define a coefficient matrix
-$A_\ell \in \C^{N \times (2\ell+1)}$ (with coefficients depending on
-$\mathbf{F}_0, \ldots, \mathbf{F}_{\ell-1}$).
-For generic lower-degree coefficients:
-$\operatorname{rank}_\C(A_\ell) = 2\ell+1$.
-\end{lemma}
-\begin{proof}
-For each $\ell$, $\det(A_\ell)$ is a polynomial in the lower-degree
-coefficients.  It is verified to be nonzero at a deterministic
-witness (seed~42, real-signal constraint) for each
-$\ell \in \{4, \ldots, 100\}$
-(script: \texttt{benchmarks/verify\_linear\_bootstrap.py}).
-Since a polynomial that is nonzero at one point is not identically zero,
-$\det(A_\ell) \not\equiv 0$ for each verified~$\ell$.
-For $\ell \ge 8$, the matrix uses the closed-form tridiagonal family
-$\mathcal{T}_\ell$ with $2\ell+1$ entries.
-\end{proof}
-
-\begin{remark}[Scope of the bootstrap verification]\label{rem:bootstrap-scope}
-The bootstrap rank is verified for each individual $\ell \le 100$.
-For any fixed band-limit $L$, the genericity condition is that
-$\det(A_\ell) \ne 0$ for $\ell = 4, \ldots, L$.
-This is the complement of $L - 3$ proper algebraic subvarieties
-(one per degree), hence a Zariski-open dense set of full Lebesgue measure.
-For $L > 100$, the verification script can be extended to cover any
-desired~$L$.  The structural form of the tridiagonal family is uniform
-for all $\ell \ge 8$, and no mechanism exists by which the determinant
-polynomial could vanish identically at a particular $\ell$ while being
-nonzero at neighbouring degrees.
-\end{remark}
-
-\begin{remark}[Complex rank suffices for uniqueness]\label{rem:complex-real}
-For real signals, $\mathbf{F}_\ell$ has $2\ell+1$ real degrees of freedom.
-The bispectral entries define two subsystems: chain entries, linear in
-$\conj{\mathbf{F}_\ell}$, and cross entries, linear in $\mathbf{F}_\ell$.
-Neither subsystem alone has full rank.
-However, combining both via the reality constraint
-$\conj{a_\ell^m} = (-1)^m a_\ell^{-m}$ yields a real linear system
-of full rank $2\ell+1$ in the $2\ell+1$ real parameters.
-Equivalently: the complex system (treating chain and cross jointly)
-has a unique solution $(\mathbf{F}_\ell, \conj{\mathbf{F}_\ell})$,
-and for a real input signal this solution automatically satisfies
-the reality constraint.
-\end{remark}
-
-\begin{lemma}[Augmented per-degree real rank]\label{lem:augmented-perdegree}
-For each $\ell \ge 4$, the full augmented entries at degree $\ell$
-(linear bispectral, self-coupling, and CG power) have real Jacobian
-rank $2\ell+1$ w.r.t.\ the $2\ell+1$ real parameters of $\mathbf{F}_\ell$.
-The CG power entries are essential at every degree: the scalar
-bispectrum alone (linear + self-coupling) achieves only partial rank.
-Verified at the witness for $\ell \in \{4, \ldots, 30\}$.
-\end{lemma}
-
-
-\section{Degree-4 Fibre Reduction and $T_R$ Resolution}
-
-\begin{lemma}[Degree-4 eliminates all spurious seed solutions]
-\label{lem:degree4-filter}
-Let $\mathcal{S}$ denote the finite seed fibre at degrees~$0$--$3$
-(Lemma~\ref{lem:ell3}).
-For generic real signals with $L \ge 4$, the degree-4 augmented entries
-(12~bispectral + 5~CG power, providing $29$ real constraints on
-$9$~real unknowns of $\mathbf{F}_4$) are simultaneously satisfiable
-for \emph{exactly one} $T_R$-pair in~$\mathcal{S}$:
-the true solution and its reflection.
-All other seed solutions are inconsistent at degree~$4$.
-\end{lemma}
-\begin{proof}
-The degree-4 augmented system is an overdetermined polynomial system
-in the 9~real parameters of $\mathbf{F}_4$, with coefficients depending
-on the seed solution $(\mathbf{F}_0, \ldots, \mathbf{F}_3)$.
-
-For each of the~10 seed solutions on the $+v$ branch at the rational
-witness, a multi-start nonlinear least-squares search (200~random starts)
-is used to find the best-fit~$\mathbf{F}_4$.
-Only the true seed solution achieves zero residual
-($\norm{r} \sim 10^{-16}$); the 9~spurious solutions all have
-$\norm{r} \ge 10^{-2}$, showing the degree-4 system is \emph{inconsistent}
-for them.
-
-The inconsistency arises because the degree-4 entries include constraints
-that depend only on $(\mathbf{F}_0, \mathbf{F}_1, \mathbf{F}_2, \mathbf{F}_4)$
-and not on $\mathbf{F}_3$---notably $\beta_{2,2,4}$, $\beta_{2,4,2}$,
-$P_{1,4,3}$, $P_{1,4,4}$, $P_{2,4,2}$, $P_{2,4,3}$,
-$\beta_{4,4,2}$, $\beta_{4,4,4}$.
-These 8~entries nearly determine $\mathbf{F}_4$ independently of $\mathbf{F}_3$.
-The remaining entries (involving $\mathbf{F}_3$) then impose
-constraints that are generically inconsistent when $\mathbf{F}_3$ is a
-spurious seed solution.
-
-Since the residual for each spurious seed is a polynomial
-function of the signal's coefficients that is nonzero at the witness,
-its vanishing locus is a proper algebraic subvariety.
-\end{proof}
-
-\begin{lemma}[The $T_R$ branch is overdetermined-inconsistent]
-\label{lem:TR-resolve}
-For $L \ge 4$, the $T_R$ partner of the true seed---i.e.,
-$(\mathbf{F}_2', \mathbf{F}_3') = T_R(\mathbf{F}_2, \mathbf{F}_3)$---is
-also inconsistent at degree~$4$ for generic signals.
-\end{lemma}
-\begin{proof}
-By Lemma~\ref{lem:degree4-filter}, the $T_R$ branch is one of the
-seed solutions that fails the degree-4 consistency check.
-The mechanism is the parity structure (Lemma~\ref{lem:parity}):
-the odd-parity entries $\beta_{2,3,4}$, $\beta_{2,4,3}$, $\beta_{3,4,2}$
-(all-distinct indices, $\im\beta \ne 0$) are part of the degree-4
-constraints.  On the $T_R$ branch these imaginary parts have the
-wrong sign, distorting $\mathbf{F}_4'$ so that the $T_R$-invariant
-self-coupling and CG power entries cannot be simultaneously satisfied.
-
-Explicitly: the degree-4 entries define rational functions of the
-lower-degree coefficients.  Let $\Delta(\mathbf{F}_0,\ldots,\mathbf{F}_3)$
-denote the sum of squared residuals of the non-bootstrap entries
-evaluated at the $T_R$-branch bootstrap solution.
-Then $\Delta$ is a rational function whose numerator is a polynomial $p$
-and whose denominator is $(\det A_4)^2$ (from the bootstrap solve).
-At the witness, $\det A_4 \ne 0$ and $p \ne 0$
-(residual ${\sim}10^{-2}$).
-Hence $p$ is not identically zero, and $\Delta \ne 0$ on the complement
-of the proper algebraic subvariety $\{p = 0\} \cup \{\det A_4 = 0\}$.
-\end{proof}
-
-\begin{proposition}[Uniqueness propagates]\label{prop:TR-propagate}
-Once the correct seed $(\mathbf{F}_0, \ldots, \mathbf{F}_3)$ is identified
-at degree~$4$ (Lemmas~\ref{lem:degree4-filter}--\ref{lem:TR-resolve}),
-the complex bootstrap at degrees $\ell \ge 5$ is unambiguous:
-given the correct $\mathbf{F}_0, \ldots, \mathbf{F}_{\ell-1}$, the
-full-rank matrix $A_\ell$ determines $\mathbf{F}_\ell$ uniquely
-(Lemma~\ref{lem:bootstrap-rank}, Remark~\ref{rem:complex-real}).
-\end{proposition}
-
-
-\section{Main Theorem}
-
-\begin{theorem}[Completeness of the augmented selective bispectrum]
-\label{thm:main}
-For $L \ge 4$, let $f, f': S^2 \to \R$ be band-limited at degree $L$
-and satisfy the genericity conditions:
-\begin{enumerate}[nosep,label=(G\arabic*)]
-\item $a_0^0 \ne 0$,
-\item $\norm{\mathbf{F}_1} \ne 0$,
-\item $a_2^1 \ne 0$ after gauge-fixing $\mathbf{F}_1$.
-\end{enumerate}
-If $\Phi_{\mathrm{aug}}(f) = \Phi_{\mathrm{aug}}(f')$
-(as complex-valued output, including both real and imaginary parts
-of each bispectral entry), then $f' = g \cdot f$ for some $g \in \SO(3)$.
-
-The generic set (satisfying (G1)--(G3) and lying outside a proper
-algebraic subvariety) has full Lebesgue measure in $V_L$.
-The augmented output has $\Theta(L^2)$ real components, achieving the
-information-theoretic lower bound up to a constant factor.
-\end{theorem}
-\begin{proof}
-\textbf{Step 1: Gauge fixing.}
-By Lemma~\ref{lem:gauge}, both $f$ and $f'$ can be gauge-fixed.
-If $\Phi_{\mathrm{aug}}(f) = \Phi_{\mathrm{aug}}(f')$, then by
-$\SO(3)$-invariance, the gauge-fixed versions $\tilde{f}$ and $\tilde{f}'$
-(possibly with different gauge rotations $g, g'$) satisfy
-$\Phi_{\mathrm{aug}}(\tilde{f}) = \Phi_{\mathrm{aug}}(\tilde{f}')$.
-It suffices to show $\tilde{f} = \tilde{f}'$ on the gauge slice.
-
-\medskip
-\textbf{Step 2: Recovery of $\mathbf{F}_0$, $\mathbf{F}_1$.}
-By Lemma~\ref{lem:F0F1}, the gauge-fixed $A_0$ and $c$ are uniquely
-determined. Hence $\tilde{f}$ and $\tilde{f}'$ agree at degrees $0$ and $1$.
-
-\medskip
-\textbf{Step 3: Seed recovery (degrees 2--3).}
-By Lemma~\ref{lem:ell2}, the degree-2 parameters $(y, x, u, |v|)$
-are uniquely determined.
-By Lemma~\ref{lem:ell3}, the augmented seed system at degrees $0$--$3$
-has full Jacobian rank~$11$, giving a $0$-dimensional (finite) fibre
-$\mathcal{S} = \{s_1, \ldots, s_K, T_R(s_1), \ldots, T_R(s_K)\}$
-consisting of $T_R$-paired points.
-The gauge-fixed $\tilde{f}'$ agrees with one of these seed solutions.
-
-\medskip
-\textbf{Step 4: Degree-4 fibre reduction.}
-By Lemma~\ref{lem:degree4-filter}, the degree-4 augmented entries
-are simultaneously satisfiable for at most one $T_R$-pair in
-$\mathcal{S}$: the true solution $s_1 = (\mathbf{F}_2, \mathbf{F}_3)$ and
-$T_R(s_1)$.  All other seed solutions are eliminated.
-By Lemma~\ref{lem:TR-resolve}, the $T_R$ partner is itself inconsistent
-at degree~$4$ due to the parity-breaking entries
-$\beta_{2,3,4}, \beta_{2,4,3}, \beta_{3,4,2}$
-(whose imaginary parts flip sign under~$T_R$).
-This leaves the unique solution
-$(\mathbf{F}_2', \mathbf{F}_3') = (\mathbf{F}_2, \mathbf{F}_3)$,
-with $\mathbf{F}_4' = \mathbf{F}_4$ determined by the augmented system.
-
-\medskip
-\textbf{Step 5: Inductive recovery ($\ell \ge 5$).}
-By Proposition~\ref{prop:TR-propagate}, given the correct
-$\mathbf{F}_0, \ldots, \mathbf{F}_{\ell-1}$, the complex bootstrap
-uniquely determines $\mathbf{F}_\ell$ at each degree $\ell \ge 5$.
-
-\medskip
-\textbf{Step 6: Conclusion.}
-By induction on $\ell$, the gauge-fixed signals $\tilde{f}$ and $\tilde{f}'$
-agree at all degrees $0, \ldots, L$.
-Hence $\tilde{f} = \tilde{f}'$, which means $f' = (g')^{-1} g \cdot f$
-with $(g')^{-1} g \in \SO(3)$.
-
-The generic set is the complement of finitely many proper algebraic
-subvarieties: (G1)--(G3), the seed Jacobian rank condition, the
-degree-4 fibre-reduction consistency for each spurious seed solution
-(each is a polynomial condition verified nonzero at the witness),
-and the bootstrap rank conditions $\det(A_\ell) \ne 0$ for
-$\ell = 4, \ldots, L$.
-A finite union of proper algebraic subvarieties has Lebesgue measure zero.
-\end{proof}
-
-\begin{corollary}[Band-limits $L \le 3$]
-For $L \le 3$, the augmented selective bispectrum has finite fibres
-that include $T_R$-pairs (by Lemma~\ref{lem:ell3}), but without
-degree-4 entries, the spurious seed solutions and the $T_R$ ambiguity
-cannot be resolved.
-The invariant separates $\OG(3)$-orbits
-(where $\OG(3) = \SO(3) \cup T_R \cdot \SO(3)$) among the true
-$T_R$-pair, but the fibre may contain additional points.
-\end{corollary}
-
-\section{Computational Certificates}
-
-The proof relies on five computational certificates, all reproducible
-from scripts in the repository:
-
-\begin{enumerate}[nosep]
-\item \textbf{Seed Jacobian rank} (Lemma~\ref{lem:ell3}):
-  The $13 \times 11$ Jacobian of the augmented seed system
-  (degrees $0$--$3$, $13$ nonzero entries) has rank $11$ at the rational
-  witness.
-  Verified by \emph{exact} rational arithmetic (sympy), providing a
-  rigorous certificate with no floating-point error.
-
-\item \textbf{Seed fibre enumeration} (Lemma~\ref{lem:ell3},
-  Remark~\ref{rem:seed-fibre-size}):
-  Exhaustive multi-start search (5000 starts per branch) finds exactly
-  $10$ solutions per $\sgn(v)$ branch ($20$ total, $10$ $T_R$-pairs).
-  Verified by \texttt{benchmarks/verify\_seed\_fibre.py}.
-
-\item \textbf{Degree-4 fibre reduction}
-  (Lemma~\ref{lem:degree4-filter}):
-  For each of the $10$ seed solutions on the $+v$ branch, a
-  multi-start nonlinear least-squares search (200 starts) attempts to
-  find $\mathbf{F}_4$ satisfying all $29$ real degree-4 constraints.
-  Only the true solution achieves zero residual; all $9$ spurious
-  solutions have residual $\ge 10^{-2}$.
-  Verified by \texttt{benchmarks/verify\_degree4\_filter.py}.
-
-\item \textbf{Complex bootstrap rank} (Lemma~\ref{lem:bootstrap-rank}):
-  For each $\ell \in \{4, \ldots, 100\}$, the bootstrap matrix $A_\ell$
-  has full \emph{complex} rank $2\ell+1$ at the deterministic witness
-  (seed~42).  The minimum singular value is well above zero
-  ($\sigma_{\min} / \sigma_{\max} \ge 10^{-4}$).
-  Verified by \texttt{benchmarks/verify\_linear\_bootstrap.py}.
-
-\item \textbf{Augmented per-degree real rank}
-  (Lemma~\ref{lem:augmented-perdegree}):
-  For each $\ell \in \{4, \ldots, 30\}$, the full augmented system
-  (bispectral + CG power) has \emph{real} Jacobian rank $2\ell+1$
-  at the witness.
-\end{enumerate}
-
-\begin{remark}[Exact vs.\ floating-point certificates]
-Certificate~1 uses exact rational arithmetic (no numerical error).
-Certificates~2--3 use IEEE~754 double precision.
-Since the quantities being tested (solution residuals, singular values)
-are well-separated from zero (residuals $\ge 10^{-2}$ for spurious
-solutions, singular values $\ge 10^{-2}$), rounding errors of order
-$10^{-15}$ do not affect the conclusions.
-Certificate~4 uses double precision with condition numbers $\le 10^4$,
-ensuring the rank determination is reliable.
-For any desired level of rigour, certificates~2--5 can be upgraded to
-interval arithmetic or exact computation at the cost of runtime.
-\end{remark}
-
-All certificates use deterministic witness points (rational or seeded
-pseudo-random), making them fully reproducible.
-The algebraic argument (``nonzero polynomial at one point implies nonzero
-polynomial generically'') upgrades each pointwise certificate to a
-generic (measure-zero exceptional set) statement.
-
-\end{document}
diff --git a/paper/proof_draft.tex b/paper/proof_draft.tex
deleted file mode 100644
index de9eae6..0000000
--- a/paper/proof_draft.tex
+++ /dev/null
@@ -1,200 +0,0 @@
-% DRAFT: Revised completeness proof for augmented selective SO(3) bispectrum
-% This file drafts the key changes to paper.tex
-
-% ============================================================
-% CHANGE 1: New definition — CG power spectrum
-% (goes after bispectrum definition, before selective set)
-% ============================================================
-
-\begin{definition}[CG power spectrum]
-\label{def:cg-power}
-For a band-limited signal $f : S^2 \to \R$ with spherical harmonic
-coefficient vectors $\mathbf{F}_\ell \in V_\ell$, the
-\emph{Clebsch--Gordan power spectrum} entry at triple
-$(\ell_1, \ell_2, \ell)$ is
-\[
-  P_{\ell_1,\ell_2,\ell}
-  := \bigl\lVert
-    (\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})\big|_\ell
-  \bigr\rVert^2
-  = \sum_{m=-\ell}^{\ell}
-    \Bigl| \sum_{\substack{m_1, m_2 \\ m_1+m_2=m}}
-    \CG{\ell_1}{m_1}{\ell_2}{m_2}{\ell}{m}\,
-    a_{\ell_1}^{m_1}\, a_{\ell_2}^{m_2} \Bigr|^2,
-\]
-where $|\ell_1-\ell_2| \le \ell \le \ell_1+\ell_2$.
-Each $P_{\ell_1,\ell_2,\ell}$ is a degree-$4$ polynomial in the
-coefficients and is $\SO(3)$-invariant (it is the squared norm of a
-CG-projected tensor product, hence preserved by the Wigner rotation).
-\end{definition}
-
-\begin{remark}
-The standard power spectrum
-$\norm{\mathbf{F}_\ell}^2 = \sum_m |a_\ell^m|^2$ is the special case
-$P_{0,\ell,\ell} = |a_0^0|^2 \norm{\mathbf{F}_\ell}^2$. The CG power
-spectrum generalises this by measuring the squared projection of
-$\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2}$ onto each
-irreducible channel~$V_\ell$.  These degree-$4$ invariants capture
-\emph{magnitude} information about the CG projections, complementing
-the degree-$3$ bispectral entries which capture \emph{phase}
-information via the contraction
-$\beta_{\ell_1,\ell_2,\ell}
-  = \ip{(\mathbf{F}_{\ell_1} \otimes \mathbf{F}_{\ell_2})|_\ell}{\mathbf{F}_\ell}$.
-\end{remark}
-
-
-% ============================================================
-% CHANGE 2: Revised theorem statement
-% ============================================================
-
-\begin{theorem}[Selective $\SO(3)$ bispectrum on $S^2$]
-\label{thm:so3-selective}
-For $f : S^2 \to \R$ band-limited at degree $L \ge 2$, there exists a
-set $\mathcal{S}_\beta$ of $O(L^2)$ bispectral triples and a set
-$\mathcal{S}_P$ of $O(L^2)$ CG power spectrum triples such that the
-combined invariant
-\[
-  \Phi(f) := \bigl(
-    \{\beta_{\ell_1,\ell_2,\ell}\}_{(\ell_1,\ell_2,\ell) \in \mathcal{S}_\beta},\;
-    \{P_{\ell_1,\ell_2,\ell}\}_{(\ell_1,\ell_2,\ell) \in \mathcal{S}_P}
-  \bigr)
-\]
-is a complete $\SO(3)$-invariant for Lebesgue-almost-every signal.
-The total number of scalar entries is $O(L^2)$, matching the
-$\Omega(L^2)$ lower bound from the orbit-space dimension
-$(L{+}1)^2 - 3$.
-\end{theorem}
-
-
-% ============================================================
-% CHANGE 3: Why the scalar bispectrum alone is insufficient
-% (new subsection in proof, before the main completeness argument)
-% ============================================================
-
-\paragraph{Why CG power augmentation is necessary.}
-For \emph{complex} signals ($f : S^2 \to \C$), the scalar bispectrum
-alone suffices: the bootstrap coefficient matrix $A$ has full complex
-rank $2\ell{+}1$ at each degree~$\ell$ (Lemma~\ref{lem:app-linear-bootstrap}),
-and the standard Zariski-density argument applies because the real
-signal space $\R^{(L+1)^2}$ is Zariski-dense in its complexification
-$\C^{(L+1)^2}$.
-
-For \emph{real} signals, however, this argument breaks down. The
-reality constraint $a_\ell^{-m} = (-1)^m \conj{a_\ell^m}$ is
-\emph{not} algebraic over $\C$ (it involves complex conjugation), so
-the set of real signals is not a complex-algebraic subvariety.
-Concretely, the constraint forces the coefficient matrix $A$---whose
-entries are polynomials in the lower-degree coefficients and their
-conjugates---to satisfy algebraic symmetries that reduce its complex
-rank when evaluated at real signals.
-
-Numerical verification confirms the severity: at $\ell = 4$, the
-selective bispectral entries give a bootstrap matrix of complex
-rank~$4$ (instead of $9 = 2\ell{+}1$), and even the \emph{full}
-bispectrum (all valid triples) achieves only rank~$4$. The rank
-deficiency is intrinsic to the scalar bispectrum for real signals
-and cannot be resolved by choosing different triples.
-
-The CG power spectrum entries $P_{\ell_1,\ell_2,\ell}$ provide the
-complementary information: they are degree-$4$ polynomials in the
-coefficients that capture the \emph{magnitudes} of the CG projections,
-which the degree-$3$ scalar bispectral entries (capturing
-\emph{phases}) miss. Together, the combined invariant achieves full
-Jacobian rank at generic signals.
-
-
-% ============================================================
-% CHANGE 4: Revised completeness proof
-% ============================================================
-
-\begin{proof}[Proof of Theorem~\ref{thm:so3-selective}]
-We construct an explicit invariant $\Phi$ and prove it separates
-$\SO(3)$-orbits for Lebesgue-almost-every signal.
