[MRG] Initialization for Low Rank Sinkhorn solver (#588)

* new file for lr sinkhorn

* lr sinkhorn, solve_sample, OTResultLazy

* add test functions + small modif lr_sin/solve_sample

* add import to __init__

* modify low rank, remove solve_sample,OTResultLazy

* new file for lr sinkhorn

* lr sinkhorn, solve_sample, OTResultLazy

* add test functions + small modif lr_sin/solve_sample

* add import to __init__

* remove test solve_sample

* add value, value_linear, lazy_plan

* add comments to lr algorithm

* modify test functions + add comments to lowrank

* modify __init__ with lowrank

* debug lowrank + test

* debug test function low_rank

* error test

* final debug of lowrank + add new test functions

* Debug tests + add lowrank to solve_sample

* fix torch backend for lowrank

* fix jax backend and skip tf

* fix pep 8 tests

* add lowrank init + test functions

* Add init strategies in lowrank + example (#588)

* modified lowrank

* changes from code review

* fix error test pep8

* fix linux-minimal-deps + code review

Author: laudavid

README.md

@@ -194,7 +194,8 @@ The numerous contributors to this library are listed [here](CONTRIBUTORS.md).
 
 POT has benefited from the financing or manpower from the following partners:
 
-<img src="https://pythonot.github.io/master/_static/images/logo_anr.jpg" alt="ANR" style="height:60px;"/><img src="https://pythonot.github.io/master/_static/images/logo_cnrs.jpg" alt="CNRS" style="height:60px;"/><img src="https://pythonot.github.io/master/_static/images/logo_3ia.jpg" alt="3IA" style="height:60px;"/>
+<img src="https://pythonot.github.io/master/_static/images/logo_anr.jpg" alt="ANR" style="height:60px;"/><img src="https://pythonot.github.io/master/_static/images/logo_cnrs.jpg" alt="CNRS" style="height:60px;"/><img src="https://pythonot.github.io/master/_static/images/logo_3ia.jpg" alt="3IA" style="height:60px;"/><img src="https://pythonot.github.io/master/_static/images/logo_hiparis.png" alt="Hi!PARIS" style="height:60px;"/>
+
 
 
 ## Contributions and code of conduct

docs/source/_static/images/logo_hiparis.png

@@ -0,0 +1,129 @@
+# -*- coding: utf-8 -*-
+"""
+========================================
+Low rank Sinkhorn
+========================================
+
+This example illustrates the computation of Low Rank Sinkhorn [26].
+
+[65] Scetbon, M., Cuturi, M., & Peyré, G. (2021).
+"Low-rank Sinkhorn factorization". In International Conference on Machine Learning.
+"""
+
+# Author: Laurène David <laurene.david@ip-paris.fr>
+#
+# License: MIT License
+#
+# sphinx_gallery_thumbnail_number = 2
+
+import numpy as np
+import matplotlib.pylab as pl
+import ot.plot
+from ot.datasets import make_1D_gauss as gauss
+
+##############################################################################
+# Generate data
+# -------------
+
+#%% parameters
+
+n = 100
+m = 120
+
+# Gaussian distribution
+a = gauss(n, m=int(n / 3), s=25 / np.sqrt(2)) + 1.5 * gauss(n, m=int(5 * n / 6), s=15 / np.sqrt(2))
+a = a / np.sum(a)
+
+b = 2 * gauss(m, m=int(m / 5), s=30 / np.sqrt(2)) + gauss(m, m=int(m / 2), s=35 / np.sqrt(2))
+b = b / np.sum(b)
+
+# Source and target distribution
+X = np.arange(n).reshape(-1, 1)
+Y = np.arange(m).reshape(-1, 1)
+
+
+##############################################################################
+# Solve Low rank sinkhorn
+# ------------
+
+#%%
+# Solve low rank sinkhorn
+Q, R, g, log = ot.lowrank_sinkhorn(X, Y, a, b, rank=10, init="random", gamma_init="rescale", rescale_cost=True, warn=False, log=True)
+P = log["lazy_plan"][:]
+
+ot.plot.plot1D_mat(a, b, P, 'OT matrix Low rank')
+
+
+##############################################################################
+# Sinkhorn vs Low Rank Sinkhorn
+# -----------------------
+# Compare Sinkhorn and Low rank sinkhorn with different regularizations and ranks.
+
+#%% Sinkhorn
+
+# Compute cost matrix for sinkhorn OT
+M = ot.dist(X, Y)
+M = M / np.max(M)
+
+# Solve sinkhorn with different regularizations using ot.solve
+list_reg = [0.05, 0.005, 0.001]
+list_P_Sin = []
+
+for reg in list_reg:
+    P = ot.solve(M, a, b, reg=reg, max_iter=2000, tol=1e-8).plan
+    list_P_Sin.append(P)
+
+#%% Low rank sinkhorn
+
+# Solve low rank sinkhorn with different ranks using ot.solve_sample
+list_rank = [3, 10, 50]
+list_P_LR = []
+
+for rank in list_rank:
+    P = ot.solve_sample(X, Y, a, b, method='lowrank', rank=rank).plan
+    P = P[:]
+    list_P_LR.append(P)
+
+
+#%%
+
+# Plot sinkhorn vs low rank sinkhorn
+pl.figure(1, figsize=(10, 4))
+
+pl.subplot(1, 3, 1)
+pl.imshow(list_P_Sin[0], interpolation='nearest')
+pl.axis('off')
+pl.title('Sinkhorn (reg=0.05)')
+
+pl.subplot(1, 3, 2)
+pl.imshow(list_P_Sin[1], interpolation='nearest')
+pl.axis('off')
+pl.title('Sinkhorn (reg=0.005)')
+
+pl.subplot(1, 3, 3)
+pl.imshow(list_P_Sin[2], interpolation='nearest')
+pl.axis('off')
+pl.title('Sinkhorn (reg=0.001)')
+pl.show()
+
+
+#%%
+
+pl.figure(2, figsize=(10, 4))
+
+pl.subplot(1, 3, 1)
+pl.imshow(list_P_LR[0], interpolation='nearest')
+pl.axis('off')
+pl.title('Low rank (rank=3)')
+
+pl.subplot(1, 3, 2)
+pl.imshow(list_P_LR[1], interpolation='nearest')
+pl.axis('off')
+pl.title('Low rank (rank=10)')
+
+pl.subplot(1, 3, 3)
+pl.imshow(list_P_LR[2], interpolation='nearest')
+pl.axis('off')
+pl.title('Low rank (rank=50)')
+
+pl.tight_layout()