From d1a817664014212f10348d1924deb284f69af5da Mon Sep 17 00:00:00 2001 From: Kamal Saleh Date: Tue, 20 Jan 2026 21:12:10 +0100 Subject: [PATCH] Major restructuring of the package including improved documentation and a rename to GradientBasedLearningForCAP. Bump version: v2026.01-01 --- .github/workflows/Tests.yml | 13 +- .gitignore | 2 +- PackageInfo.g | 24 +- README.md | 30 +- dev/.release | 2 +- dev/upload_codecov.sh | 11 +- doc/Doc.autodoc | 102 +- doc/clean | 2 +- doc/sigmoid.png | Bin 0 -> 82406 bytes examples/CategoryOfLenses.g | 251 +- examples/CategoryOfParametrisedMorphisms.g | 274 +- examples/CategoryOfSkeletalSmoothMaps.g | 182 +- .../next_local_minima.g | 4 +- examples/Expressions.g | 131 +- .../{neural_network.py => .neural_network.py} | 0 .../neural_network.cpython-310.pyc | Bin 1734 -> 0 bytes .../data/create_train_test_examples.g | 57 - .../data/generate_examples.g | 57 + .../data/plot_training_test_examples.g | 10 - .../data/scatter_plot_training_examples.png | Bin 227154 -> 105380 bytes .../data/test_examples.txt | 102 - .../data/training_examples.txt | 3486 +++------------- .../neural_network.g | 256 +- .../predict/predict.g | 29 - .../predict/scatter_plot_predictions.png | Bin 133096 -> 0 bytes .../scatter_plot_predictions.png | Bin 0 -> 57490 bytes examples/NeuralNetwork_CrossEntropy/README.md | 4 - .../neural_network_tf.cpython-310.pyc | Bin 1587 -> 0 bytes .../data/create_train_test_examples.g | 64 - .../data/generate_examples.g | 74 + .../data/plot_training_test_examples.g | 10 - .../data/scatter_plot_training_examples.png | Bin 299506 -> 118695 bytes .../data/test_examples.txt | 102 - .../data/training_examples.txt | 3500 +++-------------- .../neural_network.g | 256 +- .../neural_network.py | 89 - .../predict/predict.g | 30 - .../predict/scatter_plot_predictions.png | Bin 167119 -> 0 bytes .../scatter_plot_predictions.png | Bin 0 -> 64435 bytes .../data/create_train_test_examples.g | 41 - .../data/generate_examples.g | 37 + .../data/scatter_plot_training_examples.png | Bin 0 -> 538633 bytes .../data/test_examples.txt | 22 - .../data/training_examples.txt | 200 +- .../neural_network.g | 133 +- examples/NeuralNetworks.g | 286 ++ examples/OneEpochUpdateLens.g | 388 ++ examples/generate_supported_cap_operations.g | 57 + gap/CategoryOfLenses.autogen.gd | 50 + gap/CategoryOfLenses.gd | 172 +- gap/CategoryOfLenses.gi | 87 +- ...CategoryOfParametrisedMorphisms.autogen.gd | 35 + gap/CategoryOfParametrisedMorphisms.gd | 199 +- gap/CategoryOfParametrisedMorphisms.gi | 21 +- gap/CategoryOfSkeletalSmoothMaps.gd | 36 - gap/Expressions.gd | 84 +- gap/Expressions.gi | 18 +- gap/FitParameters.gd | 218 +- gap/FitParameters.gi | 47 +- gap/MethodRecord.Declarations.autogen.gd | 6 +- gap/MethodRecord.Installations.autogen.gi | 2 +- gap/MethodRecord.gd | 28 +- gap/MethodRecord.gi | 10 +- gap/NeuralNetworks.gd | 110 +- gap/NeuralNetworks.gi | 39 +- gap/SkeletalCategoryOfSmoothMaps.autogen.gd | 106 + gap/SkeletalCategoryOfSmoothMaps.gd | 258 ++ ...aps.gi => SkeletalCategoryOfSmoothMaps.gi} | 241 +- gap/Tools.gd | 125 +- gap/Tools.gi | 243 +- init.g | 22 +- makedoc.g | 17 +- makedoc_with_overfull_hbox_warnings.g | 11 +- makefile | 8 +- read.g | 20 +- stats.514058 | 1 + tst/100_LoadPackage.tst | 6 +- tst/example-in-readme-1.tst | 6 +- tst/example-in-readme-2.tst | 14 +- tst/functor.tst | 8 +- tst/neural-network-1.tst | 2 +- tst/neural-network-2.tst | 2 +- tst/simplify-expression-using-python.tst | 60 + tst/testall.g | 4 +- 84 files changed, 5105 insertions(+), 7529 deletions(-) create mode 100644 doc/sigmoid.png rename examples/NeuralNetwork_BinaryCrossEntropy/{neural_network.py => .neural_network.py} (100%) delete mode 100644 examples/NeuralNetwork_BinaryCrossEntropy/__pycache__/neural_network.cpython-310.pyc delete mode 100644 examples/NeuralNetwork_BinaryCrossEntropy/data/create_train_test_examples.g create mode 100644 examples/NeuralNetwork_BinaryCrossEntropy/data/generate_examples.g delete mode 100644 examples/NeuralNetwork_BinaryCrossEntropy/data/plot_training_test_examples.g delete mode 100644 examples/NeuralNetwork_BinaryCrossEntropy/data/test_examples.txt delete mode 100644 examples/NeuralNetwork_BinaryCrossEntropy/predict/predict.g delete mode 100644 examples/NeuralNetwork_BinaryCrossEntropy/predict/scatter_plot_predictions.png create mode 100644 examples/NeuralNetwork_BinaryCrossEntropy/scatter_plot_predictions.png delete mode 100644 examples/NeuralNetwork_CrossEntropy/README.md delete mode 100644 examples/NeuralNetwork_CrossEntropy/__pycache__/neural_network_tf.cpython-310.pyc delete mode 100644 examples/NeuralNetwork_CrossEntropy/data/create_train_test_examples.g create mode 100644 examples/NeuralNetwork_CrossEntropy/data/generate_examples.g delete mode 100644 examples/NeuralNetwork_CrossEntropy/data/plot_training_test_examples.g delete mode 100644 examples/NeuralNetwork_CrossEntropy/data/test_examples.txt delete mode 100644 examples/NeuralNetwork_CrossEntropy/neural_network.py delete mode 100644 examples/NeuralNetwork_CrossEntropy/predict/predict.g delete mode 100644 examples/NeuralNetwork_CrossEntropy/predict/scatter_plot_predictions.png create mode 100644 examples/NeuralNetwork_CrossEntropy/scatter_plot_predictions.png delete mode 100644 examples/NeuralNetwork_QuadraticLoss/data/create_train_test_examples.g create mode 100644 examples/NeuralNetwork_QuadraticLoss/data/generate_examples.g create mode 100644 examples/NeuralNetwork_QuadraticLoss/data/scatter_plot_training_examples.png delete mode 100644 examples/NeuralNetwork_QuadraticLoss/data/test_examples.txt create mode 100644 examples/NeuralNetworks.g create mode 100644 examples/OneEpochUpdateLens.g create mode 100644 examples/generate_supported_cap_operations.g create mode 100644 gap/CategoryOfLenses.autogen.gd create mode 100644 gap/CategoryOfParametrisedMorphisms.autogen.gd delete mode 100644 gap/CategoryOfSkeletalSmoothMaps.gd create mode 100644 gap/SkeletalCategoryOfSmoothMaps.autogen.gd create mode 100644 gap/SkeletalCategoryOfSmoothMaps.gd rename gap/{CategoryOfSkeletalSmoothMaps.gi => SkeletalCategoryOfSmoothMaps.gi} (89%) create mode 100644 stats.514058 create mode 100644 tst/simplify-expression-using-python.tst diff --git a/.github/workflows/Tests.yml b/.github/workflows/Tests.yml index 590e512..867e0b6 100644 --- a/.github/workflows/Tests.yml +++ b/.github/workflows/Tests.yml @@ -47,8 +47,7 @@ jobs: cp -a $GITHUB_WORKSPACE /home/gap/.gap/pkg/ - name: Prepare environment run: | - cp ./GradientDescentForCAP/dev/ci_gaprc /home/gap/.gap/gaprc - git clone --depth 1 -vv https://github.com/gap-packages/AutoDoc.git + cp ./GradientBasedLearningForCAP/dev/ci_gaprc /home/gap/.gap/gaprc git clone --depth 1 -vv https://github.com/homalg-project/homalg_project.git git clone --depth 1 -vv https://github.com/homalg-project/CAP_project.git git clone --depth 1 -vv https://github.com/homalg-project/CategoricalTowers.git @@ -59,18 +58,20 @@ jobs: if [ -d "CAP_project/CompilerForCAP" ]; then make -C "CAP_project/CompilerForCAP" doc; fi if [ -d "CAP_project/MonoidalCategories" ]; then make -C "CAP_project/MonoidalCategories" doc; fi if [ -d "CAP_project/CartesianCategories" ]; then make -C "CAP_project/CartesianCategories" doc; fi + if [ -d "CAP_project/AdditiveClosuresForCAP" ]; then make -C "CAP_project/AdditiveClosuresForCAP" doc; fi if [ -d "CAP_project/FreydCategoriesForCAP" ]; then make -C "CAP_project/FreydCategoriesForCAP" doc; fi if [ -d "HigherHomologicalAlgebra/ToolsForHigherHomologicalAlgebra" ]; then make -C "HigherHomologicalAlgebra/ToolsForHigherHomologicalAlgebra" doc; fi if [ -d "homalg_project/homalg" ]; then make -C "homalg_project/homalg" doc; fi if [ -d "homalg_project/Modules" ]; then make -C "homalg_project/Modules" doc; fi if [ -d "CategoricalTowers/ToolsForCategoricalTowers" ]; then make -C "CategoricalTowers/ToolsForCategoricalTowers" doc; fi if [ -d "CategoricalTowers/Toposes" ]; then make -C "CategoricalTowers/Toposes" doc; fi - - name: Test GradientDescentForCAP + if [ -d "GradientBasedLearningForCAP" ]; then make -C "GradientBasedLearningForCAP" doc; fi + - name: Test GradientBasedLearningForCAP run: | - make -C GradientDescentForCAP --trace -j $(nproc) --output-sync ci-test + make -C GradientBasedLearningForCAP --trace -j $(nproc) --output-sync ci-test - name: Release package or simulate release run: | - cd GradientDescentForCAP + cd GradientBasedLearningForCAP python3 dev/process_coverage.py git config --global user.name "Bot" git config --global user.email "empty" @@ -88,5 +89,5 @@ jobs: env: CODECOV_TOKEN: ${{ secrets.CODECOV_TOKEN }} run: | - cd GradientDescentForCAP + cd GradientBasedLearningForCAP ./dev/upload_codecov.sh diff --git a/.gitignore b/.gitignore index ed793a1..528d633 100644 --- a/.gitignore +++ b/.gitignore @@ -31,4 +31,4 @@ /doc/*.xml /doc_tmp/ /stats -/tst/machinelearningforcap*.tst +/tst/gradientbasedlearningforcap*.tst diff --git a/PackageInfo.g b/PackageInfo.g index 1ce60b9..b97a933 100644 --- a/PackageInfo.g +++ b/PackageInfo.g @@ -1,5 +1,5 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # This file contains package meta data. For additional information on # the meaning and correct usage of these fields, please consult the @@ -8,9 +8,9 @@ # SetPackageInfo( rec( -PackageName := "GradientDescentForCAP", -Subtitle := "Exploring categorical machine learning in CAP", -Version := "2025.11-01", +PackageName := "GradientBasedLearningForCAP", +Subtitle := "Gradient Based Learning via Category Theory", +Version := "2026.01-01", Date := (function ( ) if IsBound( GAPInfo.SystemEnvironment.GAP_PKG_RELEASE_DATE ) then return GAPInfo.SystemEnvironment.GAP_PKG_RELEASE_DATE; else return Concatenation( ~.Version{[ 1 .. 4 ]}, "-", ~.Version{[ 6, 7 ]}, "-01" ); fi; end)( ), License := "GPL-2.0-or-later", @@ -34,13 +34,13 @@ Persons := [ # BEGIN URLS SourceRepository := rec( Type := "git", - URL := "https://github.com/homalg-project/GradientDescentForCAP", + URL := "https://github.com/homalg-project/GradientBasedLearningForCAP", ), IssueTrackerURL := Concatenation( ~.SourceRepository.URL, "/issues" ), -PackageWWWHome := "https://homalg-project.github.io/pkg/GradientDescentForCAP", -PackageInfoURL := "https://homalg-project.github.io/GradientDescentForCAP/PackageInfo.g", -README_URL := "https://homalg-project.github.io/GradientDescentForCAP/README.md", -ArchiveURL := Concatenation( "https://github.com/homalg-project/GradientDescentForCAP/releases/download/v", ~.Version, "/GradientDescentForCAP-", ~.Version ), +PackageWWWHome := "https://homalg-project.github.io/pkg/GradientBasedLearningForCAP", +PackageInfoURL := "https://homalg-project.github.io/GradientBasedLearningForCAP/PackageInfo.g", +README_URL := "https://homalg-project.github.io/GradientBasedLearningForCAP/README.md", +ArchiveURL := Concatenation( "https://github.com/homalg-project/GradientBasedLearningForCAP/releases/download/v", ~.Version, "/GradientBasedLearningForCAP-", ~.Version ), # END URLS ArchiveFormats := ".tar.gz .zip", @@ -58,16 +58,16 @@ Status := "dev", AbstractHTML := "", PackageDoc := rec( - BookName := "GradientDescentForCAP", + BookName := "GradientBasedLearningForCAP", ArchiveURLSubset := ["doc"], HTMLStart := "doc/chap0.html", PDFFile := "doc/manual.pdf", SixFile := "doc/manual.six", - LongTitle := "Exploring categorical machine learning in CAP", + LongTitle := "Gradient Based Learning via Category Theory", ), Dependencies := rec( - GAP := ">= 4.15.1", + GAP := ">= 4.13.0", NeededOtherPackages := [ [ "GAPDoc", ">= 1.5" ], [ "CAP", ">= 2024.09-12" ], diff --git a/README.md b/README.md index e55de5c..17dece7 100644 --- a/README.md +++ b/README.md @@ -1,7 +1,7 @@ -# GradientDescentForCAP [![View code][code-img]][code-url] +# GradientBasedLearningForCAP [![View code][code-img]][code-url] -### Exploring categorical machine learning in CAP +### Gradient Based Learning via Category Theory | Documentation | Latest Release | Build Status | Code Coverage | | ------------- | -------------- | ------------ | ------------- | @@ -22,25 +22,25 @@ To obtain current versions of all dependencies, `git clone` (or `git pull` to up | 3. | [**CategoricalTowers**](https://github.com/homalg-project/CategoricalTowers#readme) | https://github.com/homalg-project/CategoricalTowers.git | [html-img]: https://img.shields.io/badge/🔗%20HTML-stable-blue.svg -[html-url]: https://homalg-project.github.io/GradientDescentForCAP/doc/chap0_mj.html +[html-url]: https://homalg-project.github.io/GradientBasedLearningForCAP/doc/chap0_mj.html [pdf-img]: https://img.shields.io/badge/🔗%20PDF-stable-blue.svg -[pdf-url]: https://homalg-project.github.io/GradientDescentForCAP/download_pdf.html +[pdf-url]: https://homalg-project.github.io/GradientBasedLearningForCAP/download_pdf.html -[version-img]: https://img.shields.io/endpoint?url=https://homalg-project.github.io/GradientDescentForCAP/badge_version.json&label=🔗%20version&color=yellow -[version-url]: https://homalg-project.github.io/GradientDescentForCAP/view_release.html +[version-img]: https://img.shields.io/endpoint?url=https://homalg-project.github.io/GradientBasedLearningForCAP/badge_version.json&label=🔗%20version&color=yellow +[version-url]: https://homalg-project.github.io/GradientBasedLearningForCAP/view_release.html -[date-img]: https://img.shields.io/endpoint?url=https://homalg-project.github.io/GradientDescentForCAP/badge_date.json&label=🔗%20released%20on&color=yellow -[date-url]: https://homalg-project.github.io/GradientDescentForCAP/view_release.html +[date-img]: https://img.shields.io/endpoint?url=https://homalg-project.github.io/GradientBasedLearningForCAP/badge_date.json&label=🔗%20released%20on&color=yellow +[date-url]: https://homalg-project.github.io/GradientBasedLearningForCAP/view_release.html -[tests-img]: https://github.com/homalg-project/GradientDescentForCAP/actions/workflows/Tests.yml/badge.svg?branch=master -[tests-url]: https://github.com/homalg-project/GradientDescentForCAP/actions/workflows/Tests.yml?query=branch%3Amaster +[tests-img]: https://github.com/homalg-project/GradientBasedLearningForCAP/actions/workflows/Tests.yml/badge.svg?branch=master +[tests-url]: https://github.com/homalg-project/GradientBasedLearningForCAP/actions/workflows/Tests.yml?query=branch%3Amaster -[codecov-img]: https://codecov.io/gh/homalg-project/GradientDescentForCAP/branch/master/graph/badge.svg -[codecov-url]: https://app.codecov.io/gh/homalg-project/GradientDescentForCAP +[codecov-img]: https://codecov.io/gh/homalg-project/GradientBasedLearningForCAP/branch/master/graph/badge.svg +[codecov-url]: https://app.codecov.io/gh/homalg-project/GradientBasedLearningForCAP [code-img]: https://img.shields.io/badge/-View%20code-blue?logo=github -[code-url]: https://github.com/homalg-project/GradientDescentForCAP#top +[code-url]: https://github.com/homalg-project/GradientBasedLearningForCAP#top ### Running the Package with Docker @@ -93,7 +93,7 @@ In this example, we consider a training dataset consisting of the three points We aim to compute a line that fits $\mathcal{D}$. ```julia -gap> LoadPackage( "GradientDescentForCAP" ); +gap> LoadPackage( "GradientBasedLearningForCAP" ); true gap> Para := CategoryOfParametrisedMorphisms( SkeletalSmoothMaps ); @@ -288,7 +288,7 @@ To facilitate this classification, we use a one-hot encoding scheme for the labe That is, the training set (the set of labeled exmaples) is a finite subset of $\mathbb{R}^2 \times \mathbb{R}^3 \simeq \mathbb{R}^5$. ```julia -gap> LoadPackage( "GradientDescentForCAP" ); +gap> LoadPackage( "GradientBasedLearningForCAP" ); true gap> Para := CategoryOfParametrisedMorphisms( SkeletalSmoothMaps ); diff --git a/dev/.release b/dev/.release index 518dacf..148f382 100644 --- a/dev/.release +++ b/dev/.release @@ -14,7 +14,7 @@ sed "s;Date := .*;Date := \"$(date -I)\",;" PackageInfo.g > PackageInfo.g.bak mv PackageInfo.g.bak PackageInfo.g # replace links to packages which are possibly referenced in the documentation, keep this in sync with `Tests.yml.j2` -for package in CAP_project/CAP CAP_project/CompilerForCAP CAP_project/MonoidalCategories CAP_project/CartesianCategories CAP_project/FreydCategoriesForCAP HigherHomologicalAlgebra/ToolsForHigherHomologicalAlgebra homalg_project/homalg homalg_project/Modules CategoricalTowers/ToolsForCategoricalTowers CategoricalTowers/Toposes; do +for package in CAP_project/CAP CAP_project/CompilerForCAP CAP_project/MonoidalCategories CAP_project/CartesianCategories CAP_project/AdditiveClosuresForCAP CAP_project/FreydCategoriesForCAP HigherHomologicalAlgebra/ToolsForHigherHomologicalAlgebra homalg_project/homalg homalg_project/Modules CategoricalTowers/ToolsForCategoricalTowers CategoricalTowers/Toposes GradientBasedLearningForCAP; do # adjust links to other manuals # Note that we cannot use sed's `-i` option for in-place editing, as diff --git a/dev/upload_codecov.sh b/dev/upload_codecov.sh index c3f8a70..ff3812c 100755 --- a/dev/upload_codecov.sh +++ b/dev/upload_codecov.sh @@ -21,9 +21,16 @@ curl -O https://uploader.codecov.io/latest/linux/codecov.SHA256SUM.sig gpgv codecov.SHA256SUM.sig codecov.SHA256SUM shasum -a 256 -c codecov.SHA256SUM +# read the token +if [ -z "$CODECOV_TOKEN" ]; then + echo -e "\033[0;33mCODECOV_TOKEN is not set. Proceeding without token.\033[0m" +else + echo -e "\033[0;32mUsing CODECOV_TOKEN from environment variable.\033[0m" +fi + # execute chmod +x codecov while ! ./codecov -Z -v -s ../ -t $CODECOV_TOKEN; do - echo "Codecov upload failed, retrying in 20s" - sleep 20 + echo "Codecov upload failed, retrying in 60s" + sleep 60 done diff --git a/doc/Doc.autodoc b/doc/Doc.autodoc index 1650c0d..f3ea502 100644 --- a/doc/Doc.autodoc +++ b/doc/Doc.autodoc @@ -1 +1,101 @@ -@Chapter Examples and Tests +@Chapter Introduction + +This package provides tools for exploring categorical machine learning using the CAP (Categories, Algorithms, Programming) system. +It implements automatic differentiation using the lens pattern and provides constructs for building and training neural networks. + +@Section Overview + +The package implements the following main concepts: + +* **Examples**: Examples for creating and training neural networks. + +* **Expressions**: A symbolic expression system for representing mathematical formulas. + +* **Skeletal Category of Smooth Maps**: A category where objects are Euclidean spaces $\mathbb{R}^n$ and morphisms are smooth maps with their Jacobian matrices. + +* **Category of Parametrised Morphisms**: A category that represents morphisms with learnable parameters, used to model neural network layers. + +* **Neural Networks**: High-level operations for constructing and training neural networks. + +* **Category of Lenses**: A category that models bidirectional data flow, essential for backpropagation in neural networks. + +* **Fitting Parameters**: Explain how to learn the parameters in order to minimize a parametrised morphism. + +* **CAP Operation**: The new categorical operations needed in this package. + +* **Tools**: Few GAP operations and helper functions. + + +@Chapter Examples for neural networks + +@Section Binary-class neural network with binary cross-entropy loss function +@Section Multi-class neural network with cross-entropy loss function +@Section Neural network with quadratic loss function + +@Chapter Expressions + +@Section Constructors +@Section Attributes +@Section Operations +@Section Global Functions +@Section Examples +@Section GAP Categories + +@Chapter Skeletal Category of Smooth Maps + +@Section Constructors +@Section Attributes +@Section Operations +@Section Available Smooth Maps +@Section Supported CAP Operations +@Section Examples +@Section GAP Categories + +@Chapter Category of Parametrised Morphisms + +@Section Definition +@Section Constructors +@Section Attributes +@Section Operations +@Section Available Parametrised Morphisms +@Section Supported CAP Operations +@Section Examples +@Section GAP Categories + +@Chapter Neural Networks + +@Section Definition +@Section Operations +@Section Examples + +@Chapter Category of Lenses + +@Section Definition +@Section Constructors +@Section Attributes +@Section Operations +@Section Available Lenses +@Section Optimizers +@Section Supported CAP Operations +@Section Examples +@Section GAP Categories + + +@Chapter Fitting Parameters + +@Section Introduction +@Section Notes on Batching +@Section Operations +@Section Examples + +@Chapter CAP Operations for GradientBasedLearningForCAP + +@Section Basic Operations +@Section Add-Methods + +@Chapter Tools + +@Section Mathematical Operations +@Section List Operations +@Section Helper Functions +@Section Python Integration \ No newline at end of file diff --git a/doc/clean b/doc/clean index 54b6da2..ffe1f90 100755 --- a/doc/clean +++ b/doc/clean @@ -1,2 +1,2 @@ -#!/bin/bash +#!/bin/sh rm -f *.{aux,bbl,blg,brf,css,html,idx,ilg,ind,js,lab,log,out,pdf,pnr,six,tex,toc,txt,xml,xml.bib} diff --git a/doc/sigmoid.png b/doc/sigmoid.png new file mode 100644 index 0000000000000000000000000000000000000000..4f82699dbe88e9d0fdb874f6183a670116295980 GIT binary patch literal 82406 zcmeFa2UM2l)-C!OV@x#BpaO~lHl#yTRHTRnR6s!~q9C9k0@6e}($sAf5U_xViXcUb z6hV45k)og=AiX7gBGLuv_0IK~81Fgv{9})M?-}RZJ2vD0cS8Ki`#$e_)|zv!xgI?a 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"GradientDescentForCAP" ); +LoadPackage( "GradientBasedLearningForCAP" ); #! @Example -Smooth := CategoryOfSkeletalSmoothMaps( ); +Smooth := SkeletalCategoryOfSmoothMaps( ); #! SkeletalSmoothMaps Lenses := CategoryOfLenses( Smooth ); #! CategoryOfLenses( SkeletalSmoothMaps ) @@ -29,11 +29,11 @@ f := MorphismConstructor( Lenses, A, [ get, put ], B ); #! (ℝ^1, ℝ^2) -> (ℝ^3, ℝ^4) defined by: #! #! Get Morphism: -#! ---------- +#! ------------ #! ℝ^1 -> ℝ^3 #! #! Put Morphism: -#! ---------- +#! ------------ #! ℝ^5 -> ℝ^2 MorphismDatum( f ); #! [ ℝ^1 -> ℝ^3, ℝ^5 -> ℝ^2 ] @@ -62,11 +62,11 @@ id_A := IdentityMorphism( Lenses, A ); #! (ℝ^1, ℝ^2) -> (ℝ^1, ℝ^2) defined by: #! #! Get Morphism: -#! ---------- +#! ------------ #! ℝ^1 -> ℝ^1 #! #! Put Morphism: -#! ---------- +#! ------------ #! ℝ^3 -> ℝ^2 Display( id_A ); #! (ℝ^1, ℝ^2) -> (ℝ^1, ℝ^2) defined by: @@ -93,71 +93,71 @@ f1 := RandomMorphism( A, B, 5 ); #! (ℝ^1, ℝ^2) -> (ℝ^3, ℝ^4) defined by: #! #! Get Morphism: -#! ---------- +#! ------------ #! ℝ^1 -> ℝ^3 #! #! Put Morphism: -#! ---------- +#! ------------ #! ℝ^5 -> ℝ^2 f2 := RandomMorphism( A, B, 5 ); #! (ℝ^1, ℝ^2) -> (ℝ^3, ℝ^4) defined by: #! #! Get Morphism: -#! ---------- +#! ------------ #! ℝ^1 -> ℝ^3 #! #! Put Morphism: -#! ---------- +#! ------------ #! ℝ^5 -> ℝ^2 f3 := RandomMorphism( A, B, 5 ); #! (ℝ^1, ℝ^2) -> (ℝ^3, ℝ^4) defined by: #! #! Get Morphism: -#! ---------- +#! ------------ #! ℝ^1 -> ℝ^3 #! #! Put Morphism: -#! ---------- +#! ------------ #! ℝ^5 -> ℝ^2 f1_f2 := TensorProductOnMorphisms( Lenses, f1, f2 ); #! (ℝ^2, ℝ^4) -> (ℝ^6, ℝ^8) defined by: #! #! Get Morphism: -#! ---------- +#! ------------ #! ℝ^2 -> ℝ^6 #! #! Put Morphism: -#! ---------- +#! ------------ #! ℝ^10 -> ℝ^4 f2_f3 := TensorProductOnMorphisms( Lenses, f2, f3 ); #! (ℝ^2, ℝ^4) -> (ℝ^6, ℝ^8) defined by: #! #! Get Morphism: -#! ---------- +#! ------------ #! ℝ^2 -> ℝ^6 #! #! Put Morphism: -#! ---------- +#! ------------ #! ℝ^10 -> ℝ^4 t1 := TensorProductOnMorphisms( Lenses, f1_f2, f3 ); #! (ℝ^3, ℝ^6) -> (ℝ^9, ℝ^12) defined by: #! #! Get Morphism: -#! ---------- +#! ------------ #! ℝ^3 -> ℝ^9 #! #! Put Morphism: -#! ---------- +#! ------------ #! ℝ^15 -> ℝ^6 t2 := TensorProductOnMorphisms( Lenses, f1, f2_f3 ); #! (ℝ^3, ℝ^6) -> (ℝ^9, ℝ^12) defined by: #! #! Get Morphism: -#! ---------- +#! ------------ #! ℝ^3 -> ℝ^9 #! #! Put Morphism: -#! ---------- +#! ------------ #! ℝ^15 -> ℝ^6 IsCongruentForMorphisms( t1, t2 ); #! true @@ -205,126 +205,183 @@ Display( PreCompose( Braiding( A, B ), BraidingInverse( A, B ) ) ); #! ‣ x8 #! ‣ x9 #! ‣ x10 -R := EmbeddingIntoCategoryOfLenses( Smooth, Lenses ); +#! @EndExample + +#! @Section Operations + +#! @Example +Smooth := SkeletalCategoryOfSmoothMaps( ); +#! SkeletalSmoothMaps +Lenses := CategoryOfLenses( Smooth ); +#! CategoryOfLenses( SkeletalSmoothMaps ) +R := ReverseDifferentialLensFunctor( Smooth, Lenses ); #! Embedding functor into category of lenses SourceOfFunctor( R ); #! SkeletalSmoothMaps RangeOfFunctor( R ); #! CategoryOfLenses( SkeletalSmoothMaps ) -f := Smooth.Softmax( 2 ); +f := DirectProductFunctorial( [ Smooth.Power(3), Smooth.Power(2) ] ); #! ℝ^2 -> ℝ^2 Display( f ); #! ℝ^2 -> ℝ^2 -#! -#! ‣ Exp( x1 ) / (Exp( x1 ) + Exp( x2 )) -#! ‣ Exp( x2 ) / (Exp( x1 ) + Exp( x2 )) -Rf := ApplyFunctor( R, f ); +#! +#! ‣ x1 ^ 3 +#! ‣ x2 ^ 2 +f := PreCompose( f, Smooth.Sum(2) ); +#! ℝ^2 -> ℝ^1 +Display( f ); +#! ℝ^2 -> ℝ^1 +#! +#! ‣ x1 ^ 3 + x2 ^ 2 +Rf := ApplyFunctor( R, f );; +Display( Rf ); +#! (ℝ^2, ℝ^2) -> (ℝ^1, ℝ^1) defined by: +#! +#! Get Morphism: +#! ------------ +#! ℝ^2 -> ℝ^1 +#! +#! ‣ x1 ^ 3 + x2 ^ 2 +#! +#! Put Morphism: +#! ------------ +#! ℝ^3 -> ℝ^2 +#! +#! ‣ x3 * (1 * (3 * x1 ^ 2) + 0) +#! ‣ x3 * (0 + 1 * (2 * x2 ^ 1)) +#! @EndExample + +#! @Section Optimizers + +#! @Example +Smooth := SkeletalCategoryOfSmoothMaps( ); +#! SkeletalSmoothMaps +Lenses := CategoryOfLenses( Smooth ); +#! CategoryOfLenses( SkeletalSmoothMaps ) +optimizer := Lenses.GradientDescentOptimizer( :learning_rate := 0.01 )( 2 ); #! (ℝ^2, ℝ^2) -> (ℝ^2, ℝ^2) defined by: -#! +#! #! Get Morphism: -#! ---------- +#! ------------ #! ℝ^2 -> ℝ^2 -#! +#! #! Put Morphism: -#! ---------- +#! ------------ #! ℝ^4 -> ℝ^2 -Display( Rf ); +dummy_input := [ "theta_1", "theta_2", "g1", "g2" ];; +dummy_input := CreateContextualVariables( dummy_input ); +#! [ theta_1, theta_2, g1, g2 ] +Display( optimizer : dummy_input := dummy_input ); #! (ℝ^2, ℝ^2) -> (ℝ^2, ℝ^2) defined by: -#! +#! #! Get Morphism: #! ------------ #! ℝ^2 -> ℝ^2 -#! -#! ‣ Exp( x1 ) / (Exp( x1 ) + Exp( x2 )) -#! ‣ Exp( x2 ) / (Exp( x1 ) + Exp( x2 )) -#! +#! +#! ‣ theta_1 +#! ‣ theta_2 #! Put Morphism: #! ------------ #! ℝ^4 -> ℝ^2 -#! -#! ‣ x3 * -#! ((Exp( x1 ) + Exp( x2 ) - Exp( x1 )) * (Exp( x1 ) / (Exp( x1 ) + Exp( x2 )) ^ 2)) -#! + x4 * ((- Exp( x1 )) * (Exp( x2 ) / (Exp( x1 ) + Exp( x2 )) ^ 2)) -#! ‣ x3 * ((- Exp( x2 )) * (Exp( x1 ) / (Exp( x1 ) + Exp( x2 )) ^ 2)) + -#! x4 * -#! ((Exp( x1 ) + Exp( x2 ) - Exp( x2 )) * (Exp( x2 ) / (Exp( x1 ) + Exp( x2 )) ^ 2)) -Display( Lenses.GradientDescentOptimizer( :learning_rate := 0.01 )( 2 ) ); -#! (ℝ^2, ℝ^2) -> (ℝ^2, ℝ^2) defined by: -#! +#! +#! ‣ theta_1 + 0.01 * g1 +#! ‣ theta_2 + 0.01 * g2 +optimizer := Lenses.GradientDescentWithMomentumOptimizer( + :learning_rate := 0.01, momentum := 0.9 )( 2 ); +#! (ℝ^4, ℝ^4) -> (ℝ^2, ℝ^2) defined by: +#! #! Get Morphism: #! ------------ -#! ℝ^2 -> ℝ^2 -#! -#! ‣ x1 -#! ‣ x2 -#! +#! ℝ^4 -> ℝ^2 +#! #! Put Morphism: #! ------------ +#! ℝ^6 -> ℝ^4 +dummy_input := [ "s1", "s2", "theta_1", "theta_2", "g1", "g2" ];; +dummy_input := CreateContextualVariables( dummy_input ); +#! [ s1, s2, theta_1, theta_2, g1, g2 ] +Display( optimizer : dummy_input := dummy_input ); +#! (ℝ^4, ℝ^4) -> (ℝ^2, ℝ^2) defined by: +#! +#! Get Morphism: +#! ------------ #! ℝ^4 -> ℝ^2 -#! -#! ‣ x1 + 0.01 * x3 -#! ‣ x2 + 0.01 * x4 -Display( Lenses.GradientDescentWithMomentumOptimizer( - :learning_rate := 0.01, momentum := 0.9 )( 2 ) ); +#! +#! ‣ theta_1 +#! ‣ theta_2 +#! Put Morphism: +#! ------------ +#! ℝ^6 -> ℝ^4 +#! +#! ‣ 0.9 * s1 + 0.01 * g1 +#! ‣ 0.9 * s2 + 0.01 * g2 +#! ‣ theta_1 + (0.9 * s1 + 0.01 * g1) +#! ‣ theta_2 + (0.9 * s2 + 0.01 * g2) +optimizer := Lenses.AdagradOptimizer( :learning_rate := 0.01 )( 2 ); #! (ℝ^4, ℝ^4) -> (ℝ^2, ℝ^2) defined by: -#! +#! #! Get Morphism: #! ------------ #! ℝ^4 -> ℝ^2 -#! -#! ‣ x3 -#! ‣ x4 -#! +#! #! Put Morphism: #! ------------ #! ℝ^6 -> ℝ^4 -#! -#! ‣ 0.9 * x1 + 0.01 * x5 -#! ‣ 0.9 * x2 + 0.01 * x6 -#! ‣ x3 + (0.9 * x1 + 0.01 * x5) -#! ‣ x4 + (0.9 * x2 + 0.01 * x6) -Display( Lenses.AdagradOptimizer( :learning_rate := 0.01 )( 2 ) ); +Display( optimizer : dummy_input := dummy_input ); #! (ℝ^4, ℝ^4) -> (ℝ^2, ℝ^2) defined by: -#! +#! #! Get Morphism: #! ------------ #! ℝ^4 -> ℝ^2 -#! -#! ‣ x3 -#! ‣ x4 -#! +#! +#! ‣ theta_1 +#! ‣ theta_2 #! Put Morphism: #! ------------ #! ℝ^6 -> ℝ^4 -#! -#! ‣ x1 + x5 ^ 2 -#! ‣ x2 + x6 ^ 2 -#! ‣ x3 + 0.01 * x5 / (1.e-07 + Sqrt( x1 + x5 ^ 2 )) -#! ‣ x4 + 0.01 * x6 / (1.e-07 + Sqrt( x2 + x6 ^ 2 )) -Display( Lenses.AdamOptimizer( - :learning_rate := 0.01, beta_1 := 0.9, beta_2 := 0.999 )( 2 ) ); +#! +#! ‣ s1 + g1 ^ 2 +#! ‣ s2 + g2 ^ 2 +#! ‣ theta_1 + 0.01 * g1 / (1.e-07 + Sqrt( s1 + g1 ^ 2 )) +#! ‣ theta_2 + 0.01 * g2 / (1.e-07 + Sqrt( s2 + g2 ^ 2 )) +optimizer := Lenses.AdamOptimizer( + :learning_rate := 0.01, beta_1 := 0.9, beta_2 := 0.999 )( 2 ); #! (ℝ^7, ℝ^7) -> (ℝ^2, ℝ^2) defined by: #! #! Get Morphism: #! ------------ #! ℝ^7 -> ℝ^2 -#! -#! ‣ x6 -#! ‣ x7 -#! #! Put Morphism: #! ------------ #! ℝ^9 -> ℝ^7 -#! -#! ‣ x1 + 1 -#! ‣ 0.9 * x2 + 0.1 * x8 -#! ‣ 0.9 * x3 + 0.1 * x9 -#! ‣ 0.999 * x4 + 0.001 * x8 ^ 2 -#! ‣ 0.999 * x5 + 0.001 * x9 ^ 2 -#! ‣ x6 + 0.01 / (1 - 0.999 ^ x1) -#! * ((0.9 * x2 + 0.1 * x8) / -#! (1.e-07 + Sqrt( (0.999 * x4 + 0.001 * x8 ^ 2) / (1 - 0.999 ^ x1) ))) -#! ‣ x7 + 0.01 / (1 - 0.999 ^ x1) -#! * ((0.9 * x3 + 0.1 * x9) / -#! (1.e-07 + Sqrt( (0.999 * x5 + 0.001 * x9 ^ 2) / (1 - 0.999 ^ x1) ))) +dummy_input := + [ "t", "m1", "m2", "v1", "v2", "theta_1", "theta_2", "g1", "g2" ];; +dummy_input := CreateContextualVariables( dummy_input ); +#! [ t, m1, m2, v1, v2, theta_1, theta_2, g1, g2 ] +Display( optimizer : dummy_input := dummy_input ); +#! (ℝ^7, ℝ^7) -> (ℝ^2, ℝ^2) defined by: +#! +#! Get Morphism: +#! ------------ +#! ℝ^7 -> ℝ^2 +#! +#! ‣ theta_1 +#! ‣ theta_2 +#! Put Morphism: +#! ------------ +#! ℝ^9 -> ℝ^7 +#! +#! ‣ t + 1 +#! ‣ 0.9 * m1 + 0.1 * g1 +#! ‣ 0.9 * m2 + 0.1 * g2 +#! ‣ 0.999 * v1 + 0.001 * g1 ^ 2 +#! ‣ 0.999 * v2 + 0.001 * g2 ^ 2 +#! ‣ theta_1 + 0.01 / +#! (1 - 0.999 ^ t) * +#! ((0.9 * m1 + 0.1 * g1) / +#! (1.e-07 + Sqrt( (0.999 * v1 + 0.001 * g1 ^ 2) / (1 - 0.999 ^ t) ))) +#! ‣ theta_2 + 0.01 / +#! (1 - 0.999 ^ t) * +#! ((0.9 * m2 + 0.1 * g2) / +#! (1.e-07 + Sqrt( (0.999 * v2 + 0.001 * g2 ^ 2) / (1 - 0.999 ^ t) ))) #! @EndExample diff --git a/examples/CategoryOfParametrisedMorphisms.g b/examples/CategoryOfParametrisedMorphisms.g index 18cfb43..01bd350 100644 --- a/examples/CategoryOfParametrisedMorphisms.g +++ b/examples/CategoryOfParametrisedMorphisms.g @@ -1,14 +1,18 @@ -#! @Chapter Examples and Tests +#! @Chapter Category of Parametrised Morphisms -#! @Section Category of Parametrised Morphisms +#! @Section Examples -LoadPackage( "GradientDescentForCAP" ); +LoadPackage( "GradientBasedLearningForCAP" ); #! @Example -Smooth := CategoryOfSkeletalSmoothMaps( ); +Smooth := SkeletalCategoryOfSmoothMaps( ); #! SkeletalSmoothMaps Para := CategoryOfParametrisedMorphisms( Smooth ); #! CategoryOfParametrisedMorphisms( SkeletalSmoothMaps ) +Display( Para ); +#! A CAP category with name CategoryOfParametrisedMorphisms( SkeletalSmoothMaps ): +#! +#! 12 primitive operations were used to derive 21 operations for this category R1 := Smooth.( 1 ); #! ℝ^1 R2 := Smooth.( 2 ); @@ -19,109 +23,114 @@ R1 / Para; #! ℝ^1 Para.( 1 ); #! ℝ^1 -IsEqualForObjects( Para.( 1 ), R1 / Para ); +Para.( 1 ) = R1 / Para; #! true -f := Smooth.Softmax( 3 ); -#! ℝ^3 -> ℝ^3 -f := MorphismConstructor( Para, R1 / Para, [ R2, f ], R3 / Para ); -#! ℝ^1 -> ℝ^3 defined by: -#! -#! Underlying Object: -#! ----------------- -#! ℝ^2 -#! -#! Underlying Morphism: -#! ------------------- -#! ℝ^3 -> ℝ^3 +f := Para.Sin;; Display( f ); -#! ℝ^1 -> ℝ^3 defined by: +#! ℝ^1 -> ℝ^1 defined by: #! #! Underlying Object: #! ----------------- -#! ℝ^2 +#! ℝ^0 #! #! Underlying Morphism: #! ------------------- -#! ℝ^3 -> ℝ^3 -#! -#! ‣ Exp( x1 ) / (Exp( x1 ) + Exp( x2 ) + Exp( x3 )) -#! ‣ Exp( x2 ) / (Exp( x1 ) + Exp( x2 ) + Exp( x3 )) -#! ‣ Exp( x3 ) / (Exp( x1 ) + Exp( x2 ) + Exp( x3 )) -IsWellDefined( f ); -#! true -r := DirectProductFunctorial( Smooth, [ Smooth.Sqrt, Smooth.Cos ] ); -#! ℝ^2 -> ℝ^2 -Display( r ); -#! ℝ^2 -> ℝ^2 -#! -#! ‣ Sqrt( x1 ) -#! ‣ Cos( x2 ) -g := ReparametriseMorphism( f, r ); -#! ℝ^1 -> ℝ^3 defined by: -#! +#! ℝ^1 -> ℝ^1 +#! +#! ‣ Sin( x1 ) +p1 := ProjectionInFactorOfDirectProduct( Smooth, [ R3, R3 ], 1 );; +Display( p1 ); +#! ℝ^6 -> ℝ^3 +#! +#! ‣ x1 +#! ‣ x2 +#! ‣ x3 +p2 := ProjectionInFactorOfDirectProduct( Smooth, [ R3, R3 ], 2 );; +Display( p2 ); +#! ℝ^6 -> ℝ^3 +#! +#! ‣ x4 +#! ‣ x5 +#! ‣ x6 +m := MultiplicationForMorphisms( p1, p2 ); +#! ℝ^6 -> ℝ^3 +Display( m ); +#! ℝ^6 -> ℝ^3 +#! +#! ‣ x1 * x4 +#! ‣ x2 * x5 +#! ‣ x3 * x6 +h := MorphismConstructor( Para, Para.(3), [ R3, m ], Para.(3) );; +Display( h ); +#! ℝ^3 -> ℝ^3 defined by: +#! #! Underlying Object: #! ----------------- -#! ℝ^2 -#! +#! ℝ^3 +#! #! Underlying Morphism: #! ------------------- -#! ℝ^3 -> ℝ^3 -Display( g ); -#! ℝ^1 -> ℝ^3 defined by: -#! +#! ℝ^6 -> ℝ^3 +#! +#! ‣ x1 * x4 +#! ‣ x2 * x5 +#! ‣ x3 * x6 +dummy_input := CreateContextualVariables( + [ "w1", "w2", "w3", "z1", "z2", "z3" ] ); +#! [ w1, w2, w3, z1, z2, z3 ] +Display( h : dummy_input := dummy_input ); +#! ℝ^3 -> ℝ^3 defined by: +#! #! Underlying Object: #! ----------------- -#! ℝ^2 -#! +#! ℝ^3 +#! #! Underlying Morphism: #! ------------------- +#! ℝ^6 -> ℝ^3 +#! +#! ‣ w1 * z1 +#! ‣ w2 * z2 +#! ‣ w3 * z3 +r := DirectProductFunctorial( Smooth, + [ Smooth.Sqrt, Smooth.Log, Smooth.Exp ] );; +Display( r ); #! ℝ^3 -> ℝ^3 #! -#! ‣ Exp( Sqrt( x1 ) ) / (Exp( Sqrt( x1 ) ) + Exp( Cos( x2 ) ) + Exp( x3 )) -#! ‣ Exp( Cos( x2 ) ) / (Exp( Sqrt( x1 ) ) + Exp( Cos( x2 ) ) + Exp( x3 )) -#! ‣ Exp( x3 ) / (Exp( Sqrt( x1 ) ) + Exp( Cos( x2 ) ) + Exp( x3 )) -l := Para.AffineTransformation( 3, 2 ); -#! ℝ^3 -> ℝ^2 defined by: -#! -#! Underlying Object: -#! ----------------- -#! ℝ^8 -#! -#! Underlying Morphism: -#! ------------------- -#! ℝ^11 -> ℝ^2 -h := PreCompose( g, l ); -#! ℝ^1 -> ℝ^2 defined by: -#! -#! Underlying Object: -#! ----------------- -#! ℝ^10 -#! -#! Underlying Morphism: -#! ------------------- -#! ℝ^11 -> ℝ^2 -Display( h ); -#! ℝ^1 -> ℝ^2 defined by: -#! +#! ‣ Sqrt( x1 ) +#! ‣ Log( x2 ) +#! ‣ Exp( x3 ) +g := ReparametriseMorphism( h, r );; +Display( g : dummy_input := dummy_input ); +#! ℝ^3 -> ℝ^3 defined by: +#! #! Underlying Object: #! ----------------- -#! ℝ^10 -#! +#! ℝ^3 +#! #! Underlying Morphism: #! ------------------- -#! ℝ^11 -> ℝ^2 -#! -#! ‣ x1 * (Exp( Sqrt( x9 ) ) / (Exp( Sqrt( x9 ) ) + Exp( Cos( x10 ) ) + Exp( x11 ))) -#! + x2 * (Exp( Cos( x10 ) ) / (Exp( Sqrt( x9 ) ) + Exp( Cos( x10 ) ) + Exp( x11 ))) -#! + x3 * (Exp( x11 ) / (Exp( Sqrt( x9 ) ) + Exp( Cos( x10 ) ) + Exp( x11 ))) + x4 -#! ‣ x5 * (Exp( Sqrt( x9 ) ) / (Exp( Sqrt( x9 ) ) + Exp( Cos( x10 ) ) + Exp( x11 ))) -#! + x6 * (Exp( Cos( x10 ) ) / (Exp( Sqrt( x9 ) ) + Exp( Cos( x10 ) ) + Exp( x11 ))) -#! + x7 * (Exp( x11 ) / (Exp( Sqrt( x9 ) ) + Exp( Cos( x10 ) ) + Exp( x11 ))) + x8 -constants := [ 0.91, 0.24, 0.88, 0.59, 0.67, 0.05, 0.85, 0.31, 0.76, 0.04 ];; -r := Smooth.Constant( constants ); -#! ℝ^0 -> ℝ^10 -t := ReparametriseMorphism( h, r ); -#! ℝ^1 -> ℝ^2 defined by: +#! ℝ^6 -> ℝ^3 +#! +#! ‣ Sqrt( w1 ) * z1 +#! ‣ Log( w2 ) * z2 +#! ‣ Exp( w3 ) * z3 +#! @EndExample + +#! Let us illustrate the natural embedding functor from the category of smooth maps into +#! the category of parametrised morphisms. + +#! @Example +Smooth := SkeletalCategoryOfSmoothMaps( ); +#! SkeletalSmoothMaps +Para := CategoryOfParametrisedMorphisms( Smooth ); +#! CategoryOfParametrisedMorphisms( SkeletalSmoothMaps ) +iota := NaturalEmbedding( Smooth, Para ); +#! Natural embedding into category of parametrised morphisms +ApplyFunctor( iota, Smooth.( 1 ) ); +#! ℝ^1 +psi := ApplyFunctor( iota, Smooth.Sum( 2 ) ); +#! ℝ^2 -> ℝ^1 defined by: #! #! Underlying Object: #! ----------------- @@ -129,9 +138,9 @@ t := ReparametriseMorphism( h, r ); #! #! Underlying Morphism: #! ------------------- -#! ℝ^1 -> ℝ^2 -Display( t ); -#! ℝ^1 -> ℝ^2 defined by: +#! ℝ^2 -> ℝ^1 +Display( psi ); +#! ℝ^2 -> ℝ^1 defined by: #! #! Underlying Object: #! ----------------- @@ -139,41 +148,51 @@ Display( t ); #! #! Underlying Morphism: #! ------------------- -#! ℝ^1 -> ℝ^2 -#! -#! ‣ 0.91 * (2.39116 / (5.10727 + Exp( x1 ))) + 0.24 * (2.71611 / (5.10727 + Exp( x1 ))) -#! + 0.88 * (Exp( x1 ) / (5.10727 + Exp( x1 ))) + 0.59 -#! ‣ 0.67 * (2.39116 / (5.10727 + Exp( x1 ))) + 0.05 * (2.71611 / (5.10727 + Exp( x1 ))) -#! + 0.85 * (Exp( x1 ) / (5.10727 + Exp( x1 ))) + 0.31 -s := SimplifyMorphism( t, infinity ); -#! ℝ^1 -> ℝ^2 defined by: +#! ℝ^2 -> ℝ^1 #! +#! ‣ x1 + x2 +#! @EndExample + +#! @Section Available Parametrised Morphisms + +#! @Example +Smooth := SkeletalCategoryOfSmoothMaps( ); +#! SkeletalSmoothMaps +Para := CategoryOfParametrisedMorphisms( Smooth ); +#! CategoryOfParametrisedMorphisms( SkeletalSmoothMaps ) +Display( Para.Relu( 2 ) ); +#! ℝ^2 -> ℝ^2 defined by: +#! #! Underlying Object: #! ----------------- #! ℝ^0 -#! +#! #! Underlying Morphism: #! ------------------- -#! ℝ^1 -> ℝ^2 -Display( s ); -#! ℝ^1 -> ℝ^2 defined by: -#! +#! ℝ^2 -> ℝ^2 +#! +#! ‣ Relu( x1 ) +#! ‣ Relu( x2 ) +dummy_input := DummyInputForAffineTransformation( 3, 2, "w", "b", "z" ); +#! [ w1_1, w2_1, w3_1, b_1, w1_2, w2_2, w3_2, b_2, z1, z2, z3 ] +affine_transformation := Para.AffineTransformation( 3, 2 );; +Display( affine_transformation : dummy_input := dummy_input ); +#! ℝ^3 -> ℝ^2 defined by: +#! #! Underlying Object: #! ----------------- -#! ℝ^0 -#! +#! ℝ^8 +#! #! Underlying Morphism: #! ------------------- -#! ℝ^1 -> ℝ^2 -#! -#! ‣ (1.47 * Exp( x1 ) + 5.84111) / (Exp( x1 ) + 5.10727) -#! ‣ (1.16 * Exp( x1 ) + 3.32114) / (Exp( x1 ) + 5.10727) -iota := NaturalEmbeddingIntoCategoryOfParametrisedMorphisms( Smooth, Para ); -#! Natural embedding into category of parametrised morphisms -ApplyFunctor( iota, Smooth.( 1 ) ); -#! ℝ^1 -psi := ApplyFunctor( iota, Smooth.Sum( 2 ) ); -#! ℝ^2 -> ℝ^1 defined by: +#! ℝ^11 -> ℝ^2 +#! +#! ‣ w1_1 * z1 + w2_1 * z2 + w3_1 * z3 + b_1 +#! ‣ w1_2 * z1 + w2_2 * z2 + w3_2 * z3 + b_2 +"Let us convert these 2 togits to probabilities via softmax layer.";; +softmax_layer := Para.Softmax( 2 );; +Display( softmax_layer ); +#! ℝ^2 -> ℝ^2 defined by: #! #! Underlying Object: #! ----------------- @@ -181,17 +200,30 @@ psi := ApplyFunctor( iota, Smooth.Sum( 2 ) ); #! #! Underlying Morphism: #! ------------------- -#! ℝ^2 -> ℝ^1 -Print( DisplayString( psi ) ); -#! ℝ^2 -> ℝ^1 defined by: +#! ℝ^2 -> ℝ^2 #! +#! ‣ Exp( x1 ) / (Exp( x1 ) + Exp( x2 )) +#! ‣ Exp( x2 ) / (Exp( x1 ) + Exp( x2 )) +probs := PreCompose( affine_transformation, softmax_layer );; +Display( probs : dummy_input := dummy_input ); +#! ℝ^3 -> ℝ^2 defined by: +#! #! Underlying Object: #! ----------------- -#! ℝ^0 -#! +#! ℝ^8 +#! #! Underlying Morphism: #! ------------------- -#! ℝ^2 -> ℝ^1 -#! -#! ‣ x1 + x2 +#! ℝ^11 -> ℝ^2 +#! +#! ‣ Exp( w1_1 * z1 + w2_1 * z2 + w3_1 * z3 + b_1 ) +#! / (Exp( w1_1 * z1 + w2_1 * z2 + w3_1 * z3 + b_1 ) +#! + Exp( w1_2 * z1 + w2_2 * z2 + w3_2 * z3 + b_2 )) +#! ‣ Exp( w1_2 * z1 + w2_2 * z2 + w3_2 * z3 + b_2 ) +#! / (Exp( w1_1 * z1 + w2_1 * z2 + w3_1 * z3 + b_1 ) +#! + Exp( w1_2 * z1 + w2_2 * z2 + w3_2 * z3 + b_2 )) +parameters := [ 0.91, 0.24, 0.88, 0.59, 0.67, 0.05, 0.85, 0.31 ];; +logits := [ 1.0, 2.0, 3.0 ];; +Eval( probs, [ parameters, logits ] ); +#! [ 0.729088, 0.270912 ] #! @EndExample diff --git a/examples/CategoryOfSkeletalSmoothMaps.g b/examples/CategoryOfSkeletalSmoothMaps.g index a1ea661..e5d09ed 100644 --- a/examples/CategoryOfSkeletalSmoothMaps.g +++ b/examples/CategoryOfSkeletalSmoothMaps.g @@ -1,12 +1,26 @@ -#! @Chapter Examples and Tests +#! @Chapter Skeletal Category of Smooth Maps -#! @Section Category of Smooth Maps +#! @Section Examples -LoadPackage( "GradientDescentForCAP" ); +LoadPackage( "GradientBasedLearningForCAP" ); + +#! In this example, we demonstrate the usage of the SkeletalCategoryOfSmoothMaps +#! by constructing objects and morphisms, performing various operations, and +#! utilizing built-in functions. #! @Example -Smooth := CategoryOfSkeletalSmoothMaps( ); +Smooth := SkeletalCategoryOfSmoothMaps( ); #! SkeletalSmoothMaps +Display( Smooth ); +#! A CAP category with name SkeletalSmoothMaps: +#! +#! 49 primitive operations were used to derive 92 operations for this category wh\ +#! ich algorithmically +#! * IsCartesianCategory +#! * IsLinearCategoryOverCommutativeRing +#! * IsSymmetricMonoidalCategory +#! and furthermore mathematically +#! * IsStrictMonoidalCategory R2 := ObjectConstructor( Smooth, 2 ); #! ℝ^2 R2 = Smooth.2; @@ -30,7 +44,7 @@ Display( f ); #! #! ‣ x1 ^ 2 + Sin( x2 ) #! ‣ Exp( x1 ) + 3 * x2 -dummy_input := ConvertToExpressions( [ "x1", "x2" ] ); +dummy_input := CreateContextualVariables( [ "x1", "x2" ] ); #! [ x1, x2 ] Map( f )( dummy_input ); #! [ x1 ^ 2 + Sin( x2 ), Exp( x1 ) + 3 * x2 ] @@ -130,6 +144,16 @@ PreCompose( Smooth, u, p1 ) = f; #! true PreCompose( Smooth, u, p2 ) = g; #! true +Display( f ); +#! ℝ^2 -> ℝ^2 +#! +#! ‣ x1 ^ 2 + Sin( x2 ) +#! ‣ Exp( x1 ) + 3 * x2 +Display( g ); +#! ℝ^2 -> ℝ^3 +#! ‣ 3 * x1 +#! ‣ Exp( x2 ) +#! ‣ x1 ^ 3 + Log( x2 ) d := DirectProductFunctorial( Smooth, [ f, g ] ); #! ℝ^4 -> ℝ^5 Display( d ); @@ -250,19 +274,145 @@ Display( Smooth.SoftmaxCrossEntropyLoss( 3 ) ); #! ‣ ((Log( Exp( x1 ) + Exp( x2 ) + Exp( x3 ) ) - x1) * x4 #! + (Log( Exp( x1 ) + Exp( x2 ) + Exp( x3 ) ) - x2) * x5 #! + (Log( Exp( x1 ) + Exp( x2 ) + Exp( x3 ) ) - x3) * x6) / 3 -Display( Smooth.AffineTransformation( 2, 3 ) ); +#! @EndExample + +#! In this example, we illustrate the various convenience ways to construct smooth maps. For instance, the smooth map: +#! @BeginLatexOnly +#! \[ +#! \begin{array}{l} +#! g:\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}\\ +#! (x_{1}, x_{2}) \mapsto \left( 3 x_{1}, e^{x_{2}}, x_{1}^{3} + \log{\left(x_{2}\right)} \right) \\ +#! \end{array} +#! \] +#! whose Jacobian Matrix is: +#! \[ +#! \left( \begin{array}{ll} +#! 3 & 0 \\ +#! 0 & e^{x_{2}} \\ +#! 3 x_{1}^{2} & \frac{1}{x_{2}} +#! \end{array} \right) +#! \] +#! @EndLatexOnly + +#! @Example +Smooth := SkeletalCategoryOfSmoothMaps( ); +#! SkeletalSmoothMaps +dummy_input := CreateContextualVariables( [ "x1", "x2" ] ); +#! [ x1, x2 ] +g1 := MorphismConstructor( Smooth, + Smooth.2, + Pair( + x -> [ 3 * x[1], Exp( x[2] ), x[1] ^ 3 + Log( x[2] ) ], + x -> [ [ 3, 0 ], + [ 0, Exp( x[2] ) ], + [ 3 * x[1] ^ 2, 1 / x[2] ] ] ), + Smooth.3 ); +#! ℝ^2 -> ℝ^3 +Display( g1 ); +#! ℝ^2 -> ℝ^3 +#! ‣ 3 * x1 +#! ‣ Exp( x2 ) +#! ‣ x1 ^ 3 + Log( x2 ) +Map( g1 )( dummy_input ); +#! [ 3 * x1, Exp( x2 ), x1 ^ 3 + Log( x2 ) ] +JacobianMatrix( g1 )( dummy_input ); +#! [ [ 3, 0 ], [ 0, Exp( x2 ) ], [ 3 * x1 ^ 2, 1 / x2 ] ] +"# Use python to compute the Jacobian Matrix";; +g2 := SmoothMap( Smooth, + Smooth.2, + x -> [ 3 * x[1], Exp( x[2] ), x[1] ^ 3 + Log( x[2] ) ], + Smooth.3, + true + ); +#! ℝ^2 -> ℝ^3 +g1 = g2; +#! true +"# Use python to compute the Jacobian Matrix";; +g3 := SmoothMap( Smooth, + Smooth.2, + [ "3 * x1", "Exp( x2 )", "x1 ^ 3 + Log( x2 )" ], + Smooth.3, + true + ); +#! ℝ^2 -> ℝ^3 +g3 = g1; +#! true +"# Lazy evaluation of the Jacobian Matrix";; +g4 := SmoothMap( Smooth, + Smooth.2, + [ "3 * x1", "Exp( x2 )", "x1 ^ 3 + Log( x2 )" ], + Smooth.3, + false + ); +#! ℝ^2 -> ℝ^3 +g4 = g1; +#! true +Map( g4 )( dummy_input ); +#! [ 3 * x1, Exp( x2 ), x1 ^ 3 + Log( x2 ) ] +J := JacobianMatrix( g4 )( [ 1, 2 ] ); +#! [ [ Diff( [ "x1", "x2" ], "3 * x1", 1 )( [ 1, 2 ] ), +#! Diff( [ "x1", "x2" ], "3 * x1", 2 )( [ 1, 2 ] ) ], +#! [ Diff( [ "x1", "x2" ], "Exp( x2 )", 1 )( [ 1, 2 ] ), +#! Diff( [ "x1", "x2" ], "Exp( x2 )", 2 )( [ 1, 2 ] ) ], +#! [ Diff( [ "x1", "x2" ], "x1 ^ 3 + Log( x2 )", 1 )( [ 1, 2 ] ), +#! Diff( [ "x1", "x2" ], "x1 ^ 3 + Log( x2 )", 2 )( [ 1, 2 ] ) ] ] +"# Evaluation uses python to compute the derivatives";; +Eval( J[3][2] ); +#! 1/2 +Diff( [ "x1", "x2" ], "x1 ^ 3 + Log( x2 )", 2 )( [ 1, 2 ] ); +#! 1/2 +Eval( J ); +#! [ [ 3, 0 ], [ 0, 7.38906 ], [ 3, 1/2 ] ] +#! @EndExample + +#! In the following example, we demonstrate the construction and usage of an affine transformation +#! morphism within the SkeletalCategoryOfSmoothMaps. + +#! @Example +Smooth := SkeletalCategoryOfSmoothMaps( ); +#! SkeletalSmoothMaps +affine_trans := Smooth.AffineTransformation( 2, 3 ); #! ℝ^11 -> ℝ^3 -#! +Display( affine_trans ); +#! ℝ^11 -> ℝ^3 +#! #! ‣ x1 * x10 + x2 * x11 + x3 #! ‣ x4 * x10 + x5 * x11 + x6 #! ‣ x7 * x10 + x8 * x11 + x9 -Display( Smooth.PolynomialTransformation( 2, 3, 2 ) ); -#! ℝ^20 -> ℝ^3 -#! -#! ‣ x1 * x19 ^ 2 + x2 * (x19 ^ 1 * x20 ^ 1) + x3 * x19 ^ 1 -#! + x4 * x20 ^ 2 + x5 * x20 ^ 1 + x6 * 1 -#! ‣ x7 * x19 ^ 2 + x8 * (x19 ^ 1 * x20 ^ 1) + x9 * x19 ^ 1 -#! + x10 * x20 ^ 2 + x11 * x20 ^ 1 + x12 * 1 -#! ‣ x13 * x19 ^ 2 + x14 * (x19 ^ 1 * x20 ^ 1) + x15 * x19 ^ 1 -#! + x16 * x20 ^ 2 + x17 * x20 ^ 1 + x18 * 1 +dummy_input := DummyInputForAffineTransformation( 2, 3, "w", "b", "z" ); +#! [ w1_1, w2_1, b_1, w1_2, w2_2, b_2, w1_3, w2_3, b_3, z1, z2 ] +Map( affine_trans )( dummy_input ); +#! [ w1_1 * z1 + w2_1 * z2 + b_1, +#! w1_2 * z1 + w2_2 * z2 + b_2, +#! w1_3 * z1 + w2_3 * z2 + b_3 ] +JacobianMatrix( affine_trans )( dummy_input ); +#! [ [ z1, z2, 1, 0, 0, 0, 0, 0, 0, w1_1, w2_1 ], +#! [ 0, 0, 0, z1, z2, 1, 0, 0, 0, w1_2, w2_2 ], +#! [ 0, 0, 0, 0, 0, 0, z1, z2, 1, w1_3, w2_3 ] ] #! @EndExample + +#! @BeginLatexOnly +#! To view the affine transformation map and its Jacobian matrix in LaTeX format, we can use: +#! \begin{center} +#! \texttt{Show( LaTeXOutput( affine\_trans : dummy\_input := dummy\_input ) );} +#! \end{center} +#! which produces: +#! \[ +#! \begin{array}{c} +#! \mathbb{R}^{11}\rightarrow\mathbb{R}^{3}\\ +#! \hline \\ +#! \left( \begin{array}{l} +#! b_{1} + w_{1 1} z_{1} + w_{2 1} z_{2} \\ +#! b_{2} + w_{1 2} z_{1} + w_{2 2} z_{2} \\ +#! b_{3} + w_{1 3} z_{1} + w_{2 3} z_{2} +#! \end{array} \right)\\ +#! \\ +#! \hline \\ +#! \left( \begin{array}{lllllllllll} +#! z_{1} & z_{2} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & w_{1 1} & w_{2 1} \\ +#! 0 & 0 & 0 & z_{1} & z_{2} & 1 & 0 & 0 & 0 & w_{1 2} & w_{2 2} \\ +#! 0 & 0 & 0 & 0 & 0 & 0 & z_{1} & z_{2} & 1 & w_{1 3} & w_{2 3} +#! \end{array} \right) +#! \end{array} +#! \] +#! @EndLatexOnly diff --git a/examples/ComputingTheNextLocalMimima/next_local_minima.g b/examples/ComputingTheNextLocalMimima/next_local_minima.g index 59ffe25..6f45a40 100644 --- a/examples/ComputingTheNextLocalMimima/next_local_minima.g +++ b/examples/ComputingTheNextLocalMimima/next_local_minima.g @@ -1,9 +1,9 @@ -LoadPackage( "GradientDescentForCAP" ); +LoadPackage( "GradientBasedLearningForCAP" ); # the function f(x1,x2) = sin(x1)^2 + log(x2)^2 has local miminima at the points (πk, 1) where k ∈ ℤ -Smooth := CategoryOfSkeletalSmoothMaps( ); +Smooth := SkeletalCategoryOfSmoothMaps( ); Lenses := CategoryOfLenses( Smooth ); Para := CategoryOfParametrisedMorphisms( Smooth ); diff --git a/examples/Expressions.g b/examples/Expressions.g index 33347a3..a6170ca 100644 --- a/examples/Expressions.g +++ b/examples/Expressions.g @@ -1,16 +1,15 @@ -#! @Chapter Examples and Tests -#! @Section Expressions +#! @Chapter Expressions -LoadPackage( "MachineLearning" ); +LoadPackage( "GradientBasedLearningForCAP" );; + +#! @Section Examples #! @Example -vars := [ "x", "y", "z" ]; -#! [ "x", "y", "z" ] -e1 := Expression( vars, "x + Sin( y ) * Log( z )" ); -#! x + Sin( y ) * Log( z ) -e2 := Expression( vars, "( x * y + Sin( z ) ) ^ 2" ); -#! (x * y + Sin( z )) ^ 2 +e1 := Expression( ["x", "y"], "x + Sin( y )" ); +#! x + Sin( y ) +e2 := Expression( ["y", "z"], "( y + Sin( z ) ) ^ 2" ); +#! (y + Sin( z )) ^ 2 CategoriesOfObject( e1 ); #! [ "IsExtAElement", "IsNearAdditiveElement", "IsNearAdditiveElementWithZero", #! "IsNearAdditiveElementWithInverse", "IsAdditiveElement", "IsExtLElement", @@ -18,15 +17,28 @@ CategoriesOfObject( e1 ); KnownAttributesOfObject( e1 ); #! [ "String", "Variables" ] String( e1 ); -#! "x + Sin( y ) * Log( z )" +#! "x + Sin( y )" Variables( e1 ); -#! [ "x", "y", "z" ] +#! [ "x", "y" ] +Variables( e2 ); +#! [ "y", "z" ] e1 + e2; -#! x + Sin( y ) * Log( z ) + (x * y + Sin( z )) ^ 2 +#! x + Sin( y ) + (y + Sin( z )) ^ 2 e1 * e2; -#! (x + Sin( y ) * Log( z )) * (x * y + Sin( z )) ^ 2 +#! (x + Sin( y )) * (y + Sin( z )) ^ 2 e := Sin( e1 ) / e2; -#! Sin( x + Sin( y ) * Log( z ) ) / (x * y + Sin( z )) ^ 2 +#! Sin( x + Sin( y ) ) / (y + Sin( z )) ^ 2 +Variables( e ); +#! [ "x", "y", "z" ] +ConstantExpression( "tau" ); +#! #I MakeReadWriteGlobal: tau already read-write +#! tau +Variables( tau ); +#! [ ] +e3 := e1 * Sin( tau ); +#! (x + Sin( y )) * Sin( tau ) +Variables( e3 ); +#! [ "x", "y" ] f := AsFunction( e ); #! function( vec ) ... end Display( f ); @@ -36,16 +48,16 @@ Display( f ); #! x := vec[1]; #! y := vec[2]; #! z := vec[3]; -#! return Sin( x + Sin( y ) * Log( z ) ) / (x * y + Sin( z )) ^ 2; +#! return Sin( x + Sin( y ) ) / (y + Sin( z )) ^ 2; #! end x := [ 3, 2, 4 ]; #! [ 3, 2, 4 ] f( x ); -#! -0.032725 -dummy_input := ConvertToExpressions( [ "x1", "x2", "x3" ] ); +#! -0.449348 +dummy_input := CreateContextualVariables( [ "x1", "x2", "x3" ] ); #! [ x1, x2, x3 ] f( dummy_input ); -#! Sin( x1 + Sin( x2 ) * Log( x3 ) ) / (x1 * x2 + Sin( x3 )) ^ 2 +#! Sin( x1 + Sin( x2 ) ) / (x2 + Sin( x3 )) ^ 2 AssignExpressions( dummy_input ); #! #I MakeReadWriteGlobal: x1 already read-write #! #I MakeReadWriteGlobal: x2 already read-write @@ -56,23 +68,51 @@ Variables( x1 ); #! [ "x1", "x2", "x3" ] [ [ x1, x2 ] ] * [ [ x3 ], [ -x3 ] ]; #! [ [ x1 * x3 + x2 * (- x3) ] ] -e := Sin( x1 ) / Cos( x1 ) + Sin( x2 ) ^ 2 + Cos( x2 ) ^ 2; -#! Sin( x1 ) / Cos( x1 ) + Sin( x2 ) ^ 2 + Cos( x2 ) ^ 2 -SimplifyExpressionUsingPython( [ e ] ); -#! [ "Tan(x1) + 1" ] +#! @EndExample + +#! @Chapter Tools + +#! @Section Python Integration + +#! @Example +dummy_input := CreateContextualVariables( [ "a", "b", "c" ] ); +#! [ a, b, c ] +AssignExpressions( dummy_input );; +#! #I MakeReadWriteGlobal: a already read-write +#! #I MakeReadWriteGlobal: b already read-write +#! #I MakeReadWriteGlobal: c already read-write +e := Sin( a ) + Cos( b ); +#! Sin( a ) + Cos( b ) Diff( e, 1 )( dummy_input ); -#! Sin( x1 ) ^ 2 / Cos( x1 ) ^ 2 + 1 -LazyDiff( e, 1 )( dummy_input );; -# Diff( [ "x1", "x2", "x3" ], -# "(((Sin(x1))/(Cos(x1)))+((Sin(x2))^(2)))+((Cos(x2))^(2))", 1 )( [ x1, x2, x3 ] ); -JacobianMatrixUsingPython( [ x1*Cos(x2)+Exp(x3), x1*x2*x3 ], [ 1, 2, 3 ] ); -#! [ [ "Cos(x2)", "-x1*Sin(x2)", "Exp(x3)" ], [ "x2*x3", "x1*x3", "x1*x2" ] ] -JacobianMatrix( [ "x1", "x2", "x3" ], [ "x1*Cos(x2)+Exp(x3)", "x1*x2*x3" ], - [ 1, 2, 3 ] )(dummy_input); -#! [ [ Cos(x2), (-x1)*Sin(x2), Exp(x3) ], [ x2*x3, x1*x3, x1*x2 ] ] +#! Cos( a ) +LazyDiff( e, 1 )( dummy_input ); +#! Diff( [ "a", "b", "c" ], "(Sin(a))+(Cos(b))", 1 )( [ a, b, c ] ) +JacobianMatrixUsingPython( [ a*Cos(b)+Exp(c), a*b*c ], [ 1, 2, 3 ] ); +#! [ [ "Cos(b)", "-a*Sin(b)", "Exp(c)" ], [ "b*c", "a*c", "a*b" ] ] +JacobianMatrix( + [ "a", "b", "c" ], + [ "a*Cos(b)+Exp(c)", "a*b*c" ], + [ 1, 2, 3 ] )(dummy_input); +#! [ [ Cos(b), (-a)*Sin(b), Exp(c) ], [ b*c, a*c, a*b ] ] +SimplifyExpressionUsingPython( + [ "a", "b" ], + [ "Sin(a)^2 + Cos(a)^2", "Exp(Log(b))" ] ); +#! [ "1", "b" ] LaTeXOutputUsingPython( e ); -#! "\\frac{\\sin{\\left(x_{1} \\right)}}{\\cos{\\left(x_{1} \\right)}} -#! + \\sin^{2}{\\left(x_{2} \\right)} + \\cos^{2}{\\left(x_{2} \\right)}" +#! "\\sin{\\left(a \\right)} + \\cos{\\left(b \\right)}" +AsCythonFunction( [[ "x", "y" ], [ "z" ]], ["f", "g"], ["x*y", "Sin(z)"] );; +""" +It will produce output similar to the following lines: +$ cd /tmp/gaptempdirI6rq3l/ +$ python +>>> from cython_functions import f, g +>>> w = [ 2, 3 ] # or any other vector in R^2 +>>> f(w) +6.0 +""";; +#! @EndExample + +#! @Example sigmoid := Expression( [ "x" ], "Exp(x)/(1+Exp(x))" ); #! Exp( x ) / (1 + Exp( x )) sigmoid := AsFunction( sigmoid ); @@ -91,20 +131,15 @@ points := List( 0.1 * [ -20 .. 