Minor corrections suggested by @agramfort + new barycenter example + test function

Author: Nicolas Courty

README.md

@@ -185,4 +185,4 @@ You can also post bug reports and feature requests in Github issues. Make sure t
 
 [11] Flamary, R., Cuturi, M., Courty, N., & Rakotomamonjy, A. (2016). [Wasserstein Discriminant Analysis](https://arxiv.org/pdf/1608.08063.pdf). arXiv preprint arXiv:1608.08063.
 
-[12] Peyré, Gabriel, Marco Cuturi, and Justin Solomon, [Gromov-Wasserstein averaging of kernel and distance matrices](http://proceedings.mlr.press/v48/peyre16.html)  International Conference on Machine Learning (ICML). 2016.
+[12] Gabriel Peyré, Marco Cuturi, and Justin Solomon, [Gromov-Wasserstein averaging of kernel and distance matrices](http://proceedings.mlr.press/v48/peyre16.html)  International Conference on Machine Learning (ICML). 2016.

data/carre.png

@@ -1,8 +1,8 @@
 # -*- coding: utf-8 -*-
 """
-====================
+==========================
 Gromov-Wasserstein example
-====================
+==========================
 This example is designed to show how to use the Gromov-Wassertsein distance
 computation in POT.
 """
@@ -14,14 +14,14 @@
 
 import scipy as sp
 import numpy as np
+import matplotlib.pylab as pl
 
 import ot
-import matplotlib.pylab as pl
 
 
 """
 Sample two Gaussian distributions (2D and 3D)
-====================
+=============================================
 The Gromov-Wasserstein distance allows to compute distances with samples that do not belong to the same metric space.
 For demonstration purpose, we sample two Gaussian distributions in 2- and 3-dimensional spaces.
 """
@@ -42,7 +42,7 @@
 
 """
 Plotting the distributions
-====================
+==========================
 """
 fig = pl.figure()
 ax1 = fig.add_subplot(121)
@@ -54,7 +54,7 @@
 
 """
 Compute distance kernels, normalize them and then display
-====================
+=========================================================
 """
 
 C1 = sp.spatial.distance.cdist(xs, xs)
@@ -72,7 +72,7 @@
 
 """
 Compute Gromov-Wasserstein plans and distance
-====================
+=============================================
 """
 
 p = ot.unif(n)

