[WIP] Add debiased barycenter (Sinkhorn + convolutional sinkhorn) (#291)

* add debiased sinkhorn barycenter + make loops pythonic

* add debiased arg in tests

* add 1d and 2d examples of debiased barycenters

* fix doctest

* fix flake8

* pep8 + make func private + add convergence warnings

* remove rel paths + add rng + pylab to pyplot

* fix stopping criterion debiased

* pass alex

* change params with new API

* add logdomain barycenters + separate debiased API

* test new API

* fix jax read-only ?

* raise error for jax

* test catch jax error

* fix pytest catch error

* fix relative path

* fix flake8

* add warn arg everywhere

* fix ref number

* catch warnings in tests

* add contrib to readme + change ref number

* fix convolution example + gallery thumbnails

* increase coverage

* fix flake

Co-authored-by: Hicham Janati <hicham.janati@inria.fr>
Co-authored-by: Rémi Flamary <remi.flamary@gmail.com>
Co-authored-by: Alexandre Gramfort <alexandre.gramfort@m4x.org>

Author: 4 people

README.md

@@ -22,7 +22,8 @@ POT provides the following generic OT solvers (links to examples):
 * [Conditional gradient](https://pythonot.github.io/auto_examples/plot_optim_OTreg.html) [6] and [Generalized conditional gradient](https://pythonot.github.io/auto_examples/plot_optim_OTreg.html) for regularized OT [7].
 * Entropic regularization OT solver with [Sinkhorn Knopp Algorithm](https://pythonot.github.io/auto_examples/plot_OT_1D.html) [2] , stabilized version [9] [10] [34], greedy Sinkhorn [22] and [Screening Sinkhorn [26] ](https://pythonot.github.io/auto_examples/plot_screenkhorn_1D.html).
 * Bregman projections for [Wasserstein barycenter](https://pythonot.github.io/auto_examples/barycenters/plot_barycenter_lp_vs_entropic.html) [3], [convolutional barycenter](https://pythonot.github.io/auto_examples/barycenters/plot_convolutional_barycenter.html) [21]  and unmixing [4].
-* Sinkhorn divergence [23] and entropic regularization OT  from empirical data.
+* Sinkhorn divergence [23] and entropic regularization OT from empirical data.
+* Debiased Sinkhorn barycenters [Sinkhorn divergence barycenter](https://pythonot.github.io/auto_examples/barycenters/plot_debiased_barycenter.html) [37]
 * [Smooth optimal transport solvers](https://pythonot.github.io/auto_examples/plot_OT_1D_smooth.html) (dual and semi-dual) for KL and squared L2 regularizations [17].
 * Non regularized [Wasserstein barycenters [16] ](https://pythonot.github.io/auto_examples/barycenters/plot_barycenter_lp_vs_entropic.html)) with LP solver (only small scale).
 * [Gromov-Wasserstein distances](https://pythonot.github.io/auto_examples/gromov/plot_gromov.html) and [GW barycenters](https://pythonot.github.io/auto_examples/gromov/plot_gromov_barycenter.html)  (exact [13] and regularized [12])
@@ -188,7 +189,7 @@ The contributors to this library are
 * [Kilian Fatras](https://kilianfatras.github.io/) (Stochastic solvers)
 * [Alain Rakotomamonjy](https://sites.google.com/site/alainrakotomamonjy/home)
 * [Vayer Titouan](https://tvayer.github.io/) (Gromov-Wasserstein -, Fused-Gromov-Wasserstein)
-* [Hicham Janati](https://hichamjanati.github.io/) (Unbalanced OT)
+* [Hicham Janati](https://hichamjanati.github.io/) (Unbalanced OT, Debiased barycenters)
 * [Romain Tavenard](https://rtavenar.github.io/) (1d Wasserstein)
 * [Mokhtar Z. Alaya](http://mzalaya.github.io/) (Screenkhorn)
 * [Ievgen Redko](https://ievred.github.io/) (Laplacian DA, JCPOT)
@@ -293,3 +294,6 @@ You can also post bug reports and feature requests in Github issues. Make sure t
 (2019, May). [Sliced-Wasserstein flows: Nonparametric generative modeling 
 via optimal transport and diffusions](http://proceedings.mlr.press/v97/liutkus19a/liutkus19a.pdf). In International Conference on 
 Machine Learning (pp. 4104-4113). PMLR.
+
+[37] Janati, H., Cuturi, M., Gramfort, A. Proceedings of the 37th International
+Conference on Machine Learning, PMLR 119:4692-4701, 2020

examples/barycenters/plot_barycenter_1D.py

@@ -18,10 +18,10 @@
 #
 # License: MIT License
 
-# sphinx_gallery_thumbnail_number = 4
+# sphinx_gallery_thumbnail_number = 1
 
 import numpy as np
-import matplotlib.pylab as pl
+import matplotlib.pyplot as plt
 import ot
 # necessary for 3d plot even if not used
 from mpl_toolkits.mplot3d import Axes3D  # noqa
@@ -50,18 +50,6 @@
 M = ot.utils.dist0(n)
 M /= M.max()
 