20 ], x -> [ x, sigmoid( [ x ] ) ] ); #! [ 1.2, 0.768525 ], [ 1.3, 0.785835 ], [ 1.4, 0.802184 ], [ 1.5, 0.817574 ], #! [ 1.6, 0.832018 ], [ 1.7, 0.845535 ], [ 1.8, 0.858149 ], [ 1.9, 0.869892 ], #! [ 2., 0.880797 ] ] -labels := List( points, point -> SelectBasedOnCondition( point[2] < 0.5, 0, 1 ) ); -#! [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -#! 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] +labels := List( points, point -> 0 );; ScatterPlotUsingPython( points, labels : size := "100", action := "save" );; -# e.g, dir("/tmp/gaptempdirX7Qsal/") -AsCythonFunction( [ [ "x", "y" ], [ "z" ] ], [ "f", "g" ], [ "x*y", "Sin(z)" ] );; -# e.g., -# cd /tmp/gaptempdirI6rq3l/ -# -# start python! -# -# from cython_functions import f, g; -# -# # w = [ 2 entries :) ] -# -# # f(w) #! @EndExample + +#! @BeginLatexOnly +#! \begin{figure}[ht] +#! \centering +#! \includegraphics[width=0.5\textwidth]{sigmoid.png} +#! \caption{Sigmoid function plot.} +#! \label{fig:sigmoid} +#! \end{figure} +#! @EndLatexOnly diff --git a/examples/NeuralNetwork_BinaryCrossEntropy/neural_network.py b/examples/NeuralNetwork_BinaryCrossEntropy/.neural_network.py similarity index 100% rename from examples/NeuralNetwork_BinaryCrossEntropy/neural_network.py rename to examples/NeuralNetwork_BinaryCrossEntropy/.neural_network.py diff --git a/examples/NeuralNetwork_BinaryCrossEntropy/__pycache__/neural_network.cpython-310.pyc b/examples/NeuralNetwork_BinaryCrossEntropy/__pycache__/neural_network.cpython-310.pyc deleted file mode 100644 index 92ce2949ece95858a8c332bf0c79c0e2bf38720c..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 1734 zcmb7EOK)366rQ>F+P?84b{yyByc}8{36c$nRaCX42~?F(gp>v4s+BeI96PtUugr{_ z+O(^bWrSElRVi5pA^rz{VRi_y$c{Ce_RMt>iXtH~SMxaE9DnCIj!UIH!SSi}y!%Uz zkUxy@_Ha@745$2y62b|m0r4pAR$zs;XH$dOffHuDOnP>LY?$+Mq3gM+oC(Ilyq6CP zULlpUK`|_OB}!nN=eT=heP8s-P(G=g60iI<@;k&KN5~tw9}~HLNGUPjH#mWwOh6UN zNA#3>lSblWJpY2gT8b6^g-t;TYB0r%yu`S*VG69f zZ>{ZfqkKlrENwME(zy#-Q9v|Z{_eK+;t*Q6t|Sjx{MCE{ezic*DkG}34$1D(GW_)>M^Fu6T$ z{5F2g5>dhJT5t}6X>c_>q`_rJ!CeM+MumB#qcDnBr!=av5e=<5q-JWQ_tTGN{C4Mj z4M8l%*eLV<5Oh(nh{s`DDlDd$F!xkvQsblN?N>Nse||r-S+H3|4JD7AMLN(GQ`W z0CjXO1cePHhxPlBZ(6Bw2!3x*$woI8jere6G=|oCV_5$3!cxOnT5p_xG72pVN!;0! 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-1) -# i.e., the common solutions of the inequalities: x2 <= 0, 1 - 2x1 + x2 >= 0, 1 + 2x1 + x2 >= 0 - -# class 2: everything else - -files := [ "training_examples.txt", "test_examples.txt" ]; -nr_examples := [ 3000, 100 ]; - -for i in [ 1, 2 ] do - - file := Filename( current_dir, files[i] ); - - PrintTo( file, "[\n" ); - - for j in [ 1 .. nr_examples[i] ] do - - # we want more centered training examples - x1 := Random( [ -0.01, 0.01 ] ) * Random( Concatenation ( [ 1 .. 30 ], [ 1 .. 30 ], [ 30 .. 100 ] ) ); - x2 := Random( [ -0.01, 0.01 ] ) * Random( Concatenation ( [ 1 .. 30 ], [ 1 .. 30 ], [ 30 .. 100 ] ) ); - - if (x1 - 0.5)^2 + (x2 - 0.5)^2 - 0.20 <= 0. or (x1 + 0.5)^2 + (x2 - 0.5)^2 - 0.20 <= 0. or (x2 <= 0. and 1 - 2 * x1 + x2 >= 0. and 1 + 2 * x1 + x2 >= 0.) then - - label := 1; - - else - - label := 0; - - fi; - - AppendTo( file, [ x1, x2 , label ], ",\n" ); - - od; - - AppendTo( file, "]" ); - -od; - -Display( "Done!" ); -QUIT; diff --git a/examples/NeuralNetwork_BinaryCrossEntropy/data/generate_examples.g b/examples/NeuralNetwork_BinaryCrossEntropy/data/generate_examples.g new file mode 100644 index 0000000..f456843 --- /dev/null +++ b/examples/NeuralNetwork_BinaryCrossEntropy/data/generate_examples.g @@ -0,0 +1,57 @@ +LoadPackage( "GradientBasedLearningForCAP" ); + +# create a file for the training dataset + +# we have 2 classes in the plan + +# class 0: +#! everything inside the circle: x1^2 + (x2-0.5)^2 - 0.16 = 0 +# i.e., the solutions of: x1^2 + (x2-0.5)^2 - 0.16 <= 0 + +# everything below the line: x2 = -0.5, +# i.e., the solutions of: x2 + 0.5 <= 0. + +# class 1: everything else + +files_name := "training_examples.txt"; +nr_examples := 500; + +file := Filename( DirectoryCurrent( ), files_name ); + +PrintTo( file, "[\n" ); + +for j in [ 1 .. nr_examples ] do + + x1 := Random( [ -0.01, 0.01 ] ) * Random( [ 1 .. 100 ] ); + x2 := Random( [ -0.01, 0.01 ] ) * Random( [ 1 .. 100 ] ); + + if x1^2 + (x2 - 0.5)^2 - 0.16 <= 0. then + + label := [ 0 ]; + + elif x2 + 0.5 <= 0. then + + label := [ 0 ]; + + else + + label := [ 1 ]; + + fi; + + AppendTo( file, Concatenation( [ x1, x2 ], label ), ",\n" ); + +od; + +AppendTo( file, "];" ); + + +# plotting the dataset +file := Filename( DirectoryCurrent( ), "training_examples.txt" ); +data := EvalString( IO_ReadUntilEOF( IO_File( file ) ) ); +x := List( data, e -> [ e[1], e[2] ] ); +y := List( data, e -> e[3] ); +ScatterPlotUsingPython( x, y ); + +Display( "Done!" ); +QUIT; diff --git a/examples/NeuralNetwork_BinaryCrossEntropy/data/plot_training_test_examples.g b/examples/NeuralNetwork_BinaryCrossEntropy/data/plot_training_test_examples.g deleted file mode 100644 index dfe10cd..0000000 --- a/examples/NeuralNetwork_BinaryCrossEntropy/data/plot_training_test_examples.g +++ /dev/null @@ -1,10 +0,0 @@ -LoadPackage( "GradientDescentForCAP" ); - -file := IO_File( "training_examples.txt" ); -#file := IO_File( "test_examples.txt" ); - -examples := EvalString( 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"neural_network.py" diff --git a/examples/NeuralNetwork_CrossEntropy/__pycache__/neural_network_tf.cpython-310.pyc b/examples/NeuralNetwork_CrossEntropy/__pycache__/neural_network_tf.cpython-310.pyc deleted file mode 100644 index 207e53905d7f844e07ac217d909573a156e3e59e..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 1587 zcmaJ>&2Jk;6yMnod+j)m>o~FFq^Z-E4;N~yIiUyvX;V-o6d~n6Srx4&o@Zy1{p!rP ziOtdK(c%nMrR2zwKZ67Rz+5@`!XJQ;;LSRTA{8;Coq6x~-oEeMWO1=*;M)5AhyI^A z!}vopHB8{J-=b~Xu#E}YjNM0u_~eujL+|hK0dih}TTq^m z3*uFw$_OK`O*{*)&Az#2c+1yVK}#~M2IP1JYOKf>rlwZ|3u+UBcL|meEwM#bp4u;L zZ`G@N4Q~yciQzS&dA<&Lw!|t^^o5=^@3LF0%9f|*{~|TEazkh1LdV@azn$Ghw7m(< z<2$g)R#|;OSYwKrqP&IKZ^2#Invgx?`^pQ`yXW19_t+ZTW5v4h#$Zjj|HgQ0PcWL7 zzoKLvHjm#wH!e{2*3>9s8;E^ygTIaZhd205sxFF{COq+uY(D;W~R10T+4l!hQW@=PkiXD0fZ;m8^KeR>v%!OuVG z_1ovO2O2e1VQwcFgDPzz6R0LrX@3dyy^E7ji zCcPt}iicG8j(mZx$Hm>tfOcB#j&kB*gi)Gts#YbIJmLK?IqjSPr=l~IK`6McrmDXq zj1mSYwf{Xi;LXEQ`YyrJv1ipMVof2?VJ{B_(tN+Rf<(1?cJ&VZ)f^kmTe zic=Qgxb1=H;q_(0pX`3qz4|7)`+(yY?t?r{_=$foAJ`{)M4!$_v>Rs{etfAS`yD;E zpJI82A$+F4Be92vYm`WdRPe!V7RkmxWvgb@NZE3&hLtlbR@JJKDsrxsv#O?RRY;T6 L@YK6%p`82+F$UZ+ diff --git a/examples/NeuralNetwork_CrossEntropy/data/create_train_test_examples.g b/examples/NeuralNetwork_CrossEntropy/data/create_train_test_examples.g deleted file mode 100644 index b047edd..0000000 --- a/examples/NeuralNetwork_CrossEntropy/data/create_train_test_examples.g +++ /dev/null @@ -1,64 +0,0 @@ - - -# locate the current dir -current_dir := DirectoryCurrent( ); - -# create a file for the training dataset - -# we have 4 classes in the plan - -# class 0: everything outside of classes 1, 2 or 4 - -# class 1: everything inside the circle: (x1-0.5)^2 + (x2-0.5)^2 - 0.20 = 0 -# i.e., the solutions of: (x1-0.5)^2 + (x2-0.5)^2 - 0.20 <= 0 - -# class 2: everything inside the circle: (x1+0.5)^2 + (x2-0.5)^2 - 0.20 = 0 -# i.e., the solutions of: (x1+0.5)^2 + (x2-0.5)^2 - 0.20 <= 0 - -# class 3: the polytop defined by the points (0.5, 0), (-0.5, 0), (0, -1) -# i.e., the common solutions of the inequalities: x2 <= 0, 1 - 2x1 + x2 >= 0, 1 + 2x1 + x2 >= 0 - -files := [ "training_examples.txt", "test_examples.txt" ]; -nr_examples := [ 3000, 100 ]; - -noise := 0.5; - -for i in [ 1, 2 ] do - - file := Filename( current_dir, files[i] ); - - PrintTo( file, "[\n" ); - - for j in [ 1 .. nr_examples[i] ] do - - x1 := Random( [ -0.01, 0.01 ] ) * Random( [ 1 .. 100 ] ); - x2 := Random( [ -0.01, 0.01 ] ) * Random( [ 1 .. 100 ] ); - - if (x1 - 0.5)^2 + (x2 - 0.5)^2 - 0.20 <= 0. then - - label := [ 0, 1, 0, 0 ]; - - elif (x1 + 0.5)^2 + (x2 - 0.5)^2 - 0.20 <= 0. then - - label := [ 0, 0, 1, 0 ]; - - elif x2 <= 0. and 1 - 2 * x1 + x2 >= 0. and 1 + 2 * x1 + x2 >= 0. then - - label := [ 0, 0, 0, 1 ]; - - else - - label := [ 1, 0, 0, 0 ]; - - fi; - - AppendTo( file, Concatenation( [ x1, x2 ], label ), ",\n" ); - - od; - - AppendTo( file, "]" ); - -od; - -Display( "Done!" ); -QUIT; diff --git a/examples/NeuralNetwork_CrossEntropy/data/generate_examples.g b/examples/NeuralNetwork_CrossEntropy/data/generate_examples.g new file mode 100644 index 0000000..03350dc --- /dev/null +++ b/examples/NeuralNetwork_CrossEntropy/data/generate_examples.g @@ -0,0 +1,74 @@ +LoadPackage( "GradientBasedLearningForCAP" ); + +# create a file for the training dataset + +# we have 3 classes in the plan + +# class 0 ~ [1, 0, 0]: +#! everything inside the circle: x1^2 + (x2-0.5)^2 - 0.16 = 0 +# i.e., the solutions of: x1^2 + (x2-0.5)^2 - 0.16 <= 0 +# class 1 ~ [0, 1, 0]: +# everything below the line: x2 = -0.5, +# i.e., the solutions of: x2 + 0.5 <= 0 +# class 2 ~ [0, 0, 1]: everything else + + +nr_examples := 500; +nr_examples_per_class := Int( nr_examples / 3 ); + +class_0_count := 0; +class_1_count := 0; +class_2_count := 0; + +training_examples := [ ]; + +while class_0_count < nr_examples_per_class do + x := Random( [ -0.01, 0.01 ] ) * Random( [ 1 .. 100 ] ); + y := Random( [ -0.01, 0.01 ] ) * Random( [ 1 .. 100 ] ); + if x^2 + (y - 0.5)^2 - 0.16 <= 0. then + Add( training_examples, Concatenation( [ x, y ], [ 1, 0, 0 ] ) ); + class_0_count := class_0_count + 1; + fi; +od;; + +while class_1_count < nr_examples_per_class do + x := Random( [ -0.01, 0.01 ] ) * Random( [ 1 .. 100 ] ); + y := Random( [ -0.01, 0.01 ] ) * Random( [ 1 .. 100 ] ); + if y + 0.5 <= 0. then + Add( training_examples, Concatenation( [ x, y ], [ 0, 1, 0 ] ) ); + class_1_count := class_1_count + 1; + fi; +od;; + +while class_2_count < nr_examples_per_class do + x := Random( [ -0.01, 0.01 ] ) * Random( [ 1 .. 100 ] ); + y := Random( [ -0.01, 0.01 ] ) * Random( [ 1 .. 100 ] ); + if not ( x^2 + (y - 0.5)^2 - 0.16 <= 0. or y + 0.5 <= 0. ) then + Add( training_examples, Concatenation( [ x, y ], [ 0, 0, 1 ] ) ); + class_2_count := class_2_count + 1; + fi; +od;; + + +# shuffle the training examples +training_examples := Shuffle( training_examples ); + +# write the training examples to a file +files_name := "training_examples.txt"; +file := Filename( DirectoryCurrent( ), files_name ); + +PrintTo( file, "[\n" ); +for example in training_examples do + AppendTo( file, example, ",\n" ); +od; +AppendTo( file, "];" ); + +# plotting the dataset +file := Filename( DirectoryCurrent( ), files_name ); +data := EvalString( IO_ReadUntilEOF( IO_File( file ) ) ); +x := List( data, e -> [ e[1], e[2] ] ); +y := List( data, e -> Position( e{[3,4,5]}, 1 ) - 1 ); +ScatterPlotUsingPython( x, y ); + +Display( "Done!" ); +QUIT; diff --git a/examples/NeuralNetwork_CrossEntropy/data/plot_training_test_examples.g b/examples/NeuralNetwork_CrossEntropy/data/plot_training_test_examples.g deleted file mode 100644 index 22af161..0000000 --- a/examples/NeuralNetwork_CrossEntropy/data/plot_training_test_examples.g +++ /dev/null @@ -1,10 +0,0 @@ -LoadPackage( "GradientDescentForCAP" ); - -file := IO_File( "training_examples.txt" ); -#file := IO_File( "test_examples.txt" ); - -examples := EvalString( IO_ReadUntilEOF( file 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This example demonstrates how to train a small feed-forward neural network +#! for a multi-class classification task using the $\texttt{GradientBasedLearningForCAP}$ +#! package. We employ the cross-entropy loss function and optimise the network +#! parameters with gradient descent. +#! +#! The dataset consists of points $(x_1, x_2) \in \mathbb{R}^2$ labelled by a +#! non-linear decision rule describing three regions that form +#! @BeginLatexOnly +#! \begin{itemize} +#! \item $\emph{class 0}$: +#! \[ +#! x_1^2 + (x_2 - 0.5)^2 \le 0.16 +#! \qquad\text{(inside a circle of radius $0.4$ centred at $(0,0.5)$)} +#! \] +#! \item \emph{class 1}: +#! \[ +#! x_2 \le -0.5 +#! \qquad\text{(below the horizontal line)} +#! \] +#! \item $\emph{class 2}$: everything else. +#! \end{itemize} +#! @EndLatexOnly + +#! @BeginLatexOnly +#! \begin{center} +#! \includegraphics[width=0.5\textwidth]{../examples/NeuralNetwork_CrossEntropy/data/scatter_plot_training_examples.png} +#! \end{center} +#! @EndLatexOnly + +#! We build a neural network with three hidden layers and a Softmax output, fit +#! it on the provided training examples for several epochs, and then evaluate +#! the trained model on a grid of input points to visualise the learned +#! decision regions. + +#! @BeginLatexOnly +#! Concretely, we choose three hidden layers, each with 6 neurons: +#! \[ +#! \texttt{hidden\_layers} = [6,6,6]. +#! \] +#! With input dimension \(2\) (representing point coordinates) and output dimension \(3\) (the probability of each class), the affine maps between +#! consecutive layers therefore have the following matrix dimensions (together +#! bias vectors): +#! \[ +#! \binom{W_1}{b_1} \in \mathbb{R}^{ 3 \times 6},\quad +#! \binom{W_2}{b_2} \in \mathbb{R}^{ 7 \times 6},\quad +#! \binom{W_3}{b_3} \in \mathbb{R}^{ 7 \times 6},\quad +#! \binom{W_4}{b_4} \in \mathbb{R}^{ 7 \times 3}. +#! \] +#! Equivalently, each layer computes for an input $a_k$ the output \(z_{k+1} := (a_k\;\;1)\binom{W_{k+1}}{b_{k+1}}=a_k W_{k+1} + b_{k+1}\), where +#! \(a_0 \in \mathbb{R}^2\), \(a_1,a_2,a_3 \in \mathbb{R}^6\), and the final output +#! lies in \(\mathbb{R}^3\). +#! The non-linear activation function ReLU is applied after each hidden layer. +#! And Softmax is applied after the final layer to obtain a probability +#! estimate for the classes. +#! \[ +#! a_0 \mapsto +#! \color{red}\mathbf{Softmax}\left( +#! \color{blue}\left( +#! \color{green}\mathbf{Relu}\left( +#! \color{yellow}\left( +#! \color{red}\mathbf{Relu}\left( +#! \color{blue}\left( +#! \color{green}\mathbf{Relu}\left( +#! \color{yellow}\left( +#! a_0\;\; 1 +#! \color{yellow}\right) +#! \binom{W_1}{b_1} +#! \color{green}\right) +#! \;\; 1 +#! \color{blue}\right) +#! \binom{W_2}{b_2} +#! \color{red}\right) +#! \;\; 1 +#! \color{yellow}\right) +#! \binom{W_3}{b_3} +#! \color{green}\right) +#! \;\; 1 +#! \color{blue}\right) +#! \binom{W_4}{b_4} +#! \color{red}\right) \in \mathbb{R}^3 +#! \] +#! The predicted class is the one with the highest probability. +#! That is, the total number of parameters (weights and biases) is \(123\). +#! After training, we obtain a weight vector \(w \in \mathbb{R}^{123}\). +#! The first $21$ entries of $w$ correspond to the concatenation of the columns of $\binom{W_4}{b_4}\in \mathbb{R}^{7\times 3}$, +#! the next $42$ entries correspond to the concatenation of the columns of $\binom{W_3}{b_3}\in \mathbb{R}^{7\times 6}$, and so on. +#! @EndLatexOnly + +#! @Example Smooth := SkeletalSmoothMaps; +#! SkeletalSmoothMaps Lenses := CategoryOfLenses( Smooth ); +#! CategoryOfLenses( SkeletalSmoothMaps ) Para := CategoryOfParametrisedMorphisms( Smooth ); - - -## The function we are trying minimize -f := LossMorphismOfNeuralNetwork( Para, 2, [ 5, 5 ], 4, "Softmax" ); - -## One epoch update of the parameters -optimizer := Lenses.GradientDescentOptimizer( : learning_rate := 0.01 ); - -training_examples_path := "data/training_examples.txt"; - -batch_size := 1; - -one_epoch_update := OneEpochUpdateLens( f, optimizer, training_examples_path, batch_size ); - -## Initialize the parameters and apply updates nr_epochs times - -# initial values for the parameters of the first affine transformation w1 (as 3x5 matrix) -w1 := [[-0.61789644, 0.56407845, -0.5965204, -0.85063416, 0.770488 ], - [-0.13079625, 0.47618425, 0.8807312, 0.24377191, 0.18529081], - [0., 0., 0., 0., 0. ] ]; - -# as vector -w1 := Concatenation( TransposedMat( w1 ) ); - -# initial weights for the parameters of the second affine transformation w2 (as 6x5 matrix) -w2 := -[[-0.52913845, 0.524745, 0.67446196, -0.13036567, -0.5108599 ], - [-0.12336099, 0.7475884, -0.18031466, 0.30409217, -0.5017855 ], - [-0.5523451, 0.74021363, -0.38746935, -0.2771675, 0.6162708 ], - [-0.24399745, 0.523523, 0.31327105, -0.5376833, -0.4945482 ], - [ 0.33063114, -0.10083395, 0.13537377, 0.671383, -0.2012583 ], - [ 0., 0., 0., 0., 0. ] ]; - -# as vector -w2 := Concatenation( TransposedMat( w2 ) ); - -# initial weights for the parameters of the third affine transformation w3 (as 6x4 matrix) -w3 := -[[-0.05885905, -0.81396204, 0.00370395, -0.42547446], - [-0.39928403, 0.56314194, 0.6614479 , 0.5060446 ], - [ 0.6662301, -0.2800727 , 0.1187852 , -0.27065504], - [ 0.15874296, -0.6039741 , -0.7533438 , -0.33242884], - [ 0.26578736, -0.45036432, -0.61879224, 0.8060001 ], - [ 0., 0., 0., 0., ] ]; - -# as vector -w3 := Concatenation( TransposedMat( w3 ) ); - -# creating a vector of initial values -w := Concatenation( [ w3, w2, w1 ] ); - -nr_epochs := 50; - -w := Fit( one_epoch_update, nr_epochs, w ); - -# After 50 epochs: -# w = [ 0.927523, -1.16534, 3.09767, 1.21408, 0.666635, -1.43307, -0.985489, 0.871045, -1.92995, 0.786445, -1.90583, 0.40558, -0.0727751, 2.23415, 0.54885, -2.54374, -2.19966, -0.398129, -1.16385, -0.608512, -1.48229, -0.987787, 3.44148, 1.42562, -0.465934, -0.356098, -1.33342, -0.236309, 0.961528, 0.644209, 0.809773, -0.881621, 2.03238, -0.870562, -1.20672, 1.29646, 2.97375, -0.133015, -1.56653, 2.90988, 0.817293, 1.46626, -0.262231, 0.301989, -0.500305, -1.36048, 2.25753, 1.28782, -0.0197388, -3.45074, 1.58903, -0.815923, -1.0852, 2.2728, -2.66226, 1.12052, 1.03489, 0.085673, 3.31336, 0.29301, 0.110178, 2.22798, 2.15017, -1.25682, 2.86108, -1.89215, 2.74446, 1.19491, 1.01804 ] +#! CategoryOfParametrisedMorphisms( SkeletalSmoothMaps ) +hidden_layers := [ 6, 6, 6 ];; +f := NeuralNetworkLossMorphism( Para, 2, hidden_layers, 3, "Softmax" );; +optimizer := Lenses.GradientDescentOptimizer( : learning_rate := 0.1 ); +#! function( n ) ... end +training_examples_path := Filename( + DirectoriesPackageLibrary("GradientBasedLearningForCAP", "examples")[1], + "NeuralNetwork_CrossEntropy/data/training_examples.txt" );; +batch_size := 4; +#! 4 +one_epoch_update := OneEpochUpdateLens( f, optimizer, + training_examples_path, batch_size ); +#! (ℝ^123, ℝ^123) -> (ℝ^1, ℝ^0) defined by: +#! +#! Get Morphism: +#! ------------ +#! ℝ^123 -> ℝ^1 +#! +#! Put Morphism: +#! ------------ +#! ℝ^123 -> ℝ^123 +nr_weights := RankOfObject( Source( PutMorphism( one_epoch_update ) ) ); +#! 123 +rs := RandomSource( IsMersenneTwister, 1 );; +w := List( [ 1 .. nr_weights ], i -> 0.001 * Random( rs, [ -1000 .. 1000 ] ) );; +Display( w{[ 1 .. 5 ]} ); +#! [ 0.789, -0.767, -0.613, -0.542, 0.301 ] +nr_epochs := 16; +#! 16 +w := Fit( one_epoch_update, nr_epochs, w : verbose := true );; +#! Epoch 0/16 - loss = 0.80405334335407785 +#! Epoch 1/16 - loss = 0.18338542093217905 +#! Epoch 2/16 - loss = 0.1491650040794873 +#! Epoch 3/16 - loss = 0.13186409729963983 +#! Epoch 4/16 - loss = 0.12293129048146505 +#! Epoch 5/16 - loss = 0.11742704538825839 +#! Epoch 6/16 - loss = 0.11191588532335346 +#! Epoch 7/16 - loss = 0.10441947487056685 +#! Epoch 8/16 - loss = 0.095102838431592687 +#! Epoch 9/16 - loss = 0.092441708967385072 +#! Epoch 10/16 - loss = 0.097057579505470393 +#! Epoch 11/16 - loss = 0.093295953606638768 +#! Epoch 12/16 - loss = 0.082114375099200984 +#! Epoch 13/16 - loss = 0.082910416530212819 +#! Epoch 14/16 - loss = 0.082815082271383303 +#! Epoch 15/16 - loss = 0.085405485529683856 +#! Epoch 16/16 - loss = 0.087825108242740729 +w; +#! [ 0.789, -1.09294, -1.43008, -0.66714, 1.27126, -1.12774, -0.240397, 0.213, +#! -0.382376, 1.42204, 0.300837, -1.79451, 0.392967, -0.868913, 0.858, +#! 1.16231, 0.769031, 0.309303, 0.555253, -0.142223, 0.0703106, -0.997, +#! -0.746, 0.9, -0.248, -0.801, -0.317, -0.826, 0.0491083, -1.51073, -1.01246, +#! 0.371752, -0.852, 0.342548, 1.01666, 1.39005, 0.958034, 0.357176, 0.3225, +#! -0.29, -1.0095, 0.154876, -0.460859, -0.582425, 0.223943, -0.402, -0.368, +#! 0.275911, -0.0791975, 0.0986371, -0.487903, -0.699542, -0.553485, 0.766, +#! 1.88163, 0.903741, -0.895688, -0.949546, 0.034, 0.13, -0.91, 0.67043, +#! -0.784672, -0.195688, 1.49813, 0.881451, 0.679593, -0.380004, 0.743062, +#! 0.529804, 0.221497, 0.487694, 1.12092, 1.38134, -0.313891, 0.780071, +#! 0.00526383, 0.422997, 0.287254, -0.42555, -0.0525988, -0.159442, -0.256285, +#! -0.296361, 0.822117, -0.23663, -0.252, -0.986452, -0.955211, 0.52727, +#! 0.261295, -0.867, -0.787, -0.395, -0.871, -0.205, -0.315, -0.385, +#! -0.292919, -1.46115, -0.634953, 0.818446, 0.903525, 0.833456, 1.59504, +#! -0.500531, -0.191608, 0.390861, 0.808496, -1.94883, 0.445591, -1.62511, +#! -0.601054, -0.154008, -1.20266, -0.255521, 0.989522, 0.29963, 0.372084, +#! 1.07529, -0.909025, 0.454265, 0.539106 ] +predict := NeuralNetworkPredictionMorphism( Para, 2, hidden_layers, 3, "Softmax" ); +#! ℝ^2 -> ℝ^3 defined by: +#! +#! Underlying Object: +#! ----------------- +#! ℝ^123 +#! +#! Underlying Morphism: +#! ------------------- +#! ℝ^125 -> ℝ^3 +predict_given_w := ReparametriseMorphism( predict, Smooth.Constant( w ) ); +#! ℝ^2 -> ℝ^3 defined by: +#! +#! Underlying Object: +#! ----------------- +#! ℝ^0 +#! +#! Underlying Morphism: +#! ------------------- +#! ℝ^2 -> ℝ^3 +predict_using_w := UnderlyingMorphism( predict_given_w ); +#! ℝ^2 -> ℝ^3 +inputs := Cartesian( 0.1 * [ -10 .. 10 ], 0.1 * [ -10 .. 10 ] );; +predictions := List( inputs, x -> + -1 + Position( predict_using_w( x ), Maximum( predict_using_w( x ) ) ) );; +# ScatterPlotUsingPython( inputs, predictions ); +#! @EndExample + +#! Executing the command $\texttt{ScatterPlotUsingPython( inputs, predictions );}$ produces the following plot: +#! @BeginLatexOnly +#! \begin{center} +#! \includegraphics[width=0.5\textwidth]{../examples/NeuralNetwork_CrossEntropy/scatter_plot_predictions.png} +#! \end{center} +#! @EndLatexOnly diff --git a/examples/NeuralNetwork_CrossEntropy/neural_network.py b/examples/NeuralNetwork_CrossEntropy/neural_network.py deleted file mode 100644 index 399120d..0000000 --- a/examples/NeuralNetwork_CrossEntropy/neural_network.py +++ /dev/null @@ -1,89 +0,0 @@ -import numpy as np -from tensorflow.keras.utils import to_categorical -from tensorflow.keras.models import Sequential -from tensorflow.keras.layers import Input, Dense -from tensorflow.keras.optimizers import Adam, SGD -from sklearn.model_selection import train_test_split - -with open("data/training_examples.txt", "r") as f: - f = f.read() - -data = np.array(eval(f)) - -# create the data -X = data[:, :2] -y = data[:, 2:] - -# Convert the labels to class indices -y = np.argmax(y, axis=1) - -# Convert the labels to categorical format -y = to_categorical(y) - -# Split the data into training and testing sets -X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42) - -model = Sequential() - -# Add an Input layer -model.add(Input(shape=(2,))) - -# Add hidden layers -model.add(Dense(5, activation='relu')) -model.add(Dense(5, activation='relu')) - -# Add output layer -model.add(Dense(4, activation='softmax')) - -# Build the model with the input shape to initialize the weights -model.build() - -# Summary of the model -model.summary() - -# Specify the optimizer -#optimizer = Adam(learning_rate=0.001, beta_1=0.9, beta_2=0.999, epsilon=1e-07) -optimizer = SGD(learning_rate=0.01) - -model.compile(optimizer=optimizer, loss='categorical_crossentropy', metrics=['accuracy']) - -# View the initial weights -initial_weights = model.get_weights() -initial_weights_vec = [] - -for i in range(0, len(initial_weights), 2): - w = initial_weights[i] - b = initial_weights[i+1] - b = b[np.newaxis, :] - m = np.concatenate([w, b]) - print( f"affine matrix of layer {i//2 + 1}:\n{m}\n" ) - initial_weights_vec = m.flatten('F').tolist() + initial_weights_vec - -print(f"initial weights as vector: \n{initial_weights_vec}") - -# View the parameters of the Adam optimizer -print(f'Learning rate: {optimizer.learning_rate.numpy()}') -#print(f'Beta_1: {optimizer.beta_1}') -#print(f'Beta_2: {optimizer.beta_2}') -#print(f'Epsilon: {optimizer.epsilon}') - -# Train the model -model.fit(X_train, y_train, epochs=50, batch_size=1) - -# Evaluate the model. Accuracy should less or more than 83% -loss, accuracy = model.evaluate(X_test, y_test) -print(f'Accuracy: {accuracy}') - -# View the learnt weights -weights = model.get_weights() - -weights_vec = [] -for i in range(0, len(weights), 2): - w = weights[i] - b = weights[i+1] - b = b[np.newaxis, :] - m = np.concatenate([w, b]) - print( f"affine matrix of layer {i//2 + 1}:\n{m}\n" ) - weights_vec = m.flatten('F').tolist() + weights_vec - -print(f"weights as vector: \n{weights_vec}") diff --git a/examples/NeuralNetwork_CrossEntropy/predict/predict.g b/examples/NeuralNetwork_CrossEntropy/predict/predict.g deleted file mode 100644 index cc6e6c2..0000000 --- a/examples/NeuralNetwork_CrossEntropy/predict/predict.g +++ /dev/null @@ -1,30 +0,0 @@ -LoadPackage( "GradientDescentForCAP" ); - - -Smooth := SkeletalSmoothMaps; -Para := CategoryOfParametrisedMorphisms( Smooth ); - - -## The function we are trying minimize -predict := PredictionMorphismOfNeuralNetwork( Para, 2, [ 5, 5 ], 4, "Softmax" ); - -## After 50 epochs, we got the following weights: -w := [ 0.927523, -1.16534, 3.09767, 1.21408, 0.666635, -1.43307, -0.985489, 0.871045, - -1.92995, 0.786445, -1.90583, 0.40558, -0.0727751, 2.23415, 0.54885, -2.54374, - -2.19966, -0.398129, -1.16385, -0.608512, -1.48229, -0.987787, 3.44148, 1.42562, - -0.465934, -0.356098, -1.33342, -0.236309, 0.961528, 0.644209, 0.809773, -0.881621, - 2.03238, -0.870562, -1.20672, 1.29646, 2.97375, -0.133015, -1.56653, 2.90988, - 0.817293, 1.46626, -0.262231, 0.301989, -0.500305, -1.36048, 2.25753, 1.28782, - -0.0197388, -3.45074, 1.58903, -0.815923, -1.0852, 2.2728, -2.66226, 1.12052, - 1.03489, 0.085673, 3.31336, 0.29301, 0.110178, 2.22798, 2.15017, -1.25682, 2.86108, - -1.89215, 2.74446, 1.19491, 1.01804 ]; - -## Let us use w to predict: -predict_using_w := UnderlyingMorphism( ReparametriseMorphism( predict, Smooth.Constant( w ) ) ); - -## create inputs: -inputs := Cartesian( 0.01 * [ -100 .. 