This example is designed to show how to use the CO-Optimal Transport [47]_ in POT. CO-Optimal Transport allows to calculate the distance between two arbitrary-size matrices, and to align their rows and columns. In this example, we consider two random matrices \(X_1\) and \(X_2\) defined by \((X_1)_{i,j} = \cos(\frac{i}{n_1} \pi) + \cos(\frac{j}{d_1} \pi) + \sigma \mathcal N(0,1)\) and \((X_2)_{i,j} = \cos(\frac{i}{n_2} \pi) + \cos(\frac{j}{d_2} \pi) + \sigma \mathcal N(0,1)\).
# Author: Remi Flamary <remi.flamary@unice.fr> # Quang Huy Tran <quang-huy.tran@univ-ubs.fr> # License: MIT License from matplotlib.patches import ConnectionPatch import matplotlib.pylab as pl import numpy as np from ot.coot import co_optimal_transport as coot from ot.coot import co_optimal_transport2 as coot2
Generating two random matrices
Visualizing the matrices
pl.figure(1, (8, 5))
pl.subplot(1, 2, 1)
pl.imshow(X1)
pl.title("$X_1$")
pl.subplot(1, 2, 2)
pl.imshow(X2)
pl.title("$X_2$")
pl.tight_layout()
Visualizing the alignments of rows and columns, and calculating the CO-Optimal Transport distance
pi_sample, pi_feature, log = coot(X1, X2, log=True, verbose=True)
coot_distance = coot2(X1, X2)
print("CO-Optimal Transport distance = {:.5f}".format(coot_distance))
fig = pl.figure(4, (9, 7))
pl.clf()
ax1 = pl.subplot(2, 2, 3)
pl.imshow(X1)
pl.xlabel("$X_1$")
ax2 = pl.subplot(2, 2, 2)
ax2.yaxis.tick_right()
pl.imshow(np.transpose(X2))
pl.title("Transpose($X_2$)")
ax2.xaxis.tick_top()
for i in range(n1):
j = np.argmax(pi_sample[i, :])
xyA = (d1 - 0.5, i)
xyB = (j, d2 - 0.5)
con = ConnectionPatch(
xyA=xyA, xyB=xyB, coordsA=ax1.transData, coordsB=ax2.transData, color="black"
)
fig.add_artist(con)
for i in range(d1):
j = np.argmax(pi_feature[i, :])
xyA = (i, -0.5)
xyB = (-0.5, j)
con = ConnectionPatch(
xyA=xyA, xyB=xyB, coordsA=ax1.transData, coordsB=ax2.transData, color="blue"
)
fig.add_artist(con)
CO-Optimal Transport cost at iteration 1: 0.10903416567681642 CO-Optimal Transport cost at iteration 2: 0.0987568083692249 CO-Optimal Transport cost at iteration 3: 0.09685398028260428 CO-Optimal Transport distance = 0.09685
Total running time of the script: (0 minutes 0.355 seconds)
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