-
-\medskip\noindent
-\textbf{Step 1: Gauge fixing.}
-By Lemmas~\ref{lem:app-F0}--\ref{lem:app-gauge}, the coefficients
-$\mathbf{F}_0$ and $\mathbf{F}_1$ are recovered from
-$\beta_{0,0,0}$ and $\beta_{0,1,1}$, and the rotation gauge is
-fixed (consuming all three $\SO(3)$ degrees of freedom). After
-gauge fixing, the signal is parametrised by
-$n := (L{+}1)^2 - 4$ real unknowns (the gauge-fixed coefficients
-$\mathbf{F}_2, \dotsc, \mathbf{F}_L$, with
-$\operatorname{Im}(a_2^1) = 0$).
-
-\medskip\noindent
-\textbf{Step 2: The augmented invariant map.}
-Define $\Phi_{\mathrm{aug}} \colon \R^n \to \R^m$ as the map sending
-the gauge-fixed coefficients to the combined vector of:
-\begin{enumerate}[nosep]
-  \item all ``live'' scalar bispectral entries
-    $\beta_{\ell_1,\ell_2,\ell}$ (those that do not vanish
-    identically for real signals), using the parity-correct
-    component ($\operatorname{Re}\beta$ if
-    $\ell_1{+}\ell_2{+}\ell$ is even,
-    $\operatorname{Im}\beta$ if odd);
-  \item a selected set of CG power spectrum entries
-    $P_{\ell_1,\ell_2,\ell}$.
-\end{enumerate}
-The CG power entries are chosen by a greedy rank-maximising procedure
-(Algorithm~\ref{alg:so3-augmented}).
-
-\medskip\noindent
-\textbf{Step 3: Full Jacobian rank.}
-We claim that $\Phi_{\mathrm{aug}}$ has Jacobian rank exactly $n$ at
-Lebesgue-almost-every gauge-fixed signal.
-
-At each of finitely many explicit rational witness signals, we
-evaluate the Jacobian $J_\Phi \in \R^{m \times n}$ numerically
-(using exact arithmetic in a verified computation framework) and
-confirm that $J_\Phi$ has rank~$n$. Since the entries of $J_\Phi$
-are polynomial functions of the signal coefficients, the set
-\[
-  \mathcal{Z} := \{x \in \R^n : \text{all } n \times n \text{ minors
-  of } J_\Phi(x) \text{ vanish}\}
-\]
-is a real algebraic variety. Since $\mathcal{Z}$ does not contain
-the witness points, $\mathcal{Z}$ is a proper subvariety and
-therefore has Lebesgue measure zero.
-
-\medskip\noindent
-\textbf{Step 4: Completeness.}
-At a generic gauge-fixed signal $x \in \R^n \setminus \mathcal{Z}$,
-the Jacobian $J_\Phi(x)$ has rank~$n$, so $\Phi_{\mathrm{aug}}$ is
-a local immersion. By the implicit function theorem, the fibre
-$\Phi_{\mathrm{aug}}^{-1}(\Phi_{\mathrm{aug}}(x))$ is locally a
-discrete set of points near~$x$.
-
-Since $\Phi_{\mathrm{aug}}$ is a polynomial map (degree $\le 4$) and
-$\SO(3)$-invariant, each fibre is a real algebraic set contained in
-the union of finitely many $\SO(3)$-orbits (a polynomial system of
-degree $\le 4$ in $n$ unknowns has at most finitely many isolated
-solutions in each bounded component, by B\'ezout's theorem applied
-to the complexification).
-
-The orbit of the true signal is always contained in the fibre.
-Since the fibre is discrete near the true signal, the orbit is
-an isolated component of the fibre. For a generic signal, no other
-orbit produces the same invariant values, and hence $\Phi_{\mathrm{aug}}$
-separates $\SO(3)$-orbits.
-\end{proof}
-
-
-% ============================================================
-% CHANGE 5: Entry count analysis
-% ============================================================
-
-\begin{proposition}[Output size of the augmented selective invariant]
-\label{prop:augmented-size}
-The augmented selective invariant $\Phi_{\mathrm{aug}}$ has
-$\Theta(L^2)$ scalar entries.
-\end{proposition}
-\begin{proof}
-The scalar bispectral component contributes at most
-$(L{+}1)^2 - 3$ entries (as in the original selective set).
-The CG power component adds at most $\sum_{\ell=2}^{L} (2\ell)$
-entries (at most two per lower degree $\ell_1 < \ell$, for each
-target $\ell$), contributing $O(L^2)$ entries.
-The total is $\Theta(L^2)$, matching the
-$\Omega(L^2)$ lower bound up to a constant factor.
-\end{proof}
diff --git a/paper/quick_rank.py b/paper/quick_rank.py
deleted file mode 100644
index ea2621b..0000000
--- a/paper/quick_rank.py
+++ /dev/null
@@ -1,83 +0,0 @@
-import sys, os
-sys.path.insert(0, os.path.join(os.path.dirname(__file__), '..', 'src'))
-import numpy as np
-from bispectrum.so3_on_s2 import _build_selective_index_map, load_cg_matrices
-
-CG = load_cg_matrices(8)
-out = []
-
-def cg(l1,m1,l2,m2,l,m):
-    if m1+m2!=m or l<abs(l1-l2) or l>l1+l2: return 0.
-    c=CG[(l1,l2)]; r=(m1+l1)*(2*l2+1)+(m2+l2); s=0
-    for v in range(abs(l1-l2),l1+l2+1):
-        if v==l: return c[r,s+m+l].item()
-        s+=2*v+1
-    return 0.
-
-def rfc(lmax,seed):
-    rng=np.random.RandomState(seed); fc={}
-    for l in range(lmax+1):
-        c=np.zeros(2*l+1,dtype=complex); c[l]=rng.randn()
-        for m in range(1,l+1):
-            x,y=rng.randn(2); c[l+m]=x+1j*y; c[l-m]=((-1)**m)*(x-1j*y)
-        fc[l]=c
-    return fc
-
-def cfc(lmax,seed):
-    rng=np.random.RandomState(seed); fc={}
-    for l in range(lmax+1):
-        c=np.zeros(2*l+1,dtype=complex)
-        for m in range(-l,l+1): c[l+m]=rng.randn()+1j*rng.randn()
-        fc[l]=c
-    return fc
-
-def bA(fc,lt,tr):
-    n=2*lt+1; rows=[]
-    for l1,l2,lv in tr:
-        if lv==lt and max(l1,l2)<lt:
-            row=np.zeros(n,dtype=complex)
-            for m in range(-lt,lt+1):
-                s=0j
-                for m1 in range(-l1,l1+1):
-                    m2=m-m1
-                    if abs(m2)>l2: continue
-                    s+=cg(l1,m1,l2,m2,lt,m)*fc[l1][l1+m1]*fc[l2][l2+m2]
-                row[lt+m]=np.conj(s)
-            rows.append(row)
-        elif l2==lt and lv<lt and l1<lt:
-            row=np.zeros(n,dtype=complex)
-            for m2 in range(-lt,lt+1):
-                s=0j
-                for m1 in range(-l1,l1+1):
-                    m=m1+m2
-                    if abs(m)>lv: continue
-                    s+=cg(l1,m1,lt,m2,lv,m)*fc[l1][l1+m1]*np.conj(fc[lv][lv+m])
-                row[lt+m2]=s
-            rows.append(row)
-    return np.array(rows) if rows else np.zeros((0,n),dtype=complex)
-
-idx=_build_selective_index_map(8)
-outf=os.path.join(os.path.dirname(__file__), 'rank_results.txt')
-with open(outf,'w') as f:
-    for label,mkfc in [("REAL",rfc),("COMPLEX",cfc)]:
-        f.write(f"{label} signals:\n")
-        for lt in range(4,9):
-            tr=[(a,b,c) for a,b,c in idx if max(a,b,c)==lt]
-            rs=[]
-            for s in [42,123,999]:
-                A=bA(mkfc(8,s),lt,tr)
-                sv=np.linalg.svd(A,compute_uv=False)
-                rs.append(int(np.sum(sv>1e-10*sv[0])) if sv[0]>0 else 0)
-            f.write(f"  l={lt} shape={A.shape} ranks={rs} need={2*lt+1}\n")
-        f.write("\n")
-    f.write("FULL bispec REAL:\n")
-    for lt in [4,5,6,7]:
-        at=[]
-        for a in range(lt+1):
-            for b in range(a,lt+1):
-                for c in range(abs(a-b),min(a+b,lt)+1):
-                    if max(a,b,c)==lt: at.append((a,b,c))
-        A=bA(rfc(lt,42),lt,at)
-        sv=np.linalg.svd(A,compute_uv=False)
-        r=int(np.sum(sv>1e-10*sv[0])) if sv[0]>0 else 0
-        f.write(f"  l={lt} {len(at)}triples A={A.shape} rank={r}/{2*lt+1}\n")
diff --git a/paper/rank_results.txt b/paper/rank_results.txt
deleted file mode 100644
index f9e1777..0000000
--- a/paper/rank_results.txt
+++ /dev/null
@@ -1,19 +0,0 @@
-REAL signals:
-  l=4 shape=(10, 9) ranks=[4, 4, 4] need=9
-  l=5 shape=(11, 11) ranks=[4, 4, 4] need=11
-  l=6 shape=(13, 13) ranks=[8, 8, 8] need=13
-  l=7 shape=(15, 15) ranks=[8, 8, 8] need=15
-  l=8 shape=(17, 17) ranks=[7, 7, 7] need=17
-
-COMPLEX signals:
-  l=4 shape=(10, 9) ranks=[9, 9, 9] need=9
-  l=5 shape=(11, 11) ranks=[11, 11, 11] need=11
-  l=6 shape=(13, 13) ranks=[13, 13, 13] need=13
-  l=7 shape=(15, 15) ranks=[15, 15, 15] need=15
-  l=8 shape=(17, 17) ranks=[17, 17, 17] need=17
-
-FULL bispec REAL:
-  l=4 19triples A=(10, 9) rank=4/9
-  l=5 27triples A=(16, 11) rank=4/11
-  l=6 37triples A=(24, 13) rank=9/13
-  l=7 48triples A=(33, 15) rank=9/15
diff --git a/paper/references.bib b/paper/references.bib
deleted file mode 100644
index f506d18..0000000
--- a/paper/references.bib
+++ /dev/null
@@ -1,351 +0,0 @@
-@article{myers2025selective,
-  title={The Selective Disk Bispectrum and Its Inversion, with Application to Multi-Reference Alignment},
-  author={Myers, Sophia and Miolane, Nina},
-  journal={arXiv preprint arXiv:2511.19706},
-  year={2025},
-  url={https://arxiv.org/abs/2511.19706},
-}
-
-@inproceedings{mataigne2024selective,
-  title={The Selective {G}-Bispectrum and its Inversion: Applications to {G}-Invariant Networks},
-  author={Mataigne, Simon and Mathe, Johan and Sanborn, Sophia and Hillar, Christopher and Miolane, Nina},
-  booktitle={Advances in Neural Information Processing Systems},
-  pages={115682--115711},
-  year={2024},
-  volume={37},
-  publisher={Neural Information Processing Systems Foundation, Inc.},
-  doi={10.52202/079017-3674},
-  url={https://proceedings.neurips.cc/paper_files/paper/2024/hash/d1a1e8713fcd5626656553c82f7c3b26-Abstract-Conference.html},
-}
-
-@phdthesis{kakarala1992triple,
-  title={Triple Correlation on Groups},
-  author={Kakarala, Ramakrishna},
-  school={University of California, Irvine},
-  year={1992},
-}
-
-@article{bandeira2017estimation,
-  title={Estimation under group actions: recovering orbits from invariants},
-  author={Bandeira, Afonso S. and Blum-Smith, Ben and Kileel, Joe and Perry, Amelia and Weed, Jonathan and Wein, Alexander S.},
-  journal={arXiv preprint arXiv:1712.10163},
-  year={2017},
-  url={https://arxiv.org/abs/1712.10163},
-}
-
-@article{kakarala2012bispectrum,
-  title={The Bispectrum as a Source of Phase-Sensitive Invariants for {F}ourier Descriptors: A Group-Theoretic Approach},
-  author={Kakarala, Ramakrishna},
-  journal={Journal of Mathematical Imaging and Vision},
-  volume={44},
-  number={3},
-  pages={341--353},
-  year={2012},
-  publisher={Springer},
-  doi={10.1007/s10851-012-0330-6},
-  url={https://arxiv.org/abs/0902.0196},
-}
-
-@inproceedings{cohen2016group,
-  title={Group Equivariant Convolutional Networks},
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-  year={2016},
-}
-
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-  year={2019},
-}
-
-@inproceedings{sanborn2023bispectral,
-  title={Bispectral Neural Networks},
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-  booktitle={International Conference on Learning Representations (ICLR)},
-  year={2023},
-}
-
-@inproceedings{oreiller2022bispectral,
-  title={Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral {U-Net}},
-  author={Oreiller, Valentin and Fageot, Julien and Andrearczyk, Vincent and Prior, John O. and Depeursinge, Adrien},
-  booktitle={Proceedings of The 5th International Conference on Medical Imaging with Deep Learning},
-  pages={929--943},
-  year={2022},
-  volume={172},
-  series={Proceedings of Machine Learning Research},
-  publisher={PMLR},
-  url={https://proceedings.mlr.press/v172/oreiller22a.html},
-}
-
-@inproceedings{veeling2018rotation,
-  title={Rotation Equivariant {CNN}s for Digital Pathology},
-  author={Veeling, Bastiaan S. and Linmans, Jasper and Winkens, Jim and Cohen, Taco and Welling, Max},
-  booktitle={Medical Image Computing and Computer Assisted Intervention (MICCAI)},
-  pages={210--218},
-  year={2018},
-  publisher={Springer},
-}
-
-@article{thomas2018tensor,
-  title={Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for {3D} Point Clouds},
-  author={Thomas, Nathaniel and Smidt, Tess and Kearnes, Steven and Yang, Lusann and Li, Li and Kohlhoff, Kai and Riley, Patrick},
-  journal={arXiv preprint arXiv:1802.08219},
-  year={2018},
-  url={https://arxiv.org/abs/1802.08219},
-}
-
-@inproceedings{batatia2022mace,
-  title={{MACE}: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields},
-  author={Batatia, Ilyes and Cs{\'a}nyi, G{\'a}bor and Kov{\'a}cs, D{\'a}vid P. and Ortner, Christoph and Simm, Gregor},
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-}
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-  title={Array programming with {NumPy}},
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\ No newline at end of file
diff --git a/paper/research_log.md b/paper/research_log.md
deleted file mode 100644
index 59469de..0000000
--- a/paper/research_log.md
+++ /dev/null
@@ -1,801 +0,0 @@
-# Research Log: Fixing the SO(3) Bispectrum Completeness Proof
-
-## Problem Statement
-
-The scalar SO(3) bispectrum β_{l1,l2,l} = <(F_{l1}⊗F_{l2})|_l, F_l> has
-fundamental Jacobian rank deficiency for real signals on S^2:
-
-- Seed Jacobian (10×11) has rank 6 (not 10 as claimed)
-- Two entries (β_{2,2,3}, β_{2,3,2}) are identically zero for real signals
-- Additional algebraic dependencies reduce effective rank further
-- Even the FULL bispectrum has rank 15 (not 22) for lmax=4
-
-Root cause: for real signals, conj(β_{l1,l2,l}) = (-1)^{l1+l2-l} β,
-making entries with odd parity purely imaginary. Entries with l_v ∈ {l1,l2}
-and the complementary degree odd vanish identically due to CG cancellation
-under the reality constraint |a_l^{-m}|^2 = |a_l^m|^2.
-
-## Approaches Under Investigation
-
-### Approach A: Augment scalar bispectrum with degree-4 "CG power spectrum" invariants
-
-Replace dead/dependent entries with:
-  P_{l1,l2,l} := ||(F_{l1}⊗F_{l2})|_l||^2
-These are degree-4 SO(3)-invariant and carry information the scalar
-bispectrum misses.
-
-### Approach B: Switch to unsummed / matrix-valued bispectrum
-
-Use vector-valued entries instead of scalar contractions. Higher rank
-but larger output.
-
-## Critical Finding: Bootstrap Rank for Real vs Complex Signals
-
-Complex rank of bootstrap matrix A (solving A conj(F_ℓ) = β):
-
-| ℓ | COMPLEX signal rank | REAL signal rank | Expected |
-|---|---|---|---|
-| 4 | 9 | 4 | 9 |
-| 5 | 11 | 4 | 11 |
-| 6 | 13 | 8 | 13 |
-| 7 | 15 | 8 | 15 |
-| 8 | 17 | 7 | 17 |
-
-Root cause: the reality constraint a_ℓ^{-m} = (-1)^m conj(a_ℓ^m) creates
-algebraic dependencies in A that are NOT present for complex signals.
-The Zariski density argument ("real signals are Zariski-dense in C^n")
-fails because the reality constraint involves conjugation, which is not
-algebraic over C.
-
-Even the FULL bispectrum (all valid triples) has the same rank deficiency.
-This is intrinsic to the scalar SO(3) bispectrum for real signals.
-
-## Solution Strategy
-
-Augment the scalar bispectrum with "CG power spectrum" entries at every
-degree:
-  P_{l1,l2,ℓ} := ||(F_{l1} ⊗ F_{l2})|_ℓ||^2  (degree-4 invariant)
-where l1 or l2 = ℓ (so that F_ℓ is involved).
-
-These are QUADRATIC in F_ℓ (when the other factor is known), providing
-additional constraints that fill the rank gap.
-
-The augmented invariant set has O(L^2) entries (still matching the
-orbit-space dimension up to constant factor).
-
-## Augmentation Verification
-
-Greedy augmentation with CG power entries achieves full Jacobian rank:
-
-### lmax=4 (20 unknowns)
-- 18 live scalar bispec entries → rank 10
-- +10 CG power entries → rank 20 (FULL RANK) ✓
-- Total: 30 entries (overhead: +8 over budget 22)
-
-### lmax=5 (31 unknowns)
-- 28 live scalar bispec entries → rank 14
-- +17 CG power entries → rank 31 (FULL RANK) ✓
-- Total: 47 entries (overhead: +14 over budget 33)
-
-Pattern: the rank deficit is ~50% of unknowns. CG power entries fill it.
-Total output: ~1.5 × (L+1)^2 entries = O(L^2), still quadratic.
-
-The augmentation entries follow a systematic rule:
-- For each degree ℓ ≥ 2 and each known degree l1 < ℓ:
-  add ||(F_{l1} ⊗ F_ℓ)|_{l_out}||^2 for l_out = |l1-ℓ|, ..., l1+ℓ
-  (up to budget constraints)
-
-## KEY DISCOVERY: Reflection Obstruction and T_R Resolution
-
-### The Reflection T_R
-
-The azimuthal reflection R: (θ,φ) → (θ,-φ) acts on SH coefficients as:
-  a_ℓ^m → (-1)^m a_ℓ^{-m} = conj(a_ℓ^m) for real signals
-
-On the gauge slice (F₁ aligned to z, Im(a₂¹) = 0):
-  T_R: v → -v, q_{ℓ,m} → -q_{ℓ,m} for all ℓ ≥ 3, m ≥ 1
-
-T_R preserves the gauge conditions and is NOT in SO(3) (det = -1).
-
-### Parity Classification (Proven Symbolically + Numerically)
-
-For real signals:
-  β_{l1,l2,l}(T_R · f) = (-1)^{l1+l2+l} · conj(β_{l1,l2,l}(f))
-
-- **Even parity** (l1+l2+l even): β is real and T_R-invariant
-- **Odd parity** (l1+l2+l odd): β is purely imaginary, Im(β) flips under T_R
-
-CG power P is always T_R-invariant (proved via CG symmetry).
-
-### Vanishing Pattern for Odd-Parity Entries
-
-Odd-parity β_{l1,l2,l} is identically ZERO whenever any two indices are equal:
-- l1 = l2: CG antisymmetry vs symmetric square
-- l1 = l or l2 = l: CG antisymmetry + reality constraint cancellation
-
-**First NONZERO odd-parity entry: β(2,3,4)** with l1+l2+l = 9, all distinct.
-
-Verified numerically at seed-42 signal (using sympy CG reference):
-  Im(β(2,3,4)) ≈ 3.26 (nonzero)
-  Im(β(2,3,4))(T_R · f) ≈ -3.26 (sign flip)
-
-Other nonzero odd-parity entries at degree 4: β(2,4,3), β(3,4,2).
-
-### Complete Classification (degrees ≤ 5)
-
-All odd-parity entries verified (60+ entries checked):
-- **ZERO** for real signals: ALL entries with l1=l2, l2=l, or l1=l
-  (includes β(1,1,1), β(2,2,3), β(1,3,3), β(2,3,2), β(3,3,1), etc.)
-- **NONZERO** (purely imaginary): entries with ALL THREE indices distinct
-  - β(2,3,4), β(2,4,3), β(3,4,2): Im ≈ ±8 to ±12
-  - β(2,4,5), β(2,5,4), β(4,5,2): Im ≈ ±1.4 to ±2.1
-
-### Implication for Completeness
-
-- **L ≤ 3**: No nonzero odd-parity entries exist. The augmented bispectrum
-  is O(3)-complete but NOT SO(3)-complete. Fibers are 2-point: {f, T_R(f)}.
-
-- **L ≥ 4**: The complex-valued β(2,3,4) breaks T_R via its imaginary part.
-  Combined with full Jacobian rank, fibers are 1-point. Full SO(3)-completeness.
-
-## Proof Strategy (Final)
-
-The augmented selective bispectrum Φ_aug: V_L → C^N × R^M separates
-SO(3) orbits on R^{(L+1)^2} / SO(3) for L ≥ 4.
-
-Proof structure:
-1. Gauge fix using F₁ (2 DOF) and Im(a₂¹) = 0 (1 DOF)
-2. Recover F₀ from β(0,0,0), F₁ from β(0,1,1)
-3. Seed recovery (ℓ ≤ 3): augmented system has full rank → 0-dim fibers
-   with 2 points {f, T_R(f)} (T_R = azimuthal reflection)
-4. T_R resolution at ℓ = 4: Im(β(2,3,4)) ≠ 0 generically, flips sign
-   under T_R, identifying the correct branch
-5. Per-degree bootstrap (ℓ ≥ 5): full-rank linear bootstrap determines
-   F_ℓ uniquely given correct lower degrees
-
-## Linear Bootstrap Verification
-
-Verified for ℓ = 4 to 100 (script: benchmarks/verify_linear_bootstrap.py):
-- All pass: rank = 2ℓ+1 at deterministic witness (seed 42)
-- Condition numbers: 10^1 to 10^4 (well-conditioned)
-- For ℓ ≥ 8: uses closed-form tridiagonal family
-
-## Seed System Analysis (Symbolic, via sympy)
-
-### Degree 2 (4 unknowns: y, x, u, v)
-
-| Invariant | Expression | Determines |
-|-----------|-----------|-----------|
-| β_{1,1,2} | √6·c²·y/3 | y (linear) |
-| P_{1,2,1} | c²(3x²+2y²)/5 | x² (then x>0 by gauge) |
-| β_{0,2,2} | A₀(2u²+2v²+2x²+y²) | S = u²+v² |
-| β_{2,2,2} | (6√14·S·y - 6√21·u·x² - ...)/7 | u (linear in u) |
-
-Remaining ambiguity: v² = S - u² gives |v| but not sign(v).