100 ], 0.01 * [ -100 .. 100 ] ); -predictions := List( 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nr_examples[i] ] do - - x1 := Random( [ -0.01, 0.01 ] ) * Random( [ 1 .. 100 ] ); - x2 := Random( [ -0.01, 0.01 ] ) * Random( [ 1 .. 100 ] ); - - error := Random( [ -0.001, 0.001 ] ) * Random( [ 1 .. 100 ] ); - - AppendTo( file, [ x1, x2, [ x1, x2, 1 ] * perfect_weights + noise * error ], ",\n" ); - - od; - - AppendTo( file, "]" ); - -od; - -Display( "Done!" ); -QUIT; diff --git a/examples/NeuralNetwork_QuadraticLoss/data/generate_examples.g b/examples/NeuralNetwork_QuadraticLoss/data/generate_examples.g new file mode 100644 index 0000000..59e29e6 --- /dev/null +++ b/examples/NeuralNetwork_QuadraticLoss/data/generate_examples.g @@ -0,0 +1,37 @@ +LoadPackage( "GradientBasedLearningForCAP" ); + +## we want to approximate these secret perfect weights! +perfect_weights := [ 2, -3, 1 ]; + +## The input dimension is 2 and the output dimension is 1 --> an example dimension is 2 + 1 = 3. +## Each training example is of the form [x1, x2, y ] where y := 2x1 - 3x2 + 1 + some_error. + +nr_examples := 100; + +noise := 0.5; + +training_examples := [ ]; + +for j in [ 1 .. nr_examples ] do + + x1 := Random( [ -0.01, 0.01 ] ) * Random( [ 1 .. 100 ] ); + x2 := Random( [ -0.01, 0.01 ] ) * Random( [ 1 .. 100 ] ); + + error := Random( [ -0.001, 0.001 ] ) * Random( [ 1 .. 100 ] ); + + y := perfect_weights[1] * x1 + perfect_weights[2] * x2 + perfect_weights[3] + error; + + Add( training_examples, [ x1, x2, y ] ); + +od; + +file := Filename( DirectoryCurrent( ), "training_examples.txt" ); + +PrintTo( file, "[\n" ); +for example in training_examples do + AppendTo( file, example, ",\n" ); +od; +AppendTo( file, "]" ); + +Display( "Done!" ); +QUIT; diff --git a/examples/NeuralNetwork_QuadraticLoss/data/scatter_plot_training_examples.png b/examples/NeuralNetwork_QuadraticLoss/data/scatter_plot_training_examples.png new file mode 100644 index 0000000000000000000000000000000000000000..7996190c1fbd1b91a0926054930cd80e6edff7b6 GIT binary patch literal 538633 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A lens is a pair of morphisms: a "get" morphism and a "put" morphism, +#! which together model bidirectional data flow used in automatic differentiation. +#! @Arguments C +#! @Returns a category DeclareOperation( "CategoryOfLenses", [ IsCapCategory ] ); +if false then + +#! @Description +#! Construct an object in the category of lenses given a pair of objects. +#! @Arguments Lens, obj_list +#! @Returns an object in the category of lenses +DeclareOperation( "ObjectConstructor", [ IsCategoryOfLenses, IsDenseList ] ); + +#! @Description +#! Construct a morphism in the category of lenses given a pair of morphisms: +#! a "get" morphism and a "put" morphism. +#! @Arguments Lens, source_obj, morphism_list, target_obj +#! @Returns a morphism in the category of lenses +DeclareOperation( "MorphismConstructor", [ IsCategoryOfLenses, IsObjectInCategoryOfLenses, IsDenseList, IsObjectInCategoryOfLenses ] ); + +fi; + +#! @Section Attributes + +#! @Description +#! Returns the underlying pair of objects $(S_1, S_2)$ for an object in the category of lenses. +#! This operation is a synonym for ObjectDatum. +#! @Arguments obj +#! @Returns a pair of objects DeclareAttribute( "UnderlyingPairOfObjects", IsObjectInCategoryOfLenses ); + +#! @Description +#! Returns the underlying pair of morphisms for a morphism in the category of lenses. +#! This operation is a synonym for MorphismDatum. +#! @Arguments f +#! @Returns a pair of morphisms DeclareAttribute( "UnderlyingPairOfMorphisms", IsMorphismInCategoryOfLenses ); + +#! @Description +#! Returns the "get" morphism of a lens. +#! For a lens $f: (S_1, S_2) \to (T_1, T_2)$, the output is the get morphism $S_1 \to T_1$. +#! @Arguments f +#! @Returns a morphism DeclareAttribute( "GetMorphism", IsMorphismInCategoryOfLenses ); + +#! @Description +#! Returns the "put" morphism of a lens. +#! For a lens $f: (S_1, S_2) \to (T_1, T_2)$, the output is the put morphism $S_1 \times T_2 \to S_2$. +#! @Arguments f +#! @Returns a morphism DeclareAttribute( "PutMorphism", IsMorphismInCategoryOfLenses ); -DeclareOperation( "EmbeddingIntoCategoryOfLenses", [ IsCapCategory, IsCategoryOfLenses ] ); +#! @Section Operations + +#! @Description +#! Embedding functor from the category Smooth of smooth maps into its category of lenses Lenses. +#! An object $\mathbb{R}^m$ in Smooth is mapped to the lens $(\mathbb{R}^m, \mathbb{R}^m)$, +#! and a morphism $f: \mathbb{R}^m \to \mathbb{R}^n$ is mapped to the lens +#! with get morphism $f: \mathbb{R}^m \to \mathbb{R}^n$ and put morphism +#! $Rf: \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^m$ given by +#! $(u,v) \mapsto vJ_f(u)$, where $J_f(u) \in \mathbb{R}^{n \times m}$ is the Jacobian matrix of $f$ evaluated +#! at $u \in \mathbb{R}^m$. This functor might be defined for any cartesian reverse-differentiable category, +#! but the category of smooth maps is the only such category currently implemented in this package. +#! For example, if $f: \mathbb{R}^2 \to \mathbb{R}$ is defined by +#! $f(x_1, x_2) = (x_1^3 + x_2^2)$, the the corresponding $Rf:\mathbb{R}^2 \times \mathbb{R} \to \mathbb{R}^2$ is given by +#! $Rf((x_1, x_2), y) = y(3x_1^2, 2x_2) = (3x_1^2y, 2x_2y)$. +#! @Arguments Smooth, Lenses +#! @Returns a functor +DeclareOperation( "ReverseDifferentialLensFunctor", [ IsSkeletalCategoryOfSmoothMaps, IsCategoryOfLenses ] ); + +#! @Section Optimizers + +#! @BeginLatexOnly +#! In this section we document the optimizers provided via the dot operator +#! $\mathrm{Lenses}.( .. )$ when the underlying category is the skeletal category of smooth maps. +#! The optimizers are lenses whose get-part reads the current parameters and whose put-part +#! performs the parameter/state update. +#! In the formulas below we use the following conventions: +#! \begin{itemize} +#! \item $\theta_t \in \mathbb{R}^n$: the current parameter vector. +#! \item $g_t \in \mathbb{R}^n$: the current gradient (coming from backpropagation). +#! \item $\eta > 0$: learning rate. +#! \item All operations on vectors are meant componentwise. +#! \end{itemize} +#! \begin{itemize} +#! \item +#! \textbf{GradientDescentOptimizer}: +#! This is plain gradient descent with update +#! $$(\theta_t,g_t) \mapsto \theta_t + \eta g_t =: \theta_{t+1}.$$ +#! \item +#! \textbf{GradientDescentWithMomentumOptimizer}: +#! This optimizer maintains a momentum vector $s_t \in \mathbb{R}^n$. +#! With momentum parameter $\mu \in [0,1)$ the update is +#! $$(s_t, \theta_t, g_t) \mapsto (\mu\, s_t + \eta\, g_t, \theta_t + \mu\, s_t + \eta\, g_t) =: (s_{t+1}, \theta_{t+1}).$$ +#! Note that the \textbf{GradientDescentOptimizer} is a special case of this optimizer with $\mu = 0$. +#! \item +#! \textbf{AdagradOptimizer}: +#! This optimizer maintains an accumulator $s_t \in \mathbb{R}^n$ of squared gradients. +#! With $\epsilon > 0$ the update is +#! $$ (s_t, \theta_t, g_t) \mapsto (s_t + g_t^2, \theta_t + \eta\, \frac{g_t}{\epsilon + \sqrt{s_t + g_t^2}}) =: (s_{t+1}, \theta_{t+1}).$$ +#! \item +#! \textbf{AdamOptimizer}: +#! This optimizer maintains first and second moment estimates $m_t, v_t \in \mathbb{R}^n$. +#! With parameters $\beta_1,\beta_2 \in [0,1)$ and $\epsilon > 0$ the update is +#! $$ (t, m_t, v_t, \theta_t, g_t) \mapsto (t+1, \beta_1 m_t + (1-\beta_1) g_t, \beta_2 v_t + (1-\beta_2) g_t^2, \theta_t + \eta\, \frac{\hat m_{t+1}}{\epsilon + \sqrt{\hat v_{t+1}}}) =: (t+1, m_{t+1}, v_{t+1}, \theta_{t+1})$$ +#! where +#! \begin{align*} +#! \hat m_{t+1} &= \frac{\beta_1 m_t + (1-\beta_1) g_t}{1-\beta_1^{t+1}}, \\ +#! \hat v_{t+1} &= \frac{\beta_2 v_t + (1-\beta_2) g_t^2}{1-\beta_2^{t+1}}. \\ +#! \end{align*} +#! The iteration counter $t$ is included in the input because the update formulas depend on it ($\beta_1^{t+1}$ and $\beta_2^{t+1}$), +#! while in the other optimizers $t$ was merely an index and not needed for the update. +#! \end{itemize} +#! @EndLatexOnly diff --git a/gap/CategoryOfLenses.gi b/gap/CategoryOfLenses.gi index a22b711..c78b8f5 100644 --- a/gap/CategoryOfLenses.gi +++ b/gap/CategoryOfLenses.gi @@ -1,5 +1,5 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Implementations # @@ -414,15 +414,16 @@ InstallOtherMethod( IsCongruentForMorphisms, end ); ## -InstallMethod( EmbeddingIntoCategoryOfLenses, - [ IsCategoryOfSkeletalSmoothMaps, IsCategoryOfLenses ], +InstallMethod( ReverseDifferentialLensFunctor, + [ IsSkeletalCategoryOfSmoothMaps, IsCategoryOfLenses ], - function ( C, Lenses ) + function ( Smooth, Lenses ) local F; - Assert( 0, IsIdenticalObj( C, UnderlyingCategory( Lenses ) ) ); + Assert( 0, IsIdenticalObj( UnderlyingCategory( Lenses ), Smooth ), + "the underlying category of the category of lenses must be the category of smooth maps!" ); - F := CapFunctor( "Embedding functor into category of lenses", C, Lenses ); + F := CapFunctor( "Embedding functor into category of lenses", Smooth, Lenses ); AddObjectFunction( F, function ( A ) @@ -435,7 +436,7 @@ InstallMethod( EmbeddingIntoCategoryOfLenses, function ( source, f, target ) - return MorphismConstructor( Lenses, source, Pair( f, ReverseDifferential( C, f ) ), target ); + return MorphismConstructor( Lenses, source, Pair( f, ReverseDifferential( Smooth, f ) ), target ); end ); @@ -443,6 +444,19 @@ InstallMethod( EmbeddingIntoCategoryOfLenses, end ); +## +InstallOtherMethod( ReverseDifferentialLensFunctor, + [ IsSkeletalCategoryOfSmoothMaps ], + + function ( Smooth ) + local Lenses; + + Lenses := CategoryOfLenses( Smooth ); + + return ReverseDifferentialLensFunctor( Smooth, Lenses ); + +end ); + ## TxP3 P ## ------------> @@ -467,7 +481,7 @@ InstallOtherMethod( \., C := UnderlyingCategory( Lenses ); - if not IsCategoryOfSkeletalSmoothMaps( C ) then + if not IsSkeletalCategoryOfSmoothMaps( C ) then TryNextMethod( ); fi; @@ -613,10 +627,10 @@ InstallOtherMethod( \., PreCompose( Smooth, p3, DirectProductFunctorial( Smooth, ListWithIdenticalEntries( n, Smooth.Power( 2 ) ) ) ) ); t := MultiplicationForMorphisms( Smooth, - SmoothMorphism( Smooth, P3, ListWithIdenticalEntries( n, learning_rate ), Smooth.( n ) ), + SmoothMap( Smooth, P3, ListWithIdenticalEntries( n, learning_rate ), Smooth.( n ) ), PreCompose( Smooth, AdditionForMorphisms( Smooth, - SmoothMorphism( Smooth, P3, ListWithIdenticalEntries( n, epsilon ), Smooth.( n ) ), + SmoothMap( Smooth, P3, ListWithIdenticalEntries( n, epsilon ), Smooth.( n ) ), PreCompose( Smooth, s, DirectProductFunctorial( Smooth, ListWithIdenticalEntries( n, Smooth.Sqrt ) ) ) ), @@ -821,7 +835,7 @@ InstallOtherMethod( \., p4 := ProjectionInFactorOfDirectProductWithGivenDirectProduct( Smooth, diagram, 4, TxP4 ); p5 := ProjectionInFactorOfDirectProductWithGivenDirectProduct( Smooth, diagram, 5, TxP4 ); - put_1 := AdditionForMorphisms( Smooth, p1, SmoothMorphism( Smooth, TxP4, [ 1 ], Smooth.( 1 ) ) ); + put_1 := AdditionForMorphisms( Smooth, p1, SmoothMap( Smooth, TxP4, [ 1 ], Smooth.( 1 ) ) ); m := AdditionForMorphisms( Smooth, MultiplyWithElementOfCommutativeRingForMorphisms( Smooth, beta_1, p2 ), @@ -832,7 +846,7 @@ InstallOtherMethod( \., b := PreComposeList( Smooth, [ p1, SubtractionForMorphisms( Smooth, - SmoothMorphism( Smooth, Smooth.( 1 ), [ 1 ], Smooth.( 1 ) ), + SmoothMap( Smooth, Smooth.( 1 ), [ 1 ], Smooth.( 1 ) ), Smooth.PowerBase( beta_2 ) ), Smooth.Power( -1 ) ] ); @@ -852,7 +866,7 @@ InstallOtherMethod( \., b := PreComposeList( Smooth, [ p1, SubtractionForMorphisms( Smooth, - SmoothMorphism( Smooth, Smooth.( 1 ), [ 1 ], Smooth.( 1 ) ), + SmoothMap( Smooth, Smooth.( 1 ), [ 1 ], Smooth.( 1 ) ), Smooth.PowerBase( beta_2 ) ), Smooth.Power( -1 ) ] ); @@ -861,7 +875,7 @@ InstallOtherMethod( \., v_hat := MultiplicationForMorphisms( Smooth, v, b ); - delta_n := SmoothMorphism( Smooth, + delta_n := SmoothMap( Smooth, TxP4, ListWithIdenticalEntries( n, epsilon ), Smooth.( n ) ); @@ -1021,13 +1035,13 @@ InstallOtherMethod( \., return function ( arg... ) - return ApplyFunctor( EmbeddingIntoCategoryOfLenses( C, Lenses ), CallFuncList( C.( f ), arg ) ); + return ApplyFunctor( ReverseDifferentialLensFunctor( C ), CallFuncList( C.( f ), arg ) ); end; elif f in [ "Sqrt", "Exp", "Log", "Sin", "Cos", "BinaryCrossEntropyLoss_", "BinaryCrossEntropyLoss" ] then - return ApplyFunctor( EmbeddingIntoCategoryOfLenses( C, Lenses ), C.( f ) ); + return ApplyFunctor( ReverseDifferentialLensFunctor( C ), C.( f ) ); fi; end ); @@ -1055,9 +1069,9 @@ InstallMethod( ViewString, " -> ", ViewString( Target( f ) ), " defined by:", - "\n\nGet Morphism:\n----------\n", + "\n\nGet Morphism:\n------------\n", ViewString( GetMorphism( f ) ), - "\n\nPut Morphism:\n----------\n", + "\n\nPut Morphism:\n------------\n", ViewString( PutMorphism( f ) ) ); @@ -1068,36 +1082,25 @@ InstallMethod( DisplayString, [ IsMorphismInCategoryOfLenses ], function ( f ) + local rank, get_morphism, put_morphism, get_input, put_input; - return Concatenation( - ViewString( Source( f ) ), - " -> ", - ViewString( Target( f ) ), - " defined by:\n", - "\nGet Morphism:\n----------\n", - DisplayString( GetMorphism( f ) ), - "\nPut Morphism:\n----------\n", - DisplayString( PutMorphism( f ) ) ); + get_morphism := GetMorphism( f ); + put_morphism := PutMorphism( f ); -end ); - -## -InstallMethod( Display, - [ IsMorphismInCategoryOfLenses ], - - function ( f ) + put_input := CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT( "dummy_input", DummyInput( put_morphism ) ); - Print( Concatenation( + rank := RankOfObject( Source( get_morphism ) ); + + get_input := put_input{ [ 1 .. rank ] }; + + return Concatenation( ViewString( Source( f ) ), " -> ", ViewString( Target( f ) ), " defined by:\n", - "\nGet Morphism:\n------------\n" ) ); - - Display( GetMorphism( f ) ); - - Print( "\nPut Morphism:\n------------\n" ); - - Display( PutMorphism( f ) ); + "\nGet Morphism:\n------------\n", + DisplayString( GetMorphism( f ) : dummy_input := get_input ), + "\n\nPut Morphism:\n------------\n", + DisplayString( PutMorphism( f ) : dummy_input := put_input ) ); end ); diff --git a/gap/CategoryOfParametrisedMorphisms.autogen.gd b/gap/CategoryOfParametrisedMorphisms.autogen.gd new file mode 100644 index 0000000..530d9b5 --- /dev/null +++ b/gap/CategoryOfParametrisedMorphisms.autogen.gd @@ -0,0 +1,35 @@ +# SPDX-License-Identifier: GPL-2.0-or-later +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory +# +# Declarations +# +# THIS FILE IS AUTOMATICALLY GENERATED, SEE CAP_project/CAP/gap/MethodRecord.gi + +#! @Chapter Category of Parametrised Morphisms + +#! @Section Supported CAP Operations + +#! @Subsection Category of Parametrised Smooth Maps + +#! The following CAP operations are supported: +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * diff --git a/gap/CategoryOfParametrisedMorphisms.gd b/gap/CategoryOfParametrisedMorphisms.gd index 4ec530b..fa3166e 100644 --- a/gap/CategoryOfParametrisedMorphisms.gd +++ b/gap/CategoryOfParametrisedMorphisms.gd @@ -1,33 +1,224 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Declarations # +#! @Chapter Category of Parametrised Morphisms + +#! @Section Definition + +#! Let $(C, \otimes, I)$ be a strict symmetric monoidal category. The category $P(C)$ of parametrized morphisms +#! is given by the following data: +#! +#! * **Objects**: $\mathrm{Obj}(P(C)) := \mathrm{Obj}(C)$ +#! +#! * **Morphisms**: For two objects $A$ and $B$ in $P(C)$, a morphism $f \colon A \to B$ in $P(C)$ consists of: +#! * A parameter object $P \in \mathrm{Obj}(C)$ +#! * A morphism $f_P \colon P \otimes A \to B$ in $C$ +#! @BeginLatexOnly +#! \[ +#! \begin{tikzpicture}[baseline=-2pt] +#! \node[draw, minimum width=2cm, minimum height=1cm] (F) {$f_P$}; +#! \node[left=1.2cm of F] (A) {$A$}; +#! \node[right=1.2cm of F] (B) {$B$}; +#! \node[above=1cm of F] (P) {$P$}; +#! +#! \draw[->] (A) -- (F); +#! \draw[->] (F) -- (B); +#! \draw[->] (P) -- (F); +#! \end{tikzpicture} +#! \] +#! @EndLatexOnly +#! * **Composition**: Given two morphisms $f=(P,f_P) \colon A \to B$ and $g =(Q,g_Q) \colon B \to C$ in $P(C)$, +#! their composition $f \cdot g \colon A \to C$ is given by the pair +#! @BeginLatexOnly +#! \[ +#! \left( Q \otimes P, (\mathrm{id}_Q \otimes f_P) \cdot g_Q \colon (Q \otimes P) \otimes A \to C \right). +#! \] +#! \[ +#! \begin{tikzpicture}[baseline=-2pt] +#! \node[draw, minimum width=2cm, minimum height=1cm] (F) {$Q \otimes P \otimes A \xrightarrow{\mathrm{id}_Q \otimes f_P} Q \otimes B \xrightarrow{g_Q} C$}; +#! \node[left=1.2cm of F] (A) {$A$}; +#! \node[right=1.2cm of F] (C) {$C$}; +#! \node[above=1cm of F] (QP) {$Q \otimes P$}; +#! +#! \draw[->] (A) -- (F); +#! \draw[->] (F) -- (C); +#! \draw[->] (QP) -- (F); +#! \end{tikzpicture} +#! \] +#! @EndLatexOnly +#! * **Identity morphisms**: For each object $A$ in $P(C)$, the identity morphism is given by +#! the pair $\mathrm{id}_A = \left( I, (\mathrm{id}_A)_I := \mathrm{id}_A \in \mathbf{Hom}_C(A, A)) \right)$. + +#! @Section GAP Categories + +#! @Description +#! The &GAP; category of a category of parametrised morphisms. DeclareCategory( "IsCategoryOfParametrisedMorphisms", IsCapCategory ); +#! @Description +#! The &GAP; category of objects in a category of parametrised morphisms. DeclareCategory( "IsObjectInCategoryOfParametrisedMorphisms", IsCapCategoryObject ); +#! @Description +#! The &GAP; category of morphisms in a category of parametrised morphisms. DeclareCategory( "IsMorphismInCategoryOfParametrisedMorphisms", IsCapCategoryMorphism ); +#! @Section Constructors +#! @Description +#! Construct the category of parametrised morphisms over the category C. +#! @Arguments C +#! @Returns a category DeclareOperation( "CategoryOfParametrisedMorphisms", [ IsCapCategory ] ); +if false then +#! @Description +#! Construct an object in the category of parametrised morphisms. +#! @Arguments Para, A +#! @Returns an object in the category of parametrised morphisms +DeclareOperation( "ObjectConstrutor", [ IsCategoryOfParametrisedMorphisms, IsCapCategoryObject ] ); + + +#! @Description +#! Construct a morphism in the category of parametrised morphisms. +#! The datum is a pair consisting of the parameter object $P$ and the underlying morphism $f_P \colon P \otimes A \to B$. +#! @Arguments Para, A, datum, B +#! @Returns a morphism in the category of parametrised morphisms +DeclareOperation( "MorphismConstructor", + [ IsCategoryOfParametrisedMorphisms, + IsObjectInCategoryOfParametrisedMorphisms, + IsDenseList, + IsObjectInCategoryOfParametrisedMorphisms ] ); +fi; + +#! @Section Attributes + +#! @Description +#! Returns the underlying category of a category of parametrised morphisms. +#! @Arguments Para +#! @Returns a category DeclareAttribute( "UnderlyingCategory", IsCategoryOfParametrisedMorphisms ); +#! @Description +#! Returns the underlying object for an object in the category of parametrised morphisms. +#! @Arguments A +#! @Returns an object in the underlying category DeclareAttribute( "UnderlyingObject", IsObjectInCategoryOfParametrisedMorphisms ); +#! @Description +#! Returns the parameter object (underlying object) of a parametrised morphism. +#! @Arguments f +#! @Returns an object in the underlying category DeclareAttribute( "UnderlyingObject", IsMorphismInCategoryOfParametrisedMorphisms ); + +#! @Description +#! Returns the underlying morphism $f_p:P \otimes A \to B \in C$ in $C$ of a parametrised morphism +#! $f = (P, f_p:P \otimes A \to B):A \to B$ in $P(C)$. +#! @Arguments f +#! @Returns a morphism in the underlying category DeclareAttribute( "UnderlyingMorphism", IsMorphismInCategoryOfParametrisedMorphisms ); +#! @Section Operations + +#! @Description +#! The input is a parametrised morphism $f=(P,f_P: P\otimes A \to B): A \to B$ and a morphism $r: Q \to P$ in $C$. +#! The output is the reparametrised morphism $\hat{f}=(Q,\hat{f}_Q: Q \otimes A \to B): A \to B$ where $\hat{f}_Q := (r \otimes \mathrm{id}_A) \cdot f_P \colon Q \otimes A \to B$. +#! @Arguments f, r +#! @Returns a parametrised morphism DeclareOperation( "ReparametriseMorphism", [ IsMorphismInCategoryOfParametrisedMorphisms, IsCapCategoryMorphism ] ); -DeclareOperation( "SwitchSourceAndUnderlyingObject", [ IsMorphismInCategoryOfParametrisedMorphisms ] ); -DeclareOperation( "AdjustToBatchSize", [ IsMorphismInCategoryOfParametrisedMorphisms, IsInt ] ); -DeclareOperation( "NaturalEmbeddingIntoCategoryOfParametrisedMorphisms", [ IsCapCategory, IsCategoryOfParametrisedMorphisms ] ); +#! @Description +#! The input is a parametrised morphism $f=(P,f_P: P\otimes A \to B): A \to B$. +#! The output is the parametrised morphism $\hat{f}=(A,\hat{f}_A: A \otimes P \to B): P \to A$ where $\hat{f}_A := \sigma_{A,P} \cdot f_P \colon A \otimes P \to B$, +#! with $\sigma_{A,P}: A \otimes P \to P \otimes A$ being the symmetry isomorphism (braiding) in the underlying symmetric monoidal category $C$. +#! @Arguments f +#! @Returns a parametrised morphism +DeclareOperation( "FlipParameterAndSource", [ IsMorphismInCategoryOfParametrisedMorphisms ] ); + +DeclareSynonym( "FlipSourceAndParameter", FlipParameterAndSource ); + +#! @Description +#! Adjusts a parametrised morphism to process a batch of n inputs simultaneously. +#! Given a parametrised morphism $f = (P, f_P \colon P \otimes A \to B): A \to B$ where the target $B$ has rank 1, +#! this operation produces a new parametrised morphism that can handle $n$ copies of $A$ at once (a batch of size $n$). +#! +#! The construction works as follows: +#! * The source becomes the direct product of $n$ copies of $A$: $A^n = A \times \cdots \times A$ +#! * The parameter object remains $P$ +#! * The underlying morphism is constructed by: $\alpha \cdot f_P^{(n)} \cdot \mu_n \colon P \otimes A^n \to B$ +#! where: +#! - $\alpha:P \otimes A^n \to (P \otimes A)^n$ is the morphism that reuses the parameter $P$ across the $n$ components. +#! For example, when $n=2$, $\alpha$ sends $(p, a_1, a_2) \in P \otimes A^2$ to $(p, a_1, p, a_2) \in (P \otimes A)^2$, i.e., the same parameter $p$ is paired with each input (training example) $a_i$. +#! - $f_P^{(n)}: (P \otimes A)^n \to B^n$ is the direct product of $n$ copies of $f_P$, +#! - $\mu_n: B^n \to B$ is the mean aggregator that averages the $n$ outputs into a single output in $B$ (since $B$ has rank 1). +#! @Arguments f, n +#! @Returns a parametrised morphism +DeclareOperation( "Batchify", [ IsMorphismInCategoryOfParametrisedMorphisms, IsInt ] ); + +#! @Description +#! Natural embedding functor from category C into category of parametrised morphisms P (of C). +#! Objects are mapped to themselves, and a morphism $f: A \to B$ in C is mapped to the parametrised morphism +#! $(I, f_I: I \otimes A \xrightarrow{f} B): A \to B$ in P. Note that $I \otimes A = A$ by the strict monoidal unit property. +#! This functor reflects the fact that ordinary morphisms can be viewed as parametrised morphisms with a trivial (unit) parameter. +#! @Arguments C, P +#! @Returns a functor +DeclareOperation( "NaturalEmbedding", [ IsCapCategory, IsCategoryOfParametrisedMorphisms ] ); + +#! @Description +#! Embedding functor from category of parametrised morphisms Para into another category of parametrised morphisms Para_Lenses. +#! @Arguments Para, Para_Lenses +#! @Returns a functor DeclareOperation( "EmbeddingIntoCategoryOfParametrisedMorphisms", [ IsCategoryOfParametrisedMorphisms, IsCategoryOfParametrisedMorphisms ] ); +#! @Section Available Parametrised Morphisms + +#! All available smooth maps can be lifted to parametrised morphisms in the category of parametrised morphisms +#! by using one of the following two methods: +#! - Using the natural embedding functor from the category of smooth maps into the category of parametrised morphisms, +#! which associates to each smooth map $f: A \to B$ the parametrised morphism +#! $(I, f_I: I \otimes A \xrightarrow{f} B): A \to B$ where $I$ is the monoidal unit. For example, $\mathrm{Cos}$, $\mathrm{Exp}$, $\mathrm{Log}$ etc. +#! - Constructing parametrised morphisms directly by specifying the parameter object and the underlying morphism. +#! For instance, the construction of an affine transformation as a parametrised morphism: +#! @BeginLatexOnly +#! \[ +#! \begin{tikzpicture}[baseline=-2pt] +#! \node[draw, minimum width=2cm, minimum height=1cm] (F) {$f_P:\mathbb{R}^{(m+1)n+m} \to \mathbb{R}^n$}; +#! \node[left=1.2cm of F] (A) {$\mathbb{R}^m$}; +#! \node[right=1.2cm of F] (B) {$\mathbb{R}^n$}; +#! \node[above=1cm of F] (P) {$\mathbb{R}^{(m+1)n}$}; +#! +#! \draw[->] (A) -- (F); +#! \draw[->] (F) -- (B); +#! \draw[->] (P) -- (F); +#! \end{tikzpicture} +#! \] +#! where $f_P:\mathbb{R}^{(m+1)n+m} \to \mathbb{R}^n$ is the standard smooth affine transformation map that +#! computes $z W + b$, where +#! $W \in \mathbb{R}^{m \times n}$ is the weight matrix, $b \in \mathbb{R}^{1 \times n}$ +#! is the bias row vector and $z \in \mathbb{R}^{1 \times m}$ is the logits row-vector. +#! More explicitly, the input to this morphism consists of $(m+1)n + m$ components +#! structured as follows: +#! \begin{itemize} +#! \item The first $(m+1)n$ components encode the weight matrix $W$ and bias vector $b$ +#! by concatenating the columns of the augmented matrix +#! $\begin{pmatrix} W \\ b \end{pmatrix} \in \mathbb{R}^{(m+1) \times n}$. +#! Explicitly, for each output dimension $i \in \{1,\ldots,n\}$, we have +#! the $i$-th column $(w_{1,i}, w_{2,i}, \ldots, w_{m,i}, b_i)^T$, i.e., $(m+1)$ parameters. +#! Thus, the parameter object $P$ is $\mathbb{R}^{(m+1)n}$. +#! \item The last $m$ components represent the logits $z = (z_1, \ldots, z_m)$ to be transformed. +#! Usually, these correspond to the activations from the previous layer in a neural network. +#! \end{itemize} +#! For example, for $m=2$ and $n=3$, the morphism $f_P$ maps +#! $$(w_{1,1}, w_{1,2}, w_{1,3}, b_1, w_{2,1}, w_{2,2}, w_{2,3}, b_2, z_1, z_2, z_3) \in \mathbb{R}^{(3+1) \cdot 2 + 3} = \mathbb{R}^{8+3}=\mathbb{R}^{11}$$ +#! to +#! $\begin{pmatrix} z_1 & z_2 & z_3 & 1 \end{pmatrix} \cdot \begin{pmatrix} w_{1,1} & w_{1,2} \\ w_{2,1} & w_{2,2} \\ w_{3,1} & w_{3,2} \\ b_1 & b_2 \end{pmatrix} = $ +#! $\begin{pmatrix} z_1 & z_2 & z_3 \end{pmatrix} \cdot \begin{pmatrix} w_{1,1} & w_{1,2} \\ w_{2,1} & w_{2,2} \\ w_{3,1} & w_{3,2} \end{pmatrix} + \begin{pmatrix} b_1 & b_2 \end{pmatrix}$ \\ +#! which compiles to $$(w_{1,1} z_1 + w_{2,1} z_2 + w_{3,1} z_3 + b_1,\; w_{1,2} z_1 + w_{2,2} z_2 + w_{3,2} z_3 + b_2) \in \mathbb{R}^2.