data/coeur.png

@@ -0,0 +1,240 @@
+# -*- coding: utf-8 -*-
+"""
+=====================================
+Gromov-Wasserstein Barycenter example
+=====================================
+This example is designed to show how to use the Gromov-Wassertsein distance
+computation in POT.
+"""
+
+# Author: Erwan Vautier <erwan.vautier@gmail.com>
+#         Nicolas Courty <ncourty@irisa.fr>
+#
+# License: MIT License
+
+
+import numpy as np
+import scipy as sp
+
+import scipy.ndimage as spi
+import matplotlib.pylab as pl
+from sklearn import manifold
+from sklearn.decomposition import PCA
+
+import ot
+
+"""
+
+Smacof MDS
+==========
+This function allows to find an embedding of points given a dissimilarity matrix
+that will be given by the output of the algorithm
+"""
+
+
+def smacof_mds(C, dim, maxIter=3000, eps=1e-9):
+    """
+    Returns an interpolated point cloud following the dissimilarity matrix C using SMACOF
+    multidimensional scaling (MDS) in specific dimensionned target space
+
+    Parameters
+    ----------
+    C : np.ndarray(ns,ns)
+        dissimilarity matrix
+    dim : Integer
+          dimension of the targeted space
+    maxIter : Maximum number of iterations of the SMACOF algorithm for a single run
+
+    eps : relative tolerance w.r.t stress to declare converge
+
+
+    Returns
+    -------
+    npos : R**dim ndarray
+           Embedded coordinates of the interpolated point cloud (defined with one isometry)
+
+
+    """
+
+    seed = np.random.RandomState(seed=3)
+
+    mds = manifold.MDS(
+        dim,
+        max_iter=3000,
+        eps=1e-9,
+        dissimilarity='precomputed',
+        n_init=1)
+    pos = mds.fit(C).embedding_
+
+    nmds = manifold.MDS(
+        2,
+        max_iter=3000,
+        eps=1e-9,
+        dissimilarity="precomputed",
+        random_state=seed,
+        n_init=1)
+    npos = nmds.fit_transform(C, init=pos)
+
+    return npos
+
+
+"""
+Data preparation
+================
+The four distributions are constructed from 4 simple images
+"""
+
+
+def im2mat(I):
+    """Converts and image to matrix (one pixel per line)"""
+    return I.reshape((I.shape[0] * I.shape[1], I.shape[2]))
+
+
+carre = spi.imread('../data/carre.png').astype(np.float64) / 256
+rond = spi.imread('../data/rond.png').astype(np.float64) / 256
+triangle = spi.imread('../data/triangle.png').astype(np.float64) / 256
+fleche = spi.imread('../data/coeur.png').astype(np.float64) / 256
+
+shapes = [carre, rond, triangle, fleche]
+
+S = 4
+xs = [[] for i in range(S)]
+
+
+for nb in range(4):
+    for i in range(8):
+        for j in range(8):
+            if shapes[nb][i, j] < 0.95:
+                xs[nb].append([j, 8 - i])
+
+xs = np.array([np.array(xs[0]), np.array(xs[1]),
+               np.array(xs[2]), np.array(xs[3])])
+
+
+"""
+Barycenter computation
+======================
+The four distributions are constructed from 4 simple images
+"""
+ns = [len(xs[s]) for s in range(S)]
+N = 30
+
+"""Compute all distances matrices for the four shapes"""
+Cs = [sp.spatial.distance.cdist(xs[s], xs[s]) for s in range(S)]
+Cs = [cs / cs.max() for cs in Cs]
+
+ps = [ot.unif(ns[s]) for s in range(S)]
+p = ot.unif(N)
+
+
+lambdast = [[float(i) / 3, float(3 - i) / 3] for i in [1, 2]]
+
+Ct01 = [0 for i in range(2)]
+for i in range(2):
+    Ct01[i] = ot.gromov.gromov_barycenters(N, [Cs[0], Cs[1]], [
+                                           ps[0], ps[1]], p, lambdast[i], 'square_loss', 5e-4, numItermax=100, stopThr=1e-3)
+
+Ct02 = [0 for i in range(2)]
+for i in range(2):
+    Ct02[i] = ot.gromov.gromov_barycenters(N, [Cs[0], Cs[2]], [
+                                           ps[0], ps[2]], p, lambdast[i], 'square_loss', 5e-4, numItermax=100, stopThr=1e-3)
+
+Ct13 = [0 for i in range(2)]
+for i in range(2):
+    Ct13[i] = ot.gromov.gromov_barycenters(N, [Cs[1], Cs[3]], [
+                                           ps[1], ps[3]], p, lambdast[i], 'square_loss', 5e-4, numItermax=100, stopThr=1e-3)
+
+Ct23 = [0 for i in range(2)]
+for i in range(2):
+    Ct23[i] = ot.gromov.gromov_barycenters(N, [Cs[2], Cs[3]], [
+                                           ps[2], ps[3]], p, lambdast[i], 'square_loss', 5e-4, numItermax=100, stopThr=1e-3)
+
+"""
+Visualization
+=============
+"""
+
+"""The PCA helps in getting consistency between the rotations"""
+
+clf = PCA(n_components=2)
+npos = [0, 0, 0, 0]
+npos = [smacof_mds(Cs[s], 2) for s in range(S)]
+
+npost01 = [0, 0]
+npost01 = [smacof_mds(Ct01[s], 2) for s in range(2)]
+npost01 = [clf.fit_transform(npost01[s]) for s in range(2)]
+
+npost02 = [0, 0]
+npost02 = [smacof_mds(Ct02[s], 2) for s in range(2)]
+npost02 = [clf.fit_transform(npost02[s]) for s in range(2)]
+
+npost13 = [0, 0]
+npost13 = [smacof_mds(Ct13[s], 2) for s in range(2)]
+npost13 = [clf.fit_transform(npost13[s]) for s in range(2)]
+
+npost23 = [0, 0]
+npost23 = [smacof_mds(Ct23[s], 2) for s in range(2)]
+npost23 = [clf.fit_transform(npost23[s]) for s in range(2)]
+
+
+fig = pl.figure(figsize=(10, 10))
+
+ax1 = pl.subplot2grid((4, 4), (0, 0))
+pl.xlim((-1, 1))
+pl.ylim((-1, 1))
+ax1.scatter(npos[0][:, 0], npos[0][:, 1], color='r')
+
+ax2 = pl.subplot2grid((4, 4), (0, 1))
+pl.xlim((-1, 1))
+pl.ylim((-1, 1))
+ax2.scatter(npost01[1][:, 0], npost01[1][:, 1], color='b')
+
+ax3 = pl.subplot2grid((4, 4), (0, 2))
+pl.xlim((-1, 1))
+pl.ylim((-1, 1))
+ax3.scatter(npost01[0][:, 0], npost01[0][:, 1], color='b')
+
+ax4 = pl.subplot2grid((4, 4), (0, 3))
+pl.xlim((-1, 1))
+pl.ylim((-1, 1))
+ax4.scatter(npos[1][:, 0], npos[1][:, 1], color='r')
+
+ax5 = pl.subplot2grid((4, 4), (1, 0))
+pl.xlim((-1, 1))
+pl.ylim((-1, 1))
+ax5.scatter(npost02[1][:, 0], npost02[1][:, 1], color='b')
+
+ax6 = pl.subplot2grid((4, 4), (1, 3))
+pl.xlim((-1, 1))
+pl.ylim((-1, 1))
+ax6.scatter(npost13[1][:, 0], npost13[1][:, 1], color='b')
+
+ax7 = pl.subplot2grid((4, 4), (2, 0))
+pl.xlim((-1, 1))
+pl.ylim((-1, 1))
+ax7.scatter(npost02[0][:, 0], npost02[0][:, 1], color='b')
+
+ax8 = pl.subplot2grid((4, 4), (2, 3))
+pl.xlim((-1, 1))
+pl.ylim((-1, 1))
+ax8.scatter(npost13[0][:, 0], npost13[0][:, 1], color='b')
+
+ax9 = pl.subplot2grid((4, 4), (3, 0))
+pl.xlim((-1, 1))
+pl.ylim((-1, 1))
+ax9.scatter(npos[2][:, 0], npos[2][:, 1], color='r')
+
+ax10 = pl.subplot2grid((4, 4), (3, 1))
+pl.xlim((-1, 1))
+pl.ylim((-1, 1))
+ax10.scatter(npost23[1][:, 0], npost23[1][:, 1], color='b')
+
+ax11 = pl.subplot2grid((4, 4), (3, 2))
+pl.xlim((-1, 1))
+pl.ylim((-1, 1))
+ax11.scatter(npost23[0][:, 0], npost23[0][:, 1], color='b')
+
+ax12 = pl.subplot2grid((4, 4), (3, 3))
+pl.xlim((-1, 1))
+pl.ylim((-1, 1))
+ax12.scatter(npos[3][:, 0], npos[3][:, 1], color='r')