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
|
# -*- coding: utf-8 -*-
"""
==============================
Plot Fused-Gromov-Wasserstein
==============================
This example first illustrates the computation of FGW for 1D measures estimated
using a Conditional Gradient solver [24].
[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.
"""
# Author: Titouan Vayer <titouan.vayer@irisa.fr>
#
# License: MIT License
# sphinx_gallery_thumbnail_number = 3
import matplotlib.pyplot as pl
import numpy as np
import ot
from ot.gromov import gromov_wasserstein, fused_gromov_wasserstein
##############################################################################
# Generate data
# -------------
# parameters
# We create two 1D random measures
n = 20 # number of points in the first distribution
n2 = 30 # number of points in the second distribution
sig = 1 # std of first distribution
sig2 = 0.1 # std of second distribution
np.random.seed(0)
phi = np.arange(n)[:, None]
xs = phi + sig * np.random.randn(n, 1)
ys = np.vstack(
(np.ones((n // 2, 1)), 0 * np.ones((n // 2, 1)))
) + sig2 * np.random.randn(n, 1)
phi2 = np.arange(n2)[:, None]
xt = phi2 + sig * np.random.randn(n2, 1)
yt = np.vstack(
(np.ones((n2 // 2, 1)), 0 * np.ones((n2 // 2, 1)))
) + sig2 * np.random.randn(n2, 1)
yt = yt[::-1, :]
p = ot.unif(n)
q = ot.unif(n2)
##############################################################################
# Plot data
# ---------
# plot the distributions
pl.figure(1, (7, 7))
pl.subplot(2, 1, 1)
pl.scatter(ys, xs, c=phi, s=70)
pl.ylabel("Feature value a", fontsize=20)
pl.title("$\mu=\sum_i \delta_{x_i,a_i}$", fontsize=25, y=1)
pl.xticks(())
pl.yticks(())
pl.subplot(2, 1, 2)
pl.scatter(yt, xt, c=phi2, s=70)
pl.xlabel("coordinates x/y", fontsize=25)
pl.ylabel("Feature value b", fontsize=20)
pl.title("$\\nu=\sum_j \delta_{y_j,b_j}$", fontsize=25, y=1)
pl.yticks(())
pl.tight_layout()
pl.show()
##############################################################################
# Create structure matrices and across-feature distance matrix
# ------------------------------------------------------------
# Structure matrices and across-features distance matrix
C1 = ot.dist(xs)
C2 = ot.dist(xt)
M = ot.dist(ys, yt)
w1 = ot.unif(C1.shape[0])
w2 = ot.unif(C2.shape[0])
Got = ot.emd([], [], M)
##############################################################################
# Plot matrices
# -------------
cmap = "Reds"
pl.figure(2, (5, 5))
fs = 15
l_x = [0, 5, 10, 15]
l_y = [0, 5, 10, 15, 20, 25]
gs = pl.GridSpec(5, 5)
ax1 = pl.subplot(gs[3:, :2])
pl.imshow(C1, cmap=cmap, interpolation="nearest")
pl.title("$C_1$", fontsize=fs)
pl.xlabel("$k$", fontsize=fs)
pl.ylabel("$i$", fontsize=fs)
pl.xticks(l_x)
pl.yticks(l_x)
ax2 = pl.subplot(gs[:3, 2:])
pl.imshow(C2, cmap=cmap, interpolation="nearest")
pl.title("$C_2$", fontsize=fs)
pl.ylabel("$l$", fontsize=fs)
pl.xticks(())
pl.yticks(l_y)
ax2.set_aspect("auto")
ax3 = pl.subplot(gs[3:, 2:], sharex=ax2, sharey=ax1)
pl.imshow(M, cmap=cmap, interpolation="nearest")
pl.yticks(l_x)
pl.xticks(l_y)
pl.ylabel("$i$", fontsize=fs)
pl.title("$M_{AB}$", fontsize=fs)
pl.xlabel("$j$", fontsize=fs)
pl.tight_layout()
ax3.set_aspect("auto")
pl.show()
##############################################################################
# Compute FGW/GW
# --------------
# Computing FGW and GW
alpha = 1e-3
ot.tic()
Gwg, logw = fused_gromov_wasserstein(
M, C1, C2, p, q, loss_fun="square_loss", alpha=alpha, verbose=True, log=True
)
ot.toc()
# reload_ext WGW
Gg, log = gromov_wasserstein(
C1, C2, p, q, loss_fun="square_loss", verbose=True, log=True
)
##############################################################################
# Visualize transport matrices
# ----------------------------
# visu OT matrix
cmap = "Blues"
fs = 15
pl.figure(3, (13, 5))
pl.clf()
pl.subplot(1, 3, 1)
pl.imshow(Got, cmap=cmap, interpolation="nearest")
pl.ylabel("$i$", fontsize=fs)
pl.xticks(())
pl.title("Wasserstein ($M$ only)")
pl.subplot(1, 3, 2)
pl.imshow(Gg, cmap=cmap, interpolation="nearest")
pl.title("Gromov ($C_1,C_2$ only)")
pl.xticks(())
pl.subplot(1, 3, 3)
pl.imshow(Gwg, cmap=cmap, interpolation="nearest")
pl.title("FGW ($M+C_1,C_2$)")
pl.xlabel("$j$", fontsize=fs)
pl.ylabel("$i$", fontsize=fs)
pl.tight_layout()
pl.show()
|