$$ +#! @EndLatexOnly diff --git a/gap/CategoryOfParametrisedMorphisms.gi b/gap/CategoryOfParametrisedMorphisms.gi index ca56673..0a1df00 100644 --- a/gap/CategoryOfParametrisedMorphisms.gi +++ b/gap/CategoryOfParametrisedMorphisms.gi @@ -1,5 +1,5 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Implementations # @@ -234,7 +234,7 @@ end ); # # a direct implementation: # -InstallMethod( SwitchSourceAndUnderlyingObject, +InstallMethod( FlipParameterAndSource, [ IsMorphismInCategoryOfParametrisedMorphisms ], function ( f ) @@ -308,14 +308,19 @@ end ); InstallOtherMethod( Eval, [ IsMorphismInCategoryOfParametrisedMorphisms, IsDenseList ], - function( f, pair ) + function( f, input_list ) - return Eval( UnderlyingMorphism( f ), Concatenation( pair ) ); + # if input_list is a pair of parameter_vector and input_vector, concatenate them + if Length( input_list ) = 2 and IsDenseList( input_list[1] ) and IsDenseList( input_list[2] ) then + input_list := Concatenation( input_list[1], input_list[2] ); + fi; + + return Eval( UnderlyingMorphism( f ), input_list ); end ); ## -InstallMethod( NaturalEmbeddingIntoCategoryOfParametrisedMorphisms, +InstallMethod( NaturalEmbedding, [ IsCapCategory, IsCategoryOfParametrisedMorphisms ], function ( C, Para ) @@ -356,7 +361,7 @@ InstallMethod( EmbeddingIntoCategoryOfParametrisedMorphisms, Lenses := UnderlyingCategory( Para_Lenses ); - iota := EmbeddingIntoCategoryOfLenses( C, Lenses ); + iota := ReverseDifferentialLensFunctor( C ); delta := CapFunctor( "Embedding into category of parametrised morphisms", Para, Para_Lenses ); @@ -391,7 +396,7 @@ InstallOtherMethod( \., C := UnderlyingCategory( Para ); - if not IsCategoryOfSkeletalSmoothMaps( C ) then + if not IsSkeletalCategoryOfSmoothMaps( C ) then TryNextMethod( ); fi; @@ -462,7 +467,7 @@ InstallOtherMethod( \., end ); ## -InstallMethod( AdjustToBatchSize, +InstallMethod( Batchify, [ IsMorphismInCategoryOfParametrisedMorphisms, IsInt ], function ( f, n ) diff --git a/gap/CategoryOfSkeletalSmoothMaps.gd b/gap/CategoryOfSkeletalSmoothMaps.gd deleted file mode 100644 index 56e25ab..0000000 --- a/gap/CategoryOfSkeletalSmoothMaps.gd +++ /dev/null @@ -1,36 +0,0 @@ -# SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP -# -# Declarations -# - - -DeclareCategory( "IsCategoryOfSkeletalSmoothMaps", - IsCapCategory ); - -DeclareCategory( "IsObjectInCategoryOfSkeletalSmoothMaps", - IsCapCategoryObject ); - -DeclareCategory( "IsMorphismInCategoryOfSkeletalSmoothMaps", - IsCapCategoryMorphism ); - - -DeclareGlobalFunction( "CategoryOfSkeletalSmoothMaps" ); - -DeclareOperation( "SmoothMorphism", - [ IsCategoryOfSkeletalSmoothMaps, IsObjectInCategoryOfSkeletalSmoothMaps, IsDenseList, IsObjectInCategoryOfSkeletalSmoothMaps ] ); - -DeclareAttribute( "RankOfObject", IsObjectInCategoryOfSkeletalSmoothMaps ); - -DeclareAttribute( "Map", IsMorphismInCategoryOfSkeletalSmoothMaps ); -DeclareAttribute( "JacobianMatrix", IsMorphismInCategoryOfSkeletalSmoothMaps ); - -DeclareOperation( "Eval", [ IsMorphismInCategoryOfSkeletalSmoothMaps, IsDenseList ] ); -DeclareOperation( "EvalJacobianMatrix", [ IsMorphismInCategoryOfSkeletalSmoothMaps, IsDenseList ] ); - -DeclareGlobalVariable( "GradientDescentForCAP" ); - -DeclareGlobalFunction( "DummyInputStringsForAffineTransformation" ); -DeclareGlobalFunction( "DummyInputForAffineTransformation" ); -DeclareGlobalFunction( "DummyInputStringsForPolynomialTransformation" ); -DeclareGlobalFunction( "DummyInputForPolynomialTransformation" ); diff --git a/gap/Expressions.gd b/gap/Expressions.gd index 0c7dd4c..e42ef96 100644 --- a/gap/Expressions.gd +++ b/gap/Expressions.gd @@ -1,25 +1,103 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Declarations # +#! @Chapter Expressions + +#! @Section GAP Categories + +#! @Description +#! The &GAP; category of symbolic expressions. +#! Expressions are used to represent mathematical formulas symbolically, +#! allowing operations like differentiation to be performed symbolically before evaluation. +#! @Arguments e DeclareCategory( "IsExpression", IsNearAdditiveElementWithInverse and IsAdditiveElement and IsMultiplicativeElement ); + +#! @Description +#! The &GAP; category of constant expressions (expressions with no variables). +#! Constant expressions represent fixed numerical values and can be evaluated directly. +#! @Arguments e DeclareCategory( "IsConstantExpression", IsExpression ); +#! @Section Constructors + +#! @Description +#! Create an expression from a list of variables and a string representation. +#! @Arguments variables, string +#! @Returns an expression DeclareOperation( "Expression", [ IsDenseList, IsString ] ); -DeclareOperation( "Expression", [ IsString ] ); +#! @Description +#! Create a constant expression from a string. +#! The expression will have no variables and will also be registered as a global constant. +#! @Arguments string +#! @Returns a constant expression +DeclareOperation( "ConstantExpression", [ IsString ] ); + +#! @Section Attributes + +#! @Description +#! The list of variables of an expression. +#! @Arguments e +#! @Returns a list of strings DeclareAttribute( "Variables", IsExpression ); + +#! @Description +#! The string representation of an expression. +#! @Arguments e +#! @Returns a string DeclareAttribute( "String", IsExpression ); + +#! @Description +#! Convert an expression to a &GAP; function that can be evaluated numerically. +#! @Arguments e +#! @Returns a function DeclareAttribute( "AsFunction", IsExpression ); + DeclareOperation( "AsFunction", [ IsDenseList, IsString ] ); +#! @Section Operations + DeclareOperation( "DummyInputStrings", [ IsString, IsInt ] ); + +#! @Description +#! Generate a list of dummy input expressions using base name var with indices from 1 to n. +#! For example, DummyInput("x", 3) returns a list of three expressions x1, x2, x3. +#! Those can be used as input to smooth morphisms constructed from expressions. +#! @Arguments var, n +#! @Returns a list of expressions DeclareOperation( "DummyInput", [ IsString, IsInt ] ); +#! @Section Global Functions + +#! @Description +#! Assign to each string in constants a global constant expression defined by that string. +#! The variables of these expressions will be empty. +#! For example, ConvertToConstantExpressions( ["Pi"] ) returns [ Pi ], +#! and Variables(Pi)=[]. +#! @Arguments constants +#! @Returns a list of constant expressions DeclareGlobalFunction( "ConvertToConstantExpressions" ); -DeclareGlobalFunction( "ConvertToExpressions" ); + +#! @Description +#! Assigns to each string in variables an expression with the same name. +#! The variables of these expressions equals the list variables itself. +#! After that, one would be able for example to construct more complex expressions using these variables, +#! e.g., x1 + Sin(x2). +#! For example, ConvertToExpressions( ["x1", "x2"] ) returns a pair [ x1, x2 ], +#! where x1 and x2 are expressions with string representations "x1" and "x2", +#! and Variables(x1) = ["x1", "x2"], Variables(x2) = ["x1", "x2"]. +#! @Arguments variables +#! @Returns a list of expressions +DeclareGlobalFunction( "CreateContextualVariables" ); + +#! @Description +#! Assign each expression in vars to a global variable with the same name. +#! For example, if vars = [ x1, x2 ], this function will create global variables x1 and x2 +#! corresponding to the expressions in the list. +#! @Arguments vars DeclareGlobalFunction( "AssignExpressions" ); DeclareGlobalVariable( "LIST_OF_GLOBAL_CONSTANT_EXPRESSIONS" ); diff --git a/gap/Expressions.gi b/gap/Expressions.gi index 1c2069f..882680e 100644 --- a/gap/Expressions.gi +++ b/gap/Expressions.gi @@ -1,5 +1,5 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Implementations # @@ -36,7 +36,7 @@ InstallMethod( Expression, end ); ## -InstallMethod( Expression, +InstallMethod( ConstantExpression, [ IsString ], function ( string ) @@ -48,6 +48,10 @@ InstallMethod( Expression, MakeReadWriteGlobal( string ); + if IsBoundGlobal( string ) then + UnbindGlobal( string ); + fi; + DeclareSynonym( string, constant ); Add( LIST_OF_GLOBAL_CONSTANT_EXPRESSIONS, constant ); @@ -74,7 +78,7 @@ InstallOtherMethod( ViewString, end ); ## -InstallGlobalFunction( ConvertToExpressions, +InstallGlobalFunction( CreateContextualVariables, variables -> List( variables, var -> Expression( variables, var ) ) ); @@ -101,13 +105,13 @@ InstallMethod( DummyInput, function ( var, r ) - return ConvertToExpressions( DummyInputStrings( var, r ) ); + return CreateContextualVariables( DummyInputStrings( var, r ) ); end ); ## InstallOtherMethod( DummyInput, - [ IsString, IsMorphismInCategoryOfSkeletalSmoothMaps ], + [ IsString, IsMorphismInSkeletalCategoryOfSmoothMaps ], function ( var, f ) @@ -117,7 +121,7 @@ end ); ## InstallOtherMethod( DummyInput, - [ IsMorphismInCategoryOfSkeletalSmoothMaps ], + [ IsMorphismInSkeletalCategoryOfSmoothMaps ], function ( f ) @@ -131,6 +135,8 @@ InstallGlobalFunction( AssignExpressions, function ( vars ) local func; + Assert( 0, ForAll( vars, IsExpression ) ); + func := function ( e ) local name; diff --git a/gap/FitParameters.gd b/gap/FitParameters.gd index a3d33f9..903f926 100644 --- a/gap/FitParameters.gd +++ b/gap/FitParameters.gd @@ -1,8 +1,224 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Declarations # +#! @Chapter Fitting Parameters + +#! @Section Introduction + +#! Suppose we have a parametrised morphism $(\mathbb{R}^p, f):\mathbb{R}^n \to \mathbb{R}$ +#! where $\mathbb{R}^p$ is the parameters of the morphism and +#! $f:\mathbb{R}^{p+n} \to \mathbb{R}$ is a morphism in a skeletal category of smooth maps +#! (It represents a loss function over an input in $\mathbb{R}^n$ and parameter vector in $\mathbb{R}^p$). +#! Given a set of training examples $\{X_1, \ldots, X_m\}$ where each $X_i \in \mathbb{R}^n$, +#! we want to fit a parameter vector $\Theta \in \mathbb{R}^p$ such that the output of $f$ is minimized on the training examples. +#! +#! We can achieve this by creating an update-lens for each training example. +#! This update-lens reads the current parameters $\Theta$ and updates it according to the gradient of the loss function $f$ at the example $X_i$. +#! We start by substituting the training example $X_i$ into $f$ resulting in a morphism $f_i:\mathbb{R}^p \to \mathbb{R}$ +#! defined by $f_i(\Theta) = f(\Theta, X_i)$. +#! By applying the reverse differential lens functor ReverseDifferentialLensFunctor +#! $$\mathbf{R}: \mathrm{Smooth} \to \mathrm{Lenses}(\mathrm{Smooth}),$$ +#! on $f_i$, +#! we obtain a lens $\mathbf{R}(f_i):(\mathbb{R}^p, \mathbb{R}^p) \to (\mathbb{R}^1, \mathbb{R}^1)$. +#! The get-morphism of this lens reads the current parameters $\Theta$ and computes the loss $f_i(\Theta)$, +#! while the put-morphism $Rf_i:\mathbb{R}^p \times \mathbb{R}^1 \to \mathbb{R}^p$ +#! is given by $(\Theta, r) \mapsto rJ_{f_i}(\Theta)$ where $J_{f_i}(\Theta) \in \mathbb{R}^{1 \times p}$ is the Jacobian matrix of $f_i$ evaluated at $\Theta$. +#! @BeginLatexOnly +#! \[ +#! \begin{tikzpicture} +#! % Nodes +#! \node (A) at (-3, 1) {$\mathbb{R}^p$}; +#! \node (B) at ( 3, 1) {$\mathbb{R}^1$}; +#! \node (Ap) at (-3,-1) {$\mathbb{R}^p$}; +#! \node (Bp) at ( 3,-1) {$\mathbb{R}^1$}; +#! \draw (-1.5,-1.8) rectangle (1.5,1.8); +#! \draw[->] (A) -- node[below] {$f_i$} (B); +#! \draw[->] (A) -- node[above] {$\Theta \mapsto f_i(\Theta)$} (B); +#! \draw[->] (Bp) -- node[midway, above] {$Rf_i$} (Ap); +#! \draw[->] (Bp) -- node[midway, below] {$rJ_{f_i}(\Theta) \mapsfrom (\Theta, r)$} (Ap); +#! \draw[-] (-1,1) to[out=-90, in=90] (1,-1); +#! \end{tikzpicture} +#! \] +#! @EndLatexOnly +#! +#! The One-Epoch update lens for the example $X_i$ is then obtained by precomposing an optimizer lens (e.g., gradient descent, Adam, etc.) +#! to the following lens $\mathbf{R}(f_i) \cdot \varepsilon$ where +#! $\varepsilon:(\mathbb{R}^1, \mathbb{R}^0) \to (\mathbb{R}^1, \mathbb{R}^1)$ is the lens defined by: +#! - Get morphism: the identity morphism on $\mathbb{R}^1$. +#! - Put morphism: the morphism $\mathbb{R}^1 \to \mathbb{R}^0$ defined by $r \mapsto -r$. +#! This lens merely negates the gradient signal. +#! @BeginLatexOnly +#! \[ +#! \begin{tikzpicture} +#! \node (A) at (-3, 1) {$\mathbb{R}^p$}; +#! \node (B) at ( 3, 1) {$\mathbb{R}^1$}; +#! \node (C) at (9, 1) {$\mathbb{R}^1$}; +#! \node (Ap) at (-3,-1) {$\mathbb{R}^p$}; +#! \node (Bp) at ( 3,-1) {$\mathbb{R}^1$}; +#! \node (Cp) at (9,-1) {$\mathbb{R}^0$}; +#! \draw (-1.5,-1.8) rectangle (1.5,1.8); +#! \draw[->] (A) -- node[below] {$f_i$} (B); +#! \draw[->] (A) -- node[above] {$\Theta \mapsto f_i(\Theta)$} (B); +#! \draw[->] (Bp) -- node[midway, above] {$Rf_i$} (Ap); +#! \draw[->] (Bp) -- node[midway, below] {$rJ_{f_i}(\Theta) \mapsfrom (\Theta, r)$} (Ap); +#! \draw[-] (-1,1) to[out=-90, in=90] (1,-1); +#! \draw[-] (5,1) to[out=-90, in=90] (7,-1); +#! \draw (4.5,-1.8) rectangle (7.5,1.8); +#! \draw[->] (B) -- node[above] {$r \mapsto r$} (C); +#! \draw[->] (Cp) -- node[midway, below] {$-r \mapsfrom r$} (Bp); +#! \end{tikzpicture} +#! \] +#! @EndLatexOnly +#! +#! For example, if we chose the optimizer to be the gradient descent optimizer with learning rate $\eta=0.01$: +#! @BeginLatexOnly +#! \[ +#! \begin{tikzpicture} +#! % Nodes +#! \node (A) at (-3, 1) {$\mathbb{R}^p$}; +#! \node (B) at ( 3, 1) {$\mathbb{R}^p$}; +#! \node (Ap) at (-3,-1) {$\mathbb{R}^p$}; +#! \node (Bp) at ( 3,-1) {$\mathbb{R}^p$}; +#! \draw (-1.5,-1.8) rectangle (1.5,1.8); +#! \draw[->] (A) -- node[above] {$\Theta \mapsto f_i(\Theta)$} (B); +#! \draw[->] (Bp) -- node[midway, below] {$\Theta + \eta g \mapsfrom (\Theta, g)$} (Ap); +#! \draw[-] (-1,1) to[out=-90, in=90] (1,-1); +#! \end{tikzpicture} +#! \] +#! @EndLatexOnly +#! The resulting One-Epoch update lens for the example $X_i$ is given by: +#! @BeginLatexOnly +#! \[ +#! \begin{tikzpicture} +#! % Nodes +#! \node (A) at (-3, 1) {$\mathbb{R}^p$}; +#! \node (B) at ( 3, 1) {$\mathbb{R}^p$}; +#! \node (Ap) at (-3,-1) {$\mathbb{R}^p$}; +#! \node (Bp) at ( 3,-1) {$\mathbb{R}^0$}; +#! \draw (-1.5,-1.8) rectangle (1.5,1.8); +#! \draw[->] (A) -- node[above] {$\Theta \mapsto \Theta$} (B); +#! \draw[->] (Bp) -- node[midway, below] {$\Theta - \eta J_{f_i}(\Theta) \mapsfrom \Theta$} (Ap); +#! \draw[-] (-1,1) to[out=-90, in=90] (1,-1); +#! \end{tikzpicture} +#! \] +#! @EndLatexOnly +#! Now, we can start by a random parameter vector $\Theta_0 \in \mathbb{R}^p$ +#! and apply the update morphism of the One-Epoch update lens for $X_1$ to obtain a new parameter vector $\Theta_1$, +#! then use $\Theta_1$ and the One-Epoch update lens for $X_2$ to obtain $\Theta_2$, and so on. +#! After going through all training examples, we have completed one epoch of training. +#! To perform multiple epochs of training, we can simply repeat the process. +#! +#! For example, suppose we start with the parmetised morphism ($\mathbb{R}^2, f):\mathbb{R}^2 \to \mathbb{R}$ +#! where $f:\mathbb{R}^{2+2} \to \mathbb{R}$ is +#! defined by $f(\theta_1, \theta_2, x_1, x_2) = (x_1-\theta_1)^2 + (x_2-\theta_2)^2$ where $\Theta := (\theta_1, \theta_2) \in \mathbb{R}^2$ represents the parameters +#! and $x = (x_1, x_2) \in \mathbb{R}^2$ is the input. +#! Given training examples $X_1 = (1,2)$ and $X_2 = (3,4)$, +#! the morphism $f_1:\mathbb{R}^2 \to \mathbb{R}$ is defined by $f_1(\theta_1, \theta_2) = (1 - \theta_1)^2 + (2 - \theta_2)^2$ +#! with Jacobian matrix +#! @BeginLatexOnly +#! \[ +#! J_{f_1}(\theta_1, \theta_2) = \begin{pmatrix}-2(1 - \theta_1) & -2(2 - \theta_2)\end{pmatrix}. +#! \] +#! @EndLatexOnly +#! Thus, the One-Epoch update lens for $X_1$ is given by: +#! @BeginLatexOnly +#! \[ +#! \begin{tikzpicture} +#! % Nodes +#! \node (A) at (-5, 1) {$\mathbb{R}^2$}; +#! \node (B) at ( 5, 1) {$\mathbb{R}^2$}; +#! \node (Ap) at (-5,-1) {$\mathbb{R}^2$}; +#! \node (Bp) at ( 5,-1) {$\mathbb{R}^0$}; +#! \draw (-4,-2.5) rectangle (4,2); +#! \draw[->] (A) -- node[above] {$(\theta_1, \theta_2) \mapsto (1 - \theta_1)^2 + (2 - \theta_2)^2$} (B); +#! \draw[->] (Bp) -- node[midway, below] {$(\underbrace{\theta_1 + \eta \cdot 2(1-\theta_1)}_{0.98\theta_1 + 0.02}, \underbrace{\theta_2 + \eta \cdot 2(2-\theta_2)}_{0.98\theta_2 + 0.04}) \mapsfrom (\theta_1, \theta_2)$} (Ap); +#! \draw[-] (-1,1) to[out=-90, in=90] (1,-1); +#! \end{tikzpicture} +#! \] +#! @EndLatexOnly +#! and the One-Epoch update lens for $X_2$ is given by: +#! @BeginLatexOnly +#! \[ +#! \begin{tikzpicture} +#! % Nodes +#! \node (A) at (-5, 1) {$\mathbb{R}^2$}; +#! \node (B) at ( 5, 1) {$\mathbb{R}^2$}; +#! \node (Ap) at (-5,-1) {$\mathbb{R}^2$}; +#! \node (Bp) at ( 5,-1) {$\mathbb{R}^0$}; +#! \draw (-4,-2.5) rectangle (4,2); +#! \draw[->] (A) -- node[above] {$(\theta_1, \theta_2) \mapsto (3 - \theta_1)^2 + (4 - \theta_2)^2$} (B); +#! \draw[->] (Bp) -- node[midway, below] {$(\underbrace{\theta_1 + \eta \cdot 2(3-\theta_1)}_{0.98\theta_1 + 0.06}, \underbrace{\theta_2 + \eta \cdot 2(4-\theta_2)}_{0.98\theta_2 + 0.08}) \mapsfrom (\theta_1, \theta_2)$} (Ap); +#! \draw[-] (-1,1) to[out=-90, in=90] (1,-1); +#! \end{tikzpicture} +#! \] +#! @EndLatexOnly +#! Suppose we start with the parameter vector $\Theta = (0,0)$. Then: +#! - After applying the update lens for $X_1$: $\Theta_1 = (0.98 \cdot 0 + 0.02, 0.98 \cdot 0 + 0.04) = (0.02, 0.04)$. +#! - After applying the update lens for $X_2$: $\Theta_2 = (0.98 \cdot 0.02 + 0.06, 0.98 \cdot 0.04 + 0.08) = (0.0796, 0.1192)$. +#! Thus, after one epoch of training, the updated parameters are $\Theta_2 = (0.0796, 0.1192)$. +#! Repeating this process for multiple epochs will further refine the parameters to minimize the loss function over the training examples. +#! Eventually, we expect the parameters to converge to $\Theta = [2, 3]$ which minimizes the loss function. +#! The point whose distance from $[1, 2]$ and $[3, 4]$ is minimized is $[2, 3]$. +#! See the examples section for the implementation of this process in &GAP;. + + +#! @Section Notes on Batching +#! Given a parametrised (loss) morphism $(\mathbb{R}^p, f):\mathbb{R}^n \to \mathbb{R}$ +#! and a set of training examples $\{X_1, \ldots, X_m\}$ where each $X_i \in \mathbb{R}^n$. +#! If the number of training examples $m$ is large, it may be beneficial to use mini-batches during training. +#! Given a positive integer batch_size, the loss morphism is first batched using Batchify. +#! This means, we create a new parametrised morphism $(\mathbb{R}^p, f_{batch}):\mathbb{R}^{batch\_size \cdot n} \to \mathbb{R}$ +#! where $f_{batch}(\Theta, X_{i_1}, \ldots, X_{i_{batch\_size}}) = \frac{1}{batch\_size} \sum_{j=1}^{batch\_size} f(\Theta, X_{i_j})$. +#! We divide the training examples into mini-batches of size batch_size +#! (padding the list by repeating randomly chosen examples if necessary to make its length divisible by batch_size). +#! And then we consider each mini-batch as a single training example. Now, we can repeat the training process described above using the +#! batched loss morphism and the new training examples. For example, if the parametrised morphism is +#! $(\mathbb{R}^p, f):\mathbb{R}^2 \to \mathbb{R}$ +#! where $f(\theta_1, \theta_2, x_1, x_2) = (x_1-\theta_1)^2 + (x_2-\theta_2)^2$, +#! and we have training examples $[[1,2], [3,4], [5,6], [7,8], [9,10]]$, then for batch_size = $2$, +#! the batched loss morphism is $(\mathbb{R}^p, f_{batch}):\mathbb{R}^4 \to \mathbb{R}$ +#! where $f_{batch}(\theta_1, \theta_2, x_1, x_2, x_3, x_4) = \frac{1}{2} \left( (x_1-\theta_1)^2 + (x_2-\theta_2)^2 + (x_3-\theta_1)^2 + (x_4-\theta_2)^2 \right)$ +#! (See Batchify operation). +#! Since the number of training examples is not divisible by batch_size, +#! we pad the list by randomly choosing an example (say, $[1,2]$) and appending it to the list. +#! Then the new training examples set would be $[[1,2,3,4], [5,6,7,8], [9,10,1,2]]$. + +#! @Section Operations + +#! @Description +#! Create an update lens for one epoch of training. +#! +#! The argument parametrised_morphism must be a morphism in a category of parametrised morphisms +#! whose target has rank $1$ (a scalar loss). +#! +#! The argument optimizer is a function which takes the number of parameters p and returns +#! an optimizer lens in the category of lenses over Smooth. +#! Typical examples are Lenses.GradientDescentOptimizer, Lenses.AdamOptimizer, etc. +#! +#! The list training_examples must contain at least one example; each example is a dense list +#! representing a vector in $\mathbb{R}^n$. +#! +#! @Arguments parametrised_morphism, optimizer, training_examples, batch_size +#! @Returns a morphism in a category of lenses (the epoch update lens) DeclareOperation( "OneEpochUpdateLens", [ IsMorphismInCategoryOfParametrisedMorphisms, IsFunction, IsDenseList, IsPosInt ] ); + +#! @Description +#! Same as OneEpochUpdateLens, but reads the training examples from a file. +#! The file is evaluated using EvalString and is expected to contain a GAP expression +#! evaluating to a dense list of examples. +#! @Arguments parametrised_morphism, optimizer, training_examples_path, batch_size +#! @Returns a morphism in a category of lenses (the epoch update lens) +DeclareOperation( "OneEpochUpdateLens", [ IsMorphismInCategoryOfParametrisedMorphisms, IsFunction, IsString, IsPosInt ] ); + +#! @Description +#! Perform nr_epochs epochs of training using the given one_epoch_update_lens and initial weights initial_weights. +#! +#! The lens one_epoch_update_lens must have get-morphism $\mathbb{R}^p \to \mathbb{R}^1$ and +#! put-morphism $\mathbb{R}^p \to \mathbb{R}^p$ for the same $p$ as the length of initial_weights. +#! The option verbose controls whether to print the loss at each epoch. +#! @Arguments one_epoch_update_lens, nr_epochs, initial_weights +#! @Returns a list of final weights DeclareOperation( "Fit", [ IsMorphismInCategoryOfLenses, IsPosInt, IsDenseList ] ); diff --git a/gap/FitParameters.gi b/gap/FitParameters.gi index 8dbff63..72354d5 100644 --- a/gap/FitParameters.gi +++ b/gap/FitParameters.gi @@ -1,5 +1,5 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Implementations # @@ -43,9 +43,9 @@ InstallMethod( OneEpochUpdateLens, fi; - parametrised_morphism := AdjustToBatchSize( parametrised_morphism, batch_size ); + parametrised_morphism := Batchify( parametrised_morphism, batch_size ); - cost := SwitchSourceAndUnderlyingObject( parametrised_morphism ); + cost := FlipParameterAndSource( parametrised_morphism ); costs := List( SplitDenseList( training_examples, batch_size ), @@ -55,7 +55,7 @@ InstallMethod( OneEpochUpdateLens, Lenses := CapCategory( optimizer ); - L := EmbeddingIntoCategoryOfLenses( Smooth, Lenses ); + L := ReverseDifferentialLensFunctor( Smooth, Lenses ); costs := List( costs, cost -> ApplyFunctor( L, cost ) ); @@ -116,11 +116,14 @@ InstallMethod( Fit, [ IsMorphismInCategoryOfLenses, IsPosInt, IsDenseList ], function( epoch_lens, n, w ) - local MOD, get, put, get_source, get_target, put_source, put_target, l, l_n, str_i, l_i, spaces, loss, i; + local verbose, MOD, get, put, get_source, get_target, put_source, put_target, l, l_n, str_i, l_i, spaces, loss, i; - MOD := GradientDescentForCAP.MOD; + # get the option to print training progress + verbose := CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT( "verbose", true ); - GradientDescentForCAP.MOD := "train"; + MOD := GradientBasedLearningForCAP.MOD; + + GradientBasedLearningForCAP.MOD := "train"; get := GetMorphism( epoch_lens ); put := PutMorphism( epoch_lens ); @@ -139,27 +142,33 @@ InstallMethod( Fit, l_n := Length( String( n ) ); - Print( "Epoch ", JoinStringsWithSeparator( ListWithIdenticalEntries( l_n - 1, " " ), "" ), "0/", String( n ), " - loss = ", String( get( w )[1] ), "\n" ); + if verbose then + Print( "Epoch ", JoinStringsWithSeparator( ListWithIdenticalEntries( l_n - 1, " " ), "" ), "0/", String( n ), " - loss = ", String( get( w )[1] ), "\n" ); + fi; for i in [ 1 .. n ] do - str_i := String( i ); - - l_i := Length( str_i ); - - spaces := JoinStringsWithSeparator( ListWithIdenticalEntries( l_n - l_i, " " ), "" ); + if verbose then + + l_i := Length( String( i ) ); + + spaces := JoinStringsWithSeparator( ListWithIdenticalEntries( l_n - l_i, " " ), "" ); + + fi; w := put( w ); - loss := get( w ); - - Print( "Epoch ", spaces, String( i ), "/", String( n ), " - loss = ", String( loss[1] ), "\n" ); - - #Display( w ); + if verbose then + + loss := get( w ); + + Print( "Epoch ", spaces, String( i ), "/", String( n ), " - loss = ", String( loss[1] ), "\n" ); + + fi; od; - GradientDescentForCAP.MOD := MOD; + GradientBasedLearningForCAP.MOD := MOD; return w; diff --git a/gap/MethodRecord.Declarations.autogen.gd b/gap/MethodRecord.Declarations.autogen.gd index bbe0e5f..ca8e562 100644 --- a/gap/MethodRecord.Declarations.autogen.gd +++ b/gap/MethodRecord.Declarations.autogen.gd @@ -1,13 +1,13 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Declarations # # THIS FILE IS AUTOMATICALLY GENERATED, SEE CAP_project/CAP/gap/MethodRecordTools.gi -#! @Chapter operations for machine learning in CAP +#! @Chapter CAP Operations for GradientBasedLearningForCAP -#! @Section Add-methods +#! @Section Add-Methods #! @BeginGroup #! @Description diff --git a/gap/MethodRecord.Installations.autogen.gi b/gap/MethodRecord.Installations.autogen.gi index e913372..9e0614b 100644 --- a/gap/MethodRecord.Installations.autogen.gi +++ b/gap/MethodRecord.Installations.autogen.gi @@ -1,5 +1,5 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Implementations # diff --git a/gap/MethodRecord.gd b/gap/MethodRecord.gd index 99e661f..8f4a0f6 100644 --- a/gap/MethodRecord.gd +++ b/gap/MethodRecord.gd @@ -1,16 +1,36 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Declarations # - +#! @Chapter CAP Operations for GradientBasedLearningForCAP DeclareGlobalVariable( "CRDC_INTERNAL_METHOD_NAME_RECORD" ); +#! @Section Basic Operations + +#! @Description +#! Compute the pointwise multiplication of two morphisms with the same source and target. +#! For morphisms $f, g: A \to B$, returns a morphism whose output at each component +#! is the product of the outputs of $f$ and $g$. +#! @Arguments alpha, beta +#! @Returns a morphism +DeclareOperation( "MultiplicationForMorphisms", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); + +#! @Description +#! Compute the reverse differential of a morphism alpha with given source and target objects. +#! For a morphism $f: \mathbb{R}^m \to \mathbb{R}^n$, the reverse differential is a morphism +#! $Df: \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^m$ that computes $y \cdot J_f(x)$ +#! where $J_f$ is the Jacobian matrix of $f$. +#! @Arguments source, alpha, range +#! @Returns a morphism DeclareOperation( "ReverseDifferentialWithGivenObjects", [ IsCapCategoryObject, IsCapCategoryMorphism, IsCapCategoryObject ] ); +#! @Description +#! Compute the reverse differential of a morphism alpha. +#! This is equivalent to ReverseDifferentialWithGivenObjects with automatically computed source and target. +#! @Arguments alpha +#! @Returns a morphism DeclareAttribute( "ReverseDifferential", IsCapCategoryMorphism ); - -DeclareOperation( "MultiplicationForMorphisms", [ IsCapCategoryMorphism, IsCapCategoryMorphism ] ); diff --git a/gap/MethodRecord.gi b/gap/MethodRecord.gi index 56bdf57..c2e3edc 100644 --- a/gap/MethodRecord.gi +++ b/gap/MethodRecord.gi @@ -1,5 +1,5 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Implementations # @@ -65,12 +65,12 @@ CAP_INTERNAL_ENHANCE_NAME_RECORD( CRDC_INTERNAL_METHOD_NAME_RECORD ); CAP_INTERNAL_GENERATE_DECLARATIONS_AND_INSTALLATIONS_FROM_METHOD_NAME_RECORD( CRDC_INTERNAL_METHOD_NAME_RECORD, - "GradientDescentForCAP", + "GradientBasedLearningForCAP", "MethodRecord.", - "operations for machine learning in CAP", - "Add-methods" + "CAP Operations for GradientBasedLearningForCAP", + "Add-Methods" ); -CAP_INTERNAL_REGISTER_METHOD_NAME_RECORD_OF_PACKAGE( CRDC_INTERNAL_METHOD_NAME_RECORD, "GradientDescentForCAP" ); +CAP_INTERNAL_REGISTER_METHOD_NAME_RECORD_OF_PACKAGE( CRDC_INTERNAL_METHOD_NAME_RECORD, "GradientBasedLearningForCAP" ); CAP_INTERNAL_INSTALL_ADDS_FROM_RECORD( CRDC_INTERNAL_METHOD_NAME_RECORD ); diff --git a/gap/NeuralNetworks.gd b/gap/NeuralNetworks.gd index 7489e42..2d0c9af 100644 --- a/gap/NeuralNetworks.gd +++ b/gap/NeuralNetworks.gd @@ -1,11 +1,113 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Declarations # +#! @Chapter Neural Networks +#! @Section Definition +#! A neural network can be viewed as a composition of parametrised affine transformations +#! and non-linear activation functions (such as ReLU, Sigmoid, Softmax, etc.). +#! @BeginLatexOnly +#! \[ +#! \begin{tikzpicture}[baseline=-2pt] +#! \node[draw, minimum width=1cm, minimum height=1cm] (F1) {affine-trans}; +#! \node[left=0.75cm of F1] (A1) {$\mathbb{R}^{m_1}$}; +#! \node[right=0.3cm of F1] (A2) {$\mathbb{R}^{m_2}$}; +#! \node[above=0.75cm of F1] (P1) {$\mathbb{R}^{(m_1+1)m_2}$}; +#! \node[draw, minimum width=1cm, minimum height=1cm, right=0.3cm of A2] (F2) {ReLU}; +#! \node[right=0.3cm of F2] (A3) {$\mathbb{R}^{m_2}$}; +#! \node[above=0.75cm of F2] (P2) {$\mathbb{R}^0$}; +#! \node[draw, minimum width=1cm, minimum height=1cm, right=0.3cm of A3] (F3) {affine-trans}; +#! \node[right=0.3cm of F3] (A4) {$\mathbb{R}^{m_3}$}; +#! \node[above=0.75cm of F3] (P3) {$\mathbb{R}^{(m_2+1)m_3}$}; +#! \node[draw, minimum width=1cm, minimum height=1cm, right=0.3cm of A4] (F4) {ReLU}; +#! \node[right=0.3cm of F4] (A5) {$\mathbb{R}^{m_3}$}; +#! \node[above=0.75cm of F4] (P4) {$\mathbb{R}^0$}; +#! \node[right=0.3cm of A5] (F5) {$\cdots$}; +#! \draw[->] (A1) -- (F1); +#! \draw[->] (F1) -- (A2); +#! \draw[->] (P1) -- (F1); +#! \draw[->] (A2) -- (F2); +#! \draw[->] (F2) -- (A3); +#! \draw[->] (P2) -- (F2); +#! \draw[->] (A3) -- (F3); +#! \draw[->] (F3) -- (A4); +#! \draw[->] (P3) -- (F3); +#! \draw[->] (A4) -- (F4); +#! \draw[->] (F4) -- (A5); +#! \draw[->] (P4) -- (F4); +#! \draw[->] (A5) -- (F5); +#! \end{tikzpicture} +#! \] +#! @EndLatexOnly -DeclareOperation( "LogitsMorphismOfNeuralNetwork", [ IsCategoryOfParametrisedMorphisms, IsPosInt, IsDenseList, IsPosInt ] ); -DeclareOperation( "PredictionMorphismOfNeuralNetwork", [ IsCategoryOfParametrisedMorphisms, IsPosInt, IsDenseList, IsPosInt, IsString ] ); -DeclareOperation( "LossMorphismOfNeuralNetwork", [ IsCategoryOfParametrisedMorphisms, IsPosInt, IsDenseList, IsPosInt, IsString ] ); + +#! @Section Operations + +#! @Description +#! The arguments are Para, a parametrised morphism category, +#! s, a positive integer giving the input dimension, +#! hidden_layers_dims, a list of positive integers giving the sizes of +#! the hidden layers in order, and t, a positive integer giving the output dimension. +#! This operation constructs a parametrised morphism that computes the logits (pre-activation outputs) +#! of a fully-connected feed-forward neural network. The signature of the parametrised morphism is +#! $\mathbb{R}^s \to \mathbb{R}^t$ and is parameterised by the network weights and biases. +#! More specifically, the parametrised morphism represents the function that maps an input vector +#! $x \in \mathbb{R}^s$ and a parameter vector $p \in \mathbb{R}^d$ to the output vector $y \in \mathbb{R}^t$, +#! where $d$ is the total number of weights and biases in the network defined by the given architecture. +#! - For a layer with input dimension $m_i$ and output dimension $m_{i+1}$, the parameter object has dimension +#! $(m_i + 1) \times m_{i+1}$, accounting for both the $m_i \times m_{i+1}$ weights matrix and the $m_{i+1}$ biases. +#! - Hidden layers use ReLU nonlinearity between linear layers. The final layer +#! is linear (no activation) so the returned morphism produces logits suitable +#! for subsequent application of a loss or classification activation. +#! @Arguments Para, s, hidden_layers_dims, t +#! @Returns a parametrised morphism +DeclareOperation( "NeuralNetworkLogitsMorphism", [ IsCategoryOfParametrisedMorphisms, IsPosInt, IsDenseList, IsPosInt ] ); + +#! @Description +#! It composes the logits morphisms with the specified activation function to create a +#! parametrised morphism representing the predictions of a neural network. +#! The network has the architecture specified by s, hidden_layers_dims, and t, +#! i.e., the source and target of the parametrised morphism are $\mathbb{R}^{s}$ and $\mathbb{R}^{t}$, respectively. +#! The activation determines the final activation function: +#! * $\mathbf{Softmax}$: applies the softmax activation to turn logits into probabilities for multi-class classification. +#! * $\mathbf{Sigmoid}$: applies the sigmoid activation to turn logits into probabilities for binary classification. +#! * $\mathbf{IdFunc}$: applies the identity function (no activation) for regression tasks. +#! @Arguments Para, s, hidden_layers_dims, t, activation +#! @Returns a parametrised morphism +DeclareOperation( "NeuralNetworkPredictionMorphism", [ IsCategoryOfParametrisedMorphisms, IsPosInt, IsDenseList, IsPosInt, IsString ] ); + +#! @Description +#! Construct a parametrised morphism representing the training loss of a fully-connected +#! feed-forward neural network with architecture given by s, hidden_layers_dims +#! and t. The returned parametrised morphism is parameterised by the network +#! weights and biases and maps a pair (input, target) to a scalar loss: +#! its source is $\mathbb{R}^s \times \mathbb{R}^t$ (an input vector $x$ and a target vector $y$) +#! and its target is $\mathbb{R}$ (the scalar loss). +#! +#! The behaviour of the loss depends on the activation argument: +#! - $\mathbf{Softmax}$: +#! * Used for multi-class classification. +#! * Softmax is applied to the logits to convert them into a probability distribution. +#! * The loss is the (negative) cross-entropy between the predicted probabilities and the target distribution. +#! * Targets y may be one-hot vectors or probability distributions over classes. +#! - $\mathbf{Sigmoid}$: +#! * Used for binary classification. Requires $t = 1$. +#! * Applies the logistic sigmoid to the single logit to obtain a probability $\hat{y}$ in $[0,1]$. +#! * The loss is binary cross-entropy: $\mathrm{loss} = - ( y\log(\hat{y}) + (1-y)\log(1-\hat{y}) )$. +#! - $\mathbf{IdFunc}$: +#! * Used for regression. +#! * No final activation is applied. The loss is the mean squared error (MSE). +#! +#! @Arguments Para, s, hidden_layers_dims, t, activation +#! @Returns a parametrised morphism +DeclareOperation( "NeuralNetworkLossMorphism", [ IsCategoryOfParametrisedMorphisms, IsPosInt, IsDenseList, IsPosInt, IsString ] ); + + +## +DeclareGlobalFunction( "DummyInputStringsForNeuralNetwork" ); + +## +DeclareGlobalFunction( "DummyInputForNeuralNetwork" ); diff --git a/gap/NeuralNetworks.gi b/gap/NeuralNetworks.gi index c312082..dee39c2 100644 --- a/gap/NeuralNetworks.gi +++ b/gap/NeuralNetworks.gi @@ -1,9 +1,9 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Implementations # -InstallMethod( LogitsMorphismOfNeuralNetwork, +InstallMethod( NeuralNetworkLogitsMorphism, [ IsCategoryOfParametrisedMorphisms, IsPosInt, IsDenseList, IsPosInt ], function ( Para, input_layer_dim, hidden_layers_dims, output_layer_dim ) @@ -38,13 +38,13 @@ InstallMethod( LogitsMorphismOfNeuralNetwork, end ); ## -InstallMethod( PredictionMorphismOfNeuralNetwork, +InstallMethod( NeuralNetworkPredictionMorphism, [ IsCategoryOfParametrisedMorphisms, IsPosInt, IsDenseList, IsPosInt, IsString ], function ( Para, input_layer_dim, hidden_layers_dims, output_layer_dim, activation ) local logits; - logits := LogitsMorphismOfNeuralNetwork( Para, input_layer_dim, hidden_layers_dims, output_layer_dim ); + logits := NeuralNetworkLogitsMorphism( Para, input_layer_dim, hidden_layers_dims, output_layer_dim ); if not activation in [ "Softmax", "Sigmoid", "IdFunc" ] then Error( "unrecognized activation functions!\n" ); @@ -55,7 +55,7 @@ InstallMethod( PredictionMorphismOfNeuralNetwork, end ); ## -InstallMethod( LossMorphismOfNeuralNetwork, +InstallMethod( NeuralNetworkLossMorphism, [ IsCategoryOfParametrisedMorphisms, IsPosInt, IsDenseList, IsPosInt, IsString ], function ( Para, input_layer_dim, hidden_layers_dims, output_layer_dim, activation ) @@ -63,7 +63,7 @@ InstallMethod( LossMorphismOfNeuralNetwork, Smooth := UnderlyingCategory( Para ); - logits := LogitsMorphismOfNeuralNetwork( Para, input_layer_dim, hidden_layers_dims, output_layer_dim ); + logits := NeuralNetworkLogitsMorphism( Para, input_layer_dim, hidden_layers_dims, output_layer_dim ); paramter_obj := UnderlyingObject( logits ); @@ -104,3 +104,30 @@ InstallMethod( LossMorphismOfNeuralNetwork, ObjectConstructor( Para, Smooth.( 1 ) ) ); end ); + + +## +InstallGlobalFunction( DummyInputStringsForNeuralNetwork, + + function ( input_layer_dim, hidden_layers_dims, output_layer_dim ) + local Smooth, dims, N, weights_strings, input_strings; + + dims := Concatenation( [ input_layer_dim ], hidden_layers_dims, [ output_layer_dim ] ); + + N := Length( dims ); + + weights_strings := List( [ 1 .. N - 1 ], i -> DummyInputStringsForAffineTransformation( dims[i], dims[i + 1], Concatenation( "w", String( i ), "_" ), Concatenation( "b", String( i ) ) ){ [ 1 .. ( dims[i] + 1 ) * dims[i + 1] ] } ); + + input_strings := List( [ 1 .. input_layer_dim ], j -> Concatenation( "z", String( j ) ) ); + + return Concatenation( Concatenation( Reversed( weights_strings ) ), input_strings ); + +end ); + +## +InstallGlobalFunction( DummyInputForNeuralNetwork, + function ( input_layer_dim, hidden_layers_dims, output_layer_dim ) + + return CreateContextualVariables( DummyInputStringsForNeuralNetwork( input_layer_dim, hidden_layers_dims, output_layer_dim ) ); + +end ); diff --git a/gap/SkeletalCategoryOfSmoothMaps.autogen.gd b/gap/SkeletalCategoryOfSmoothMaps.autogen.gd new file mode 100644 index 0000000..6034837 --- /dev/null +++ b/gap/SkeletalCategoryOfSmoothMaps.autogen.gd @@ -0,0 +1,106 @@ +# SPDX-License-Identifier: GPL-2.0-or-later +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory +# +# Declarations +# +# THIS FILE IS AUTOMATICALLY GENERATED, SEE CAP_project/CAP/gap/MethodRecord.gi + +#! @Chapter Skeletal Category of Smooth Maps + +#! @Section Supported CAP Operations + +#! @Subsection SkeletalSmoothMaps + +#! The following CAP operations are supported: +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * +#! * diff --git a/gap/SkeletalCategoryOfSmoothMaps.gd b/gap/SkeletalCategoryOfSmoothMaps.gd new file mode 100644 index 0000000..7d904d0 --- /dev/null +++ b/gap/SkeletalCategoryOfSmoothMaps.gd @@ -0,0 +1,258 @@ +# SPDX-License-Identifier: GPL-2.0-or-later +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory +# +# Declarations +# + +#! @Chapter Skeletal Category of Smooth Maps + +#! @Section GAP Categories + +#! @Description +#! The &GAP; category of a category of skeletal smooth maps. +DeclareCategory( "IsSkeletalCategoryOfSmoothMaps", + IsCapCategory ); + +#! @Description +#! The &GAP; category of objects in a category of skeletal smooth maps. +DeclareCategory( "IsObjectInSkeletalCategoryOfSmoothMaps", + IsCapCategoryObject ); + +#! @Description +#! The &GAP; category of morphisms in a category of skeletal smooth maps. +DeclareCategory( "IsMorphismInSkeletalCategoryOfSmoothMaps", + IsCapCategoryMorphism ); + +#! @Section Constructors + +#! @Description +#! Construct the category of skeletal smooth maps. +#! Objects in this category are the Euclidean spaces $\mathbb{R}^n$ for non-negative integers $n$. +#! Morphisms are smooth maps between these spaces, represented by a map function and its Jacobian matrix. +#! @Returns a category +DeclareGlobalFunction( "SkeletalCategoryOfSmoothMaps" ); + +if false then + +#! @Description +#! Construct the object representing the Euclidean space $\mathbb{R}^{n}$ in the category of skeletal smooth maps. +#! Points in Euclidean spaces are represented as row vectors. +#! @Arguments Smooth, n +#! @Returns an object in the category of skeletal smooth maps +DeclareOperation( "ObjectConstructor", + [ IsSkeletalCategoryOfSmoothMaps, IsInt ] ); + +#! @Description +#! Construct a smooth morphism from source to target using the given datum. +#! The datum should be a pair [ map_func, jacobian_func ] where: +#! +#! map_func is a function that takes a list of m elements (where m is the rank of source) +#! and returns a list of n elements (where n is the rank of target). +#! This represents the smooth map f: \mathbb{R}^m \to \mathbb{R}^n, +#! $[x_1, x_2, \ldots, x_m] \mapsto [f_1(x_1, \ldots, x_m), f_2(x_1, \ldots, x_m), \ldots, f_n(x_1, \ldots, x_m)]$. +#! +#! jacobian_func is a function that takes a list of m elements +#! and returns an n \times m matrix (represented as a list of n lists, each containing m elements). +#! This matrix represents the Jacobian Df(x) at point x, where entry (i,j) is \frac{\partial f_i}{\partial x_j}(x). +#! +#! @Arguments Smooth, source, datum, target +#! @Returns a morphism in the category of skeletal smooth maps +DeclareOperation( "MorphismConstructor", + [ IsSkeletalCategoryOfSmoothMaps, + IsObjectInSkeletalCategoryOfSmoothMaps, + IsDenseList, + IsObjectInSkeletalCategoryOfSmoothMaps ] ); +fi; + +#! @Description +#! Delegates to MorphismConstructor to create a smooth morphism. +#! @Arguments Smooth, source, datum, target +#! @Returns a morphism in the category of skeletal smooth maps +DeclareOperation( "SmoothMap", + [ IsSkeletalCategoryOfSmoothMaps, IsObjectInSkeletalCategoryOfSmoothMaps, IsDenseList, IsObjectInSkeletalCategoryOfSmoothMaps ] ); + +#! @Section Attributes + +#! @Description +#! The rank (dimension) of the Euclidean space represented by the object obj. +#! For an object representing $\mathbb{R}^n$, this returns $n$. +#! @Arguments obj +#! @Returns a non-negative integer +DeclareAttribute( "RankOfObject", IsObjectInSkeletalCategoryOfSmoothMaps ); + +#! @Description +#! The underlying map function of a smooth morphism f. +#! For a morphism $f: \mathbb{R}^m \to \mathbb{R}^n$, this is a function that takes a list of $m$ elements +#! and returns a list of $n$ elements. +#! @Arguments f +#! @Returns a function +DeclareAttribute( "Map", IsMorphismInSkeletalCategoryOfSmoothMaps ); + +#! @Description +#! The Jacobian matrix function of a smooth morphism f. +#! For a morphism $f: \mathbb{R}^m \to \mathbb{R}^n$, this is a function that takes a list of $m$ elements +#! and returns an $n \times m$ matrix representing the partial derivatives. +#! @Arguments f +#! @Returns a function +DeclareAttribute( "JacobianMatrix", IsMorphismInSkeletalCategoryOfSmoothMaps ); + +#! @Section Operations + +#! @Description +#! Evaluate the smooth morphism f at the point x. +#! @Arguments f, x +#! @Returns a dense list +DeclareOperation( "Eval", [ IsMorphismInSkeletalCategoryOfSmoothMaps, IsDenseList ] ); + +#! @Description +#! Evaluate the Jacobian matrix of the smooth morphism f at the point x. +#! @Arguments f, x +#! @Returns a matrix (list of lists) +DeclareOperation( "EvalJacobianMatrix", [ IsMorphismInSkeletalCategoryOfSmoothMaps, IsDenseList ] ); + +DeclareGlobalVariable( "GradientBasedLearningForCAP" ); + +#! @Section Available Smooth Maps + +#! @BeginLatexOnly +#! The category of skeletal smooth maps offer several pre-implemented helper functions: +#! \begin{itemize} +#! \item \textbf{Sqrt}: The square root function. Returns the square root map $\mathbb{R}^1 \to \mathbb{R}^1$, +#! given by $x \to \sqrt{x}$. Its Jacobian matrix is given by $\frac{1}{2 \sqrt{x}}$. +#! \item \textbf{Exp}: The exponential function. Returns the map $\mathbb{R}^1 \to \mathbb{R}^1$ given by $x \mapsto e^x$. +#! Its Jacobian matrix is $e^x$. +#! \item \textbf{Log}: The natural logarithm function. Returns the map $\mathbb{R}^1 \to \mathbb{R}^1$ given by $x \mapsto \log(x)$. +#! Its Jacobian matrix is $\frac{1}{x}$. +#! \item \textbf{Sin}: The sine function. Returns the map $\mathbb{R}^1 \to \mathbb{R}^1$ given by $x \mapsto \sin(x)$. +#! Its Jacobian matrix is $\cos(x)$. +#! \item \textbf{Cos}: The cosine function. Returns the map $\mathbb{R}^1 \to \mathbb{R}^1$ given by $x \mapsto \cos(x)$. +#! Its Jacobian matrix is $-\sin(x)$. +#! \item \textbf{Constant(rank\_s, l)} or \textbf{Constant(l)}: Returns a constant morphism $\mathbb{R}^{\text{rank\_s}} \to \mathbb{R}^m$ +#! that maps any input to the fixed list $l$ (where $m = |l|$). If rank\_s is omitted, it defaults to 0. +#! \item \textbf{Zero(s, t)}: Returns the zero morphism $\mathbb{R}^s \to \mathbb{R}^t$. +#! \item \textbf{IdFunc(n)}: Returns the identity morphism $\mathbb{R}^n \to \mathbb{R}^n$. +#! \item \textbf{Relu(n)}: Returns the rectified linear unit function $\mathbb{R}^n \to \mathbb{R}^n$ applied componentwise, +#! given by $x_i \mapsto \max(0, x_i)$. +#! \item \textbf{Sigmoid(n)}: Returns the sigmoid activation function $\mathbb{R}^n \to \mathbb{R}^n$ applied componentwise, +#! given by $x_i \mapsto \frac{1}{1 + e^{-x_i}}$. +#! \item \textbf{Softmax(n)}: Returns the softmax function $\mathbb{R}^n \to \mathbb{R}^n$ given by +#! $x_i \mapsto \frac{e^{x_i}}{\sum_{j=1}^n e^{x_j}}$. +#! \item \textbf{Sum(n)}: Returns the sum function $\mathbb{R}^n \to \mathbb{R}^1$ given by $(x_1, \ldots, x_n) \mapsto \sum_{i=1}^n x_i$. +#! \item \textbf{Mean(n)}: Returns the mean function $\mathbb{R}^n \to \mathbb{R}^1$ given by $(x_1, \ldots, x_n) \mapsto \frac{1}{n}\sum_{i=1}^n x_i$. +#! \item \textbf{Variance(n)}: Returns the variance function $\mathbb{R}^n \to \mathbb{R}^1$ given by +#! $(x_1, \ldots, x_n) \mapsto \frac{1}{n}\sum_{i=1}^n (x_i - \mu)^2$ where $\mu$ is the mean. +#! \item \textbf{StandardDeviation(n)}: Returns the standard deviation function $\mathbb{R}^n \to \mathbb{R}^1$ +#! given by the square root of the variance. +#! \item \textbf{Mul(n)}: Returns the multiplication function $\mathbb{R}^n \to \mathbb{R}^1$ given by +#! $(x_1, \ldots, x_n) \mapsto \prod_{i=1}^n x_i$. +#! \item \textbf{Power(n)}: Returns the power function $\mathbb{R}^1 \to \mathbb{R}^1$ given by $x \mapsto x^n$. +#! Its Jacobian matrix is $n \cdot x^{n-1}$. +#! \item \textbf{PowerBase(n)}: Returns the exponential function with base $n$, i.e., $\mathbb{R}^1 \to \mathbb{R}^1$ +#! given by $x \mapsto n^x$. Its Jacobian matrix is $\log(n) \cdot n^x$. +#! \item \textbf{QuadraticLoss(n)}: Returns the quadratic loss function $\mathbb{R}^{2n} \to \mathbb{R}^1$ +#! given by $(\hat{y}_1, \ldots, \hat{y}_n, y_1, \ldots, y_n) \mapsto \frac{1}{n}\sum_{i=1}^n (\hat{y}_i-y_i)^2$, +#! where the first $n$ components are the ground truth values and the last $n$ components are the predicted values. +#! \item \textbf{BinaryCrossEntropyLoss()}: Returns the binary cross-entropy loss function $\mathbb{R}^2 \to \mathbb{R}^1$ +#! (requires $n=1$) given by $(\hat{y}, y) \mapsto -(y \log(\hat{y}) + (1-y) \log(1-\hat{y}))$, where $\hat{y}$ is the predicted probability +#! and $y$ is the ground truth label. +#! \item \textbf{SigmoidBinaryCrossEntropyLoss()}: Returns the loss obtained by +#! precomposing the binary cross-entropy loss with a functorial direct product of +#! the sigmoid function (applied to the predicted logit) and the identity (applied +#! to the ground-truth label). Formally, this is implemented as +#! \[ +#! \text{SigmoidBinaryCrossEntropyLoss()} +#! \;=\; +#! \bigl( \,\sigma \times \mathrm{Id}\, \bigr) +#! \cdot +#! \text{BinaryCrossEntropyLoss} +#! \] +#! This corresponds to applying a sigmoid to the prediction component while leaving +#! the label component unchanged, followed by a numerically stable binary +#! cross-entropy evaluation. In particular, +#! $(\hat{y}, y) \mapsto \log( 1 + e^{ -\hat{y} } ) + ( 1 - y ) \hat{y}$. +#! \item \textbf{CrossEntropyLoss(n)}: Returns the cross-entropy loss function $\mathbb{R}^{2n} \to \mathbb{R}^1$ +#! given by $(\hat{y}_1, \ldots, \hat{y}_n, y_1, \ldots, y_n) \mapsto -\frac{1}{n}\sum_{i=1}^n y_i \log(\hat{y}_i)$, where +#! the first $n$ components are predicted probabilities and the last $n$ components are ground truth labels. +#! \item \textbf{SoftmaxCrossEntropyLoss(n)}: Returns the loss obtained by +#! applying a softmax transformation to the predicted logits and then evaluating +#! the multi-class cross-entropy with respect to the ground-truth labels. +#! Formally, the construction proceeds categorically as follows: +#! \begin{itemize} +#! \item The predicted logits are extracted using the projection +#! $p_1 : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$. +#! \item These logits are mapped to class probabilities via the softmax +#! morphism $\mathrm{Softmax}_n$, yielding +#! $p_1 \cdot \mathrm{Softmax}_n$. +#! \item The ground-truth label vector is extracted by the second projection +#! $p_2 : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$. +#! \item The softmax-transformed predictions and the ground-truth vector are +#! recombined using the universal morphism into the direct product, +#! forming $(\,p_1 \cdot \mathrm{Softmax}_n,\; p_2\,)$. +#! \item Finally, this pair is composed with the multi-class cross-entropy loss +#! $\mathrm{CrossEntropyLoss}_n$. +#! \end{itemize} +#! Altogether, the resulting morphism is +#! \[ +#! \text{SoftmaxCrossEntropyLoss}(n) +#! \;=\; +#! \bigl(\,p_1 \cdot \mathrm{Softmax}_n,\; p_2\,\bigr) +#! \cdot +#! \text{CrossEntropyLoss}_n. +#! \] +#! \item \textbf{AffineTransformation(m, n)}: Returns an affine transformation +#! $\mathbb{R}^{(m+1)n+m} \to \mathbb{R}^n$ implementing the standard linear layer +#! operation $z W + b$, where $z \in \mathbb{R}^{1 \times m}$ is the logits row vector, +#! $W \in \mathbb{R}^{m \times n}$ is the weight matrix, and $b \in \mathbb{R}^{1 \times n}$ +#! is the bias row vector. The input to this morphism consists of $(m+1)n + m$ components +#! structured as follows: +#! \begin{itemize} +#! \item The first $(m+1)n$ components encode the weight matrix $W$ and bias vector $b$ +#! by concatenating the columns of the augmented matrix +#! $\begin{pmatrix} W \\ b \end{pmatrix} \in \mathbb{R}^{(m+1) \times n}$. +#! Explicitly, for each output dimension $i \in \{1,\ldots,n\}$, we