-##############################################################################
-# Plot data
-# ---------
-
-#%% plot the distributions
-
-pl.figure(1, figsize=(6.4, 3))
-for i in range(n_distributions):
-    pl.plot(x, A[:, i])
-pl.title('Distributions')
-pl.tight_layout()
-
 ##############################################################################
 # Barycenter computation
 # ----------------------
@@ -78,24 +66,20 @@
 reg = 1e-3
 bary_wass = ot.bregman.barycenter(A, M, reg, weights)
 
-pl.figure(2)
-pl.clf()
-pl.subplot(2, 1, 1)
-for i in range(n_distributions):
-    pl.plot(x, A[:, i])
-pl.title('Distributions')
+f, (ax1, ax2) = plt.subplots(2, 1, tight_layout=True, num=1)
+ax1.plot(x, A, color="black")
+ax1.set_title('Distributions')
 
-pl.subplot(2, 1, 2)
-pl.plot(x, bary_l2, 'r', label='l2')
-pl.plot(x, bary_wass, 'g', label='Wasserstein')
-pl.legend()
-pl.title('Barycenters')
-pl.tight_layout()
+ax2.plot(x, bary_l2, 'r', label='l2')
+ax2.plot(x, bary_wass, 'g', label='Wasserstein')
+ax2.set_title('Barycenters')
+
+plt.legend()
+plt.show()
 
 ##############################################################################
 # Barycentric interpolation
 # -------------------------
-
 #%% barycenter interpolation
 
 n_alpha = 11
@@ -106,24 +90,23 @@
 
 B_wass = np.copy(B_l2)
 
-for i in range(0, n_alpha):
+for i in range(n_alpha):
     alpha = alpha_list[i]
     weights = np.array([1 - alpha, alpha])
     B_l2[:, i] = A.dot(weights)
     B_wass[:, i] = ot.bregman.barycenter(A, M, reg, weights)
 
 #%% plot interpolation
+plt.figure(2)
 
-pl.figure(3)
-
-cmap = pl.cm.get_cmap('viridis')
+cmap = plt.cm.get_cmap('viridis')
 verts = []
 zs = alpha_list
 for i, z in enumerate(zs):
     ys = B_l2[:, i]
     verts.append(list(zip(x, ys)))
 
-ax = pl.gcf().gca(projection='3d')
+ax = plt.gcf().gca(projection='3d')
 
 poly = PolyCollection(verts, facecolors=[cmap(a) for a in alpha_list])
 poly.set_alpha(0.7)
@@ -134,18 +117,18 @@
 ax.set_ylim3d(0, 1)
 ax.set_zlabel('')
 ax.set_zlim3d(0, B_l2.max() * 1.01)
-pl.title('Barycenter interpolation with l2')
-pl.tight_layout()
+plt.title('Barycenter interpolation with l2')
+plt.tight_layout()
 
-pl.figure(4)
-cmap = pl.cm.get_cmap('viridis')
+plt.figure(3)
+cmap = plt.cm.get_cmap('viridis')
 verts = []
 zs = alpha_list
 for i, z in enumerate(zs):
     ys = B_wass[:, i]
     verts.append(list(zip(x, ys)))
 
-ax = pl.gcf().gca(projection='3d')
+ax = plt.gcf().gca(projection='3d')
 
 poly = PolyCollection(verts, facecolors=[cmap(a) for a in alpha_list])
 poly.set_alpha(0.7)
@@ -156,7 +139,7 @@
 ax.set_ylim3d(0, 1)
 ax.set_zlabel('')
 ax.set_zlim3d(0, B_l2.max() * 1.01)
-pl.title('Barycenter interpolation with Wasserstein')
-pl.tight_layout()
+plt.title('Barycenter interpolation with Wasserstein')
+plt.tight_layout()
 
-pl.show()
+plt.show()

examples/barycenters/plot_barycenter_lp_vs_entropic.py

@@ -1,7 +1,7 @@
 # -*- coding: utf-8 -*-
 """
 =================================================================================
-1D Wasserstein barycenter comparison between exact LP and entropic regularization
+1D Wasserstein barycenter: exact LP vs entropic regularization
 =================================================================================
 