-
-### Degree 3 (7 unknowns: t, p₁, q₁, p₂, q₂, p₃, q₃)
-
-Key finding: the 3 scalar bispectral entries β(1,2,3), β(1,3,2), β(2,3,1)
-are in the Jacobian row span of each other and contribute rank 1 collectively.
-This is much worse than the complex case.
-
-The entries β(2,3,2) and β(2,2,3) are identically zero (odd parity, repeated indices).
-
-Full augmented seed system (13 nonzero entries, 11 unknowns):
-  - Bispectrum alone (8 nonzero entries): rank 6
-  - Adding ALL 5 CG power entries: rank 11 (FULL)
-  - CG power entries essential for rank: P(1,2,1), P(1,3,2), P(2,3,1), P(2,3,2), P(3,3,2)
-  - Each CG power entry contributes exactly one additional rank direction
-Verified at rational witness using exact sympy arithmetic.
-
-## Experiment Log
-
-### 2025-04-13: Parity breakthrough
-
-Discovered and proved the complete parity structure:
-- Even-parity β and CG power P are T_R-invariant
-- Odd-parity β vanishes iff any two indices coincide
-- First nonzero odd-parity entry at degree 4: β(2,3,4)
-- This gives all-L proof for L ≥ 4 (no numerical ceiling)
-
-Wrote complete proof in paper/proof_completeness.tex.
-
-### 2026-04-13: Exact implementation verification
-
-Verified proof_completeness.tex against exact implementation entries in so3_on_s2.py:
-
-1. **CG bug found & fixed**: Custom Racah-formula CG had wrong s_min/s_max bounds.
-   Switched to sympy.physics.wigner.clebsch_gordan as reference. All parity claims
-   confirmed: odd-parity entries ARE purely imaginary (Re ~ 1e-16).
-
-2. **Exact seed system match**: The implementation uses exactly these entries at lmax=3:
-   - Bispectrum: β(1,1,2), β(0,2,2), β(2,2,2), β(1,2,3), β(2,3,2)[=0],
-     β(1,3,2), β(2,3,1), β(2,2,3)[=0], β(0,3,3), β(3,3,2)
-   - CG power: P(1,2,1), P(1,3,2), P(2,3,1), P(2,3,2), P(3,3,2)
-   - 13 nonzero entries, 2 identically zero → 13×11 Jacobian, rank 11
-
-3. **Revised rank structure**: Bispectrum alone achieves rank 6 (not 8 as earlier
-   claimed with wrong entries). ALL 5 CG power entries are rank-essential,
-   each contributing 1 rank direction.
-
-4. **T_R resolution confirmed**: β(2,3,4) is in the selective index map at lmax=4.
-   Im(β(2,3,4)) ≈ 3.26 at seed-42 witness, flips sign under T_R.
-   Also β(2,4,3) and β(3,4,2) provide redundant parity-breaking.
-
-5. **Output size**: Total augmented output is ~1.6× orbit-space dimension,
-   well within Θ(L²).
-
-6. **Critical: Per-degree REAL rank deficiency**: The "linear bootstrap" has
-   COMPLEX rank 2ℓ+1 but REAL rank only ~ℓ for real signals! The scalar
-   bispectrum ALONE cannot determine F_ℓ for real signals at ANY degree.
-   CG power entries are essential at EVERY degree (not just the seed).
-   Per-degree real rank structure:
-   - ℓ=4: linear=4/9, +self=6/9, +CG=9/9
-   - ℓ=5: linear=4/11, +self=6/11, +CG=11/11
-   - ℓ=6: linear=8/13, +self=11/13, +CG=13/13
-
-7. **T_R resolution logic**: The proof's T_R argument was corrected.
-   The T_R branch doesn't produce T_R(f) at higher degrees; instead,
-   the bootstrap on the T_R branch produces a DIFFERENT signal g that
-   satisfies all bootstrap entries but FAILS non-bootstrap entries.
-   Verified at ℓ=4: self-coupling β(4,4,2) differs by ~20, CG power
-   P(2,4,3) differs by ~35.
-
-8. **Parity formula corrected**: β(T_R·f) = (-1)^{l1+l2+l} β(f) for ALL
-   signals (not just real). Separate reality constraint gives
-   conj(β) = (-1)^{l1+l2+l} β for real signals.
-
-### 2026-04-16: STRUCTURAL AUDIT and rescue plan
-
-Auditing the entire `appendix:so3-completeness` proof for non-structural
-content. Goal: identify whether the proof can be made fully structural
-while preserving O(L²) augmented selective invariant.
-
-#### Catalogue of non-structural claims
-
-| # | Claim | Where | Type of evidence |
-|---|-------|-------|------------------|
-| C1 | Seed Jacobian (13×11) has rank 11 at rational witness | Lem app-ell3 | sympy at one rational point |
-| C2 | Seed fibre has 10 real solutions per v-branch | Rem app-seed-fibre-size | Multi-start NLS, 5000 starts |
-| C3 | Degree-4 NLS finds zero residual ONLY for true seed (9 spurious eliminated) | Lem app-degree4-filter | Multi-start NLS, 200 starts × 10 seeds |
-| C4 | Complex bootstrap rank = 2ℓ+1 for ℓ ∈ [4, 100] | Lem app-bootstrap-rank | Per-ℓ numerical witness, seed 42 |
-| C5 | Real per-degree augmented rank = 2ℓ+1 for ℓ ∈ [4, 30] | Lem app-augmented-perdegree | Per-ℓ numerical witness |
-| C6 | T_R branch is degree-4 inconsistent | Lem app-TR-resolve | Same witness as C3, rational-function argument |
-
-#### Structural status per claim
-
-- **C1 (seed Jacobian rank)**: Easily upgradable. A nonzero polynomial
-  evaluated at one rational point is generically nonzero. The proof
-  already invokes this density argument; only the witness is required
-  in the appendix. STRUCTURAL.
-
-- **C2 (10 solutions/branch)**: Not used in the proof; only in the
-  remark. Bezout gives a structural upper bound (≤ 32 complex per
-  branch from 1·1·1·2·2·2·2·4 = 128 over both branches, or 32 per
-  branch using 7 of the 8 entries). Replace the "10" by "≤ 32" in
-  the proof. STRUCTURAL after edit.
-
-- **C3 (degree-4 filter)**: This is **the** critical gap. The proof
-  enumerates all spurious seeds NUMERICALLY, then verifies each has
-  nonzero degree-4 residual via NLS. Two failure modes:
-    (a) NLS may miss spurious solutions (no a priori enumeration).
-    (b) The polynomial-density argument applies to each spurious
-        branch individually, but proves at most that "generically each
-        of the 9 KNOWN branches has nonzero residual"; says nothing
-        about possible OTHER spurious branches not enumerated.
-  Status: NOT STRUCTURAL.
-
-- **C4 (complex bootstrap)**: Per-ℓ verification is a sound
-  polynomial-nonzero-at-witness argument FOR EACH FIXED ℓ. The
-  generic set is the complement of L−3 proper subvarieties. Sound
-  for any FIXED L ≤ 100, but the script must be re-run for each new L.
-  Status: STRUCTURAL FOR FIXED L, but lacks a uniform-in-ℓ argument.
-
-- **C5 (real per-degree rank)**: Same as C4 plus the complex→real
-  density argument (Remark app-complex-real). The complex rank
-  certificate at the deterministic witness extends to a real-rank
-  statement once the gauge-fixed reality structure is plugged in.
-  Status: same as C4.
-
-- **C6 (T_R branch inconsistency)**: Inherits the C3 gap (it's a
-  special case). The "rational function with denominator (det A4)²"
-  argument is structurally sound IF the numerator polynomial is
-  nonzero at the witness. So this part IS structural-modulo-witness.
-  Status: STRUCTURAL given C3 witness.
-
-#### Empirical investigation: can extra CG power entries reduce the
-seed fibre below 10?
-
-Tested at the rational witness (`paper/test_seed_augment.py`,
-800 starts/branch). Augmenting the seed CG power set with each of:
-  - {P(1,3,3)}: still 10/10
-  - {P(2,3,3)}: still 10/10
-  - {P(1,3,3), P(2,3,3)}: still 10/10
-  - {P(2,2,3)} or {P(1,1,3)}: still 10/10 (these don't even involve F3)
-  - all four together: still 10/10
-
-**Conclusion**: degree-≤4 SO(3)-invariants on V_0..3 cannot resolve
-the 10-fold cover. The 10:1 ramification is intrinsic to the
-invariant theory of (V_0..3, SO(3)) at this Hilbert-series level.
-ANY structural reduction to a 2-point seed fibre MUST use degree-≥4
-input data (i.e., F_4 or higher). This validates the paper's
-strategy of pushing fibre reduction to degree 4.
-
-#### Why C3 cannot be patched by adding more low-degree entries
-
-The 10 spurious seeds are real points of the algebraic variety cut
-out by all SO(3) invariants of degree ≤4 in F_0..3 (modulo SO(3)).
-Equivalently, the SO(3) GIT-quotient V_0..3//SO(3) has a self-cover
-of degree (at least) 10 visible at the level of bispectrum+P
-invariants. To resolve we MUST use info beyond degree 3.
-
-#### Structural rescue plan
-
-Replace the "numerically-eliminate-9-spurious-seeds" argument with a
-classical algebraic-geometry argument applied to the JOINT system at
-degrees 0-4. Specifically:
-
-1. **Use the FULL bispectrum at L=4** as the seed. This is a constant
-   number of entries (≤ 30), so the asymptotic budget Θ(L²) is
-   unaffected. By Edidin–Satriano (2024), the full bispectrum at any
-   fixed L separates generic O(L^3) orbits of SO(3) on real
-   band-limited signals. Specialised to L=4: full bispectrum at L=4
-   determines (F_0..4) up to T_R generically.
-   - Status: cite as a known structural theorem.
-   - Output cost: O(L²) since we add O(1) extra entries at L=4.
-
-2. **T_R resolved at degree 4** by parity-odd entries β(2,3,4),
-   β(2,4,3), β(3,4,2). Their imaginary parts flip sign under T_R
-   and are nonzero generically (Cor app-first-odd). STRUCTURAL.
-
-3. **Bootstrap for ℓ ≥ 5** using the selective family T_ℓ.
-   Structural rank claim: **for every ℓ ≥ 5, det(A_ℓ) is a nonzero
-   polynomial in F_0..ℓ−1**. The current per-ℓ witness shows this for
-   ℓ ≤ 100. Need either:
-   (a) An explicit closed-form determinant or rank argument uniform
-       in ℓ, OR
-   (b) A "uniform witness" — a single signal sequence (F_a)_{a≥0}
-       (computable in closed form) such that det(A_ℓ) ≠ 0 at this
-       sequence for every ℓ. The latter is at most a one-time
-       structural lemma.
-
-#### Status of the structural rescue
-
-- Steps 1–2 give a STRUCTURAL replacement for the entire
-  "seed + degree-4 filter + T_R resolution" block (Lemmas
-  app-ell2, app-ell3, app-degree4-filter, app-TR-resolve), at the
-  cost of citing Edidin–Satriano as a black-box theorem.
-
-- Step 3 still requires a structural rank argument uniform in ℓ.
-  This is the **only remaining gap** after the rescue.
-
-#### Open structural problem: uniform bootstrap rank
-
-For each ℓ ≥ 5, define
-  A_ℓ(F_0,...,F_{ℓ-1}) ∈ C^{(2ℓ+1) × (2ℓ+1)}
-to be the bootstrap matrix from the closed-form selective family
-T_ℓ (Prop app-entry-counts). We need
-
-  Q_ℓ(F_0,...,F_{ℓ-1}) := det A_ℓ(F_0,...,F_{ℓ-1}) ≢ 0 in C[F_0..ℓ-1].
-
-PROPOSED STRUCTURAL ARGUMENT (uniform witness):
-
-Define the deterministic "geometric" witness
-  F_a^m = ζ^{a+m},   for ζ ∈ C generic,  a ≥ 1, |m| ≤ a.
-
-Then Q_ℓ becomes a Laurent polynomial p_ℓ(ζ) in ζ. By the structure
-of the closed-form family (chain rows (a,ℓ,ℓ-a), offset rows
-(a,ℓ,ℓ-a+1), C-rows (a,ℓ-a,ℓ)), p_ℓ has a leading term
-proportional to ∏_a CG[a,*,ℓ,*|...] which is a known Wigner 3j
-product.
-
-To finish: show that for some specific ζ_0 (e.g. ζ_0 = 1 or a small
-integer) and every ℓ ≥ 5, p_ℓ(ζ_0) ≠ 0. This is a single statement
-parametrised by ℓ; it can be reduced to a Wigner-3j non-vanishing
-identity provable by (e.g.) the Racah formula.
-
-This reduction was NOT executed in the present session; flagged as
-the next structural step.
-
-### 2026-04-13: Seed fibre is NOT 2-point — critical proof restructuring
-
-9. **Seed fibre has 10 solutions per v-branch (20 total)**: Exhaustive
-   multi-start search (5000 starts per branch) finds EXACTLY 10 distinct
-   real solutions per sign(v) branch, forming 10 T_R pairs.
-   All 20 solutions have full Jacobian rank 11/11 (simple zeros).
-   The degree-3 subsystem (7 real equations in 7 unknowns, degrees
-   1+2^5+4, Bezout bound 128) has more real solutions than the
-   previously claimed 1 per branch.
-   Script: benchmarks/verify_seed_fibre.py
-
-10. **Degree-4 eliminates all spurious seed solutions**: For each of the
-    10 seed solutions on the +v branch, multi-start NLS (200 starts)
-    tries to find F4 satisfying all 29 real degree-4 constraints
-    (12 bispectral × 2 + 5 CG power). ONLY the true seed solution
-    achieves zero residual (||r|| ~ 1e-16). All 9 spurious solutions
-    have residual >= 1e-2.
-    Key mechanism: degree-4 entries that DON'T involve F3 (β(2,2,4),
-    β(2,4,2), P(1,4,3), P(1,4,4), P(2,4,2), P(2,4,3), β(4,4,2),
-    β(4,4,4)) nearly determine F4 independently. Entries involving
-    F3 then impose constraints that are inconsistent for spurious F3.
-    Script: benchmarks/verify_degree4_filter.py
-
-11. **Proof restructured**: The seed lemma (lem:ell3) now correctly claims
-    only a FINITE fibre (not exactly 2 points). A new lemma
-    (lem:degree4-filter) handles fibre reduction at degree 4.
-    The T_R resolution (lem:TR-resolve) is now stated as a special case
-    of degree-4 elimination, with the rational function structure
-    (numerator p, denominator det(A4)^2) made explicit.
-
-12. **Bootstrap scope clarified**: The bootstrap rank verification
-    covers ℓ=4..100. For any fixed L, the generic set is the complement
-    of L-3 proper algebraic subvarieties (one per degree). The theorem
-    is stated for fixed L with L-dependent genericity conditions.
-
-### 2026-04-16 (cont.): RESCUE COMPLETE via Kakarala 1992/2012 Theorem 7
-
-After re-reading Edidin-Satriano 2024 carefully, my earlier rescue plan
-was based on a misreading: Edidin-Satriano's `R \ge L+2` theorem is for
-the MULTI-SHELL case (signals on R^3 sampled on R radial shells), NOT
-for single-S^2 signals. Their single-S^2 statement only cites
-[Bandeira-Blum-Smith-Kileel-Perry-Weed-Wein, 1712.10163] as a
-COMPUTATIONAL verification for L <= 15. So Edidin-Satriano cannot be
-used as a structural black-box for our seed.
-
-HOWEVER: a sharper structural result is already available.
-
-#### Kakarala 2012, Theorem 7 (the actual structural seed)
-
-Kakarala (JMIV 2012, derived from his 1992 PhD thesis) proves:
-
-> Let G be compact, H closed in G, N_H the normalizer of H in G. Let
-> r, s ∈ L^1(G) be left-H-invariant with maximal H-rank Fourier
-> coefficients. Then a3_r = a3_s iff s(g) = r(xg) for some x ∈ N_H.
-
-Specialised to G = SO(3), H = SO(2)_z (so G/H = S^2), the normalizer
-N_H = O(2) is generated by H plus the 180° rotation R_x(π) about the
-x-axis. Both lie in SO(3) (R_x(π) is a proper rotation, det = +1).
-Maximal H-rank for S^2 functions means F_l ≠ 0 for all l ≤ L
-(every Fourier vector is nonzero).
-
-CONSEQUENCE: For generic real-valued S^2 functions band-limited at L,
-the FULL bispectrum {β_{l1,l2,l} : 0 ≤ l1 ≤ l2 ≤ L, |l1-l2| ≤ l ≤ l1+l2}
-is a COMPLETE SO(3)-invariant. NO reflection ambiguity, NO finite
-fibre, NO T_R issue. Edidin-Satriano's "Kakarala recovers up to
-reflection" remark in their Section 1 is misleading: the "reflection"
-they reference is the N_H/H = Z/2 quotient, but the non-trivial coset
-representative R_x(π) is a *proper* SO(3) rotation, so it produces no
-genuine residual ambiguity for S^2 functions.
-
-#### Rescue using Kakarala Theorem 7
-
-Replace the entire seed block (degrees 0-4) with a single citation
-to Kakarala. Concretely:
-
-(a) Augment the selective bispectrum with the FULL bispectrum at a
-    fixed low degree L_0 ≥ 2 (we use L_0 = 4). The number of
-    additional entries is bounded by
-        |full bispectrum at L_0| ≤ (L_0+1)^3 / 6 + O(L_0^2)
-    which is O(1) (for L_0 = 4: ~30 entries). The asymptotic O(L^2)
-    output cost is preserved.
-
-(b) By Kakarala Thm 7, the full bispectrum at L_0 = 4 uniquely
-    determines (F_0, ..., F_4) up to SO(3), for every signal with
-    F_l ≠ 0 for l ∈ {0, 1, 2, 3, 4}. In particular, after gauge
-    fixing, (F_0, ..., F_4) is uniquely determined as a point in
-    the gauge slice. STRUCTURAL.
-
-This eliminates Lemmas app-ell2, app-ell3, app-degree4-filter,
-app-TR-resolve, Remark app-seed-fibre-size, and the entire degree-4
-NLS computational certificate. Eight pages of the proof collapse to
-"by Kakarala 1992/2012, Theorem 7".
-
-#### What survives
-
-Only the bootstrap rank for l ≥ 5 remains. The current per-l
-verification (Lem app-bootstrap-rank for complex rank, l ≤ 100;
-Lem app-augmented-perdegree for real rank, l ≤ 30) is structural
-"for any fixed L", which is the standard guarantee. The script can
-be re-run to extend coverage. A truly uniform-in-l proof remains
-open but is not necessary for the headline theorem.
-
-Key empirical observation (test_uniform_witness.py with chain-row
-conjugation bug fixed):
-
-  - Random complex signal (no reality constraint), seed 42:
-      FULL complex rank 2l+1 verified for l ∈ {8, 10, 15, 20, 30, 40,
-      50, 70, 100}. Condition number stays at 10^2 - 10^3 (excellent).
-  - Random real signal (with reality constraint), seed 42:
-      Linear bootstrap alone has real rank ≈ l (deficient by ≈ l).
-      Confirms Remark app-complex-real: chain+cross alone is
-      insufficient for real signals; CG power augmentation is
-      mandatory at every l.
-  - Closed-form analytic witnesses (W1: F_a^m = 1; W2: F_a^m = 1/(1+m^2);
-      W3: F_a^m = (1+m)*(1+a); W4: F_a^m = (-1)^a*(1+abs(m));
-      W5: F_a^m = exp(i m); — all REAL-projected) FAIL at moderate l.
-      No simple closed-form witness gives uniform-in-l rank.
-
-Conclusion: a "uniform witness" structural proof in closed form
-appears to require an actual Wigner-3j non-vanishing identity for
-the determinant of the closed-form bootstrap family, which is a
-genuine computer-algebra task beyond the scope of this session.
-
-#### Final structural map of the proof
-
-| Step | Status |
-|------|--------|
-| Gauge fixing (lem:app-gauge) | STRUCTURAL |
-| F_0, F_1 recovery (lem:app-F0F1) | STRUCTURAL |
-| Parity of bispectrum (lem:app-parity, lem:app-P-invariant, prop:app-odd-vanishing, cor:app-first-odd) | STRUCTURAL |
-| Seed (degrees 0-4) via FULL bispectrum at L_0=4 + Kakarala Thm 7 | STRUCTURAL (cite Kakarala 1992/2012) |
-| Bootstrap rank l ≥ 5 (lem:app-bootstrap-rank, lem:app-augmented-perdegree) | STRUCTURAL FOR EACH FIXED l |
-| Inductive recovery (prop:app-TR-propagate, main thm) | STRUCTURAL |
-
-The augmented selective bispectrum, augmented further with the full
-bispectrum at L_0 = 4, has output size
-
-  |selective at L| + |full at L_0=4| = Θ(L^2) + Θ(1) = Θ(L^2)
-
-and is a complete SO(3)-invariant on generic real S^2 signals,
-with the only "computer-verified per fixed L" component being the
-bootstrap rank at l ∈ [5, L]. The seed is fully structural.
-
-#### Concrete LaTeX edit plan for paper.tex
-
-1. In the proof overview (around line 1171), drop "(3) fibre reduction
-   at degree 4 via parity-breaking entries" and rephrase as:
-   "(3) seed completeness for degrees 0-4 via Kakarala's bispectrum
-   completeness theorem applied to the full L_0=4 bispectrum block".
-
-2. Add a new lemma `lem:app-seed-kakarala` immediately after
-   `lem:app-F0F1`:
-   "For generic real signals satisfying Assumption G1-G3 with F_l ≠ 0
-   for l ≤ 4, the full bispectrum at L_0=4 uniquely determines
-   (F_0,...,F_4) on the gauge slice. (Kakarala 1992/2012 Theorem 7
-   applied to G=SO(3), H=SO(2)_z, with maximal-H-rank coefficients
-   = nonzero F_l vectors.)"
-
-3. Delete or downgrade Lemmas app-ell2, app-ell3, app-degree4-filter,
-   app-TR-resolve, Remark app-seed-fibre-size to "Historical /
-   constructive remark" — the constructive seed solver remains valid
-   as an algorithm even though the completeness proof no longer
-   depends on it.
-
-4. Update Definition of `Phi_aug` to include the full bispectrum
-   at L_0=4 (in addition to the existing selective entries at all l).
-   The output count becomes
-       N_aug(L) = N_selective(L) + N_full(L_0=4) = Θ(L^2) + 30.
-
-5. Update `thm:app-main` proof Step 4 to read:
-   "By Lemma app-seed-kakarala, the seed (F_0,...,F_4) is uniquely
-   determined on the gauge slice. The reflection R_x(π) ∈ N_H \ H
-   acts on the gauge slice but is itself a proper SO(3) rotation;
-   the gauge-fixed parameterisation already breaks this Z/2."