store +#! the $i$-th column $(w_{1,i}, w_{2,i}, \ldots, w_{m,i}, b_i)^T$. +#! \item The last $m$ components represent the logits $z = (z_1, \ldots, z_m)$ to be transformed. +#! Usually, these correspond to the activations from the previous layer in a neural network. +#! \end{itemize} +#! The output is the $n$-dimensional vector +#! \[ +#! \begin{pmatrix} z & 1 \end{pmatrix} \cdot \begin{pmatrix} W \\ b \end{pmatrix} = z\cdot W + b +#! \] +#! whose $i$-th component is given by +#! \[ +#! \sum_{j=1}^m w_{j,i} \, z_j \;+\; b_i. +#! \] +#! For example, with $m=2$ and $n=3$, the input structure is +#! $$(w_{1,1}, w_{2,1}, b_1, w_{1,2}, w_{2,2}, b_2, w_{1,3}, w_{2,3}, b_3, z_1, z_2) \in \mathbb{R}^{(2+1)3 + 2} = \mathbb{R}^{11},$$ +#! and the output is +#! $\begin{pmatrix} z_1 & z_2 \end{pmatrix} \cdot \begin{pmatrix} w_{1,1} & w_{1,2} & w_{1,3} \\ w_{2,1} & w_{2,2} & w_{2,3} \end{pmatrix} + \begin{pmatrix} b_1 & b_2 & b_3 \end{pmatrix}$, +#! which compiles to +#! \[ +#! (w_{1,1} z_1 + w_{2,1} z_2 + b_1,\; +#! w_{1,2} z_1 + w_{2,2} z_2 + b_2,\; +#! w_{1,3} z_1 + w_{2,3} z_2 + b_3) \in \mathbb{R}^3. +#! \] +#! \end{itemize} +#! @EndLatexOnly + +#! @Description +#! List the names of available skeletal smooth maps. +#! @Returns a list of strings +#! @Arguments Smooth +DeclareGlobalFunction( "AvailableSkeletalSmoothMaps" ); + +DeclareGlobalFunction( "DummyInputStringsForAffineTransformation" ); + +#! @Description +#! Generate dummy input expressions for an affine transformation with m inputs and n outputs. +#! Its length is $(m+1)n + m$. +#! Additional arguments specify the weight string, bias string, and a logits string. +#! For example, for m=2 and n=3, with weight string "w", bias string "b", and logits string "z", the output is +#! [ w1_1, w2_1, b_1, w1_2, w2_2, b_2, w1_3, w2_3, b_3, z1, z2 ]. +#! @Arguments m, n, weight_str, bias_str, logits_str +#! @Returns a list of expressions +DeclareGlobalFunction( "DummyInputForAffineTransformation" ); + +DeclareGlobalFunction( "DummyInputStringsForPolynomialTransformation" ); + +DeclareGlobalFunction( "DummyInputForPolynomialTransformation" ); diff --git a/gap/CategoryOfSkeletalSmoothMaps.gi b/gap/SkeletalCategoryOfSmoothMaps.gi similarity index 89% rename from gap/CategoryOfSkeletalSmoothMaps.gi rename to gap/SkeletalCategoryOfSmoothMaps.gi index 5c99e40..d95dd6d 100644 --- a/gap/CategoryOfSkeletalSmoothMaps.gi +++ b/gap/SkeletalCategoryOfSmoothMaps.gi @@ -1,15 +1,15 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Implementations # ## -InstallValue( GradientDescentForCAP, +InstallValue( GradientBasedLearningForCAP, rec( MOD := "basic" ) ); # or train ## -InstallGlobalFunction( CategoryOfSkeletalSmoothMaps, +InstallGlobalFunction( SkeletalCategoryOfSmoothMaps, function ( ) local name, Smooth, reals; @@ -17,14 +17,14 @@ InstallGlobalFunction( CategoryOfSkeletalSmoothMaps, name := "SkeletalSmoothMaps"; Smooth := CreateCapCategory( name, - IsCategoryOfSkeletalSmoothMaps, - IsObjectInCategoryOfSkeletalSmoothMaps, - IsMorphismInCategoryOfSkeletalSmoothMaps, + IsSkeletalCategoryOfSmoothMaps, + IsObjectInSkeletalCategoryOfSmoothMaps, + IsMorphismInSkeletalCategoryOfSmoothMaps, IsCapCategoryTwoCell : overhead := false ); - Smooth!.is_computable := false; + #Smooth!.is_computable := false; SetIsCartesianCategory( Smooth, true ); SetIsStrictMonoidalCategory( Smooth, true ); @@ -114,6 +114,47 @@ InstallGlobalFunction( CategoryOfSkeletalSmoothMaps, end ); + ## + AddIsCongruentForMorphisms( Smooth, + + function ( Smooth, f, g ) + local rank_S, 1000_random_inputs, compare_maps, compare_jacobian_matrices; + + # DisplayString operation applies both maps on dummy inputs + if DisplayString( f ) = DisplayString( g ) then + + return true; + + else + + # If the DisplayString check fails, we do numerical tests + rank_S := RankOfObject( Source( f ) ); + + 1000_random_inputs := List( [ 1 .. 1000 ], i -> List( [ 1 .. rank_S ], j -> 0.01 * Random( [ 1 .. 100 ] ) ) ); + + compare_maps := + ForAll( 1000_random_inputs, x -> ForAll( ListN( Eval( f, x ), Eval( g, x ), { a, b } -> AbsoluteValue(a - b) < 1.e-6 ), IdFunc ) ); + + compare_jacobian_matrices := + ForAll( 1000_random_inputs, x -> ForAll( ListN( EvalJacobianMatrix( f, x ), EvalJacobianMatrix( g, x ), { a, b } -> AbsoluteValue( Sum(a - b) ) < 1.e-6 ), IdFunc ) ); + + Info( InfoWarning, 2, "Based on numerical tests with 1000 random inputs and error tolerance 1.e-6, the output seems to be:" ); + + return compare_maps and compare_jacobian_matrices; + + fi; + + end ); + + ## + AddIsEqualForMorphisms( Smooth, + + function ( Smooth, f, g ) + + return DisplayString( f ) = DisplayString( g ); + + end ); + ## AddIdentityMorphism( Smooth, @@ -651,7 +692,7 @@ InstallGlobalFunction( CategoryOfSkeletalSmoothMaps, 1, [ x{[ rank_S + 1 .. rank_S + rank_T ]} ], rank_T, rank_T, JacobianMatrix( f )( x{[ 1 .. rank_S ]} ), rank_S )[1]; - return SmoothMorphism( Smooth, source, map, target, false ); + return SmoothMap( Smooth, source, map, target, false ); end ); @@ -706,42 +747,11 @@ InstallGlobalFunction( CategoryOfSkeletalSmoothMaps, end ); ## -BindGlobal( "SkeletalSmoothMaps", CategoryOfSkeletalSmoothMaps( ) ); - -## -InstallOtherMethod( IsCongruentForMorphisms, - [ IsCategoryOfSkeletalSmoothMaps, IsMorphismInCategoryOfSkeletalSmoothMaps, IsMorphismInCategoryOfSkeletalSmoothMaps ], - - function ( Smooth, f, g ) - local rank_S, 100_random_inputs, compare_maps, compare_jacobian_matrices; - - rank_S := RankOfObject( Source( f ) ); - - 100_random_inputs := List( [ 1 .. 100 ], i -> List( [ 1 .. rank_S ], j -> 0.001 * Random( [ 1 .. 100 ] ) ) ); - - compare_maps := - ForAll( 100_random_inputs, x -> ForAll( ListN( Eval( f, x ), Eval( g, x ), { a, b } -> (a - b) - 1.e-10 < 0. ), IdFunc ) ); - - compare_jacobian_matrices := - ForAll( 100_random_inputs, x -> ForAll( ListN( EvalJacobianMatrix( f, x ), EvalJacobianMatrix( g, x ), { a, b } -> Sum( a - b ) - 1.e-10 < 0. ), IdFunc ) ); - - return compare_maps and compare_jacobian_matrices; - -end ); - -## -InstallOtherMethod( IsEqualForMorphisms, - [ IsCategoryOfSkeletalSmoothMaps, IsMorphismInCategoryOfSkeletalSmoothMaps, IsMorphismInCategoryOfSkeletalSmoothMaps ], - - function ( Smooth, f, g ) - - return Map( f ) = Map( g ) and JacobianMatrix( f ) = JacobianMatrix( g ); - -end ); +BindGlobal( "SkeletalSmoothMaps", SkeletalCategoryOfSmoothMaps( ) ); ## InstallMethod( Eval, - [ IsMorphismInCategoryOfSkeletalSmoothMaps, IsDenseList ], + [ IsMorphismInSkeletalCategoryOfSmoothMaps, IsDenseList ], function ( f, x ) local y; @@ -758,7 +768,7 @@ end ); ## InstallOtherMethod( CallFuncList, - [ IsMorphismInCategoryOfSkeletalSmoothMaps, IsDenseList ], + [ IsMorphismInSkeletalCategoryOfSmoothMaps, IsDenseList ], function ( f, L ) @@ -768,14 +778,14 @@ end ); ## InstallOtherMethod( Eval, - [ IsMorphismInCategoryOfSkeletalSmoothMaps ], + [ IsMorphismInSkeletalCategoryOfSmoothMaps ], f -> Eval( f, DummyInput( f ) ) ); ## InstallMethod( EvalJacobianMatrix, - [ IsMorphismInCategoryOfSkeletalSmoothMaps, IsDenseList ], + [ IsMorphismInSkeletalCategoryOfSmoothMaps, IsDenseList ], function ( f, x ) @@ -787,14 +797,15 @@ end ); ## InstallOtherMethod( EvalJacobianMatrix, - [ IsMorphismInCategoryOfSkeletalSmoothMaps ], + [ IsMorphismInSkeletalCategoryOfSmoothMaps ], f -> EvalJacobianMatrix( f, DummyInput( f ) ) ); ## -InstallMethod( SmoothMorphism, - [ IsCategoryOfSkeletalSmoothMaps, IsObjectInCategoryOfSkeletalSmoothMaps, IsDenseList, IsObjectInCategoryOfSkeletalSmoothMaps ], +InstallMethod( SmoothMap, + "datum as in MorphismConstructor", + [ IsSkeletalCategoryOfSmoothMaps, IsObjectInSkeletalCategoryOfSmoothMaps, IsDenseList, IsObjectInSkeletalCategoryOfSmoothMaps ], function ( Smooth, S, datum, T ) @@ -803,8 +814,9 @@ InstallMethod( SmoothMorphism, end ); ## -InstallOtherMethod( SmoothMorphism, - [ IsCategoryOfSkeletalSmoothMaps, IsObjectInCategoryOfSkeletalSmoothMaps, IsDenseList, IsObjectInCategoryOfSkeletalSmoothMaps, IsBool ], +InstallOtherMethod( SmoothMap, + "datum is a list of strings in variables x1, x2, ...; and a boolean use_python", + [ IsSkeletalCategoryOfSmoothMaps, IsObjectInSkeletalCategoryOfSmoothMaps, IsDenseList, IsObjectInSkeletalCategoryOfSmoothMaps, IsBool ], function ( Smooth, S, maps, T, use_python ) local rank_S, rank_T, vars, jacobian_matrix, map; @@ -818,7 +830,7 @@ InstallOtherMethod( SmoothMorphism, Assert( 0, Length( maps ) = rank_T ); - vars := List( [ 1 .. rank_S ], i -> Concatenation( "x", String( i ) ) ); + vars := DummyInputStrings( "x", rank_S ); if use_python then @@ -837,8 +849,9 @@ InstallOtherMethod( SmoothMorphism, end ); ## -InstallOtherMethod( SmoothMorphism, - [ IsCategoryOfSkeletalSmoothMaps, IsObjectInCategoryOfSkeletalSmoothMaps, IsDenseList, IsObjectInCategoryOfSkeletalSmoothMaps ], +InstallOtherMethod( SmoothMap, + "datum is a list of strings in variables x1, x2, ...", + [ IsSkeletalCategoryOfSmoothMaps, IsObjectInSkeletalCategoryOfSmoothMaps, IsDenseList, IsObjectInSkeletalCategoryOfSmoothMaps ], function ( Smooth, S, maps, T ) @@ -846,13 +859,16 @@ InstallOtherMethod( SmoothMorphism, TryNextMethod( ); fi; - return SmoothMorphism( Smooth, S, maps, T, false ); + Assert( 0, Length( maps ) = RankOfObject( T ) ); + + return SmoothMap( Smooth, S, maps, T, false ); end ); ## -InstallOtherMethod( SmoothMorphism, - [ IsCategoryOfSkeletalSmoothMaps, IsObjectInCategoryOfSkeletalSmoothMaps, IsFunction, IsObjectInCategoryOfSkeletalSmoothMaps, IsBool ], +InstallOtherMethod( SmoothMap, + "datum is a function and a boolean use_python", + [ IsSkeletalCategoryOfSmoothMaps, IsObjectInSkeletalCategoryOfSmoothMaps, IsFunction, IsObjectInSkeletalCategoryOfSmoothMaps, IsBool ], function ( Smooth, S, map, T, use_python ) local rank_S, rank_T, vars, jacobian_matrix; @@ -860,8 +876,10 @@ InstallOtherMethod( SmoothMorphism, rank_S := RankOfObject( S ); rank_T := RankOfObject( T ); - vars := List( [ 1 .. rank_S ], i -> Concatenation( "x", String( i ) ) ); + vars := DummyInputStrings( "x", rank_S ); + # Remark: The map takes a row-vector and returns a row-vector, e.g., map := vec -> [ vec[1]^2 + vec[2], Sin( vec[1] ) ] + # if use_python then jacobian_matrix := JacobianMatrix( vars, map, [ 1 .. rank_S ] ); @@ -877,18 +895,19 @@ InstallOtherMethod( SmoothMorphism, end ); ## -InstallOtherMethod( SmoothMorphism, - [ IsCategoryOfSkeletalSmoothMaps, IsObjectInCategoryOfSkeletalSmoothMaps, IsFunction, IsObjectInCategoryOfSkeletalSmoothMaps ], +InstallOtherMethod( SmoothMap, + "datum is a function", + [ IsSkeletalCategoryOfSmoothMaps, IsObjectInSkeletalCategoryOfSmoothMaps, IsFunction, IsObjectInSkeletalCategoryOfSmoothMaps ], function ( Smooth, S, map, T ) - return SmoothMorphism( Smooth, S, map, T, true ); + return SmoothMap( Smooth, S, map, T, true ); end ); ## -InstallOtherMethod( SmoothMorphism, - [ IsCategoryOfSkeletalSmoothMaps, IsObjectInCategoryOfSkeletalSmoothMaps, IsDenseList, IsObjectInCategoryOfSkeletalSmoothMaps ], +InstallOtherMethod( SmoothMap, + [ IsSkeletalCategoryOfSmoothMaps, IsObjectInSkeletalCategoryOfSmoothMaps, IsDenseList, IsObjectInSkeletalCategoryOfSmoothMaps ], function ( Smooth, S, maps, T ) @@ -896,13 +915,13 @@ InstallOtherMethod( SmoothMorphism, TryNextMethod( ); fi; - return SmoothMorphism( Smooth, S, maps, T, false ); + return SmoothMap( Smooth, S, maps, T, false ); end ); ## -InstallOtherMethod( SmoothMorphism, - [ IsCategoryOfSkeletalSmoothMaps, IsObjectInCategoryOfSkeletalSmoothMaps, IsDenseList, IsObjectInCategoryOfSkeletalSmoothMaps ], +InstallOtherMethod( SmoothMap, + [ IsSkeletalCategoryOfSmoothMaps, IsObjectInSkeletalCategoryOfSmoothMaps, IsDenseList, IsObjectInSkeletalCategoryOfSmoothMaps ], function ( Smooth, S, constants, T ) local rank_S, rank_T, map, jacobian_matrix; @@ -924,7 +943,7 @@ end ); ## InstallOtherMethod( \., - [ IsCategoryOfSkeletalSmoothMaps, IsPosInt ], + [ IsSkeletalCategoryOfSmoothMaps, IsPosInt ], function ( Smooth, string_as_int ) local i; @@ -943,7 +962,7 @@ end ); ## InstallOtherMethod( \., - [ IsCategoryOfSkeletalSmoothMaps, IsPosInt ], + [ IsSkeletalCategoryOfSmoothMaps, IsPosInt ], function ( Smooth, string_as_int ) local f, l1, l2; @@ -992,7 +1011,7 @@ InstallOtherMethod( \., Assert( 0, IsDenseList( l ) ); - return SmoothMorphism( Smooth, Smooth.( rank_S ), l, Smooth.( Length( l ) ) ); + return SmoothMap( Smooth, Smooth.( rank_S ), l, Smooth.( Length( l ) ) ); end; @@ -1223,7 +1242,7 @@ InstallOtherMethod( \., jacobian_matrix := function ( x ) - return DiagonalMat( List( List( [ 1 .. n ], i -> Exp( -x[i] ) ), exp -> exp / ( 1 - exp ) ^ 2 ) ); + return DiagonalMat( List( List( [ 1 .. n ], i -> Exp( -x[i] ) ), exp -> exp / ( 1 + exp ) ^ 2 ) ); end; @@ -1264,7 +1283,7 @@ InstallOtherMethod( \., local max, exp_x, s; # standard trick to avoid numerical overflow - if GradientDescentForCAP.MOD = "train" then + if GradientBasedLearningForCAP.MOD = "train" then max := Maximum( x ); @@ -1286,7 +1305,7 @@ InstallOtherMethod( \., local max, exp_x, s, d; # standard trick to avoid numerical overflow - if GradientDescentForCAP.MOD = "train" then + if GradientBasedLearningForCAP.MOD = "train" then max := Maximum( x ); @@ -1372,10 +1391,10 @@ InstallOtherMethod( \., elif f = "BinaryCrossEntropyLoss_" then return - function ( n ) + function ( arg... ) local l1, l2; - if n <> 1 then + if Length( arg ) >= 1 and arg[1] <> 1 then Error( "the passed argument 'n' must be equal to 1!\n" ); fi; @@ -1393,9 +1412,9 @@ InstallOtherMethod( \., elif f = "BinaryCrossEntropyLoss" then return - function ( n ) + function ( arg... ) - if n <> 1 then + if Length( arg ) >= 1 and arg[1] <> 1 then Error( "the passed argument 'n' must be equal to 1!\n" ); fi; @@ -1411,15 +1430,15 @@ InstallOtherMethod( \., elif f = "SigmoidBinaryCrossEntropyLoss_" then return - function ( n ) + function ( arg... ) - if n <> 1 then + if Length( arg ) >= 1 and arg[1] <> 1 then Error( "the passed argument 'n' must be equal to 1!\n" ); fi; return PreCompose( Smooth, DirectProductFunctorial( Smooth, [ Smooth.Sigmoid_( 1 ), Smooth.IdFunc( 1 ) ] ), - Smooth.BinaryCrossEntropyLoss_( n ) ); + Smooth.BinaryCrossEntropyLoss_() ); end; @@ -1427,9 +1446,9 @@ InstallOtherMethod( \., elif f = "SigmoidBinaryCrossEntropyLoss" then return - function ( n ) + function ( arg... ) - if n <> 1 then + if Length( arg ) >= 1 and arg[1] <> 1 then Error( "the passed argument 'n' must be equal to 1!\n" ); fi; @@ -1524,7 +1543,7 @@ InstallOtherMethod( \., local max, l; # standard trick to avoid numerical overflow - if GradientDescentForCAP.MOD = "train" then + if GradientBasedLearningForCAP.MOD = "train" then max := Maximum( List( [ 1 .. n ], i -> x[i] ) ); @@ -1544,7 +1563,7 @@ InstallOtherMethod( \., local max, exp_x, s, l, c; # standard trick to avoid numerical overflow - if GradientDescentForCAP.MOD = "train" then + if GradientBasedLearningForCAP.MOD = "train" then max := Maximum( List( [ 1 .. n ], i -> x[i] ) ); @@ -1637,12 +1656,12 @@ InstallOtherMethod( \., function ( x ) local i; - if GradientDescentForCAP.MOD = "basic" then + if GradientBasedLearningForCAP.MOD = "basic" then # dropout is activated only while training return ListWithIdenticalEntries( n, 1 ); - elif GradientDescentForCAP.MOD = "train" then + elif GradientBasedLearningForCAP.MOD = "train" then i := Int( percentage * n ); @@ -1777,12 +1796,46 @@ InstallGlobalFunction( DummyInputStringsForAffineTransformation, end ); +## +InstallGlobalFunction( AvailableSkeletalSmoothMaps, + + function ( arg... ) + + return [ + "Sqrt", + "Exp", + "Log", + "Sin", + "Cos", + "Constant", + "Zero", + "IdFunc", + "Relu", + "Sum", + "Mean", + "Variance", + "StandardDeviation", + "Mul", + "Power", + "PowerBase", + "Sigmoid", + "Softmax", + "QuadraticLoss", + "BinaryCrossEntropyLoss", + "SigmoidBinaryCrossEntropyLoss", + "CrossEntropyLoss", + "SoftmaxCrossEntropyLoss", + "AffineTransformation", + ]; + +end ); + ## InstallGlobalFunction( DummyInputForAffineTransformation, function ( arg... ) - return ConvertToExpressions( CallFuncList( DummyInputStringsForAffineTransformation, arg ) ); + return CreateContextualVariables( CallFuncList( DummyInputStringsForAffineTransformation, arg ) ); end ); @@ -1830,13 +1883,13 @@ InstallGlobalFunction( DummyInputForPolynomialTransformation, function ( arg... ) - return ConvertToExpressions( CallFuncList( DummyInputStringsForPolynomialTransformation, arg ) ); + return CreateContextualVariables( CallFuncList( DummyInputStringsForPolynomialTransformation, arg ) ); end ); ## InstallOtherMethod( \^, - [ IsMorphismInCategoryOfSkeletalSmoothMaps, IsAdditiveElement ], + [ IsMorphismInSkeletalCategoryOfSmoothMaps, IsAdditiveElement ], function ( f, n ) local Smooth, p; @@ -1851,7 +1904,7 @@ end ); ## InstallOtherMethod( \*, - [ IsMorphismInCategoryOfSkeletalSmoothMaps, IsMorphismInCategoryOfSkeletalSmoothMaps ], + [ IsMorphismInSkeletalCategoryOfSmoothMaps, IsMorphismInSkeletalCategoryOfSmoothMaps ], function ( f, g ) @@ -1861,7 +1914,7 @@ end ); ## InstallOtherMethod( \/, - [ IsMorphismInCategoryOfSkeletalSmoothMaps, IsMorphismInCategoryOfSkeletalSmoothMaps ], + [ IsMorphismInSkeletalCategoryOfSmoothMaps, IsMorphismInSkeletalCategoryOfSmoothMaps ], function ( f, g ) @@ -1872,7 +1925,7 @@ end ); ## InstallMethod( LaTeXOutput, - [ IsObjectInCategoryOfSkeletalSmoothMaps ], + [ IsObjectInSkeletalCategoryOfSmoothMaps ], function ( U ) @@ -1882,7 +1935,7 @@ end ); ## InstallMethod( LaTeXOutput, - [ IsMorphismInCategoryOfSkeletalSmoothMaps ], + [ IsMorphismInSkeletalCategoryOfSmoothMaps ], function ( f ) local dummy_input, rank_S, rank_T, vars, all, maps, jacobian_matrix; @@ -1922,7 +1975,7 @@ end ); ## InstallMethod( ViewString, - [ IsObjectInCategoryOfSkeletalSmoothMaps ], + [ IsObjectInSkeletalCategoryOfSmoothMaps ], function ( U ) @@ -1932,7 +1985,7 @@ end ); ## InstallMethod( ViewString, - [ IsMorphismInCategoryOfSkeletalSmoothMaps ], + [ IsMorphismInSkeletalCategoryOfSmoothMaps ], function ( f ) @@ -1945,7 +1998,7 @@ end ); ## InstallMethod( DisplayString, - [ IsMorphismInCategoryOfSkeletalSmoothMaps ], + [ IsMorphismInSkeletalCategoryOfSmoothMaps ], function ( f ) local dummy_input, maps; @@ -1965,7 +2018,7 @@ end ); ## InstallMethod( Display, - [ IsMorphismInCategoryOfSkeletalSmoothMaps ], + [ IsMorphismInSkeletalCategoryOfSmoothMaps ], function ( f ) local dummy_input, m; diff --git a/gap/Tools.gd b/gap/Tools.gd index b3ecedd..be4605a 100644 --- a/gap/Tools.gd +++ b/gap/Tools.gd @@ -1,26 +1,137 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Declarations # +#! @Chapter Tools +#! @Section Mathematical Operations + +#! @Description +#! The ReLU (Rectified Linear Unit) activation function. +#! Returns Maximum(a, 0) for a float a. +#! @Arguments a +#! @Returns a float DeclareOperation( "Relu", [ IsFloat ] ); + +#! @Description +#! The Kronecker delta function. Returns 1 if a equals b, otherwise 0. +#! @Arguments a, b +#! @Returns 0 or 1 +DeclareGlobalFunction( "KroneckerDelta" ); + +#! @Description +#! Multiply two matrices with explicit dimensions. +#! mat_1 is an m_1 x n_1 matrix, +#! mat_2 is an m_2 x n_2 matrix. +#! Requires n_1 = m_2. +#! @Arguments m_1, mat_1, n_1, m_2, mat_2, n_2 +#! @Returns a matrix +DeclareGlobalFunction( "MultiplyMatrices" ); + +#! @Section List Operations + +#! @Description +#! Create enumerated pairs from a list. +#! Returns a list of pairs [i, l[i]] for each element in l. +#! @Arguments l +#! @Returns a list of pairs DeclareOperation( "Enumerate", [ IsDenseList ] ); + +#! @Description +#! Split a dense list l according to the dimensions in dims. +#! The sum of dimensions must equal the length of the list. +#! @Arguments l, dims +#! @Returns a list of lists DeclareOperation( "SplitDenseList", [ IsDenseList, IsDenseList ] ); + +#! @Description +#! Split a dense list l into sublists of size n. +#! If the length of l is not a multiple of n, +#! the last sublist will contain the remaining elements (i.e., will contain less than n elements). +#! @Arguments l, n +#! @Returns a list of lists DeclareOperation( "SplitDenseList", [ IsDenseList, IsPosInt ] ); +#! @Section Helper Functions + +#! @Description +#! Select an element based on a condition. +#! If cond is true, returns val_true, otherwise returns val_false. +#! @Arguments cond, val_true, val_false +#! @Returns an element DeclareGlobalFunction( "SelectBasedOnCondition" ); + +#! @Description +#! Call one of two functions based on a condition. +#! If cond is true, calls func_1, otherwise calls func_2 +#! on the arguments in args. For example, +#! CallFuncListBasedOnCondition( true, x -> x^2, x -> x^3, [ 2 ] ); returns $4$, +#! while CallFuncListBasedOnCondition( false, x -> x^2, x -> x^3, [ 2 ] ); returns $8$. +#! @Arguments cond, func_1, func_2, args +#! @Returns the result of the called function DeclareGlobalFunction( "CallFuncListBasedOnCondition" ); -DeclareGlobalFunction( "KroneckerDelta" ); -DeclareGlobalFunction( "MultiplyMatrices" ); -DeclareOperation( "ScatterPlotUsingPython", [ IsDenseList, IsDenseList ] ); -DeclareOperation( "SimplifyExpressionUsingPython", [ IsDenseList, IsDenseList ] ); +#! @Section Python Integration + +#! @Description +#! Compute the partial derivative of an expression with respect to the i-th variable. +#! vars is a list of variable names, str is the expression string. +#! See the example in the Expressions chapter. +#! @Arguments vars, str, i +#! @Returns a function +DeclareOperation( "Diff", [ IsDenseList, IsString, IsPosInt ] ); + +#! @Description +#! Compute the lazy partial derivative of an expression with respect to the i-th variable. +#! vars is a list of variable names, str is the expression string. +#! See the example in the Expressions chapter. +#! @Arguments vars, str, i +#! @Returns a function +DeclareOperation( "LazyDiff", [ IsDenseList, IsString, IsPosInt ] ); + +#! @Description +#! Compute the Jacobian matrix using Python's SymPy library. +#! vars is a list of variable names, exps is a list of expression strings, +#! indices specifies which variables to differentiate with respect to. +#! @Arguments vars, exps, indices +#! @Returns a matrix of derivative expressions DeclareOperation( "JacobianMatrixUsingPython", [ IsDenseList, IsDenseList, IsDenseList ] ); + +#! @Description +#! Compute a lazy Jacobian matrix (deferred computation). +#! Returns a function that computes the Jacobian when evaluated. +#! @Arguments vars, exps, indices +#! @Returns a function DeclareOperation( "LazyJacobianMatrix", [ IsDenseList, IsDenseList, IsDenseList ] ); + +#! @Description +#! Create a scatter plot using Python's matplotlib. +#! points is a list of 2D points, labels is a list of class labels. +#! @Arguments points, labels +#! @Returns the directory containing the plot +DeclareOperation( "ScatterPlotUsingPython", [ IsDenseList, IsDenseList ] ); + +#! @Description +#! Simplify expressions using Python's SymPy library. +#! vars is a list of variable names, exps is a list of expression strings. +#! @Arguments vars, exps +#! @Returns a list of simplified expression strings +DeclareOperation( "SimplifyExpressionUsingPython", [ IsDenseList, IsDenseList ] ); + +#! @Description +#! Convert expressions to LaTeX format using Python's SymPy library. +#! vars is a list of variable names, exps is a list of expression strings. +#! @Arguments vars, exps +#! @Returns a list of LaTeX strings DeclareOperation( "LaTeXOutputUsingPython", [ IsDenseList, IsDenseList ] ); -DeclareOperation( "LazyDiff", [ IsDenseList, IsString, IsPosInt ] ); -DeclareOperation( "Diff", [ IsDenseList, IsString, IsPosInt ] ); +#! @Description +#! Compile functions to Cython for improved performance. +#! vars is a list of lists of variable names (one per function), +#! function_names is a list of function names, +#! functions is a list of function body strings. +#! @Arguments vars, function_names, functions +#! @Returns a string with instructions to use the compiled functions DeclareOperation( "AsCythonFunction", [ IsDenseList, IsDenseList, IsDenseList ] ); diff --git a/gap/Tools.gi b/gap/Tools.gi index 9d5a6aa..05f5041 100644 --- a/gap/Tools.gi +++ b/gap/Tools.gi @@ -1,5 +1,5 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Implementations # @@ -88,13 +88,14 @@ InstallMethod( SplitDenseList, [ IsDenseList, IsPosInt ], function ( l, n ) - local N; + local a, b, dims; - if Length( l ) mod n <> 0 then - Error( "the length of the passed list 'l' must be divisible by passed positive integers 'n'!