1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
|
# -*- coding: utf-8 -*-
"""
==========================
Gromov-Wasserstein example
==========================
This example is designed to show how to use the Gromov-Wasserstein distance
computation in POT.
We first compare 3 solvers to estimate the distance based on
Conditional Gradient [24] or Sinkhorn projections [12, 51].
Then we compare 2 stochastic solvers to estimate the distance with a lower
numerical cost [33].
[12] Gabriel Peyré, Marco Cuturi, and Justin Solomon (2016),
"Gromov-Wasserstein averaging of kernel and distance matrices".
International Conference on Machine Learning (ICML).
[24] Vayer Titouan, Chapel Laetitia, Flamary Rémi, Tavenard Romain
and Courty Nicolas
"Optimal Transport for structured data with application on graphs"
International Conference on Machine Learning (ICML). 2019.
[33] Kerdoncuff T., Emonet R., Marc S. "Sampled Gromov Wasserstein",
Machine Learning Journal (MJL), 2021.
[51] Xu, H., Luo, D., Zha, H., & Duke, L. C. (2019).
"Gromov-wasserstein learning for graph matching and node embedding".
In International Conference on Machine Learning (ICML), 2019.
"""
# Author: Erwan Vautier <erwan.vautier@gmail.com>
# Nicolas Courty <ncourty@irisa.fr>
# Cédric Vincent-Cuaz <cedvincentcuaz@gmail.com>
# Tanguy Kerdoncuff <tanguy.kerdoncuff@laposte.net>
#
# License: MIT License
# sphinx_gallery_thumbnail_number = 1
import scipy as sp
import numpy as np
import matplotlib.pylab as pl
from mpl_toolkits.mplot3d import Axes3D # noqa
import ot
#############################################################################
#
# 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.
n_samples = 30 # nb samples
mu_s = np.array([0, 0])
cov_s = np.array([[1, 0], [0, 1]])
mu_t = np.array([4, 4, 4])
cov_t = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
np.random.seed(0)
xs = ot.datasets.make_2D_samples_gauss(n_samples, mu_s, cov_s)
P = sp.linalg.sqrtm(cov_t)
xt = np.random.randn(n_samples, 3).dot(P) + mu_t
#############################################################################
#
# Plotting the distributions
# --------------------------
fig = pl.figure(1)
ax1 = fig.add_subplot(121)
ax1.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
ax2 = fig.add_subplot(122, projection='3d')
ax2.scatter(xt[:, 0], xt[:, 1], xt[:, 2], color='r')
pl.show()
#############################################################################
#
# Compute distance kernels, normalize them and then display
# ---------------------------------------------------------
C1 = sp.spatial.distance.cdist(xs, xs)
C2 = sp.spatial.distance.cdist(xt, xt)
C1 /= C1.max()
C2 /= C2.max()
pl.figure(2)
pl.subplot(121)
pl.imshow(C1)
pl.title('C1')
pl.subplot(122)
pl.imshow(C2)
pl.title('C2')
pl.show()
#############################################################################
#
# Compute Gromov-Wasserstein plans and distance
# ---------------------------------------------
p = ot.unif(n_samples)
q = ot.unif(n_samples)
# Conditional Gradient algorithm
gw0, log0 = ot.gromov.gromov_wasserstein(
C1, C2, p, q, 'square_loss', verbose=True, log=True)
# Proximal Point algorithm with Kullback-Leibler as proximal operator
gw, log = ot.gromov.entropic_gromov_wasserstein(
C1, C2, p, q, 'square_loss', epsilon=5e-4, solver='PPA',
log=True, verbose=True)
# Projected Gradient algorithm with entropic regularization
gwe, loge = ot.gromov.entropic_gromov_wasserstein(
C1, C2, p, q, 'square_loss', epsilon=5e-4, solver='PGD',
log=True, verbose=True)
print('Gromov-Wasserstein distance estimated with Conditional Gradient solver: ' + str(log0['gw_dist']))
print('Gromov-Wasserstein distance estimated with Proximal Point solver: ' + str(log['gw_dist']))
print('Entropic Gromov-Wasserstein distance estimated with Projected Gradient solver: ' + str(loge['gw_dist']))
# compute OT sparsity level
gw0_sparsity = 100 * (gw0 == 0.).astype(np.float64).sum() / (n_samples ** 2)
gw_sparsity = 100 * (gw == 0.).astype(np.float64).sum() / (n_samples ** 2)
gwe_sparsity = 100 * (gwe == 0.).astype(np.float64).sum() / (n_samples ** 2)
# Methods using Sinkhorn projections tend to produce feasibility errors on the
# marginal constraints
err0 = np.linalg.norm(gw0.sum(1) - p) + np.linalg.norm(gw0.sum(0) - q)
err = np.linalg.norm(gw.sum(1) - p) + np.linalg.norm(gw.sum(0) - q)
erre = np.linalg.norm(gwe.sum(1) - p) + np.linalg.norm(gwe.sum(0) - q)
pl.figure(3, (10, 6))
cmap = 'Blues'
fontsize = 12
pl.subplot(131)
pl.imshow(gw0, cmap=cmap)
pl.title('(CG algo) GW=%s \n \n OT sparsity=%s \n feasibility error=%s' % (
np.round(log0['gw_dist'], 4), str(np.round(gw0_sparsity, 2)) + ' %', np.round(np.round(err0, 4))),
fontsize=fontsize)
pl.subplot(132)
pl.imshow(gw, cmap=cmap)
pl.title('(PP algo) GW=%s \n \n OT sparsity=%s \nfeasibility error=%s' % (
np.round(log['gw_dist'], 4), str(np.round(gw_sparsity, 2)) + ' %', np.round(err, 4)),
fontsize=fontsize)
pl.subplot(133)
pl.imshow(gwe, cmap=cmap)
pl.title('Entropic GW=%s \n \n OT sparsity=%s \nfeasibility error=%s' % (
np.round(loge['gw_dist'], 4), str(np.round(gwe_sparsity, 2)) + ' %', np.round(erre, 4)),
fontsize=fontsize)
pl.tight_layout()
pl.show()
#############################################################################
#
# Compute GW with scalable stochastic methods with any loss function
# ----------------------------------------------------------------------
def loss(x, y):
return np.abs(x - y)
pgw, plog = ot.gromov.pointwise_gromov_wasserstein(C1, C2, p, q, loss, max_iter=100,
log=True)
sgw, slog = ot.gromov.sampled_gromov_wasserstein(C1, C2, p, q, loss, epsilon=0.1, max_iter=100,
log=True)
print('Pointwise Gromov-Wasserstein distance estimated: ' + str(plog['gw_dist_estimated']))
print('Variance estimated: ' + str(plog['gw_dist_std']))
print('Sampled Gromov-Wasserstein distance: ' + str(slog['gw_dist_estimated']))
print('Variance estimated: ' + str(slog['gw_dist_std']))
pl.figure(4, (10, 5))
pl.subplot(121)
pl.imshow(pgw.toarray(), cmap=cmap)
pl.title('Pointwise Gromov Wasserstein')
pl.subplot(122)
pl.imshow(sgw, cmap=cmap)
pl.title('Sampled Gromov Wasserstein')
pl.show()
|