-   (No more parity-breaking argument needed for T_R; T_R was already
-   resolved by Kakarala because his "ambiguity up to N_H" is
-   ambiguity up to a proper SO(3) rotation, not an O(3) reflection.)
-
-6. Drop the "Computational certificates" items 2 (seed fibre
-   enumeration) and 3 (degree-4 fibre reduction).
-
-7. In Table around line 1716, update "Edidin & Satriano (2024)" row
-   to also mention "(multi-shell only)" since their R ≥ L+2 result
-   does not apply to single-shell S^2.
-
-The user-visible cost of this rescue is exactly: a fixed Θ(1)
-additional output entries at low degree, and one extra citation
-to Kakarala. In return, the proof becomes structurally complete
-modulo per-fixed-L bootstrap rank verification.
-
-## Update: dual-citation strengthening of structural seed (post-rescue)
-
-Refined `lem:app-seed-kakarala` to invoke TWO independent
-structural arguments instead of relying on Kakarala alone:
-
-  (i)  Representation-theoretic via Kakarala 1992/2012 Theorem 7
-       (proved by Iwahori-Sugiura duality on G/H), with N_H ⊂ SO(3)
-       so no O(3)-reflection residue.
-
-  (ii) Symbolic Jacobian-rank certificate of Bandeira et al. 2017
-       (their orbit-recovery paper, valid for all L₀ ≤ 15, hence
-       certainly for L₀ = 4). One-time computer-algebra check
-       over a number field, NOT a numerical witness — fully
-       structural.
-
-This insulates the seed step from any reader who is uncomfortable
-trusting the Kakarala 1992 thesis as the sole source for Theorem 7.
-
-Also added `rem:app-edidin-reflection` explicitly explaining that
-Edidin-Satriano's "up to reflection" caveat refers to the
-*algorithm* in Kakarala Section 5 (positive-square-root step
-introduces an O(3) ambiguity that the recursion propagates), NOT
-to the bispectrum data. The full bispectrum at any L₀ ≥ 4 contains
-β_{2,3,4} (smallest all-distinct odd-parity entry), which flips
-sign under T_R, separating f from T_R · f. So Edidin-Satriano's
-caveat is correct *for the algorithm* but does not weaken the
-completeness statement of Theorem 7.
-
-Bibtex `bandeira2017estimation` added to references.bib.
-Compilation clean (tectonic, no undefined references/citations).
-
-## Update: deep rewrite to a CONDITIONAL completeness theorem (`proof_clean.tex`)
-
-Outcome: the proof in `paper/proof_clean.tex` is now defensible as a
-conditional theorem on two explicitly named hypotheses, with no
-remaining "Kakarala-proves-the-seed" overclaim and no "local Jacobian
-rank ⇒ global uniqueness" inference.
-
-### Why the previous "rescue" was still too strong
-
-After re-reading Kakarala 2012 Theorem 7 carefully against a strict
-reviewer pass, the rescue strategy above (apply Kakarala to the
-finite L₀=4 block) is NOT a valid proof:
-
-- Theorem 7 requires the Fourier coefficients to have maximal H-rank
-  for ALL irreducible representations α of G, equivalently at least
-  one nonzero coefficient at every degree.
-- A band-limited truncation has a_α = 0 for every α > L_seed by
-  construction, so the maximal-H-rank hypothesis fails immediately.
-- Kakarala 1992/2012 therefore cannot be invoked as an unconditional
-  proof of finite-band seed injectivity. The 30-entry block buys
-  inspiration only.
-
-The Bandeira et al. 2017 symbolic Jacobian is computer-algebra
-EVIDENCE for the seed map, not a published theorem about scalar
-S² bispectrum injectivity at L_seed ≤ 7. It cannot stand in for the
-seed proof either, only support it.
-
-### What the rewritten proof actually proves
-
-`thm:main` in `proof_clean.tex` is now stated as
-
-  Conditional on
-    Hyp 1 (finite seed injectivity for L_seed ≤ 7) and
-    Hyp 2 (Δ_ℓ ≢ 0 for every ℓ ≥ 8),
-  Φ_aug separates SO(3)-orbits of generic real-valued band-limited
-  signals, with |Φ_aug| = Θ(L²).
-
-The two hypotheses are isolated, named, and explicitly listed as open
-problems. The proof itself is unconditional given the hypotheses.
-
-### Per-degree bootstrap is now a SQUARE COMPLEX LINEAR system
-
-For ℓ ≥ 8, the selected block T_ℓ has 2ℓ+1 entries:
-
-  T_ℓ = {(a,ℓ,ℓ-a) : 1 ≤ a ≤ ℓ-1}                 (chain rows, conj F_ℓ)
-      ∪ {(a,ℓ,ℓ-a+1) : 2 ≤ a ≤ ℓ-1}               (cross rows, F_ℓ)
-      ∪ {(a,ℓ-a,ℓ) : 1 ≤ a ≤ 4}                   (chain rows, conj F_ℓ)
-
-After conjugating the chain equations, all rows are linear in F_ℓ
-(over C), giving a square system A_ℓ(F_<ℓ) F_ℓ = b_ℓ in
-C^{(2ℓ+1)×(2ℓ+1)}. Inductive uniqueness needs only
-
-  Δ_ℓ := det A_ℓ ≢ 0 as a polynomial in real coords of F_<ℓ.
-
-That is Hypothesis 2. No more nonlinear CG-power systems, no more
-"real Jacobian rank ≥ 2ℓ+1 implies F_ℓ = F'_ℓ" overclaim. The
-induction step is now literally:
-
-  Φ_aug(f)=Φ_aug(f') ⇒ F_<ℓ = F'_<ℓ
-                     ⇒ A_ℓ(F_<ℓ) F_ℓ = A_ℓ(F_<ℓ) F'_ℓ
-                     ⇒ F_ℓ = F'_ℓ.
-
-### CG-power is no longer load-bearing in the proof
-
-Definition `def:Phi-aug` sets S_P = ∅ in the formal theorem.
-CG-power entries are kept ONLY as optional implementation diagnostics
-(`rem:cg-optional`). Section `sec:cgpower` was rewritten:
-"Why CG power augmentation is necessary" → "CG-power diagnostics".
-The earlier real-Jacobian-rank deficit table (rank-4 vs target 9 etc.)
-is now attributed to "an earlier real-valued implementation" of the
-selected bispectrum, not to the proof.
-
-The corrected logic: invertibility of A_ℓ over C already forces
-uniqueness of F_ℓ on the real-signal slice. There is no real-rank gap
-inside the proof itself, so CG-power need not close any gap. Adding
-extra invariants cannot destroy completeness, so CG-power is harmless
-to ship in the implementation.
-
-### Numerical "L ≤ 100 certificate" downgraded to "evidence"
-
-Floating-point checks of det A_ℓ ≠ 0 for ℓ = 8…100 (singular-value
-ratio ≥ 10⁻⁴) are NOT a mathematical certificate. The text now says:
-
-- C2 "Bootstrap determinant evidence":  strong numerical evidence,
-  not a certificate.
-- "Verified band-limits" corollary removed; replaced by
-  `rem:numerical-evidence` saying rigorous proof requires exact
-  arithmetic, interval arithmetic, or a perturbation bound certifying
-  distance from singularity.
-
-This is honest about what `verify_linear_bootstrap.py` actually
-proves: per-(ℓ, fixed witness) numerical full rank. It does NOT
-prove Δ_ℓ ≢ 0 as a polynomial — that requires symbolic / interval
-work that we have not done.
-
-### Kakarala's role in the document
-
-Demoted from "proves the seed" to "motivates the structural
-template". Body text now explicitly says Theorem 7 is invoked as
-inspiration for "constant-size full low-degree block + per-degree
-recurrence", not as a proof of `hyp:seed`. The bandeira2017estimation
-remark is kept as "possible symbolic certificate" — open.
-
-### β_{2,3,4} parity corollary made symbolic and exact
-
-Old draft used a NumPy seed witness (β_{2,3,4} ≈ -0.9149i at seed 42)
-and incorrectly cited CG(2,2; 3,2 | 4,4) = 2√(2/5).
-
-Corrected:
-- CG(2,2; 3,2 | 4,4) = √(2/5) exactly (verified, every CG ≤ 1).
-- Closed-form witness:  a_2^2 = a_3^2 = 1, a_4^4 = i, all other
-  degree-2/3/4 coefficients zero, negative-m fixed by reality.
-  Only (m_1,m_2,m) ∈ {(2,2,4),(-2,-2,-4)} contribute.
-  β_{2,3,4} = √(2/5)·1·1·(-i) + (-√(2/5))·1·1·(+i) = -2i√(2/5) ≠ 0.
-  Verified numerically: -1.2649i = -2·0.6325·i. ✓
-
-### Repeated-index vanishing generalised via the symmetric tensor
-
-Old "self-coupling CG symmetry" argument only handled (a, a, ℓ).
-Replaced with the full symmetric tensor
-
-  T_{ℓ_1,ℓ_2,ℓ_3} := Σ (3j-symbol)·a_{ℓ_1}^{m_1} a_{ℓ_2}^{m_2} a_{ℓ_3}^{m_3}.
-
-Conversion 3j → CG plus reality identity gives
-
-  β_{ℓ_1,ℓ_2,ℓ} = (-1)^{ℓ_1+ℓ_2}·√(2ℓ+1)·T_{ℓ_1,ℓ_2,ℓ}.
-
-3j is invariant under even column permutations, picks up
-(-1)^{ℓ_1+ℓ_2+ℓ_3} under odd ones. Two columns with equal degree
-insert the SAME signal coefficients, so swap → T = ±T → T = 0 when
-parity is odd. This handles (a,a,ℓ), (a,ℓ,a), (ℓ,a,a) uniformly.
-
-### Genericity phrased on the gauge slice
-
-(G1)–(G5) are polynomial inequalities in gauge-fixed real coords, so
-they cut out a Zariski-open V ⊂ G in the gauge slice G of `lem:gauge`.
-The generic set in V_L is the SO(3)-saturation SO(3)·V. The previous
-phrasing "Zariski-open subset of V_L" was sloppy because aligning F_1
-with ẑ is not a global polynomial map on V_L.
-
-### Index-set deduplication
-
-Old `def:Phi-aug` had S_β AND a separate "full low-degree seed block"
-in the concatenation, while the Index-sets paragraph said S_β already
-contained the seed block — a formal duplicate. Fixed by introducing
-S_seed and defining S_β := S_seed ∪ ⋃_{ℓ≥8} T_ℓ. Φ_aug is now a
-single concatenation indexed by S_β.
-
-### Title / naming honesty
-
-Old: "Completeness of the Augmented Selective SO(3)-Bispectrum on S²".
-New: "Completeness of a Selective SO(3)-Bispectrum on S²
-(with optional Clebsch–Gordan power diagnostics)".
-
-The body and abstract clarify: the proof is for S_P = ∅; "augmented"
-refers to the implementation variant that ships the optional CG-power
-block.
-
-### What is OPEN after this round
-
-1. Hyp 1: Finite seed injectivity for L_seed ≤ 7. Needs a real
-   theorem (e.g. closed-form Gröbner / resultant non-vanishing on the
-   gauge slice). Bandeira 2017 is supportive evidence only.
-
-2. Hyp 2: Δ_ℓ ≢ 0 uniformly for all ℓ ≥ 8. Needs an exact closed-form
-   identity, or interval-arithmetic checks per ℓ. Floating-point
-   passes for ℓ ≤ 100 do not constitute proof.
-
-3. The proof is correct CONDITIONAL on (1) and (2). No further gaps
-   identified by the latest reviewer pass:
-   "Yes — as a conditional proof, this version is basically correct."
-
-### Files
-
-- `paper/proof_clean.tex` — current canonical proof (152 KB PDF, builds
-  cleanly under tectonic, no undefined refs, no overfull hboxes).
-- `benchmarks/verify_linear_bootstrap.py` — produces ℓ=4..12 numerical
-  full-rank evidence for Hyp 2 at fixed seeds. NOT a certificate.
-- `paper/exact_certificates.py` — mpmath-based Jacobian rank checks
-  for L ∈ {4,5}. Provides triangular local uniqueness only; does NOT
-  prove Hyp 1 globally.
diff --git a/paper/review_feedback.md b/paper/review_feedback.md
deleted file mode 100644
index 6b170a6..0000000
--- a/paper/review_feedback.md
+++ /dev/null
@@ -1,372 +0,0 @@
-# Review of `paper.tex`
-
-> **Status update (April 2026):** This review was written against the older
-> `paper/paper.tex`. The canonical paper source is now
-> `paper/paper_scaffold_pi_review.tex`. Several findings below are resolved:
->
-> - **Finding 6 (SO3onS2 selective):** `so3_on_s2.py` now has selective mode
->   fully implemented (`_build_selective_index_map`, O(L²) output), tested
->   (Jacobian rank tests in `tests/test_so3_on_s2.py`), and documented
->   (`docs/selective_so3_on_s2_completeness.md`). The claim on line 136 that
->   "selective mode is reserved for future use" is **stale**.
-> - **Finding 5 (steerable-to-Fourier bridge):** Dropped from the paper scope.
-> - **Finding 8 (RotMNIST contrastive):** Dropped from the paper scope.
-> - **Experiments:** The scaffold now covers PCam + OrganMNIST3D + Spherical
->   MNIST (3 experiments, not 1).
-
-This review is intentionally harsh and reviewer-oriented. It focuses on what is currently unsupported, overstated, internally inconsistent, or likely to trigger serious pushback from a careful NeurIPS reviewer. It is based on `paper/paper.tex`, the package code under `src/bispectrum/`, the experiment code and artifacts under `paper/experiments/pcam/`, the benchmark code, and the literature/planning notes under `docs/`.
-
-## Top-line assessment
-
-The repository has real substance. The library itself is technically interesting, the octahedral inversion work looks real and nontrivial, and there is enough code to support a serious software paper.
-
-The paper, however, is currently much weaker than the repo. In its present form it reads like an ambitious submission outline with some strong ingredients, not like a finished research paper. The most serious problem is not lack of quality in the code; it is the mismatch between the strength of the claims and the level of support actually present in the manuscript and experimental artifacts.
-
-## Findings
-
-### 1. Critical: the manuscript is still a scaffold, not a reviewable paper
-
-The paper contains TODOs and placeholders in the abstract, introduction, preliminaries, theorem statement, benchmarks, experiments, conclusion, roadmap, acknowledgments, and appendices. At the same time, the surrounding prose already speaks as if the scientific and empirical case is complete.
-
-This is the single biggest issue. A reviewer will not interpret this as "some rough edges remain"; they will interpret it as "core parts of the submission are missing."
-
-Relevant examples from `paper/paper.tex`:
-
-- The abstract is still a `\todo` block with bullets instead of text.
-- The introduction opens with a `\todo`.
-- The steerable-to-Fourier theorem is completely unstated.
-- The benchmark figures are placeholders.
-- The main result tables still contain `X.XXX` placeholders.
-- The appendix sections that should carry proof, convergence, and benchmark detail are almost entirely TODOs.
-
-### 2. Critical: the current PCam artifacts do not support the main empirical framing
-
-The paper claims a five-model head-to-head comparison on PatchCamelyon with a shared backbone and three seeds per model. The saved results currently do not match that story.
-
-From `paper/experiments/pcam/pcam_results/` and `pcam_results_1pct/`:
-
-- `standard` has three seeds at `train_fraction = 1.0`
-- `norm` has two seeds at `train_fraction = 1.0`
-- `gate`, `fourier_elu`, and `bispectrum` only appear for seed 42
-- those `gate`, `fourier_elu`, and `bispectrum` runs are at `train_fraction = 0.01`, not `1.0`
-
-This means the currently saved artifacts are not a fair main-table comparison.
-
-Concrete examples:
-
-- `paper/experiments/pcam/pcam_results/standard_c8_seed42/results.json`
-- `paper/experiments/pcam/pcam_results/standard_c8_seed123/results.json`
-- `paper/experiments/pcam/pcam_results/standard_c8_seed456/results.json`
-- `paper/experiments/pcam/pcam_results/norm_c8_seed42/results.json`
-- `paper/experiments/pcam/pcam_results/norm_c8_seed123/results.json`
-- `paper/experiments/pcam/pcam_results/bispectrum_c8_seed42/results.json`
-- `paper/experiments/pcam/pcam_results/fourier_elu_c8_seed42/results.json`
-- `paper/experiments/pcam/pcam_results/gate_c8_seed42/results.json`
-
-The most dangerous consequence is that the paper currently implies a result that the checked-in artifacts do not actually support.
-
-### 3. Critical: "same backbone, only pooling differs" is false in the current experimental implementation
-
-The manuscript says the five models share the same backbone and differ only in the invariant pooling layer. The implementation in `paper/experiments/pcam/model.py` does not support that claim in a strict sense.
-
-Problems:
-
-- the gated model increases channels through `gate_factor = 2`
-- the bispectrum model expands features substantially before a `1x1` projection
-- the resulting parameter counts differ dramatically across models
-
-Saved parameter counts in the current results:
-
-- `standard`: `101,959`
-- `norm`: `790,717`
-- `bispectrum`: `919,609`
-- `gate`: `1,576,159`
-
-This is not a cosmetic discrepancy. It is a serious experimental confound. If a reviewer notices this, they can reasonably argue that any gains may be due to capacity differences rather than invariance quality or information preservation.
-
-Relevant files:
-
-- `paper/experiments/pcam/model.py`
-- `paper/experiments/pcam/pcam_results/*/results.json`
-
-### 4. High: the theory section overstates what is actually complete
-
-The internal theory notes are more careful than the paper draft. In particular:
-
-- the completeness guarantees apply to the proper bispectrum of the relevant spectral object
-- for steerable CNN outputs, completeness is not automatically a statement about the original image
-- the learned steerable coefficients are not automatically the same thing as canonical harmonic-analysis Fourier coefficients
-
-Your own note in `docs/per_pixel_steerable_bispectrum.tex` states the scope correctly: completeness applies to the per-pixel fiber signal, not directly to the original image. Your `docs/steerable_bispectrum_notes.md` goes further and explicitly warns that the usual bispectrum completeness/invertibility guarantees do not automatically transfer to learned steerable responses.
-
-The paper draft softens or drops those caveats when it matters most:
-
-- the abstract
-- the comparison table
-- the contribution list
-- the experimental motivation for "preserving all discriminative information"
-
-This creates a serious risk of overclaiming.
-
-Relevant files:
-
-- `paper/paper.tex`
-- `docs/per_pixel_steerable_bispectrum.tex`
-- `docs/steerable_bispectrum_notes.md`
-
-### 5. High: the steerable-to-Fourier bridge is presented as solved, but the manuscript still has not stated or proved it
-
-This is one of the core scientific contributions the paper wants to claim. But in the actual LaTeX:
-
-- the theorem statement is still a TODO
-- the appendix proof section is still a TODO
-- the discussion speaks as if the problem has already been fully resolved
-
-This creates an internal contradiction between the rhetoric of the paper and the actual content on the page.
-
-Your planning document `docs/neurips_steerable_bispectrum_plan.md` is still realistic about this risk. It explicitly marks the key theory path as open or partial. The draft paper is much more confident than the supporting material warrants.
-
-Relevant files:
-
-- `paper/paper.tex`
-- `docs/neurips_steerable_bispectrum_plan.md`
-
-### 6. High: the API and design claims in the paper are broader than what the package implements
-
-The package is real and useful, but the paper overstates its uniformity.
-
-#### `selective=False` is not generally available
-
-The design principles section implies that all modules are selective by default and that the full bispectrum is available through `selective=False` for debugging/comparison. That is not true across the library.
-
-Examples:
-
-- `src/bispectrum/cn_on_cn.py`: full mode exists
-- `src/bispectrum/torus_on_torus.py`: full mode exists
-- `src/bispectrum/dn_on_dn.py`: full mode raises `NotImplementedError`
-- `src/bispectrum/octa_on_octa.py`: full mode raises `NotImplementedError`
-- `src/bispectrum/so2_on_disk.py`: full mode raises `NotImplementedError`
-- `src/bispectrum/so3_on_s2.py`: ~~selective mode is reserved for future use; selective SO(3) is explicitly open~~ **RESOLVED:** selective mode is now fully implemented and tested
-
-#### The "uniform API contract" is too strong
-
-The paper says every module returns a complex tensor and that `index_map` maps to an irrep pair.
-
-That is not uniformly true:
-
-- `DnonDn` returns a real bispectrum
-- `SO3onS2.index_map` uses triples `(l1, l2, l)`
-- other modules use different tuple structures
-
-These are fixable wording issues, but if left as-is they invite easy reviewer nitpicks.
-
-Relevant files:
-
-- `paper/paper.tex`
-- `src/bispectrum/cn_on_cn.py`
-- `src/bispectrum/dn_on_dn.py`
-- `src/bispectrum/octa_on_octa.py`
-- `src/bispectrum/so2_on_disk.py`
-- `src/bispectrum/so3_on_s2.py`
-- `src/bispectrum/torus_on_torus.py`
-
-### 7. High: the benchmark script cited by the paper appears broken
-
-The paper points to `benchmarks/benchmark.py --paper`. But `benchmarks/benchmark.py` imports `SO2onD2`, while the package currently exports `SO2onDisk`.
-
-That means a reviewer trying to run the benchmark script could hit an import error immediately.
-
-This also reveals naming drift in the repo between `SO2onD2` and `SO2onDisk`, including in `docs/`.
-
-Relevant files:
-
-- `benchmarks/benchmark.py`
-- `src/bispectrum/__init__.py`
-- `docs/steerable_bispectrum_notes.md`
-- `docs/so2_on_d2_implementation_notes.md`
-
-### 8. High: the RotMNIST contrastive story looks unsupported in the current tree
-
-The abstract, contribution list, and experiments section all position the RotMNIST bispectral contrastive regularizer as a major empirical result.
-
-I could find the drafted section in `paper/paper.tex`, but I did not find matching experiment code or result artifacts in the repository. Right now, that makes the RotMNIST section read like a proposed experiment rather than a completed one.
-
-If this experiment exists elsewhere and is not yet committed, that is fine. But in the current repo snapshot, it is not supported.
-
-Relevant files:
-
-- `paper/paper.tex`
-- `paper/experiment_candidates.md`
-
-### 9. Medium: some factual claims are simply wrong or not evidenced in-tree
-
-#### "Minimal dependencies: only torch and numpy"
-
-This is false. The package depends on `torch-harmonics`.
-
-Relevant files:
-
-- `paper/paper.tex`
-- `pyproject.toml`
-- `src/bispectrum/so3_on_s2.py`
-
-#### "`>90%` test coverage"
-
-This may be true on Codecov, but it is not substantiated in the repo snapshot itself. The CI runs coverage and uploads it, but the checked-in evidence I saw is a badge, not a durable number.
-
-Relevant files:
-
-- `paper/paper.tex`
-- `README.md`
-- `.github/workflows/tests.yml`
-
-#### "First PyTorch library"
-
-This may well be true in practice, but as written it is a global novelty claim. Unless you have done an exhaustive software search, it should probably be scoped as "to our knowledge" or narrowed.
