Gromov-Wasserstein 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 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]])


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()
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()
pl.subplot(121)
pl.imshow(C1)
pl.subplot(122)
pl.imshow(C2)
pl.show()

Compute Gromov-Wasserstein plans and distance

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()

Out:

It.  |Loss        |Relative loss|Absolute loss
------------------------------------------------
    0|1.040183e-01|0.000000e+00|0.000000e+00
    1|5.680045e-02|8.312927e-01|4.721780e-02
    2|4.810735e-02|1.807022e-01|8.693103e-03
    3|4.810735e-02|0.000000e+00|0.000000e+00
/home/circleci/project/ot/bregman.py:517: UserWarning: Sinkhorn did not converge. You might want to increase the number of iterations `numItermax` or the regularization parameter `reg`.
  warnings.warn("Sinkhorn did not converge. You might want to "
It.  |Err
-------------------
    0|9.071265e-02|
   10|3.833502e-04|
   20|3.306601e-08|
   30|2.414528e-12|
Gromov-Wasserstein distances: 0.048107347795629904
Entropic Gromov-Wasserstein distances: 0.04097168403053386

Compute GW with a scalable stochastic method 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(1, (10, 5))

pl.subplot(1, 2, 1)
pl.imshow(pgw.toarray(), cmap='jet')
pl.title('Pointwise Gromov Wasserstein')

pl.subplot(1, 2, 2)
pl.imshow(sgw, cmap='jet')
pl.title('Sampled Gromov Wasserstein')

pl.show()

Out:

/home/circleci/project/ot/bregman.py:517: UserWarning: Sinkhorn did not converge. You might want to increase the number of iterations `numItermax` or the regularization parameter `reg`.
  warnings.warn("Sinkhorn did not converge. You might want to "
Pointwise Gromov-Wasserstein distance estimated: 0.18827653307627662
Variance estimated: 0.0
Sampled Gromov-Wasserstein distance: 0.1534777663986762
Variance estimated: 0.001626689010280435