 This example illustrates the computation of regularized Wasserstein Barycenter

examples/barycenters/plot_convolutional_barycenter.py

@@ -6,17 +6,18 @@
 Convolutional Wasserstein Barycenter example
 ============================================
 
-This example is designed to illustrate how the Convolutional Wasserstein Barycenter
-function of POT works.
+This example is designed to illustrate how the Convolutional Wasserstein
+Barycenter function of POT works.
 """
 
 # Author: Nicolas Courty <ncourty@irisa.fr>
 #
 # License: MIT License
-
+import os
+from pathlib import Path
 
 import numpy as np
-import pylab as pl
+import matplotlib.pyplot as plt
 import ot
 
 ##############################################################################
@@ -25,22 +26,19 @@
 #
 # The four distributions are constructed from 4 simple images
 
+this_file = os.path.realpath('__file__')
+data_path = os.path.join(Path(this_file).parent.parent.parent, 'data')
 
-f1 = 1 - pl.imread('../../data/redcross.png')[:, :, 2]
-f2 = 1 - pl.imread('../../data/duck.png')[:, :, 2]
-f3 = 1 - pl.imread('../../data/heart.png')[:, :, 2]
-f4 = 1 - pl.imread('../../data/tooth.png')[:, :, 2]
+f1 = 1 - plt.imread(os.path.join(data_path, 'redcross.png'))[:, :, 2]
+f2 = 1 - plt.imread(os.path.join(data_path, 'tooth.png'))[:, :, 2]
+f3 = 1 - plt.imread(os.path.join(data_path, 'heart.png'))[:, :, 2]
+f4 = 1 - plt.imread(os.path.join(data_path, 'duck.png'))[:, :, 2]
 
-A = []
 f1 = f1 / np.sum(f1)
 f2 = f2 / np.sum(f2)
 f3 = f3 / np.sum(f3)
 f4 = f4 / np.sum(f4)
-A.append(f1)
-A.append(f2)
-A.append(f3)
-A.append(f4)
-A = np.array(A)
+A = np.array([f1, f2, f3, f4])
 
 nb_images = 5
 
@@ -57,14 +55,13 @@
 # ----------------------------------------
 #
 
-pl.figure(figsize=(10, 10))
-pl.title('Convolutional Wasserstein Barycenters in POT')
+fig, axes = plt.subplots(nb_images, nb_images, figsize=(7, 7))
+plt.suptitle('Convolutional Wasserstein Barycenters in POT')
 cm = 'Blues'
 # regularization parameter
 reg = 0.004
 for i in range(nb_images):
     for j in range(nb_images):
-        pl.subplot(nb_images, nb_images, i * nb_images + j + 1)
         tx = float(i) / (nb_images - 1)
         ty = float(j) / (nb_images - 1)
 
@@ -74,19 +71,19 @@
         weights = (1 - ty) * tmp1 + ty * tmp2
 
         if i == 0 and j == 0:
-            pl.imshow(f1, cmap=cm)
-            pl.axis('off')
+            axes[i, j].imshow(f1, cmap=cm)
         elif i == 0 and j == (nb_images - 1):
-            pl.imshow(f3, cmap=cm)
-            pl.axis('off')
+            axes[i, j].imshow(f3, cmap=cm)
         elif i == (nb_images - 1) and j == 0:
-            pl.imshow(f2, cmap=cm)
-            pl.axis('off')
+            axes[i, j].imshow(f2, cmap=cm)
         elif i == (nb_images - 1) and j == (nb_images - 1):
-            pl.imshow(f4, cmap=cm)
-            pl.axis('off')
+            axes[i, j].imshow(f4, cmap=cm)
         else:
             # call to barycenter computation
-            pl.imshow(ot.bregman.convolutional_barycenter2d(A, reg, weights), cmap=cm)
-            pl.axis('off')
-pl.show()
+            axes[i, j].imshow(
+                ot.bregman.convolutional_barycenter2d(A, reg, weights),
+                cmap=cm
+            )
+        axes[i, j].axis('off')
+plt.tight_layout()
+plt.show()