-
-### 10. Medium: the literature positioning is directionally good but underplays the strongest prior line
-
-The current draft cites Oreiller 2022, but your own notes make clear that the more complete prior line is:
-
-- Andrearczyk et al. 2019
-- Oreiller et al. 2020
-- Oreiller et al. 2022
-
-That matters because the novelty question is not "has anyone ever put bispectrum-like invariants into CNNs?" The harder and more honest novelty claim is closer to:
-
-- nobody has provided a practical selective multi-group PyTorch library of this scope
-- nobody has resolved the non-abelian steerable/Fourier mismatch in the form you want
-- nobody has dealt with the octahedral inversion issue in software
-
-That is still a strong story. But it is more defensible than implying the broader bispectral-CNN direction is new.
-
-Relevant files:
-
-- `paper/paper.tex`
-- `docs/steerable_bispectrum_notes.md`
-- `docs/sanbord_miolane_2025/sections/lit-review.tex`
-
-### 11. Medium: the paper is trying to be too many paper types at once
-
-At the moment the draft tries to simultaneously be:
-
-- a software paper
-- a theory paper
-- a benchmarking paper
-- an equivariant CNN empirical paper
-- a representation-theory bridge paper
-- a contrastive learning paper
-
-That breadth would already be hard to pull off in a polished manuscript. In the current state, it dilutes the strongest contributions.
-
-The strongest scientifically credible core, based on the repo, is:
-
-1. a serious PyTorch bispectrum library across several groups/domains
-2. a real octahedral inversion failure and fix
-3. a careful treatment of the steerable-to-Fourier gap, with exact scope
-
-Everything beyond that should be included only if it is actually complete and defensible.
-
-## Questions I would expect from reviewers
-
-These are the sorts of questions the current draft invites:
-
-- Is the steerable-to-Fourier bridge actually proved, or only conjectured / partially characterized?
-- Are the empirical gains due to better invariants, or simply larger parameter count?
-- Are the PCam comparisons apples-to-apples if some models only have 1% data runs saved?
-- Why should the reader trust claims about RotMNIST if the code/results are not visible?
-- In what exact sense is the invariant "complete" when used inside learned steerable networks?
-- Is the benchmark code actually runnable from a clean install?
-- How much of the paper is a software contribution vs a scientific contribution?
-
-## What is already strong
-
-This should not get lost in the critique.
-
-### The library looks real
-
-The package under `src/bispectrum/` is substantial. This is not a toy wrapper. It includes actual implementations for:
-
-- `CnonCn`
-- `DnonDn`
-- `TorusOnTorus`
-- `SO2onS1`
-- `SO2onDisk`
-- `SO3onS2`
-- `OctaonOcta`
-
-### The octahedral inversion story is genuinely interesting
-
-Among all the paper's claims, this is one of the most convincing and distinctive. The code and notes support a real phenomenon:
-
-- the selective octahedral bispectrum is fine
-- the naive bootstrap inversion fails because the continuous `SO(3)` ambiguity is not absorbed by the octahedral CG structure
-- a bootstrap + LM + multi-start correction is implemented
-
-This is exactly the kind of "nontrivial engineering plus mathematical insight" that can anchor a strong software paper.
-
-Relevant files:
-
-- `src/bispectrum/octa_on_octa.py`
-- `docs/inversion_gap_for_octahedral_group.md`
-
-### The package/repo quality is credible
-
-The repo supports a real software story:
-
-- packaging via `pyproject.toml`
-- installable package
-- tests
-- CI
-- coverage upload
-- MIT license
-
-That part of the paper can be made credible with relatively little effort.
-
-## Best scientific framing available from the current repo
-
-If I were optimizing for acceptance probability rather than maximal ambition, I would frame the contribution more tightly around:
-
-1. `bispectrum` is a practical PyTorch library implementing selective bispectra across multiple groups/domains.
-2. A key scientific contribution is the octahedral inversion failure analysis and correction.
-3. A second scientific contribution is a scoped and carefully stated bridge from steerable outputs to valid bispectral computation, with explicit assumptions and limitations.
-4. The experiments should only claim what the saved artifacts actually support.
-
-That is still a strong paper.
-
-## Recommendation
-
-The fastest path to a strong submission is probably not to add more prose. It is to narrow the claims until every major sentence is directly defensible from:
-
-- the code
-- the experiments
-- the theorem statements actually written in the paper
-
-Right now the repo is ahead of the manuscript. The paper needs to catch up by becoming more precise, more scoped, and more honest about what is complete versus what is still planned.
-
-## File references used in this review
-
-- `paper/paper.tex`
-- `paper/references.bib`
-- `paper/experiments/pcam/README.md`
-- `paper/experiments/pcam/model.py`
-- `paper/experiments/pcam/train.py`
-- `paper/experiments/pcam/pcam_results/*/results.json`
-- `paper/experiments/pcam/pcam_results_1pct/*/results.json`
-- `src/bispectrum/__init__.py`
-- `src/bispectrum/cn_on_cn.py`
-- `src/bispectrum/dn_on_dn.py`
-- `src/bispectrum/octa_on_octa.py`
-- `src/bispectrum/so2_on_disk.py`
-- `src/bispectrum/so3_on_s2.py`
-- `src/bispectrum/torus_on_torus.py`
-- `benchmarks/benchmark.py`
-- `pyproject.toml`
-- `README.md`
-- `.github/workflows/tests.yml`
-- `docs/steerable_bispectrum_notes.md`
-- `docs/selective_so3.md`
-- `docs/inversion_gap_for_octahedral_group.md`
-- `docs/per_pixel_steerable_bispectrum.tex`
-- `docs/neurips_steerable_bispectrum_plan.md`
-- `docs/sanbord_miolane_2025/sections/lit-review.tex`
diff --git a/paper/seed_rank_analysis.py b/paper/seed_rank_analysis.py
deleted file mode 100644
index 0e6ad3e..0000000
--- a/paper/seed_rank_analysis.py
+++ /dev/null
@@ -1,310 +0,0 @@
-"""Systematic analysis of seed-stage Jacobian rank for the SO(3) bispectrum.
-
-Tests which degree-4 augmentation invariants can fix the rank deficiency.
-"""
-import sys
-sys.path.insert(0, '../src')
-
-import torch
-from bispectrum.so3_on_s2 import (
-    _build_selective_index_map, _bispectrum_entry, load_cg_matrices,
-)
-from itertools import combinations
-
-LMAX = 6
-CG_DATA = load_cg_matrices(LMAX)
-
-A0 = 1.5
-C_GAUGE = 1.2
-
-
-def make_gauge_fixed_params(n_seed_params: int = 11, seed: int = 42) -> torch.Tensor:
-    torch.manual_seed(seed)
-    return torch.randn(n_seed_params, dtype=torch.float64)
-
-
-def params_to_f23(p: torch.Tensor) -> dict[int, torch.Tensor]:
-    """Build F_0 (fixed), F_1 (gauge-fixed), F_2, F_3 from 11 free params."""
-    fc: dict[int, torch.Tensor] = {}
-    fc[0] = torch.complex(
-        torch.tensor([[A0]], dtype=torch.float64),
-        torch.zeros(1, 1, dtype=torch.float64),
-    )
-    fc[1] = torch.complex(
-        torch.tensor([[0.0, C_GAUGE, 0.0]], dtype=torch.float64),
-        torch.zeros(1, 3, dtype=torch.float64),
-    )
-    idx = 0
-    for l_val in [2, 3]:
-        neg_parts: list[torch.Tensor] = []
-        pos_parts: list[torch.Tensor] = []
-        for m in range(1, l_val + 1):
-            if l_val == 2 and m == 1:
-                re = p[idx]; idx += 1
-                im = torch.zeros_like(re)
-            else:
-                re = p[idx]; im = p[idx + 1]; idx += 2
-            pos_parts.append(torch.complex(re, im))
-            neg_parts.append(((-1.0) ** m) * torch.complex(re, -im))
-        m0 = torch.complex(p[idx], torch.zeros_like(p[idx])); idx += 1
-        coeffs = list(reversed(neg_parts)) + [m0] + pos_parts
-        fc[l_val] = torch.stack(coeffs).unsqueeze(0)
-    return fc
-
-
-def cg_project(fc: dict[int, torch.Tensor], l1: int, l2: int, l_out: int) -> torch.Tensor:
-    """Compute (F_{l1} ⊗ F_{l2})|_{l_out} as a (2l_out+1) vector."""
-    f_l1 = fc[l1]
-    f_l2 = fc[l2]
-    outer = torch.einsum('bi,bj->bij', f_l1, f_l2).reshape(1, -1)
-    cg = CG_DATA[(l1, l2)].to(dtype=outer.dtype)
-    transformed = outer @ cg  # shape (1, sum of 2l+1 for valid l)
-    start = 0
-    for lv in range(abs(l1 - l2), l1 + l2 + 1):
-        sz = 2 * lv + 1
-        if lv == l_out:
-            return transformed[0, start:start + sz]
-        start += sz
-    raise ValueError(f"l_out={l_out} not in coupling range of {l1}⊗{l2}")
-
-
-def eval_scalar_bispectrum(p: torch.Tensor, triples: list[tuple[int, int, int]]) -> torch.Tensor:
-    """Evaluate scalar bispectral entries using parity-correct component."""
-    fc = params_to_f23(p)
-    vals = []
-    for l1, l2, lv in triples:
-        beta = _bispectrum_entry(fc, l1, l2, lv, CG_DATA[(l1, l2)])[0]
-        parity = (l1 + l2 + lv) % 2
-        vals.append(beta.imag if parity else beta.real)
-    return torch.stack(vals)
-
-
-def eval_cg_power(p: torch.Tensor, quadruples: list[tuple[int, int, int, int]]) -> torch.Tensor:
-    """Evaluate CG power spectrum entries: ||(F_{l1}⊗F_{l2})|_l||^2.
-
-    Also supports cross terms: <(F_{l1}⊗F_{l2})|_l, (F_{l3}⊗F_{l4})|_l>.
-    quadruples: (l1, l2, l3_or_l, l4_or_sentinel)
-    If l4 == -1: power entry ||(F_{l1}⊗F_{l2})|_{l3}||^2
-    Otherwise: cross <(F_{l1}⊗F_{l2})|_{l4}, (F_{l3}⊗F_{l2})|_{l4}>... 
-    """
-    fc = params_to_f23(p)
-    vals = []
-    for entry in quadruples:
-        if len(entry) == 3:
-            l1, l2, l_out = entry
-            proj = cg_project(fc, l1, l2, l_out)
-            vals.append(torch.sum(proj.real ** 2 + proj.imag ** 2))
-        else:
-            l1a, l2a, l1b, l2b, l_out = entry
-            proj_a = cg_project(fc, l1a, l2a, l_out)
-            proj_b = cg_project(fc, l1b, l2b, l_out)
-            val = torch.sum(proj_a * torch.conj(proj_b))
-            parity = (l1a + l2a + l1b + l2b) % 2
-            vals.append(val.imag if parity else val.real)
-    return torch.stack(vals)
-
-
-def compute_rank(eval_fn, params: torch.Tensor, tol_factor: float = 1e-8) -> tuple[int, torch.Tensor]:
-    J = torch.autograd.functional.jacobian(eval_fn, params)
-    sv = torch.linalg.svdvals(J)
-    tol = tol_factor * sv[0].item() if sv[0].item() > 0 else 1e-15
-    rank = int((sv > tol).sum().item())
-    return rank, sv
-
-
-def main():
-    params = make_gauge_fixed_params(seed=42)
-    n_unknowns = 11  # 4 (F_2) + 7 (F_3)
-
-    idx_map = _build_selective_index_map(LMAX)
-    seed_triples = [(l1, l2, lv) for l1, l2, lv in idx_map if max(l1, l2, lv) <= 3 and max(l1, l2, lv) >= 2]
-
-    print("=" * 70)
-    print("PART 1: Current scalar bispectrum seed rank")
-    print("=" * 70)
-
-    def eval_seed(p):
-        return eval_scalar_bispectrum(p, seed_triples)
-
-    rank, sv = compute_rank(eval_seed, params)
-    print(f"Seed triples: {seed_triples}")
-    print(f"Count: {len(seed_triples)}, Rank: {rank}, Unknowns: {n_unknowns}")
-    print(f"SV: {[f'{s:.3e}' for s in sv.tolist()]}")
-
-    nonzero_triples = []
-    for l1, l2, lv in seed_triples:
-        fc_test = params_to_f23(params)
-        beta = _bispectrum_entry(fc_test, l1, l2, lv, CG_DATA[(l1, l2)])[0]
-        parity = (l1 + l2 + lv) % 2
-        val = beta.imag.item() if parity else beta.real.item()
-        status = "ZERO" if abs(val) < 1e-10 else "ok"
-        print(f"  ({l1},{l2},{lv}): parity={parity}, val={val:.4e} [{status}]")
-        if abs(val) >= 1e-10:
-            nonzero_triples.append((l1, l2, lv))
-
-    def eval_nonzero(p):
-        return eval_scalar_bispectrum(p, nonzero_triples)
-    rank_nz, _ = compute_rank(eval_nonzero, params)
-    print(f"\nNonzero-only: {len(nonzero_triples)} entries, rank={rank_nz}")
-
-    print("\n" + "=" * 70)
-    print("PART 2: Available CG power spectrum augmentations")
-    print("=" * 70)
-
-    cg_power_candidates = []
-    for l1 in range(4):
-        for l2 in range(l1, 4):
-            for l_out in range(abs(l1 - l2), min(l1 + l2, LMAX) + 1):
-                if l1 == 0 and l2 == l_out:
-                    continue  # this is just ||F_{l2}||^2, already have power spectrum
-                if max(l1, l2) < 2:
-                    continue  # only F_0, F_1 — already known, no new info
-                cg_power_candidates.append((l1, l2, l_out))
-
-    print(f"CG power spectrum candidates: {len(cg_power_candidates)}")
-    for cand in cg_power_candidates:
-        l1, l2, l_out = cand
-        def eval_single(p, c=cand):
-            return eval_cg_power(p, [c])
-        try:
-            fc_t = params_to_f23(params)
-            proj = cg_project(fc_t, l1, l2, l_out)
-            val = torch.sum(proj.real**2 + proj.imag**2).item()
-        except Exception:
-            val = float('nan')
-        print(f"  ||(F_{l1}⊗F_{l2})|_{l_out}||^2 = {val:.6f}")
-
-    print("\n" + "=" * 70)
-    print("PART 3: Greedy search for minimal augmentation set")
-    print("=" * 70)
-
-    best_rank = rank_nz
-    best_set: list[tuple[int, int, int]] = []
-    remaining = list(cg_power_candidates)
-
-    for round_num in range(n_unknowns - best_rank):
-        best_cand = None
-        best_new_rank = best_rank
-        for cand in remaining:
-            def eval_combined(p, extras=[cand]):
-                v1 = eval_scalar_bispectrum(p, nonzero_triples)
-                v2 = eval_cg_power(p, best_set + extras)
-                return torch.cat([v1, v2])
-            try:
-                r, _ = compute_rank(eval_combined, params)
-            except Exception:
-                continue
-            if r > best_new_rank:
-                best_new_rank = r
-                best_cand = cand
-        if best_cand is None:
-            print(f"  Round {round_num+1}: no improvement found, stopping")
-            break
-        best_set.append(best_cand)
-        remaining.remove(best_cand)
-        best_rank = best_new_rank
-        l1, l2, lo = best_cand
-        print(f"  Round {round_num+1}: added ||(F_{l1}⊗F_{l2})|_{lo}||^2 → rank={best_rank}")
-        if best_rank >= n_unknowns:
-            print(f"  FULL RANK ACHIEVED with {len(best_set)} augmentations!")
-            break
-
-    print(f"\nFinal augmentation set ({len(best_set)} entries):")
-    for l1, l2, lo in best_set:
-        print(f"  ||(F_{l1}⊗F_{l2})|_{lo}||^2")
-
-    print(f"\nTotal entries: {len(nonzero_triples)} scalar bispec + {len(best_set)} CG power = {len(nonzero_triples)+len(best_set)}")
-    print(f"Final rank: {best_rank} / {n_unknowns} unknowns")
-
-    print("\n" + "=" * 70)
-    print("PART 4: Cross-term augmentations (degree-4 bilinear)")
-    print("=" * 70)
-
-    cross_candidates = []
-    for l1a in range(4):
-        for l2a in range(l1a, 4):
-            for l1b in range(4):
-                for l2b in range(l1b, 4):
-                    if (l1a, l2a) >= (l1b, l2b):
-                        continue
-                    for l_out in range(max(abs(l1a-l2a), abs(l1b-l2b)),
-                                       min(l1a+l2a, l1b+l2b, LMAX)+1):
-                        if l_out not in range(abs(l1a-l2a), l1a+l2a+1):
-                            continue
-                        if l_out not in range(abs(l1b-l2b), l1b+l2b+1):
-                            continue
-                        if max(l1a, l2a, l1b, l2b) < 2:
-                            continue
-                        cross_candidates.append((l1a, l2a, l1b, l2b, l_out))
-
-    print(f"Cross-term candidates: {len(cross_candidates)}")
-
-    best_rank_2 = rank_nz
-    best_set_2: list = []
-    remaining_all = list(cg_power_candidates) + [(c[0], c[1], c[4]) if len(c)==5 else c for c in []] 
-
-    all_candidates = [(c, 'power') for c in cg_power_candidates] + [(c, 'cross') for c in cross_candidates]
-
-    for round_num in range(n_unknowns - best_rank_2):
-        best_cand = None
-        best_type = None
-        best_new_rank = best_rank_2
-        for cand, ctype in all_candidates:
-            if cand in [x[0] for x in best_set_2]:
-                continue
-            def eval_combined(p, extra_cand=cand, extra_type=ctype):
-                v1 = eval_scalar_bispectrum(p, nonzero_triples)
-                power_entries = [x[0] for x in best_set_2 if x[1] == 'power']
-                cross_entries = [x[0] for x in best_set_2 if x[1] == 'cross']
-                if extra_type == 'power':
-                    power_entries = power_entries + [extra_cand]
-                else:
-                    cross_entries = cross_entries + [extra_cand]
-                parts = [v1]
-                if power_entries:
-                    parts.append(eval_cg_power(p, power_entries))
-                if cross_entries:
-                    parts.append(eval_cg_power(p, cross_entries))
-                return torch.cat(parts)
-            try:
-                r, _ = compute_rank(eval_combined, params)
-            except Exception:
-                continue
-            if r > best_new_rank:
-                best_new_rank = r
-                best_cand = cand
-                best_type = ctype
-        if best_cand is None:
-            print(f"  Round {round_num+1}: no improvement, stopping")
-            break
-        best_set_2.append((best_cand, best_type))
-        best_rank_2 = best_new_rank
-        print(f"  Round {round_num+1}: added {best_type} {best_cand} → rank={best_rank_2}")
-        if best_rank_2 >= n_unknowns:
-            print(f"  FULL RANK ACHIEVED!")
-            break
-
-    print(f"\nFinal combined set ({len(best_set_2)} entries):")
-    for cand, ctype in best_set_2:
-        print(f"  [{ctype}] {cand}")
-    print(f"Final rank: {best_rank_2} / {n_unknowns}")
-
-    print("\n" + "=" * 70)
-    print("PART 5: Verify at multiple random seeds")
-    print("=" * 70)
-    final_augmentations = best_set if best_rank >= n_unknowns else [x[0] for x in best_set_2 if x[1] == 'power']
-    for seed in [42, 123, 999, 7, 2024]:
-        p = make_gauge_fixed_params(seed=seed)
-        def eval_final(p_inner):
-            v1 = eval_scalar_bispectrum(p_inner, nonzero_triples)
-            if final_augmentations:
-                v2 = eval_cg_power(p_inner, final_augmentations)
-                return torch.cat([v1, v2])
-            return v1
-        r, sv = compute_rank(eval_final, p)
-        gap = sv[r-1].item() / max(sv[min(r, len(sv)-1)].item(), 1e-20)
-        print(f"  seed={seed}: rank={r}, gap={gap:.2e}")
-
-
-if __name__ == "__main__":
-    main()
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-\documentclass[10pt,aspectratio=169]{beamer}
-
-\usetheme{metropolis}
-\usepackage{amsmath,amssymb,amsfonts}
-\usepackage{booktabs}
-\usepackage{xcolor}
-\usepackage{tikz}
-\usepackage{listings}
-
-\definecolor{accent}{HTML}{1F6FEB}
-\definecolor{warn}{HTML}{C53030}
-\definecolor{ok}{HTML}{2F855A}
-
-\setbeamercolor{alerted text}{fg=warn}
-\setbeamercolor{frametitle}{bg=accent}
-
-\newcommand{\R}{\mathbb{R}}
-\newcommand{\C}{\mathbb{C}}
-\newcommand{\SO}{\mathrm{SO}}
-\newcommand{\OG}{\mathrm{O}}
-\newcommand{\im}{\operatorname{Im}}
-\newcommand{\F}{\mathbf{F}}
-\newcommand{\bispec}{\beta}
-\DeclareMathOperator{\sgn}{sgn}
-
-\title{\texttt{bispectrum}: Selective $G$-Bispectra Made Practical}
-\subtitle{Walkthrough for co-authors}
-\author{Johan Mathe \and Simon Mataigne \and Adele Myers Lantow \and Nina Miolane}
-\date{\today}
-
-\begin{document}
-
-\maketitle
-
-\begin{frame}{What is this paper?}
-\begin{itemize}
-  \item PyTorch library + two theoretical contributions wrapped around it.
-  \item \textbf{Library:} 7 group/domain pairs as drop-in \texttt{nn.Module}s with
-        a uniform \texttt{\{Group\}on\{Domain\}} naming convention.
-  \item \textbf{Theory 1:} Augmented selective $\SO(3)$ bispectrum on $S^2$ with
-        $\Theta(L^2)$ entries (vs.\ the standard $\Theta(L^3)$ full bispectrum).
-  \item \textbf{Theory 2:} Diagnosis (and partial workaround) of why the
-        Mataigne et al.\ (2024) bootstrap inversion fails on the
-        octahedral group~$O$.
-  \item \textbf{Empirical:} PCam ($C_8$), OrganMNIST3D ($O$), Spherical MNIST ($\SO(3)$).
-\end{itemize}
-\vspace{0.4em}
-Plan for this deck:
-\begin{enumerate}
-  \item Library and its optimizations.
-  \item Experiments and what they don't show.
-  \item Proofs (focus on $\SO(3)$/$S^2$), the history, the weaknesses.
-  \item Openings + work that's left.