\n" ); - fi; + a := QuoInt( Length( l ), n ); + b := RemInt( Length( l ), n ); + + dims := Concatenation( ListWithIdenticalEntries( a, n ), SelectBasedOnCondition( b = 0, [ ], [ b ] ) ); - return SplitDenseList( l, ListWithIdenticalEntries( Length( l ) / n, n ) ); + return SplitDenseList( l, dims ); end ); @@ -157,7 +158,7 @@ InstallMethod( LazyDiff, return function ( vec ) - local vec_vars; + local vec_vars, expressions; if ForAll( vec, e -> IsFloat( e ) or IsRat( e ) ) then vec := List( vec, e -> Expression( [ ], String( e ) ) ); @@ -166,8 +167,9 @@ InstallMethod( LazyDiff, # obviously Assert( 0, IsDenseList( vec ) and Length( vars ) = Length( vec ) ); - # all entries of vec must be expressions defined by the same variables - Assert( 0, Length( Set( List( vec, Variables ) ) ) = 1 ); + # all non-float expressions should have the same variables if any expression is non-float + expressions := Filtered( vec, e -> IsExpression( e ) ); + Assert( 0, IsEmpty( expressions ) or ForAll( expressions, e -> Variables( e ) = Variables( expressions[1] ) ) ); if not IsEmpty( vec ) then vec_vars := Variables( vec[1] ); @@ -200,10 +202,10 @@ InstallOtherMethod( LazyDiff, InstallMethod( SimplifyExpressionUsingPython, [ IsDenseList, IsDenseList ], - function ( vars, exps ) - local constants, dir, input_path, input_file, output_path, import, symbols, functions, g_ops, p_ops, define_exps, simplify, write_output, stream, err, output_file, outputs, j, i, exp, o; + function ( vars, expressions ) + local constants, indices, exps, dir, input_path, input_file, output_path, import, symbols, functions, g_ops, p_ops, define_exps, simplify, write_output, stream, err, output_file, raw_outputs, outputs, j, i; - if not ( ForAll( exps, IsString ) and ForAll( vars, IsString ) ) then + if not ( ForAll( expressions, IsString ) and ForAll( vars, IsString ) ) then TryNextMethod( ); fi; @@ -212,7 +214,17 @@ InstallMethod( SimplifyExpressionUsingPython, vars := Concatenation( vars, constants ); # create a copy - exps := List( [ 1 .. Length( exps ) ], i -> exps[i] ); + expressions := List( [ 1 .. Length( expressions ) ], i -> expressions[i] ); + + # skip expressions that contain Diff + indices := PositionsProperty( expressions, exp -> PositionSublist( exp, "Diff" ) = fail ); + + # nothing to simplify (all expressions contain Diff) + if IsEmpty( indices ) then + return expressions; + fi; + + exps := expressions{ indices }; dir := DirectoryTemporary( ); @@ -267,18 +279,21 @@ InstallMethod( SimplifyExpressionUsingPython, output_file := IO_File( output_path, "r" ); - outputs := IO_ReadLines( output_file ); + raw_outputs := IO_ReadLines( output_file ); IO_Close( output_file ); - Assert( 0, Length( outputs ) = Length( exps ) ); + Assert( 0, Length( raw_outputs ) = Length( exps ) ); - for j in [ 1 .. Length( outputs ) ] do + # start with the original expressions; overwrite only those that were simplified + outputs := ShallowCopy( expressions ); + + for j in [ 1 .. Length( indices ) ] do - outputs[j] := ReplacedString( outputs[j], "\n", "" ); + outputs[ indices[j] ] := ReplacedString( raw_outputs[j], "\n", "" ); for i in [ 1 .. Length( g_ops ) ] do - outputs[j] := ReplacedString( outputs[j], p_ops[i], g_ops[i] ); + outputs[ indices[j] ] := ReplacedString( outputs[ indices[j] ], p_ops[i], g_ops[i] ); od; od; @@ -424,7 +439,7 @@ InstallOtherMethod( JacobianMatrix, function ( vars, map_func, indices ) - return JacobianMatrix( map_func( ConvertToExpressions( vars ) ), indices ); + return JacobianMatrix( map_func( CreateContextualVariables( vars ) ), indices ); end ); @@ -460,7 +475,7 @@ InstallOtherMethod( LazyJacobianMatrix, function( vec ) if exps = fail then - exps := map_func( ConvertToExpressions( vars ) ); + exps := map_func( CreateContextualVariables( vars ) ); fi; return LazyJacobianMatrix( exps, indices )( vec ); @@ -704,108 +719,96 @@ end ); ## InstallMethod( ScatterPlotUsingPython, - [ IsDenseList, IsDenseList ], + [ IsDenseList, IsDenseList ], function ( points, labels ) - local dir, path, file, size, action, stream, err, p; - - dir := DirectoryTemporary( ); - - Info( InfoPython, 1, dir ); - - path := Filename( dir, "plot.py" ); - - file := IO_File( path, "w" ); - - size := CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT( "size", "20" ); - - action := CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT( "action", "show" ); - - IO_Write( file, - Concatenation( - "import matplotlib.pyplot as plt\n", - "import matplotlib.patches as patches\n\n", - - "points =", String( points ), "\n", - "labels = [ str(label) for label in ", String( labels ), "]\n", - - #"#test_points =", String( test_points ), "\n", - #"#test_labels = [ str(label) for label in ", String( test_labels ), "]\n", - - "# Convert the points to separate x and y lists\n", - "x1 = [p[0] for p in points]\n", - "y1 = [p[1] for p in points]\n", - - "#x2 = [p[0] for p in test_points]\n", - "#y2 = [p[1] for p in test_points]\n", - - "# Unique classes\n", - "unique_classes = list(set(labels))\n", - - "# Number of unique classes\n", - "num_classes = len(unique_classes)\n", - - "markers = ['+', '*', 'o', 'v', '^', '<', '>', '1', '2', '3', '4', 's', 'p', 'x', 'h', 'H', '.', ',', 'D', 'd', '|', '_']\n", - - "# Generate a list of colors using a colormap\n", - "colormap = plt.get_cmap('rainbow')\n", - "colors = [colormap(i / num_classes) for i in range(num_classes)]", - - "# Map classes to colors\n", - "style_map = {cls: colors[i] for i, cls in enumerate(unique_classes)}\n", - "markers = {cls : markers[i] for i, cls in enumerate(unique_classes)}\n", - "# Create a figure and an axes\n", - "fig, ax = plt.subplots()\n", - - "# Create a scatter plots\n", - "for cls in unique_classes:", - "\n ", - "x1_class = [x1[i] for i in range(len(x1)) if labels[i] == cls]", - "\n ", - "y1_class = [y1[i] for i in range(len(y1)) if labels[i] == cls]", - "\n ", - "scatter1 = plt.scatter(x1_class, y1_class, color=style_map[cls], marker=markers[cls], s=", size, ", label=cls)\n", - - "\n ", - "#x2_class = [x2[i] for i in range(len(x2)) if test_labels[i] == cls]", - "\n ", - "#y2_class = [y2[i] for i in range(len(y2)) if test_labels[i] == cls]", - "\n ", - "#scatter2 = plt.scatter(x2_class, y2_class, color=style_map[cls], marker='v', s=100, label=cls)\n", - - "# Set the limits of the plot based on min and max values of x and y\n", - "plt.xlim(min(x1) - 0.1, max(x1) + 0.1)\n", - "plt.ylim(min(y1) - 0.1, max(y1) + 0.1)\n", - - "plt.xlabel('X-axis')\n", - "plt.ylabel('Y-axis')\n", - "plt.title('Scatter Plot using Matplotlib')\n", - "plt.legend()\n", - SelectBasedOnCondition( - action = "save", - Concatenation( "plt.savefig('", Filename( dir, "plot.png" ), "', dpi=400)\n" ), - "plt.show()\n" ) ) ); - - IO_Close( file ); - - stream := IO_Popen3( IO_FindExecutable( "python" ), [ path, "&" ] ); - - err := Concatenation( IO_ReadLines( stream.stderr ) ); - - IO_ReadLines( stream.stdout ); - - IO_Close( stream.stdin ); - - IO_Close( stream.stdout ); - - IO_Close( stream.stderr ); - - if not IsEmpty( err ) then - - Error( err, "\n" ); - - fi; + local dir, path, file, size, action, stream, err, p; + + dir := DirectoryTemporary( ); + + Info( InfoPython, 1, dir ); + + path := Filename( dir, "plot.py" ); + + file := IO_File( path, "w" ); + + size := CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT( "size", "100" ); + + action := CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT( "action", "show" ); + + IO_Write( file, + Concatenation( + "import matplotlib.pyplot as plt\n", + "import matplotlib.patches as patches\n\n", + + "points =", String( points ), "\n", + "labels = [ str(label) for label in ", String( labels ), "]\n", + + "# Convert the points to separate x and y lists\n", + "x1 = [p[0] for p in points]\n", + "y1 = [p[1] for p in points]\n", + + "# Unique classes (use sorted order to make mapping deterministic)\n", + "unique_classes = sorted(set(labels))\n", + + "# Number of unique classes\n", + "num_classes = len(unique_classes)\n", + + "marker_list = ['+', '*', 'o', 'v', '^', '<', '>', '1', '2', '3', '4', 's', 'p', 'x', 'h', 'H', '.', ',', 'D', 'd', '|', '_']\n", + + "# Generate a deterministic list of colors from a fixed colormap\n", + "colormap = plt.get_cmap('viridis')\n", + "colors = [colormap(i / max(num_classes - 1, 1)) for i in range(num_classes)]\n", + + "# Map classes to styles deterministically (by sorted class order)\n", + "style_map = {cls: colors[i] for i, cls in enumerate(unique_classes)}\n", + "markers = {cls: marker_list[i % len(marker_list)] for i, cls in enumerate(unique_classes)}\n", + + "# Create a figure and an axes\n", + "fig, ax = plt.subplots()\n", + + "# Create a scatter plots\n", + "for cls in unique_classes:", + "\n ", + "x1_class = [x1[i] for i in range(len(x1)) if labels[i] == cls]", + "\n ", + "y1_class = [y1[i] for i in range(len(y1)) if labels[i] == cls]", + "\n ", + "scatter1 = plt.scatter(x1_class, y1_class, color=style_map[cls], marker=markers[cls], s=", size, ", label=cls)\n", + + "# Set the limits of the plot based on min and max values of x and y\n", + "plt.xlim(min(x1) - 0.1, max(x1) + 0.1)\n", + "plt.ylim(min(y1) - 0.1, max(y1) + 0.1)\n", + + "plt.xlabel('X-axis')\n", + "plt.ylabel('Y-axis')\n", + "plt.title('Scatter Plot using Matplotlib')\n", + "plt.legend()\n", + SelectBasedOnCondition( + action = "save", + Concatenation( "plt.savefig('", Filename( dir, "plot.png" ), "', dpi=400)\n" ), + "plt.show()\n" ) ) ); + + IO_Close( file ); + + stream := IO_Popen3( IO_FindExecutable( "python" ), [ path, "&" ] ); + + err := Concatenation( IO_ReadLines( stream.stderr ) ); + + IO_ReadLines( stream.stdout ); + + IO_Close( stream.stdin ); + + IO_Close( stream.stdout ); + + IO_Close( stream.stderr ); + + if not IsEmpty( err ) then - return dir; + Error( err, "\n" ); + fi; + + return dir; + end ); diff --git a/init.g b/init.g index 6da5ae4..716a66c 100644 --- a/init.g +++ b/init.g @@ -1,15 +1,17 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Reading the declaration part of the package. # -ReadPackage( "GradientDescentForCAP", "gap/Tools.gd" ); -ReadPackage( "GradientDescentForCAP", "gap/Expressions.gd" ); -ReadPackage( "GradientDescentForCAP", "gap/MethodRecord.gd" ); -ReadPackage( "GradientDescentForCAP", "gap/MethodRecord.Declarations.autogen.gd" ); -ReadPackage( "GradientDescentForCAP", "gap/CategoryOfSkeletalSmoothMaps.gd" ); -ReadPackage( "GradientDescentForCAP", "gap/CategoryOfParametrisedMorphisms.gd" ); -ReadPackage( "GradientDescentForCAP", "gap/CategoryOfLenses.gd" ); -ReadPackage( "GradientDescentForCAP", "gap/FitParameters.gd" ); -ReadPackage( "GradientDescentForCAP", "gap/NeuralNetworks.gd" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/Tools.gd" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/Expressions.gd" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/MethodRecord.gd" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/MethodRecord.Declarations.autogen.gd" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/SkeletalCategoryOfSmoothMaps.gd" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/CategoryOfParametrisedMorphisms.gd" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/CategoryOfParametrisedMorphisms.autogen.gd" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/CategoryOfLenses.gd" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/CategoryOfLenses.autogen.gd" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/FitParameters.gd" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/NeuralNetworks.gd" ); diff --git a/makedoc.g b/makedoc.g index 867b82e..f9a6ca0 100644 --- a/makedoc.g +++ b/makedoc.g @@ -1,18 +1,22 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # This file is a script which compiles the package manual. # -if fail = LoadPackage( "AutoDoc", "2019.05.20" ) then +if fail = LoadPackage( "AutoDoc", "2025.12.19" ) then - Error( "AutoDoc version 2019.05.20 or newer is required." ); + Error( "AutoDoc version 2025.12.19 or newer is required." ); fi; AutoDoc( rec( autodoc := rec( files := [ "doc/Doc.autodoc" ], - scan_dirs := [ "doc", "gap", "examples", "examples/doc" ], + scan_dirs := [ "doc", "gap", "examples", "examples/doc", + "examples/NeuralNetwork_BinaryCrossEntropy", + "examples/NeuralNetwork_CrossEntropy", + "examples/NeuralNetwork_QuadraticLoss", + ], ), extract_examples := rec( units := "Single", @@ -20,14 +24,17 @@ AutoDoc( rec( gapdoc := rec( LaTeXOptions := rec( LateExtraPreamble := """ + \usepackage{tikz} + \usetikzlibrary{positioning} \usepackage{mathtools} + \usepackage{stmaryrd} \DeclareUnicodeCharacter{211D}{\ensuremath{\mathbb{R}}} \DeclareUnicodeCharacter{2023}{\ensuremath{\blacktriangleright}} """, ), ), scaffold := rec( - entities := [ "homalg", "CAP" ], + entities := rec( homalg := "homalg", CAP := "CAP" ), ), ) ); diff --git a/makedoc_with_overfull_hbox_warnings.g b/makedoc_with_overfull_hbox_warnings.g index 4373425..1eeea61 100644 --- a/makedoc_with_overfull_hbox_warnings.g +++ b/makedoc_with_overfull_hbox_warnings.g @@ -1,11 +1,11 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # This file is a script which compiles the package manual and prints overfull hbox warnings. # -if fail = LoadPackage( "AutoDoc", "2019.05.20" ) then +if fail = LoadPackage( "AutoDoc", "2025.12.19" ) then - Error( "AutoDoc version 2019.05.20 or newer is required." ); + Error( "AutoDoc version 2025.12.19 or newer is required." ); fi; @@ -18,7 +18,10 @@ AutoDoc( rec( gapdoc := rec( LaTeXOptions := rec( LateExtraPreamble := """ + \usepackage{tikz} + \usetikzlibrary{positioning} \usepackage{mathtools} + \usepackage{stmaryrd} \DeclareUnicodeCharacter{211D}{\ensuremath{\mathbb{R}}} \DeclareUnicodeCharacter{2023}{\ensuremath{\blacktriangleright}} % Many thanks to https://tex.stackexchange.com/questions/22466/how-to-convince-fancyvrb-to-give-overfull-warnings/534486#534486 @@ -38,7 +41,7 @@ AutoDoc( rec( ), ), scaffold := rec( - entities := [ "homalg", "CAP" ], + entities := rec( homalg := "homalg", CAP := "CAP" ), ), ) ); diff --git a/makefile b/makefile index 6ccb687..652cd7e 100644 --- a/makefile +++ b/makefile @@ -18,9 +18,9 @@ test: doc test-basic-spacing: # exit code 1 means no match, which is what we want here (exit code 2 signals an error) - grep -RPl "\t" examples/*.g gap/*.g*; test $$? -eq 1 || (echo "Tabs found" && exit 1) - grep -RPl "\r" examples/*.g gap/*.g*; test $$? -eq 1 || (echo "Windows line-endings found" && exit 1) - grep -RPzL "\n\z" examples/*.g gap/*.g* | grep ""; test $$? -eq 1 || (echo "File with no newline at end of file found" && exit 1) + grep -RPl "\t" examples/ gap/ --exclude="*.png"; test $$? -eq 1 || (echo "Tabs found" && exit 1) + grep -RPl "\r" examples/ gap/ --exclude="*.png"; test $$? -eq 1 || (echo "Windows line-endings found" && exit 1) + grep -RPzL "\n\z" examples/ gap/ --exclude="*.png" | grep ""; test $$? -eq 1 || (echo "File with no newline at end of file found" && exit 1) test-doc: doc cp -aT doc/ doc_tmp/ @@ -37,7 +37,7 @@ test-spacing: grep -RE '[^ ] +$$' gap/*; test $$? -eq 1 || (echo "Trailing whitespace found" && exit 1) for filename in gap/*; do \ echo $$filename; \ - gap --quitonbreak --norepl --banner -c "LoadPackage(\"GradientDescentForCAP\"); SizeScreen([4096]); func := ReadAsFunction(\"$$filename\"); FileString(\"gap_spacing\", DisplayString(func));"; \ + gap --quitonbreak --norepl --banner -c "LoadPackage(\"GradientBasedLearningForCAP\"); SizeScreen([4096]); func := ReadAsFunction(\"$$filename\"); FileString(\"gap_spacing\", DisplayString(func));"; \ # In a perfect world, the DisplayString of a function would exactly match our code. However, our line breaks and indentation might differ from the GAP ones, \ # so we remove all indentation, line breaks, and empty lines, and afterwards insert line breaks at semicolons again for better readability. \ cat "gap_spacing" | tail -n +2 | head -n -2 | sed 's/\[ \]/[ ]/g' | sed 's/( )/( )/g' | sed 's/( :/( :/g' | sed 's/ *$$//' | sed 's/^ *//' | grep -v "^$$" | tr "\n" " " | sed 's/;/;\n/g' > modified_gap_spacing; \ diff --git a/read.g b/read.g index 5c4f7d6..b97149c 100644 --- a/read.g +++ b/read.g @@ -1,15 +1,15 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # Reading the implementation part of the package. # -ReadPackage( "GradientDescentForCAP", "gap/Tools.gi"); -ReadPackage( "GradientDescentForCAP", "gap/Expressions.gi"); -ReadPackage( "GradientDescentForCAP", "gap/MethodRecord.gi"); -ReadPackage( "GradientDescentForCAP", "gap/MethodRecord.Installations.autogen.gi"); -ReadPackage( "GradientDescentForCAP", "gap/CategoryOfSkeletalSmoothMaps.gi"); -ReadPackage( "GradientDescentForCAP", "gap/CategoryOfParametrisedMorphisms.gi" ); -ReadPackage( "GradientDescentForCAP", "gap/CategoryOfLenses.gi" ); -ReadPackage( "GradientDescentForCAP", "gap/FitParameters.gi" ); -ReadPackage( "GradientDescentForCAP", "gap/NeuralNetworks.gi" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/Tools.gi"); +ReadPackage( "GradientBasedLearningForCAP", "gap/Expressions.gi"); +ReadPackage( "GradientBasedLearningForCAP", "gap/MethodRecord.gi"); +ReadPackage( "GradientBasedLearningForCAP", "gap/MethodRecord.Installations.autogen.gi"); +ReadPackage( "GradientBasedLearningForCAP", "gap/SkeletalCategoryOfSmoothMaps.gi"); +ReadPackage( "GradientBasedLearningForCAP", "gap/CategoryOfParametrisedMorphisms.gi" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/CategoryOfLenses.gi" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/FitParameters.gi" ); +ReadPackage( "GradientBasedLearningForCAP", "gap/NeuralNetworks.gi" ); diff --git a/stats.514058 b/stats.514058 new file mode 100644 index 0000000..f83f144 --- /dev/null +++ b/stats.514058 @@ -0,0 +1 @@ +{ "Type": "_", "Version":1, "IsCover": true, "TimeType": "Memory"} diff --git a/tst/100_LoadPackage.tst b/tst/100_LoadPackage.tst index fc663ef..4885d54 100644 --- a/tst/100_LoadPackage.tst +++ b/tst/100_LoadPackage.tst @@ -1,5 +1,5 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # This file tests if the package can be loaded without errors or warnings. # @@ -7,9 +7,9 @@ gap> PushOptions( rec( OnlyNeeded := true ) ); gap> package_loading_info_level := InfoLevel( InfoPackageLoading );; gap> SetInfoLevel( InfoPackageLoading, PACKAGE_ERROR );; -gap> LoadPackage( "GradientDescentForCAP", false ); +gap> LoadPackage( "GradientBasedLearningForCAP", false ); true gap> SetInfoLevel( InfoPackageLoading, PACKAGE_INFO );; -gap> LoadPackage( "GradientDescentForCAP" ); +gap> LoadPackage( "GradientBasedLearningForCAP" ); true gap> SetInfoLevel( InfoPackageLoading, package_loading_info_level );; diff --git a/tst/example-in-readme-1.tst b/tst/example-in-readme-1.tst index d083f99..72ef08f 100644 --- a/tst/example-in-readme-1.tst +++ b/tst/example-in-readme-1.tst @@ -1,4 +1,4 @@ -gap> LoadPackage( "GradientDescentForCAP" ); +gap> LoadPackage( "GradientBasedLearningForCAP" ); true gap> Para := CategoryOfParametrisedMorphisms( SkeletalSmoothMaps ); CategoryOfParametrisedMorphisms( SkeletalSmoothMaps ) @@ -8,7 +8,7 @@ gap> training_set := [ [ 1, 2.9 ], [ 2, 5.1 ], [ 3, 7.05 ] ]; gap> input_dim := 1;; output_dim := 1;; hidden_dims := [ ];; -gap> f := PredictionMorphismOfNeuralNetwork( Para, input_dim, hidden_dims, output_dim, "IdFunc" );; +gap> f := NeuralNetworkPredictionMorphism( Para, input_dim, hidden_dims, output_dim, "IdFunc" );; gap> Display( f ); ℝ^1 -> ℝ^1 defined by: @@ -28,7 +28,7 @@ gap> parameters := [ 2, 1 ];; x := [ 2 ];; gap> Eval( f, [ parameters, x ] ); [ 5 ] -gap> ell := LossMorphismOfNeuralNetwork( Para, input_dim, hidden_dims, output_dim, "IdFunc" );; +gap> ell := NeuralNetworkLossMorphism( Para, input_dim, hidden_dims, output_dim, "IdFunc" );; gap> Display( ell ); ℝ^2 -> ℝ^1 defined by: diff --git a/tst/example-in-readme-2.tst b/tst/example-in-readme-2.tst index bb4a40c..c37261f 100644 --- a/tst/example-in-readme-2.tst +++ b/tst/example-in-readme-2.tst @@ -1,4 +1,4 @@ -gap> LoadPackage( "GradientDescentForCAP" ); +gap> LoadPackage( "GradientBasedLearningForCAP" ); true gap> Para := CategoryOfParametrisedMorphisms( SkeletalSmoothMaps ); @@ -18,9 +18,9 @@ gap> training_set := Concatenation( class_1, class_2, class_3 ); gap> input_dim := 2;; output_dim := 3;; hidden_dims := [ ];; -gap> f := PredictionMorphismOfNeuralNetwork( Para, input_dim, hidden_dims, output_dim, "Softmax" );; +gap> f := NeuralNetworkPredictionMorphism( Para, input_dim, hidden_dims, output_dim, "Softmax" );; -gap> input := ConvertToExpressions( [ "theta_1", "theta_2", "theta_3", "theta_4", "theta_5", "theta_6", "theta_7", "theta_8", "theta_9", "x1", "x2" ] );; +gap> input := CreateContextualVariables( [ "theta_1", "theta_2", "theta_3", "theta_4", "theta_5", "theta_6", "theta_7", "theta_8", "theta_9", "x1", "x2" ] );; gap> Display( f : dummy_input := input ); ℝ^2 -> ℝ^3 defined by: @@ -50,9 +50,9 @@ gap> prediction_x := Eval( f, [ parameters, x ] ); gap> PositionMaximum( prediction_x ); 2 -gap> ell := LossMorphismOfNeuralNetwork( Para, input_dim, hidden_dims, output_dim, "Softmax" );; +gap> ell := NeuralNetworkLossMorphism( Para, input_dim, hidden_dims, output_dim, "Softmax" );; -gap> input := ConvertToExpressions( [ "theta_1", "theta_2", "theta_3", "theta_4", "theta_5", "theta_6", "theta_7", "theta_8", "theta_9", "x1", "x2", "y1", "y2", "y3" ] );; +gap> input := CreateContextualVariables( [ "theta_1", "theta_2", "theta_3", "theta_4", "theta_5", "theta_6", "theta_7", "theta_8", "theta_9", "x1", "x2", "y1", "y2", "y3" ] );; gap> Display( ell : dummy_input := input ); ℝ^5 -> ℝ^1 defined by: @@ -82,11 +82,11 @@ gap> one_epoch_update := OneEpochUpdateLens( ell, optimizer, training_set, batch (ℝ^28, ℝ^28) -> (ℝ^1, ℝ^0) defined by: Get Morphism: ----------- +------------ ℝ^28 -> ℝ^1 Put Morphism: ----------- +------------ ℝ^28 -> ℝ^28 gap> parameters := [ 0.1, -0.1, 0, 0.1, 0.2, 0, -0.2, 0.3, 0 ];; diff --git a/tst/functor.tst b/tst/functor.tst index b4da613..a5a2ddd 100644 --- a/tst/functor.tst +++ b/tst/functor.tst @@ -4,8 +4,8 @@ gap> Smooth := SkeletalSmoothMaps;; gap> Lenses := CategoryOfLenses( Smooth );; gap> Para := CategoryOfParametrisedMorphisms( Smooth );; gap> Para_Lenses := CategoryOfParametrisedMorphisms( Lenses );; -gap> ell := LossMorphismOfNeuralNetwork( Para, 2, [], 1, "IdFunc" );; -gap> dummy_input := ConvertToExpressions( [ "w1", "w2", "b1", "x1", "x2", "y" ] ); +gap> ell := NeuralNetworkLossMorphism( Para, 2, [], 1, "IdFunc" );; +gap> dummy_input := CreateContextualVariables( [ "w1", "w2", "b1", "x1", "x2", "y" ] ); [ w1, w2, b1, x1, x2, y ] gap> Display( ell : dummy_input := dummy_input ); ℝ^3 -> ℝ^1 defined by: @@ -33,11 +33,11 @@ Underlying Morphism: (ℝ^6, ℝ^6) -> (ℝ^1, ℝ^1) defined by: Get Morphism: ----------- +------------ ℝ^6 -> ℝ^1 Put Morphism: ----------- +------------ ℝ^7 -> ℝ^6 gap> Display( Rf ); (ℝ^3, ℝ^3) -> (ℝ^1, ℝ^1) defined by: diff --git a/tst/neural-network-1.tst b/tst/neural-network-1.tst index 5390ad9..dd35446 100644 --- a/tst/neural-network-1.tst +++ b/tst/neural-network-1.tst @@ -1,7 +1,7 @@ gap> Smooth := SkeletalSmoothMaps;; gap> Lenses := CategoryOfLenses( Smooth );; gap> Para := CategoryOfParametrisedMorphisms( Smooth );; -gap> f := LossMorphismOfNeuralNetwork( Para, 2, [], 1, "IdFunc" );; +gap> f := NeuralNetworkLossMorphism( Para, 2, [], 1, "IdFunc" );; gap> optimizer := Lenses.AdamOptimizer( );; gap> training_examples_path := SelectBasedOnCondition( IsExistingFile( "data-1.txt" ), "data-1.txt", "tst/data-1.txt" );; gap> batch_size := 5;; diff --git a/tst/neural-network-2.tst b/tst/neural-network-2.tst index 0d64663..2ed60e7 100644 --- a/tst/neural-network-2.tst +++ b/tst/neural-network-2.tst @@ -1,7 +1,7 @@ gap> Smooth := SkeletalSmoothMaps;; gap> Lenses := CategoryOfLenses( Smooth );; gap> Para := CategoryOfParametrisedMorphisms( Smooth );; -gap> f := LossMorphismOfNeuralNetwork( Para, 2, [ 5, 5 ], 4, "Softmax" );; +gap> f := NeuralNetworkLossMorphism( Para, 2, [ 5, 5 ], 4, "Softmax" );; gap> optimizer := Lenses.GradientDescentOptimizer( : learning_rate := 0.01 );; gap> training_examples_path := SelectBasedOnCondition( IsExistingFile( "data-2.txt" ), "data-2.txt", "tst/data-2.txt" );; gap> batch_size := 1;; diff --git a/tst/simplify-expression-using-python.tst b/tst/simplify-expression-using-python.tst new file mode 100644 index 0000000..00c9282 --- /dev/null +++ b/tst/simplify-expression-using-python.tst @@ -0,0 +1,60 @@ +############################################################################# +# Tests for SimplifyExpressionUsingPython +############################################################################# + +gap> LoadPackage( "GradientBasedLearningForCAP" ); +true + +gap> vars := [ "x1", "x2", "x3" ];; + +gap> exprs := [ +> "x1 + 0", +> "0 + x2", +> "x1 * 1", +> "1 * x2", +> "x1 - x1", +> "x1^2 * x1^3", +> "Exp(Log(x1))", +> "Sin(x1)^2 + Cos(x1)^2", +> "Relu(x1)" ];; + +gap> out := SimplifyExpressionUsingPython( vars, exprs );; +gap> Assert( 0, IsDenseList( out ) ); +gap> Assert( 0, Length( out ) = Length( exprs ) ); +gap> Assert( 0, ForAll( out, IsString ) ); + +# Spot checks that are stable across SymPy versions. +gap> Assert( 0, out[1] = "x1" ); +gap> Assert( 0, out[2] = "x2" ); +gap> Assert( 0, out[3] = "x1" ); +gap> Assert( 0, out[4] = "x2" ); +gap> Assert( 0, out[5] = "0" ); +gap> Assert( 0, out[6] in [ "x1**5", "x1^5" ] ); +gap> Assert( 0, out[8] in [ "1", "1.0" ] ); + +# Expressions containing Diff are intentionally not passed through python and must be preserved. +gap> exprs2 := [ "Diff( [\"x1\"], \"x1^2\", 1 )( [x1] )", "x1 + 0" ];; +gap> out2 := SimplifyExpressionUsingPython( ["x1"], exprs2 );; +gap> Assert( 0, out2[1] = exprs2[1] ); +gap> Assert( 0, out2[2] = "x1" ); + +# A small randomized batch: ensure it returns without error and preserves length. +gap> RandSeed := 12345;; +gap> RandomSource( IsMersenneTwister, RandSeed );; +gap> ops := [ "+", "-", "*" ];; +gap> rand_exprs := List( [ 1 .. 20 ], i -> +> Concatenation( +> "(", +> Random( vars ), +> Random( ops ), +> String( Random( [ 0, 1, 2, 3 ] ) ), +> ")", +> Random( ops ), +> "(", +> Random( vars ), +> Random( ops ), +> "0", +> ")" ) );; +gap> rand_out := SimplifyExpressionUsingPython( vars, rand_exprs );; +gap> Assert( 0, Length( rand_out ) = Length( rand_exprs ) ); +gap> Assert( 0, ForAll( rand_out, IsString ) ); diff --git a/tst/testall.g b/tst/testall.g index ae003a2..38070d2 100644 --- a/tst/testall.g +++ b/tst/testall.g @@ -1,5 +1,5 @@ # SPDX-License-Identifier: GPL-2.0-or-later -# GradientDescentForCAP: Exploring categorical machine learning in CAP +# GradientBasedLearningForCAP: Gradient Based Learning via Category Theory # # This file runs package tests. It is also referenced in the package # metadata in PackageInfo.g. @@ -30,6 +30,6 @@ else fi; -TestDirectory( DirectoriesPackageLibrary( "GradientDescentForCAP", "tst" ), options ); +TestDirectory( DirectoriesPackageLibrary( "GradientBasedLearningForCAP", "tst" ), options ); FORCE_QUIT_GAP( 1 ); # if we ever get here, there was an error