examples/barycenters/plot_debiased_barycenter.py

@@ -0,0 +1,131 @@
+# -*- coding: utf-8 -*-
+"""
+=================================
+Debiased Sinkhorn barycenter demo
+=================================
+
+This example illustrates the computation of the debiased Sinkhorn barycenter
+as proposed in [37]_.
+
+
+.. [37] Janati, H., Cuturi, M., Gramfort, A. Proceedings of the 37th
+        International Conference on Machine Learning, PMLR 119:4692-4701, 2020
+"""
+
+# Author: Hicham Janati <hicham.janati100@gmail.com>
+#
+# License: MIT License
+# sphinx_gallery_thumbnail_number = 3
+
+import os
+from pathlib import Path
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+import ot
+from ot.bregman import (barycenter, barycenter_debiased,
+                        convolutional_barycenter2d,
+                        convolutional_barycenter2d_debiased)
+
+##############################################################################
+# Debiased barycenter of 1D Gaussians
+# ------------------------------------
+
+#%% parameters
+
+n = 100  # nb bins
+
+# bin positions
+x = np.arange(n, dtype=np.float64)
+
+# Gaussian distributions
+a1 = ot.datasets.make_1D_gauss(n, m=20, s=5)  # m= mean, s= std
+a2 = ot.datasets.make_1D_gauss(n, m=60, s=8)
+
+# creating matrix A containing all distributions
+A = np.vstack((a1, a2)).T
+n_distributions = A.shape[1]
+
+# loss matrix + normalization
+M = ot.utils.dist0(n)
+M /= M.max()
+
+#%% barycenter computation
+
+alpha = 0.2  # 0<=alpha<=1
+weights = np.array([1 - alpha, alpha])
+
+epsilons = [5e-3, 1e-2, 5e-2]
+
+
+bars = [barycenter(A, M, reg, weights) for reg in epsilons]
+bars_debiased = [barycenter_debiased(A, M, reg, weights) for reg in epsilons]
+labels = ["Sinkhorn barycenter", "Debiased barycenter"]
+colors = ["indianred", "gold"]
+
+f, axes = plt.subplots(1, len(epsilons), tight_layout=True, sharey=True,
+                       figsize=(12, 4), num=1)
+for ax, eps, bar, bar_debiased in zip(axes, epsilons, bars, bars_debiased):
+    ax.plot(A[:, 0], color="k", ls="--", label="Input data", alpha=0.3)
+    ax.plot(A[:, 1], color="k", ls="--", alpha=0.3)
+    for data, label, color in zip([bar, bar_debiased], labels, colors):
+        ax.plot(data, color=color, label=label, lw=2)
+    ax.set_title(r"$\varepsilon = %.3f$" % eps)
+plt.legend()
+plt.show()
+
+
+##############################################################################
+# Debiased barycenter of 2D images
+# ---------------------------------
+this_file = os.path.realpath('__file__')
+data_path = os.path.join(Path(this_file).parent.parent.parent, 'data')
+f1 = 1 - plt.imread(os.path.join(data_path, 'heart.png'))[:, :, 2]
+f2 = 1 - plt.imread(os.path.join(data_path, 'duck.png'))[:, :, 2]
+
+A = np.asarray([f1, f2]) + 1e-2
+A /= A.sum(axis=(1, 2))[:, None, None]
+
+##############################################################################
+# Display the input images
+
+fig, axes = plt.subplots(1, 2, figsize=(7, 4), num=2)
+for ax, img in zip(axes, A):
+    ax.imshow(img, cmap="Greys")
+    ax.axis("off")
+fig.tight_layout()
+plt.show()
+
+
+##############################################################################
+# Barycenter computation and visualization
+# ----------------------------------------
+#
+
+bars_sinkhorn, bars_debiased = [], []
+epsilons = [5e-3, 7e-3, 1e-2]
+for eps in epsilons:
+    bar = convolutional_barycenter2d(A, eps)
+    bar_debiased, log = convolutional_barycenter2d_debiased(A, eps, log=True)
+    bars_sinkhorn.append(bar)
+    bars_debiased.append(bar_debiased)
+
+titles = ["Sinkhorn", "Debiased"]
+all_bars = [bars_sinkhorn, bars_debiased]
+fig, axes = plt.subplots(2, 3, figsize=(8, 6), num=3)
+for jj, (method, ax_row, bars) in enumerate(zip(titles, axes, all_bars)):
+    for ii, (ax, img, eps) in enumerate(zip(ax_row, bars, epsilons)):
+        ax.imshow(img, cmap="Greys")
+        if jj == 0:
+            ax.set_title(r"$\varepsilon = %.3f$" % eps, fontsize=13)
+        ax.set_xticks([])
+        ax.set_yticks([])
+        ax.spines['top'].set_visible(False)
+        ax.spines['right'].set_visible(False)
+        ax.spines['bottom'].set_visible(False)
+        ax.spines['left'].set_visible(False)
+        if ii == 0:
+            ax.set_ylabel(method, fontsize=15)
+fig.tight_layout()
+plt.show()