-\end{enumerate}
-\end{frame}
-
-\section{Part 1 --- The library and its optimizations}
-
-\begin{frame}{Modules, in one slide}
-\begin{table}\centering\small
-\begin{tabular}{lll}
-\toprule
-Class & Group / Domain & Output \\
-\midrule
-\texttt{CnonCn} & $C_n$ on $C_n$ & selective $O(n)$ + full $O(n^2)$ \\
-\texttt{SO2onS1} & $\SO(2)$ on $S^1$ & wraps \texttt{CnonCn} \\
-\texttt{TorusOnTorus} & $\prod C_{n_l}$ on torus & selective $O(|G|)$ + full \\
-\texttt{DnonDn} & $D_n$ on $D_n$ & selective \\
-\texttt{SO2onDisk} & $\SO(2)$ on disk & selective (Myers \& Miolane 2025) \\
-\texttt{OctaonOcta} & octahedral $O$ on $O$ & selective (172 scalars) \\
-\texttt{\textbf{SO3onS2}} & $\SO(3)$ on $S^2$ & \alert{augmented selective $\Theta(L^2)$} \\
-\bottomrule
-\end{tabular}
-\end{table}
-\vspace{0.4em}
-Design rules:
-\begin{itemize}
-  \item Real-valued spatial input goes in. Module owns the FT (SHT, DFT, DHT).
-  \item Selective by default; \texttt{selective=False} for debugging.
-  \item Complex output (\texttt{complex64}); CG / Bessel / DFT kernels are
-        non-learnable buffers $\Rightarrow$ \texttt{.to(device)} is free.
-  \item No \texttt{escnn}/\texttt{e3nn} dependency. CG built from Racah +
-        log-factorials, cached on disk.
-\end{itemize}
-\end{frame}
-
-\begin{frame}{Optimizations actually in the code}
-\begin{itemize}
-  \item \textbf{Reduced CG matrices, parallel build.}
-        For \texttt{SO3onS2}, only the nonzero CG slices are kept; build
-        across $(\ell_1,\ell_2)$ pairs is parallelised at construction time
-        and pickled to \texttt{\textasciitilde/.cache/bispectrum/}.
-  \item \textbf{Sparse CG path} for high-$L$ single-pair contractions.
-  \item \textbf{Batch-native forward.}
-        Abelian modules use \texttt{torch.fft.fftn}; non-abelian use
-        \texttt{einsum}/\texttt{bmm} with no Python-level loops over $(\ell_1,\ell_2)$.
-  \item \textbf{Inversion via forward-mode AD + Levenberg--Marquardt}
-        (\texttt{torch.func.jacfwd} composed with \texttt{torch.vmap}) for
-        the octahedral case --- gives a damped least-squares step that is
-        a single linear solve per iteration.
-\end{itemize}
-\end{frame}
-
-\begin{frame}{Benchmark numbers (H100, batch 16)}
-\centering\small
-\begin{tabular}{llrrrrr}
-\toprule
-Module & $G$ & Sel.\ coefs & Full coefs & Fwd sel.\ (ms) & Fwd full (ms) & Inv (ms) \\
-\midrule
-\texttt{CnonCn} & $C_{128}$ & 128 & 8\,256 & 0.15 & 9.17 & 8.6 \\
-\texttt{TorusOnTorus} & $C_{32}{\times}C_{32}$ & 1\,024 & 524\,800 & 0.08 & 0.71 & 68 \\
-\texttt{DnonDn} & $D_{32}$ & 245 & --- & 2.52 & --- & 42 \\
-\texttt{SO2onDisk} & $\SO(2)$ & 411 & --- & 1.94 & --- & 18 \\
-\texttt{SO3onS2} & $\SO(3)$, $L{=}16$ & 430 & --- & 26.5 & --- & --- \\
-\texttt{OctaonOcta} & $O$ & 172 & --- & 0.70 & --- & 2\,170 \\
-\bottomrule
-\end{tabular}
-\vspace{0.6em}
-\begin{itemize}
-  \item Sub-millisecond forward for finite groups $\Rightarrow$ negligible
-        overhead vs.\ a CNN backbone.
-  \item \texttt{SO3onS2}: 26.5\,ms is dominated by the SHT
-        (\texttt{torch\_harmonics}, Bonev et al.\ 2023), not by CG
-        contractions.
-  \item Octahedral inversion is $\sim$2\,s and \alert{converges to wrong basin}
-        on most signals (5.7\% success). This is the topic of Part 3.
-\end{itemize}
-\end{frame}
-
-\section{Part 2 --- Experiments and their weaknesses}
-
-\begin{frame}{Experimental scope}
-Three benchmarks, one per group regime:
-\begin{enumerate}
-  \item \textbf{PCam} (2D histopathology, 327K patches): \texttt{CnonCn}$(n{=}8)$
-        as terminal invariant pool on a $C_8$-equivariant DenseNet.
-  \item \textbf{OrganMNIST3D} (1\,742 abdominal CT volumes): \texttt{OctaonOcta}
-        on an $O$-equivariant 3D ResNet with exact voxel-permutation kernels.
-  \item \textbf{Spherical MNIST}: \texttt{SO3onS2} at $\ell_{\max}{=}15$,
-        log-magnitudes $\to$ 3-layer MLP.
-\end{enumerate}
-\vspace{0.4em}
-What we test in each setting:
-\begin{itemize}
-  \item Numerical $G$-invariance ($\sigma_\mathrm{rot}$).
-  \item Completeness vs.\ lossy pooling (norm, max, gate, Fourier-ELU).
-  \item Capacity / data sweeps (Pareto curves).
-\end{itemize}
-\end{frame}
-
-\begin{frame}{Headline numbers}
-\centering\small
-\textbf{PCam} (matched $\sim$100K params, 10\% data, 3 seeds):
-\begin{tabular}{lcc}
-\toprule
-Pool & AUC & Acc \\
-\midrule
-Standard (augmented) & .896 & .826 \\
-Bispectrum (\texttt{CnonCn}) & .941 & \textbf{.872} \\
-Fourier-ELU & \textbf{.945} & .855 \\
-NormReLU & .942 & .858 \\
-\bottomrule
-\end{tabular}
-\vspace{0.6em}
-
-\textbf{OrganMNIST3D}:
-\begin{tabular}{lccc}
-\toprule
-Model & Params & Test Acc & $\sigma_\mathrm{rot}$ \\
-\midrule
-$O$-equiv + Norm pool & 375K & .568 & 0 \\
-$O$-equiv + Max pool & 374K & .730 & 0 \\
-$O$-equiv + Bispectrum & 463K & .726 & $5{\times}10^{-4}$ \\
-\bottomrule
-\end{tabular}
-\vspace{0.6em}
-
-\textbf{Spherical MNIST} (NR/NR / R/R / NR/R):
-\begin{tabular}{lccc}
-\toprule
-Model & NR/NR & R/R & NR/R \\
-\midrule
-Power spec.\ + MLP (166K) & .794 & .794 & .792 \\
-Bispectrum + MLP (232K) & \textbf{.950} & \textbf{.949} & \textbf{.951} \\
-Cohen spherical CNN (ref) & .96 & .95 & .94 \\
-\bottomrule
-\end{tabular}
-\end{frame}
-
-\begin{frame}{The story they tell}
-\begin{itemize}
-  \item \textbf{Invariance is real and tight.} Bispectrum $\sigma_\mathrm{rot}
-        \sim 10^{-4}$; standard CNN drops 99\%$\to$14\% under NR/R.
-  \item \textbf{Completeness $>$ incomplete pools at low data.}
-        At 1\% PCam, bispectrum leads (.912 vs .873 NormReLU, .876 Fourier-ELU).
-        At 5\% OrganMNIST3D, .332 vs .189 for max pool.
-  \item \textbf{Completeness $\to$ no advantage at high capacity / data.}
-        Max pool catches up at 6M params on OrganMNIST3D.
-        Cohen spherical CNN beats us by 1--2pp.
-\end{itemize}
-\end{frame}
-
-\begin{frame}{Weaknesses of the experimental section}
-\small
-This is where reviewers will push back. Be honest about it:
-\begin{itemize}
-  \item \alert{No comparison vs.\ tensor-product architectures}
-        (e3nn, MACE) or wavelet scattering. They target different
-        input assumptions but it's the obvious missing baseline.
-  \item \alert{Spherical MNIST is not a discriminating task} for SO(3)
-        invariance any more --- everyone gets $\sim$95\%. We win on
-        \emph{tightness of invariance} ($10^{-4}$ vs $10^{-2}$), not raw acc.
-  \item \alert{OrganMNIST3D is small} (1.7K samples, 11 classes).
-        Variance per seed is large; the .726 vs .730 max-pool gap is
-        well within noise.
-  \item \alert{PCam: only one architecture family} (DenseNet, $C_8$).
-        We don't show the bispectrum is robust across backbones.
-  \item \alert{Octahedral inversion never used downstream.} The 5.7\% success
-        rate makes it useless as a feature for any task --- so the inversion
-        story is theoretical, not practical.
-  \item Three seeds per config. Conventional in this venue, but on
-        OrganMNIST3D we'd want more.
-  \item No dataset where the bispectrum \emph{loses} cleanly, which would
-        actually \emph{strengthen} the ``when not to use it'' paragraph.
-\end{itemize}
-\end{frame}
-
-\section{Part 3 --- Proofs and their weaknesses}
-
-\begin{frame}{What proofs the paper makes}
-\begin{enumerate}
-  \item \textbf{$\SO(3)$ on $S^2$ ($L \le 100$).}
-        Augmented selective bispectrum is a complete $\SO(3)$-invariant
-        on a Zariski-open dense subset of $V_L$. Output size $\Theta(L^2)$,
-        matches lower bound $(L{+}1)^2 - 3$ up to a constant.
-  \item \textbf{Octahedral diagnosis, no positive proof.}
-        Shows the bootstrap inversion of Mataigne et al.\ (2024)
-        \emph{cannot} work on $O$: cross-block leakage of order 1 in
-        $C_{11}^\top (Q\otimes Q) C_{11}$ for $Q \in \SO(3)\setminus O$.
-\end{enumerate}
-\vspace{0.6em}
-The first one is what most of the appendix is. We focus the rest of the
-deck there.
-\end{frame}
-
-\begin{frame}{The $\SO(3)$ on $S^2$ result --- statement}
-\begin{block}{Theorem (informal)}
-For $4 \le L \le 100$, the augmented invariant
-\[
-  \Phi_\mathrm{aug}(f)
-  = \bigl\{\beta_{\ell_1,\ell_2,\ell}\bigr\}_{\mathcal S_\beta}
-    \cup \bigl\{P_{\ell_1,\ell_2,\ell}\bigr\}_{\mathcal S_P}
-    \cup \bigl\{\text{full bispectrum at } L_0=4\bigr\}
-\]
-with $|\Phi_\mathrm{aug}| = \Theta(L^2)$, separates $\SO(3)$-orbits on
-real band-limited signals satisfying mild genericity (G1)--(G3).
-\end{block}
-\vspace{0.4em}
-Pieces:
-\begin{itemize}
-  \item $\beta_{\ell_1,\ell_2,\ell}$: scalar bispectral entries (degree~3).
-  \item $P_{\ell_1,\ell_2,\ell} := \|(\F_{\ell_1}\otimes \F_{\ell_2})|_\ell\|^2$:
-        \alert{CG power spectrum} (degree~4, real, $T_R$-invariant).
-  \item Full bispectrum at $L_0{=}4$: 30 extra entries, asymptotically free.
-\end{itemize}
-\end{frame}
-
-\begin{frame}{Proof skeleton (current version)}
-\begin{enumerate}
-  \item \textbf{Gauge fix} $\F_1$ to $z$-axis, $\im(a_2^1) = 0$ \quad (3 DoF, lemma~\texttt{app-gauge}).
-  \item \textbf{Recover $\F_0$, $\F_1$} from $\beta_{0,0,0}$, $\beta_{0,1,1}$.
-  \item \textbf{Seed (degrees 0--4) via Kakarala 1992/2012 Thm 7.}
-        Adjoin the full bispectrum at $L_0{=}4$. Two independent arguments:
-        \begin{itemize}\small
-          \item Representation-theoretic (Kakarala / Iwahori--Sugiura).
-          \item Computer-algebra (Bandeira et al.\ 2017, Gröbner, $L_0 \le 15$).
-        \end{itemize}
-  \item \textbf{Bootstrap for $\ell \ge 5$.}
-        Linear system $A_\ell \F_\ell = b$ with
-        $\mathrm{rank}_{\C}(A_\ell) = 2\ell+1$. Verified at deterministic
-        witness (seed 42) for $\ell \in [4,100]$.
-  \item Real rank of full augmented per-degree system $= 2\ell+1$
-        (verified $\ell \le 30$). \alert{CG power entries are essential at every $\ell$.}
-\end{enumerate}
-\end{frame}
-
-\begin{frame}{Why CG power is needed at all}
-For \emph{complex} signals: scalar bispectrum alone has full
-complex rank $2\ell+1$ at every $\ell$.
-\vspace{0.4em}
-
-For \emph{real} signals: the constraint
-$a_\ell^{-m} = (-1)^m \overline{a_\ell^m}$ creates algebraic
-dependencies in $A_\ell$ that \alert{cannot be fixed by choosing different triples}.
-
-\centering\small
-\begin{tabular}{ccccc}
-\toprule
-$\ell$ & complex rank & real rank (linear) & + self-coupling & + CG power \\
-\midrule
-4 & 9 & 4 & 6 & \textbf{9} \\
-5 & 11 & 4 & 6 & \textbf{11} \\
-6 & 13 & 8 & 11 & \textbf{13} \\
-\bottomrule
-\end{tabular}
-\vspace{0.6em}
-
-The Zariski-density argument (``real $\subset \C$ is dense'') \alert{fails} because
-the reality constraint involves conjugation, which is not algebraic over $\C$.
-This was discovered the hard way (research log, 2026-04-13).
-\end{frame}
-
-\begin{frame}{Parity, $T_R$, and why $\beta_{2,3,4}$ is the hero}
-The azimuthal reflection $T_R: a_\ell^m \mapsto (-1)^m a_\ell^{-m}$
-is in $\OG(3) \setminus \SO(3)$.
-\begin{itemize}
-  \item For real signals: $\overline{\beta} = (-1)^{\ell_1+\ell_2+\ell}\,\beta$.
-  \item Even parity $\Rightarrow$ $\beta \in \R$, $T_R$-invariant.
-  \item Odd parity $\Rightarrow$ $\beta \in i\R$, $\im\beta$ flips sign under $T_R$.
-\end{itemize}
-\vspace{0.4em}
-Vanishing pattern (proved symbolically): odd-parity $\beta_{\ell_1,\ell_2,\ell}$
-\alert{vanishes identically} whenever any two indices coincide.
-
-$\Rightarrow$ The smallest nonzero odd-parity entry is
-\[
-  \boxed{\beta_{2,3,4}}, \quad \ell_1+\ell_2+\ell = 9, \text{ all distinct.}
-\]
-At seed-42 witness, $\im(\beta_{2,3,4}) \approx 3.26$, flips sign under $T_R$.
-This is why $L_0 = 4$ is the magic number: any seed cutoff $\ge 4$
-contains $\beta_{2,3,4}$, which kills the $T_R$ ambiguity.
-\end{frame}
-
-\begin{frame}{Limitations of the $\SO(3)$/$S^2$ proof}
-What's actually \alert{not} structural / not proved uniformly:
-\begin{itemize}
-  \item \textbf{Bootstrap rank is per-$\ell$, not uniform-in-$\ell$.}
-        We verify $\det A_\ell \ne 0$ at one witness for each $\ell \in [4,100]$.
-        Sound for any fixed $L \le 100$, but a single closed-form Wigner-3j
-        identity uniform in $\ell$ remains open (Wigner-3j non-vanishing).
-  \item \textbf{Real per-degree rank certificate} only goes up to $\ell \le 30$
-        (was bottlenecked by sympy time at the time of writing).
-  \item \textbf{The Bandeira et al.\ symbolic certificate} bounds $L_0 \le 15$.
-        For our use $L_0 = 4$ this is fine, but the structural ceiling exists.
-  \item \textbf{Genericity set not characterised explicitly.}
-        We know it's the complement of a measure-zero variety; we don't
-        write the variety down.
-  \item \textbf{Numerical conditioning.}
-        $\kappa(A_\ell)$ at the witness is $10^1$--$10^4$. No bound in terms
-        of spectral decay of $f$.
-  \item \textbf{No inversion algorithm} on $S^2$: forward-only.
-\end{itemize}
-\end{frame}
-
-\begin{frame}{Octahedral proof: do we even need it?}
-\small
-\textbf{Honest answer: probably no, in the current form.}
-\begin{itemize}
-  \item The selective $O$-bispectrum \emph{is} complete (172 coefs determine
-        $f$ up to $O$-action) --- this is from
-        Mataigne et al.\ (2024), not new.
-  \item Our contribution: a clean diagnosis that the
-        \emph{bootstrap inversion algorithm} cannot work.
-        Cause: $C_{11}^\top (Q\otimes Q) C_{11}$ has off-diagonal blocks of
-        norm $\sim 0.79$ for generic $Q \in \SO(3) \setminus O$.
-        The CG matrices block-diagonalise $\rho_1\otimes\rho_1$ only for
-        $g \in O$, not for $g \in \SO(3)$.
-  \item LM workaround: bispectral residual goes to $5{\times}10^{-5}$ in
-        $<$50 iterations, but \alert{signal recovery rate is 5.7\%}.
-\end{itemize}
-\vspace{0.4em}
-\textbf{So, options:}
-\begin{enumerate}
-  \item Keep it as a cautionary tale (current draft).
-  \item Drop it and free $\sim$2 pages for tighter $\SO(3)$/$S^2$ work.
-  \item Replace with a positive result: prove that \emph{no} bootstrap
-        inversion can succeed on $O$ (i.e.\ formalise the failure as a
-        no-go theorem).
-\end{enumerate}
-My take: option 3 is the only version that's worth its space.
-\end{frame}
-
-\begin{frame}{History: what we tried before landing on the current proof}
-From \texttt{paper/research\_log.md}.
-\vspace{0.4em}
-
-\textbf{Attempt A: Augment with $P_{\ell_1,\ell_2,\ell}$.}
-Started with the empirical observation that the seed Jacobian (10$\times$11)
-had rank 6 instead of 10 for real signals. Added CG power entries one at a
-time; greedy augmentation hits full rank.
-
-\textbf{Attempt B: Switch to matrix-valued bispectrum.} Higher rank but
-$O(L^3)$ output. Rejected.
-
-\textbf{Attempt C: Edidin--Satriano (2024) as black-box seed.}
-Misread their $R \ge L+2$ theorem --- it's the multi-shell case, not
-single-$S^2$. Rescue plan failed.
-
-\textbf{Attempt D (current): Kakarala 1992/2012 Theorem 7 + Bandeira et al.\ 2017.}
-After re-reading: $N_H \subset \SO(3)$, so the ``up to reflection''
-caveat in Edidin--Satriano refers to Kakarala's
-\emph{algorithm} (positive square root step), not to the bispectrum
-\emph{data}. The full bispectrum at $L_0 = 4$ already separates $f$
-from $T_R \cdot f$ via $\beta_{2,3,4}$.
-\end{frame}
-
-\begin{frame}{Findings on the way (the bits worth re-using)}
-\begin{itemize}
-  \item \textbf{Seed fibre is 10-fold, not 2-fold.}
-        Multi-start NLS (5000 starts/branch) finds 10 real solutions per
-        $\sgn(v)$ branch. Adding more degree-$\le 4$ CG-power entries
-        \emph{does not} reduce the cover.
-        $\Rightarrow$ The 10:1 ramification is intrinsic to the GIT-quotient
-        $V_{0..3}//\SO(3)$ at degree-4 invariants. Any structural reduction
-        \emph{must} use degree $\ge 4$ data. This validated the Kakarala route.
-  \item \textbf{Reflection is killed by data, not by gauge.}
-        $T_R$ flips $\im\beta_{2,3,4}$; gauge fixing only handles the
-        $\SO(3)$ part.
-  \item \textbf{No simple uniform witness exists.}
-        Tested closed-form sequences W1--W5 ($F_a^m = 1$, $1/(1+m^2)$,
-        $(1+m)(1+a)$, etc.). All fail at moderate $\ell$. A uniform
-        bootstrap-rank proof needs an actual Wigner-3j non-vanishing identity.
-\end{itemize}
-\end{frame}
-
-\begin{frame}{Side-by-side: where the proof is structural vs.\ computational}
-\centering\small
-\begin{tabular}{p{0.55\linewidth}p{0.35\linewidth}}
-\toprule
-Step & Status \\
-\midrule
-Gauge fixing & structural \\
-$\F_0, \F_1$ recovery & structural \\
-Parity / $T_R$ classification ($\ell \le 5$) & structural (sympy-symbolic) \\
-Seed (degrees 0--4) via Kakarala Thm 7 & structural (citation) \\
-Seed via Bandeira et al.\ Gröbner & structural (one-time CA) \\
-Constructive seed (Lemmas \texttt{app-ell2}--\texttt{app-TR-resolve}) & numerical witness, kept as \emph{algorithm}, not as proof \\
-Bootstrap rank $\ell \in [4,100]$ & per-$\ell$ witness, structural for fixed $L$ \\
-Augmented real per-degree rank $\ell \in [4, 30]$ & per-$\ell$ witness \\
-Uniform-in-$\ell$ bootstrap rank & \alert{open} \\
-\bottomrule
-\end{tabular}
-\end{frame}
-
-\section{Part 4 --- Openings and what's left}
-
-\begin{frame}{Concrete things left to do}
-\textbf{Theory:}
-\begin{itemize}
-  \item \textbf{Uniform-in-$\ell$ bootstrap rank.}
-        Reduce $\det A_\ell \ne 0$ to a Wigner-3j non-vanishing identity
-        over the closed-form chain/offset/C-row family $\mathcal T_\ell$.
-        Current best lead: evaluate at $F_a^m = \zeta^{a+m}$ and study the
-        leading Laurent term in $\zeta$.
-  \item \textbf{Inversion algorithm for $\SO(3)$/$S^2$.}
-        Forward map exists; inversion does not. Closed form for $\ell=0,1$;
-        nonlinear for $\ell \in \{2,3\}$ (10:1 fibre); linear pseudoinverse
-        for $\ell \ge 4$.
-  \item \textbf{Octahedral no-go.}
-        Make the failure of bootstrap inversion on $O$ a theorem,
-        not a numerical observation.
-  \item \textbf{Conditioning bound.}
-        $\kappa(A_\ell)$ as a function of spectral decay of $f$.
-\end{itemize}
-\textbf{Library:}
-\begin{itemize}
-  \item Selective bispectrum for general homogeneous spaces $G/H$.
-  \item \texttt{escnn}/\texttt{e3nn} interop.
-\end{itemize}
-\end{frame}
-
-\begin{frame}{Where this could go}
-\begin{itemize}
-  \item \textbf{Cryo-EM.}
-        Single-particle reconstruction is exactly the
-        $\SO(3)$/$S^2$ orbit-recovery problem with noise.
-        $\Theta(L^2)$ invariant + provable completeness $\to$ candidate
-        loss / preprocessing layer.
-  \item \textbf{Molecular property prediction.}
-        Bispectral pooling over $\SO(3)$-equivariant features as
-        an alternative to gated nonlinearities; data-efficiency
-        story is the right one for medium-size datasets (QM9, MD17 small).
-  \item \textbf{Multi-reference alignment.}
-        Myers \& Miolane (2025) already do disk; sphere is the natural next step.
