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
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
|
.. _sphx_glr_auto_examples_plot_gromov.py:
==========================
Gromov-Wasserstein example
==========================
This example is designed to show how to use the Gromov-Wassertsein distance
computation in POT.
.. code-block:: python
# Author: Erwan Vautier <erwan.vautier@gmail.com>
# Nicolas Courty <ncourty@irisa.fr>
#
# License: MIT License
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.
.. code-block:: python
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]])
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
--------------------------
.. code-block:: python
fig = pl.figure()
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()
.. image:: /auto_examples/images/sphx_glr_plot_gromov_001.png
:align: center
Compute distance kernels, normalize them and then display
---------------------------------------------------------
.. code-block:: python
C1 = sp.spatial.distance.cdist(xs, xs)
C2 = sp.spatial.distance.cdist(xt, xt)
C1 /= C1.max()
C2 /= C2.max()
pl.figure()
pl.subplot(121)
pl.imshow(C1)
pl.subplot(122)
pl.imshow(C2)
pl.show()
.. image:: /auto_examples/images/sphx_glr_plot_gromov_002.png
:align: center
Compute Gromov-Wasserstein plans and distance
---------------------------------------------
.. code-block:: python
p = ot.unif(n_samples)
q = ot.unif(n_samples)
gw0, log0 = ot.gromov.gromov_wasserstein(
C1, C2, p, q, 'square_loss', verbose=True, log=True)
gw, log = ot.gromov.entropic_gromov_wasserstein(
C1, C2, p, q, 'square_loss', epsilon=5e-4, log=True, verbose=True)
print('Gromov-Wasserstein distances: ' + str(log0['gw_dist']))
print('Entropic Gromov-Wasserstein distances: ' + str(log['gw_dist']))
pl.figure(1, (10, 5))
pl.subplot(1, 2, 1)
pl.imshow(gw0, cmap='jet')
pl.title('Gromov Wasserstein')
pl.subplot(1, 2, 2)
pl.imshow(gw, cmap='jet')
pl.title('Entropic Gromov Wasserstein')
pl.show()
.. image:: /auto_examples/images/sphx_glr_plot_gromov_003.png
:align: center
.. rst-class:: sphx-glr-script-out
Out::
It. |Loss |Delta loss
--------------------------------
0|4.328711e-02|0.000000e+00
1|2.281369e-02|-8.974178e-01
2|1.843659e-02|-2.374139e-01
3|1.602820e-02|-1.502598e-01
4|1.353712e-02|-1.840179e-01
5|1.285687e-02|-5.290977e-02
6|1.284537e-02|-8.952931e-04
7|1.284525e-02|-8.989584e-06
8|1.284525e-02|-8.989950e-08
9|1.284525e-02|-8.989949e-10
It. |Err
-------------------
0|7.263293e-02|
10|1.737784e-02|
20|7.783978e-03|
30|3.399419e-07|
40|3.751207e-11|
Gromov-Wasserstein distances: 0.012845252089244688
Entropic Gromov-Wasserstein distances: 0.013543882352191079
**Total running time of the script:** ( 0 minutes 1.916 seconds)
.. only :: html
.. container:: sphx-glr-footer
.. container:: sphx-glr-download
:download:`Download Python source code: plot_gromov.py <plot_gromov.py>`
.. container:: sphx-glr-download
:download:`Download Jupyter notebook: plot_gromov.ipynb <plot_gromov.ipynb>`
.. only:: html
.. rst-class:: sphx-glr-signature
`Gallery generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_
|