-  \item \textbf{Open question of independent interest:}
-        what is the smallest $L_0$ such that the full bispectrum at $L_0$
-        determines the orbit? Bandeira et al.\ verified $L_0 \le 15$;
-        can we get a closed-form $L_0$ as a function of any structural
-        parameter?
-\end{itemize}
-\end{frame}
-
-\begin{frame}{Concrete asks for prospective collaborators}
-Pick whichever is closest to your taste:
-\begin{enumerate}
-  \item \textbf{(Algebra / rep-theory)}
-        Wigner-3j non-vanishing identity for the closed-form
-        bootstrap family $\mathcal T_\ell$. One identity, kills the only
-        remaining structural gap.
-  \item \textbf{(Numerical analysis)}
-        Conditioning bound for $A_\ell$ in terms of spectral decay.
-  \item \textbf{(Algorithms)}
-        Inversion for $\SO(3)$/$S^2$ from $\Phi_\mathrm{aug}$.
-  \item \textbf{(ML / applications)}
-        A real downstream task where bispectral invariance is the
-        bottleneck (cryo-EM, MRA, molecular). PCam / Spherical MNIST
-        are too easy; we need a benchmark where this matters.
-  \item \textbf{(Theory cleanup)}
-        Octahedral no-go theorem, or: extension of selective bispectrum
-        to a non-trivial $G/H$.
-\end{enumerate}
-\end{frame}
-
-\begin{frame}[standout]
-Questions?
-\end{frame}
-
-\appendix
-
-\begin{frame}{References (selected)}
-\small
-\begin{itemize}
-  \item Kakarala (1992) \emph{Triple Correlation on Groups}, PhD thesis.
-  \item Kakarala (2012) JMIV --- Theorem 7, our seed.
-  \item Mataigne et al.\ (2024) NeurIPS --- selective bispectrum, finite groups.
-  \item Bandeira et al.\ (2017) arXiv:1712.10163 --- symbolic Jacobian
-        certificate for $L_0 \le 15$.
-  \item Edidin \& Satriano (2024) --- multi-shell completeness ($R \ge L+2$).
-  \item Cohen \& Welling (2016) --- $G$-equivariant CNNs.
-  \item Bonev et al.\ (2023) --- \texttt{torch\_harmonics}.
-\end{itemize}
-\end{frame}
-
-\end{document}
diff --git a/paper/test_sc_fiber.py b/paper/test_sc_fiber.py
deleted file mode 100644
index 7187829..0000000
--- a/paper/test_sc_fiber.py
+++ /dev/null
@@ -1,95 +0,0 @@
-"""Test: does adding self-coupling bispectral entries resolve the fiber ambiguity?"""
-from __future__ import annotations
-import sys
-import os
-import time
-sys.path.insert(0, os.path.join(os.path.dirname(__file__), '..', 'src'))
-import numpy as np
-from paper.fiber_search import (
-    _init_cg, get_live_bispec, param_indices_for_ell,
-    params_to_fc_np, bispectrum_entry_np, cg_power_entry_np,
-    damped_gauss_newton,
-)
-from bispectrum.so3_on_s2 import _build_cg_power_index_map
-
-
-def main():
-    lmax = 4
-    _init_cg(lmax)
-    n = sum(2 * l + 1 for l in range(2, lmax + 1)) - 1
-    w = np.array([(i + 1) / 3.0 for i in range(n)])
-    live = get_live_bispec(lmax, w)
-    cgp = _build_cg_power_index_map(lmax)
-
-    sc_bispec = []
-    fc_test = params_to_fc_np(w, lmax)
-    for ell in range(2, lmax + 1):
-        for l_out in range(0, min(2 * ell, lmax) + 1):
-            if l_out > 2 * ell:
-                continue
-            try:
-                beta = bispectrum_entry_np(fc_test, ell, ell, l_out)
-                parity = (2 * ell + l_out) % 2
-                val = float(np.imag(beta)) if parity else float(np.real(beta))
-                if abs(val) > 1e-10:
-                    already = (ell, ell, l_out) in live
-                    if not already:
-                        sc_bispec.append((ell, ell, l_out))
-            except Exception:
-                pass
-
-    print(f"Added {len(sc_bispec)} self-coupling entries:")
-    for t in sc_bispec:
-        print(f"  B{t}")
-
-    def eval_full(p):
-        fc = params_to_fc_np(p, lmax)
-        vals = []
-        for l1, l2, lv in live:
-            beta = bispectrum_entry_np(fc, l1, l2, lv)
-            par = (l1 + l2 + lv) % 2
-            vals.append(float(np.imag(beta)) if par else float(np.real(beta)))
-        for l1, l2, lv in sc_bispec:
-            beta = bispectrum_entry_np(fc, l1, l2, lv)
-            par = (l1 + l2 + lv) % 2
-            vals.append(float(np.imag(beta)) if par else float(np.real(beta)))
-        for l1, l2, lo in cgp:
-            vals.append(cg_power_entry_np(fc, l1, l2, lo))
-        return np.array(vals)
-
-    target = eval_full(w)
-    print(f"Total: {len(target)} constraints")
-    sys.stdout.flush()
-
-    def res(x):
-        return eval_full(x) - target
-
-    rng = np.random.RandomState(42)
-    found = []
-    t0 = time.time()
-    for trial in range(2000):
-        eps = rng.choice([0.01, 0.02, 0.05, 0.1, 0.15, 0.2, 0.3])
-        x0 = w + eps * rng.randn(n)
-        x, r = damped_gauss_newton(res, x0, max_iter=80)
-        if r < 1e-22:
-            dup = any(np.linalg.norm(x - s) < 1e-5 for s in found)
-            if not dup:
-                found.append(x.copy())
-                d = np.linalg.norm(x - w)
-                print(f"  t{trial}: sol #{len(found)}, dist={d:.2e}")
-                sys.stdout.flush()
-        if trial % 500 == 499:
-            print(f"  {trial+1}/2000 ({time.time()-t0:.1f}s)")
-            sys.stdout.flush()
-
-    elapsed = time.time() - t0
-    print(f"\nRESULT: {len(found)} solutions in {elapsed:.1f}s")
-    for i, s in enumerate(found):
-        d = np.linalg.norm(s - w)
-        label = "WITNESS" if d < 1e-5 else f"DISTINCT (dist={d:.2e})"
-        print(f"  Sol {i}: {label}")
-    sys.stdout.flush()
-
-
-if __name__ == "__main__":
-    main()
diff --git a/paper/test_seed_augment.py b/paper/test_seed_augment.py
deleted file mode 100644
index b08bc6c..0000000
--- a/paper/test_seed_augment.py
+++ /dev/null
@@ -1,161 +0,0 @@
-"""Test: does adding more CG power entries shrink the seed fibre?
-
-Baseline: 4 CG power entries at degree 3 gives 10 solutions per v-branch.
-Candidate extra entries: P(1,3,3), P(2,3,3).
-Goal: see if augmenting reduces the fibre count to 1 per branch.
-"""
-from __future__ import annotations
-
-import sys
-from pathlib import Path
-
-sys.path.insert(0, str(Path(__file__).parent.parent))
-
-import numpy as np
-from scipy.optimize import least_squares
-
-from bispectrum._cg import clebsch_gordan
-
-
-A0_VAL = 1.0
-C_VAL = 1.0
-WITNESS_D2 = (3 / 7, 2 / 5, 1 / 3, 4 / 9)
-WITNESS_D3 = (1 / 2, 3 / 8, 2 / 7, 5 / 11, 1 / 4, 3 / 13, 7 / 17)
-
-BISP_D2: list[tuple[int, int, int]] = [(1, 1, 2), (0, 2, 2), (2, 2, 2)]
-CGP_D2: list[tuple[int, int, int]] = [(1, 2, 1)]
-BISP_D3: list[tuple[int, int, int]] = [(1, 2, 3), (1, 3, 2), (2, 3, 1), (0, 3, 3), (3, 3, 2)]
-
-BASELINE_CGP_D3: list[tuple[int, int, int]] = [
-    (1, 3, 2), (2, 3, 1), (2, 3, 2), (3, 3, 2),
-]
-EXTRA_CANDIDATES: list[tuple[int, int, int]] = [
-    (1, 3, 3), (2, 3, 3), (2, 2, 3), (1, 1, 3),
-]
-
-
-def precompute_cg(triples: list[tuple[int, int, int]]) -> dict:
-    cache = {}
-    for l1, l2, l in triples:
-        if (l1, l2, l) in cache:
-            continue
-        mat = np.zeros((2 * l1 + 1, 2 * l2 + 1, 2 * l + 1))
-        for m1 in range(-l1, l1 + 1):
-            for m2 in range(-l2, l2 + 1):
-                m = m1 + m2
-                if abs(m) > l:
-                    continue
-                val = clebsch_gordan(l1, m1, l2, m2, l, m)
-                if val != 0.0:
-                    mat[m1 + l1, m2 + l2, m + l] = val
-        cache[(l1, l2, l)] = mat
-    return cache
-
-
-def make_F2(y, x, u, v):
-    F = np.zeros(5, dtype=complex)
-    F[2] = y
-    F[3] = x
-    F[1] = -x
-    F[4] = u + 1j * v
-    F[0] = u - 1j * v
-    return F
-
-
-def make_F3(params):
-    t, p1, q1, p2, q2, p3, q3 = params
-    F = np.zeros(7, dtype=complex)
-    F[3] = t
-    F[4] = p1 + 1j * q1
-    F[2] = -(p1 - 1j * q1)
-    F[5] = p2 + 1j * q2
-    F[1] = p2 - 1j * q2
-    F[6] = p3 + 1j * q3
-    F[0] = -(p3 - 1j * q3)
-    return F
-
-
-def beta(Fs, l1, l2, l, cg):
-    C = cg[(l1, l2, l)]
-    return complex(np.einsum('ijk,i,j,k->', C, Fs[l1], Fs[l2], np.conj(Fs[l])))
-
-
-def cgp(Fs, l1, l2, l, cg):
-    C = cg[(l1, l2, l)]
-    proj = np.einsum('ijk,i,j->k', C, Fs[l1], Fs[l2])
-    return float(np.sum(np.abs(proj) ** 2))
-
-
-def build_residual(bisp_d3, cgp_d3, targets_b, targets_p, F0, F1, F2, cg):
-    def resid(p):
-        F3 = make_F3(p)
-        Fs = {0: F0, 1: F1, 2: F2, 3: F3}
-        r = []
-        for i, t in enumerate(bisp_d3):
-            r.append(beta(Fs, *t, cg).real - targets_b[i])
-        for i, t in enumerate(cgp_d3):
-            r.append(cgp(Fs, *t, cg) - targets_p[i])
-        return np.array(r)
-    return resid
-
-
-def count_solutions(cgp_d3, n_starts=2000, seed=42):
-    all_triples = list(set(BISP_D2 + CGP_D2 + BISP_D3 + cgp_d3))
-    cg = precompute_cg(all_triples)
-
-    F0 = np.array([A0_VAL], dtype=complex)
-    F1 = np.zeros(3, dtype=complex)
-    F1[1] = C_VAL
-
-    y0, x0, u0, v0 = WITNESS_D2
-    F2_pos = make_F2(y0, x0, u0, v0)
-    F2_neg = make_F2(y0, x0, u0, -v0)
-    F3_w = make_F3(np.array(WITNESS_D3))
-    Fs_w = {0: F0, 1: F1, 2: F2_pos, 3: F3_w}
-
-    targets_b = np.array([beta(Fs_w, *t, cg).real for t in BISP_D3])
-    targets_p = np.array([cgp(Fs_w, *t, cg) for t in cgp_d3])
-
-    total_counts = {}
-    for name, F2 in [('+v', F2_pos), ('-v', F2_neg)]:
-        resid = build_residual(BISP_D3, cgp_d3, targets_b, targets_p, F0, F1, F2, cg)
-        sols = []
-        rng = np.random.RandomState(seed)
-        for i in range(n_starts):
-            x0_ = rng.randn(7) * 2.0
-            try:
-                r = least_squares(resid, x0_, method='lm', max_nfev=500)
-            except Exception:
-                continue
-            if r.cost < 1e-20:
-                is_new = all(np.max(np.abs(r.x - s)) > 1e-6 for s in sols)
-                if is_new:
-                    sols.append(r.x.copy())
-        total_counts[name] = len(sols)
-    return total_counts
-
-
-def main():
-    N = 800
-    print('=' * 60)
-    print('Baseline CGP augmentation:', BASELINE_CGP_D3, flush=True)
-    counts = count_solutions(BASELINE_CGP_D3, n_starts=N)
-    print(f'  Solutions per branch: {counts}', flush=True)
-
-    for extra in [
-        [(1, 3, 3)],
-        [(2, 3, 3)],
-        [(1, 3, 3), (2, 3, 3)],
-        [(2, 2, 3)],
-        [(1, 1, 3)],
-        [(1, 3, 3), (2, 3, 3), (2, 2, 3), (1, 1, 3)],
-    ]:
-        aug = BASELINE_CGP_D3 + extra
-        print('=' * 60)
-        print(f'Augmented with extra: {extra}', flush=True)
-        counts = count_solutions(aug, n_starts=N)
-        print(f'  Solutions per branch: {counts}', flush=True)
-
-
-if __name__ == '__main__':
-    main()
diff --git a/paper/test_uniform_witness.py b/paper/test_uniform_witness.py
deleted file mode 100644
index 1f75253..0000000
--- a/paper/test_uniform_witness.py
+++ /dev/null
@@ -1,180 +0,0 @@
-"""Test: uniform structural witness for bootstrap rank uniform in ℓ.
-
-Hypothesis: for the closed-form selective family T_ℓ (ℓ ≥ 8), the
-bootstrap matrix A_ℓ has full complex rank 2ℓ+1 at a specific signal
-F_a^m = something simple (ζ^{a+m}, or similar).
-
-If this works at one explicit ζ for ALL ℓ tested, then by
-density the polynomial det A_ℓ(F_0,...,F_{ℓ-1}) is generically nonzero.
-
-We test several candidates:
-  W1: F_a^m = 1 for all a, m (constant)
-  W2: F_a^m = (m+1) (varying with m)
-  W3: F_a^m = ζ^{a+m} for ζ = 2, 3
-  W4: F_a^m = (a+1)(m+ℓ+1)
-"""
-from __future__ import annotations
-
-import sys
-from pathlib import Path
-
-sys.path.insert(0, str(Path(__file__).parent.parent))
-
-import numpy as np
-from bispectrum._cg import clebsch_gordan
-
-
-def cg_array(l1: int, l2: int, l: int) -> np.ndarray:
-    mat = np.zeros((2 * l1 + 1, 2 * l2 + 1, 2 * l + 1), dtype=float)
-    for m1 in range(-l1, l1 + 1):
-        for m2 in range(-l2, l2 + 1):
-            m = m1 + m2
-            if abs(m) > l:
-                continue
-            v = clebsch_gordan(l1, m1, l2, m2, l, m)
-            if v != 0.0:
-                mat[m1 + l1, m2 + l2, m + l] = v
-    return mat
-
-
-def closed_form_block(ell: int) -> list[tuple[int, int, int]]:
-    block: list[tuple[int, int, int]] = []
-    for a in range(1, ell):
-        block.append((a, ell, ell - a))
-    for a in range(2, ell):
-        block.append((a, ell, ell - a + 1))
-    for a in range(1, 5):
-        block.append((a, ell - a, ell))
-    return block
-
-
-def witness_F(name: str, lmax: int, zeta: complex = 2.0 + 0.3j) -> dict[int, np.ndarray]:
-    Fs: dict[int, np.ndarray] = {}
-    for a in range(lmax + 1):
-        size = 2 * a + 1
-        F = np.zeros(size, dtype=complex)
-        for m_off in range(size):
-            m = m_off - a
-            if name == 'W1':
-                F[m_off] = 1.0
-            elif name == 'W2':
-                F[m_off] = m + 1
-            elif name == 'W3':
-                F[m_off] = zeta ** (a + m)
-            elif name == 'W4':
-                F[m_off] = (a + 1) * (m + a + 1)
-            elif name == 'W5':
-                F[m_off] = (1 + 0.1j) ** (a + m + 1)
-        Fs[a] = F
-    return Fs
-
-
-def build_bootstrap_matrix(
-    ell: int, Fs: dict[int, np.ndarray], block: list[tuple[int, int, int]]
-) -> np.ndarray:
-    n_unk = 2 * ell + 1
-    rows = []
-    for trip in block:
-        l1, l2, l_out = trip
-        C = cg_array(l1, l2, l_out)
-        if l1 == ell:
-            row = np.zeros(n_unk, dtype=complex)
-            for m1 in range(-l1, l1 + 1):
-                for m2 in range(-l2, l2 + 1):
-                    m = m1 + m2
-                    if abs(m) > l_out:
-                        continue
-                    coef = C[m1 + l1, m2 + l2, m + l_out] * Fs[l2][m2 + l2] * np.conj(Fs[l_out][m + l_out])
-                    row[m1 + l1] += coef
-        elif l2 == ell:
-            row = np.zeros(n_unk, dtype=complex)
-            for m1 in range(-l1, l1 + 1):
-                for m2 in range(-l2, l2 + 1):
-                    m = m1 + m2
-                    if abs(m) > l_out:
-                        continue
-                    coef = C[m1 + l1, m2 + l2, m + l_out] * Fs[l1][m1 + l1] * np.conj(Fs[l_out][m + l_out])
-                    row[m2 + l2] += coef
-        elif l_out == ell:
-            row = np.zeros(n_unk, dtype=complex)
-            for m1 in range(-l1, l1 + 1):
-                for m2 in range(-l2, l2 + 1):
-                    m = m1 + m2
-                    if abs(m) > l_out:
-                        continue
-                    coef = C[m1 + l1, m2 + l2, m + l_out] * Fs[l1][m1 + l1] * Fs[l2][m2 + l2]
-                    row[m + l_out] += coef
-        else:
-            raise RuntimeError(f'ell {ell} not in triple {trip}')
-        rows.append(row)
-    return np.array(rows)
-
-
-def test_witness(name: str, l_range: list[int]) -> None:
-    print(f'\n=== Witness {name} ===')
-    for ell in l_range:
-        block = closed_form_block(ell)
-        Fs = witness_F(name, ell)
-        A = build_bootstrap_matrix(ell, Fs, block)
-        sv = np.linalg.svd(A, compute_uv=False)
-        rank = int(np.sum(sv > 1e-9 * sv[0]))
-        cond = sv[0] / max(sv[-1], 1e-300)
-        marker = 'OK ' if rank == 2 * ell + 1 else 'BAD'
-        print(f'  {marker} ell={ell:3d}  rank={rank:3d}/{2 * ell + 1:3d}  cond={cond:.2e}')
-
-
-def witness_random(lmax: int, seed: int = 42) -> dict[int, np.ndarray]:
-    rng = np.random.RandomState(seed)
-    Fs: dict[int, np.ndarray] = {}
-    for a in range(lmax + 1):
-        size = 2 * a + 1
-        F = rng.randn(size) + 1j * rng.randn(size)
-        Fs[a] = F
-    return Fs
-
-
-def witness_real_random(lmax: int, seed: int = 42) -> dict[int, np.ndarray]:
-    """Gaussian random with reality F^{-m} = (-1)^m conj(F^m)."""
-    rng = np.random.RandomState(seed)
-    Fs: dict[int, np.ndarray] = {}
-    for a in range(lmax + 1):
-        size = 2 * a + 1
-        F = np.zeros(size, dtype=complex)
-        F[a] = rng.randn()
-        for m in range(1, a + 1):
-            re = rng.randn()
-            im = rng.randn()
-            F[a + m] = re + 1j * im
-            F[a - m] = ((-1) ** m) * (re - 1j * im)
-        Fs[a] = F
-    return Fs
-
-
-def test_witness_named(label: str, Fs: dict[int, np.ndarray], l_list: list[int]) -> None:
-    print(f'\n=== {label} ===')
-    for ell in l_list:
-        block = closed_form_block(ell)
-        A = build_bootstrap_matrix(ell, Fs, block)
-        sv = np.linalg.svd(A, compute_uv=False)
-        scale = sv[0]
-        for thr in [1e-9, 1e-12, 1e-14]:
-            rank = int(np.sum(sv > thr * scale))
-            if rank == 2 * ell + 1:
-                break
-        cond = sv[0] / max(sv[-1], 1e-300)
-        marker = 'OK ' if rank == 2 * ell + 1 else 'BAD'
-        print(f'  {marker} ell={ell:3d}  rank={rank:3d}/{2 * ell + 1:3d}  cond={cond:.2e}  thr={thr:.0e}')
-
-
-def main():
-    l_list = [8, 10, 15, 20, 30, 40, 50, 70, 100]
-    for name in ['W1', 'W2', 'W3', 'W4', 'W5']:
-        test_witness(name, l_list)
-    Fs_rng = witness_random(100, seed=42)
-    test_witness_named('Random complex (seed 42)', Fs_rng, l_list)
-    Fs_real = witness_real_random(100, seed=42)
-    test_witness_named('Random real-signal (seed 42)', Fs_real, l_list)
-
-
-if __name__ == '__main__':
-    main()
diff --git a/paper/verify_bootstrap_complex.py b/paper/verify_bootstrap_complex.py
deleted file mode 100644
index 036fb57..0000000
--- a/paper/verify_bootstrap_complex.py
+++ /dev/null
@@ -1,153 +0,0 @@
-"""Verify the COMPLEX rank of the bootstrap coefficient matrix for real signals.
-
-Key insight: the Jacobian rank deficit is about the REAL map β: R^n → R^n.
-But the bootstrap uses COMPLEX linear algebra: A F_ℓ = b where A ∈ C^{N×(2ℓ+1)}.
-The complex rank of A can be 2ℓ+1 even when the real Jacobian is rank-deficient.
-"""
-import sys
-sys.path.insert(0, '../src')
-import torch
-from bispectrum.so3_on_s2 import (
-    _build_selective_index_map, _bispectrum_entry, load_cg_matrices,
-)
-
-CG_DATA = load_cg_matrices(8)
-A0, C_GAUGE = 1.5, 1.2
-
-
-def make_real_fc(lmax, seed=42):
-    torch.manual_seed(seed)
-    fc = {}
-    for l in range(lmax + 1):
-        pp, pn = [], []
-        for m in range(1, l + 1):
-            re = torch.randn(1, dtype=torch.float64)
-            im = torch.randn(1, dtype=torch.float64)
-            pp.append(torch.complex(re, im))
-            pn.append(((-1.0) ** m) * torch.complex(re, -im))
-        m0 = torch.complex(torch.randn(1, dtype=torch.float64), torch.zeros(1, dtype=torch.float64))
-        fc[l] = torch.stack(list(reversed(pn)) + [m0] + pp, dim=-1)
-    return fc
-
-
-def build_bootstrap_matrix(fc, l_target, triples):
-    """Build the complex coefficient matrix A for the bootstrap at l_target.
-
-    For chain entries (l1,l2,l_target): F_{l_target} appears as conj(a_l^m).
-    For cross entries (l1,l_target,l_v): F_{l_target} appears as a_{l_target}^{m2}.
-    """
-    n_cols = 2 * l_target + 1
-    rows_conj = []  # entries where conj(F_l) appears
-    rows_direct = []  # entries where F_l appears directly
-
-    for l1, l2, lv in triples:
-        if lv == l_target and l2 < l_target:
-            # Chain: (l1, l2, l_target), linear in conj(a_{l_target}^m)
-            row = torch.zeros(n_cols, dtype=torch.complex128)
-            for m in range(-l_target, l_target + 1):
-                col_idx = m + l_target
-                val = torch.tensor(0.0, dtype=torch.complex128)
-                for m1 in range(-l1, l1 + 1):
-                    m2 = m - m1
-                    if abs(m2) > l2:
-                        continue
-                    cg_val = _get_cg(l1, m1, l2, m2, l_target, m)
-                    val += cg_val * fc[l1][0, m1 + l1] * fc[l2][0, m2 + l2]
-                row[col_idx] = val
-            rows_conj.append(row)
-
-        elif l2 == l_target and lv < l_target:
-            # Cross: (l1, l_target, lv), linear in a_{l_target}^{m2}
-            row = torch.zeros(n_cols, dtype=torch.complex128)
-            for m2 in range(-l_target, l_target + 1):
-                col_idx = m2 + l_target
-                val = torch.tensor(0.0, dtype=torch.complex128)
-                for m1 in range(-l1, l1 + 1):
-                    m = m1 + m2
-                    if abs(m) > lv:
-                        continue
-                    cg_val = _get_cg(l1, m1, l_target, m2, lv, m)
-                    val += cg_val * fc[l1][0, m1 + l1] * torch.conj(fc[lv][0, m + lv])
-                row[col_idx] = val
-            rows_direct.append(row)
-
-        elif lv == l_target and l2 >= l_target:
-            pass  # self-coupling, skip for now
-
-    # For chain entries, the system is: row · conj(F_l) = beta
-    # For cross entries, the system is: row · F_l = beta
-    # These are different: one uses conj(F_l), the other F_l.
-    # For a unified system, convert conj rows: if row · conj(F) = b,
-    # then conj(row) · F = conj(b), so we can use conj(row) for direct system.
-    A_rows = []
-    for row in rows_conj:
-        A_rows.append(torch.conj(row))
-    for row in rows_direct:
-        A_rows.append(row)
-
-    if not A_rows:
-        return torch.zeros(0, n_cols, dtype=torch.complex128)
-    return torch.stack(A_rows)
-
-
-def _get_cg(l1, m1, l2, m2, l, m):
-    """Get CG coefficient from precomputed data."""
-    if m1 + m2 != m:
-        return 0.0
-    if l < abs(l1 - l2) or l > l1 + l2:
-        return 0.0
-    cg_mat = CG_DATA[(l1, l2)]
-    row = (m1 + l1) * (2 * l2 + 1) + (m2 + l2)
-    start = 0
-    for lv in range(abs(l1 - l2), l1 + l2 + 1):
-        if lv == l:
-            col = start + m + l
-            return cg_mat[row, col].item()
-        start += 2 * lv + 1
-    return 0.0
-
-
-def main():
-    for lmax in [4, 5, 6, 7, 8]:
-        idx_map = _build_selective_index_map(lmax)
-
-        print(f"lmax={lmax}:")
-        for l_target in range(4, lmax + 1):
-            triples = [(l1, l2, lv) for l1, l2, lv in idx_map
-                       if (lv == l_target and l2 < l_target) or
-                       (l2 == l_target and lv < l_target)]
-            expected_rank = 2 * l_target + 1
-
-            # Test at multiple seeds
-            ranks = []
-            for seed in [42, 123, 999]:
-                fc = make_real_fc(lmax, seed)
-                A = build_bootstrap_matrix(fc, l_target, triples)
-                if A.shape[0] == 0:
-                    ranks.append(0)
-                    continue
-                sv = torch.linalg.svdvals(A)
-                tol = 1e-8 * sv[0].item() if sv[0] > 0 else 1e-15
-                rank = int((sv > tol).sum().item())
-                ranks.append(rank)
-
-            min_r = min(ranks)
-            status = "✓" if min_r >= expected_rank else "✗"
-            print(f"  ℓ={l_target}: {len(triples)} entries, "
-                  f"complex rank={min_r}/{expected_rank} {status}")
-
-    # Also check: does the dead entry contribute a zero row?
-    print("\nDead entry row check:")
-    for l_target, dead_triple in [(5, (4, 5, 4)), (7, (6, 7, 6))]:
-        if l_target > 8: continue
-        fc = make_real_fc(l_target, 42)
-        A = build_bootstrap_matrix(fc, l_target, [dead_triple])
-        if A.shape[0] > 0:
-            row_norm = torch.linalg.norm(A[0]).item()
-            print(f"  {dead_triple}: ||row||={row_norm:.6e}")
-        else:
-            print(f"  {dead_triple}: not a bootstrap entry for ℓ={l_target}")
-
-
-if __name__ == "__main__":
-    main()
diff --git a/paper/verify_bootstrap_v2.py b/paper/verify_bootstrap_v2.py
deleted file mode 100644
index 4f2148d..0000000
--- a/paper/verify_bootstrap_v2.py
+++ /dev/null
@@ -1,106 +0,0 @@
-"""Verify bootstrap rank using autograd — no manual matrix construction."""
-import sys
-sys.path.insert(0, '../src')
-import torch
-from bispectrum.so3_on_s2 import (
-    _build_selective_index_map, _bispectrum_entry, load_cg_matrices,
-)
-
-CG_DATA = load_cg_matrices(8)
-A0, C_GAUGE = 1.5, 1.2
-
-
-def params_to_fc(p_fixed, p_target, l_target, lmax):
-    """Build coefficients: p_fixed for degrees 0..l_target-1 (frozen),
-    p_target for degree l_target (free)."""
-    fc = {}
-    fc[0] = torch.complex(torch.tensor([[A0]], dtype=torch.float64), torch.zeros(1,1, dtype=torch.float64))
-    fc[1] = torch.complex(torch.tensor([[0.,C_GAUGE,0.]], dtype=torch.float64), torch.zeros(1,3, dtype=torch.float64))
-    idx = 0
-    for lv in range(2, lmax+1):
-        n_real = 2*lv+1 if lv != 2 else 2*lv  # gauge: Im(a_2^1)=0
-        if lv < l_target:
-            raw = p_fixed[idx:idx+n_real]
-            idx += n_real
-        elif lv == l_target:
-            raw = p_target
-        else:
-            raw = p_fixed[idx:idx+n_real]
-            idx += n_real
-        neg, pos = [], []
-        j = 0
-        for m in range(1, lv+1):
-            if lv==2 and m==1:
-                re = raw[j]; j += 1; im = torch.zeros_like(re)
-            else:
-                re = raw[j]; im = raw[j+1]; j += 2
-            pos.append(torch.complex(re,im)); neg.append(((-1.)**m)*torch.complex(re,-im))
-        m0 = torch.complex(raw[j], torch.zeros_like(raw[j])); j += 1
-        fc[lv] = torch.stack(list(reversed(neg))+[m0]+pos).unsqueeze(0)
-    return fc
-
-
-def main():
-    for lmax in [4, 5, 6, 7]:
-        idx_map = _build_selective_index_map(lmax)
-        print(f"lmax={lmax}:")
-
-        for l_target in range(4, lmax+1):
-            n_target = 2*l_target+1 if l_target != 2 else 2*l_target
-            n_fixed = sum(2*l+1 if l!=2 else 2*l for l in range(2, lmax+1)) - n_target
-
-            torch.manual_seed(42)
-            p_fixed = torch.randn(n_fixed, dtype=torch.float64)
-            p_target = torch.randn(n_target, dtype=torch.float64)
-
-            triples = [(l1,l2,lv) for l1,l2,lv in idx_map if max(l1,l2,lv)==l_target]
-
-            def eval_bootstrap(p_t, trips=triples, lt=l_target, lm=lmax, pf=p_fixed):
-                fc = params_to_fc(pf, p_t, lt, lm)
-                vals = []
-                for l1,l2,lv in trips:
-                    beta = _bispectrum_entry(fc, l1, l2, lv, CG_DATA[(l1,l2)])[0]
-                    # Use BOTH real and imag (full complex info)
-                    vals.append(beta.real)
-                    vals.append(beta.imag)
-                return torch.stack(vals)
-
-            def eval_bootstrap_parity(p_t, trips=triples, lt=l_target, lm=lmax, pf=p_fixed):
-                fc = params_to_fc(pf, p_t, lt, lm)
-                vals = []
-                for l1,l2,lv in trips:
-                    beta = _bispectrum_entry(fc, l1, l2, lv, CG_DATA[(l1,l2)])[0]
-                    parity = (l1+l2+lv) % 2
-                    vals.append(beta.imag if parity else beta.real)
-                return torch.stack(vals)
-
-            # Complex Jacobian (real+imag parts)
-            J_full = torch.autograd.functional.jacobian(eval_bootstrap, p_target)
-            sv_full = torch.linalg.svdvals(J_full)
-            tol = 1e-8 * sv_full[0].item() if sv_full[0]>0 else 1e-15
-            rank_full = int((sv_full > tol).sum().item())
-
-            # Parity-corrected Jacobian (real part)
-            J_par = torch.autograd.functional.jacobian(eval_bootstrap_parity, p_target)
-            sv_par = torch.linalg.svdvals(J_par)
-            tol2 = 1e-8 * sv_par[0].item() if sv_par[0]>0 else 1e-15
-            rank_par = int((sv_par > tol2).sum().item())
-
-            # Check for dead entries
-            fc_test = params_to_fc(p_fixed, p_target, l_target, lmax)
-            dead = []
-            for l1,l2,lv in triples:
-                beta = _bispectrum_entry(fc_test, l1, l2, lv, CG_DATA[(l1,l2)])[0]
-                if abs(beta.item()) < 1e-10:
-                    dead.append((l1,l2,lv))
-
-            print(f"  ℓ={l_target}: {len(triples)} entries, "
-                  f"full(Re+Im) rank={rank_full}/{n_target}, "
-                  f"parity rank={rank_par}/{n_target}, "
-                  f"dead={dead}")
-
-        print()
-
-
-if __name__ == "__main__":
-    main()
diff --git a/paper/verify_full_rank.py b/paper/verify_full_rank.py
deleted file mode 100644
index 2f4c05c..0000000
--- a/paper/verify_full_rank.py
+++ /dev/null
@@ -1,130 +0,0 @@
-"""Verify the augmented selective bispectrum has full Jacobian rank.
-
-Key insight: the scalar bispectrum has dead entries for real signals.
-We augment with 5 CG-power-spectrum entries at the seed stage:
-  P_{l1,l2,l} := ||(F_{l1} ⊗ F_{l2})|_l||^2
-These are degree-4 SO(3)-invariants.
-
-Budget analysis: we DROP the 2 dead entries and replace with 5 CG power
-entries, giving a net increase of 3 at the seed stage. To compensate,
-we reduce l=4 from 10 to 9 (standard budget) and also check if any l=5+
-entries can be trimmed.
-"""
-import sys
-sys.path.insert(0, '../src')
-import torch
-from bispectrum.so3_on_s2 import (
-    _build_selective_index_map, _bispectrum_entry, load_cg_matrices,
-)
-
-CG_DATA = load_cg_matrices(8)
-A0, C_GAUGE = 1.5, 1.2
-
-AUG_ENTRIES = [(1,2,1), (1,3,2), (2,3,1), (2,3,2), (3,3,2)]
-
-
-def params_to_fc(p, lmax):
-    fc = {}
-    fc[0] = torch.complex(torch.tensor([[A0]],dtype=torch.float64), torch.zeros(1,1,dtype=torch.float64))
-    fc[1] = torch.complex(torch.tensor([[0.,C_GAUGE,0.]],dtype=torch.float64), torch.zeros(1,3,dtype=torch.float64))
-    idx = 0
-    for lv in range(2, lmax+1):
-        neg, pos = [], []
-        for m in range(1, lv+1):
-            if lv==2 and m==1: re=p[idx]; idx+=1; im=torch.zeros_like(re)
-            else: re=p[idx]; im=p[idx+1]; idx+=2
-            pos.append(torch.complex(re,im)); neg.append(((-1.)**m)*torch.complex(re,-im))
-        m0 = torch.complex(p[idx], torch.zeros_like(p[idx])); idx+=1
-        fc[lv] = torch.stack(list(reversed(neg))+[m0]+pos).unsqueeze(0)
-    return fc
-
-
-def cg_project(fc, l1, l2, l_out):
-    outer = torch.einsum('bi,bj->bij', fc[l1], fc[l2]).reshape(1,-1)
-    cg = CG_DATA[(l1,l2)].to(dtype=outer.dtype)
-    tr = outer @ cg
-    start = 0
-    for lv in range(abs(l1-l2), l1+l2+1):
-        sz = 2*lv+1
-        if lv == l_out: return tr[0, start:start+sz]
-        start += sz
-
-
-def eval_augmented(p, lmax, bispec_triples, aug_triples):
-    fc = params_to_fc(p, lmax)
-    vals = []
-    for l1, l2, lv in bispec_triples:
-        beta = _bispectrum_entry(fc, l1, l2, lv, CG_DATA[(l1,l2)])[0]
-        parity = (l1+l2+lv) % 2
-        vals.append(beta.imag if parity else beta.real)
-    for l1, l2, lo in aug_triples:
-        proj = cg_project(fc, l1, l2, lo)
-        vals.append(torch.sum(proj.real**2 + proj.imag**2))
-    return torch.stack(vals)
-
-
-def check_dead(fc, l1, l2, lv):
-    beta = _bispectrum_entry(fc, l1, l2, lv, CG_DATA[(l1,l2)])[0]
-    parity = (l1+l2+lv) % 2
-    val = beta.imag.item() if parity else beta.real.item()
-    return abs(val) < 1e-10
-
-
-def main():
-    for lmax in [4, 5, 6, 7, 8]:
-        n_free = sum(2*l+1 for l in range(2, lmax+1)) - 1
-        torch.manual_seed(42)
-        p = torch.randn(n_free, dtype=torch.float64)
-
-        idx_map = _build_selective_index_map(lmax)
-        fc_test = params_to_fc(p, lmax)
-
-        live = []
-        dead = []
-        for l1, l2, lv in idx_map:
-            if max(l1,l2,lv) < 2: continue
-            if check_dead(fc_test, l1, l2, lv):
-                dead.append((l1,l2,lv))
-            else:
-                live.append((l1,l2,lv))
-
-        def eval_orig(p_in, trips=live, lm=lmax):
-            return eval_augmented(p_in, lm, trips, [])
-
-        def eval_aug(p_in, trips=live, aug=AUG_ENTRIES, lm=lmax):
-            return eval_augmented(p_in, lm, trips, aug)
-
-        Jo = torch.autograd.functional.jacobian(eval_orig, p)
-        svo = torch.linalg.svdvals(Jo)
-        ro = int((svo > 1e-8*svo[0]).sum().item())
-
-        Ja = torch.autograd.functional.jacobian(eval_aug, p)
-        sva = torch.linalg.svdvals(Ja)
-        ra = int((sva > 1e-8*sva[0]).sum().item())
-
-        total_orig = len(live) + 2  # +2 for l=0,1
-        total_aug = total_orig + 5
-        budget = (lmax+1)**2 - 3
-
-        print(f"lmax={lmax}: unknowns={n_free}, dead={len(dead)}, live_entries={len(live)}")
-        print(f"  Dead entries: {dead}")
-        print(f"  Original:  {len(live)} entries → rank {ro}/{n_free}")
-        print(f"  +5 CG aug: {len(live)+5} entries → rank {ra}/{n_free}")
-        print(f"  Full rank: {'YES' if ra==n_free else 'NO'}")
-        print(f"  Budget: orig={(lmax+1)**2-3}, with aug={total_aug}, overhead={total_aug-budget}")
-
-        # Verify at 2 more seeds
-        ranks_ok = True
-        for s in [123, 999]:
-            torch.manual_seed(s)
-            p2 = torch.randn(n_free, dtype=torch.float64)
-            Ja2 = torch.autograd.functional.jacobian(lambda pp: eval_aug(pp), p2)
-            sva2 = torch.linalg.svdvals(Ja2)
-            ra2 = int((sva2 > 1e-8*sva2[0]).sum().item())
-            if ra2 != n_free: ranks_ok = False
-        print(f"  Multi-seed: {'all full rank' if ranks_ok else 'FAILED'}")
-        print()
-
-
-if __name__ == "__main__":
-    main()
diff --git a/paper/witness_results.txt b/paper/witness_results.txt
deleted file mode 100644
index 5f81a4f..0000000
--- a/paper/witness_results.txt
+++ /dev/null
@@ -1,49 +0,0 @@
-lmax=4: 20 unknowns
-  Live bispec: 18, dead: [(2, 3, 2), (2, 2, 3)]
-  Bispec-only rank: 10/20
-  Augmentation: 10 CG power entries
-    P_{1,2,1}
-    P_{1,3,2}
-    P_{1,4,3}
-    P_{1,4,4}
-    P_{2,3,1}
-    P_{2,3,2}
-    P_{2,4,2}
-    P_{2,4,3}
-    P_{3,3,2}
-    P_{3,4,1}
-  Final rank: 20/20
-  seed=42: rank=20
-  seed=123: rank=20
-  seed=999: rank=20
-
-  Witness signal: p = [0.3333, 0.6667, 1.0000, 1.3333, 1.6667, 2.0000, 2.3333, 2.6667, 3.0000, 3.3333, 3.6667, 4.0000, 4.3333, 4.6667, 5.0000, 5.3333, 5.6667, 6.0000, 6.3333, 6.6667]
-
-lmax=5: 31 unknowns
-  Live bispec: 28, dead: [(2, 3, 2), (2, 2, 3), (4, 5, 4)]
-  Bispec-only rank: 14/31
-  Augmentation: 17 CG power entries
-    P_{1,2,1}
-    P_{1,3,2}
-    P_{1,4,3}
-    P_{1,4,4}
-    P_{1,5,4}
-    P_{1,5,5}
-    P_{2,3,1}
-    P_{2,3,2}
-    P_{2,4,2}
-    P_{2,4,3}
-    P_{2,5,3}
-    P_{2,5,4}
-    P_{3,3,2}
-    P_{3,4,1}
-    P_{3,5,2}
-    P_{3,5,3}
-    P_{3,5,4}
-  Final rank: 31/31
-  seed=42: rank=31
-  seed=123: rank=31
-  seed=999: rank=31
-
-  Witness signal: p = [0.3333, 0.6667, 1.0000, 1.3333, 1.6667, 2.0000, 2.3333, 2.6667, 3.0000, 3.3333, 3.6667, 4.0000, 4.3333, 4.6667, 5.0000, 5.3333, 5.6667, 6.0000, 6.3333, 6.6667, 7.0000, 7.3333, 7.6667, 8.0000, 8.3333, 8.6667, 9.0000, 9.3333, 9.6667, 10.0000, 10.3333]
-
diff --git a/pyproject.toml b/pyproject.toml
index 937cb3a..262a577 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -21,7 +21,7 @@ description = "Bispectrum analysis for machine learning"
 readme = "README.md"
 license = "MIT"
 requires-python = ">=3.12,<3.13"
-authors = [{ name = "Johan Mathe", email = "johan.mathe@gmail.com" }]
+authors = [{ name = "Anonymous" }]
 keywords = [
     "deep learning",
     "neural networks",
@@ -57,14 +57,10 @@ optional-dependencies.dev = [
 ]
 optional-dependencies.plotting = ["matplotlib>=3.7.0"]
 
-urls.Homepage = "https://github.com/geometric-intelligence/bispectrum"
-urls.Repository = "https://github.com/geometric-intelligence/bispectrum"
-urls.Issues = "https://github.com/geometric-intelligence/bispectrum/issues"
+urls.Homepage = "https://anonymous.4open.science/r/bispectrum"
+urls.Repository = "https://anonymous.4open.science/r/bispectrum"
 
-# Pin torch and torchvision to the cu128 wheels so they match systems with
-# NVIDIA driver >= 12.8 (e.g. the local 8x H100 box). PyPI's default torch
-# wheels target a newer CUDA runtime and crash with
-# "NVIDIA driver on your system is too old (found version 12080)".
+# Pin torch and torchvision to the cu128 wheels for CUDA 12.8 compatibility.
 [[tool.uv.index]]
 name = "pytorch-cu128"
 url = "https://download.pytorch.org/whl/cu128"
diff --git a/src/bispectrum/so3_on_s2.py b/src/bispectrum/so3_on_s2.py
index 7b0e0e8..17de397 100644
--- a/src/bispectrum/so3_on_s2.py
+++ b/src/bispectrum/so3_on_s2.py
@@ -1003,9 +1003,7 @@ def invert(self, beta: torch.Tensor, **kwargs: object) -> torch.Tensor:
             NotImplementedError: Selective bispectrum and inversion for
                 continuous groups remain open mathematical problems.
         """
-        raise NotImplementedError(
-            'Inversion for SO(3) on S^2 is an open mathematical problem. See DESIGN.md TODO-M1 and TODO-M4.'
-        )
+        raise NotImplementedError('Inversion for SO(3) on S^2 is an open mathematical problem.')
 
     @property
     def output_size(self) -> int:
diff --git a/uv.lock b/uv.lock
new file mode 100644
index 0000000..a6c7564
--- /dev/null
+++ b/uv.lock
@@ -0,0 +1,830 @@
+version = 1
+revision = 3
+requires-python = "==3.12.*"
+
+[[package]]
+name = "bispectrum"
+version = "0.2.0"
+source = { editable = "." }
+dependencies = [
+    { name = "numpy" },
+    { name = "torch" },
+    { name = "torch-harmonics" },
+]
+
+[package.optional-dependencies]
+dev = [
+    { name = "mypy" },
+    { name = "pre-commit" },
+    { name = "pytest" },
+    { name = "pytest-cov" },
+    { name = "pytest-xdist" },
+    { name = "ruff" },
+]
+plotting = [
+    { name = "matplotlib" },
+]
+
+[package.metadata]
+requires-dist = [
+    { name = "matplotlib", marker = "extra == 'plotting'", specifier = ">=3.7.0" },
+    { name = "mypy", marker = "extra == 'dev'", specifier = ">=1.5.0" },
+    { name = "numpy", specifier = ">=1.22.4" },
+    { name = "pre-commit", marker = "extra == 'dev'", specifier = ">=3.3.3" },
+    { name = "pytest", marker = "extra == 'dev'", specifier = ">=8" },
+    { name = "pytest-cov", marker = "extra == 'dev'", specifier = ">=4.0.0" },
+    { name = "pytest-xdist", marker = "extra == 'dev'", specifier = ">=3.5.0" },
+    { name = "ruff", marker = "extra == 'dev'", specifier = ">=0.4.0" },
+    { name = "torch", specifier = ">=2.4.0" },
+    { name = "torch-harmonics" },
+]
+provides-extras = ["dev", "plotting"]
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