Note

Go to the end to download the full example code.

Note

Example added in release: 0.9.2.

This example illustrates the computation of Low Rank Sinkhorn [26].

[65] Scetbon, M., Cuturi, M., & Peyré, G. (2021). “Low-rank Sinkhorn factorization”. In International Conference on Machine Learning.

# Author: Laurène David <laurene.david@ip-paris.fr>
#
# License: MIT License
#
# sphinx_gallery_thumbnail_number = 2

import numpy as np
import matplotlib.pylab as pl
import ot.plot
from ot.datasets import make_1D_gauss as gauss

Generate data

n = 100
m = 120

# Gaussian distribution
a = gauss(n, m=int(n / 3), s=25 / np.sqrt(2)) + 1.5 * gauss(
    n, m=int(5 * n / 6), s=15 / np.sqrt(2)
)
a = a / np.sum(a)

b = 2 * gauss(m, m=int(m / 5), s=30 / np.sqrt(2)) + gauss(
    m, m=int(m / 2), s=35 / np.sqrt(2)
)
b = b / np.sum(b)

# Source and target distribution
X = np.arange(n).reshape(-1, 1)
Y = np.arange(m).reshape(-1, 1)

Solve Low rank sinkhorn

Solve low rank sinkhorn

Q, R, g, log = ot.lowrank_sinkhorn(
    X,
    Y,
    a,
    b,
    rank=10,
    init="random",
    gamma_init="rescale",
    rescale_cost=True,
    warn=False,
    log=True,
)
P = log["lazy_plan"][:]

ot.plot.plot1D_mat(a, b, P, "OT matrix Low rank")
plot lowrank sinkhorn
(<Axes: >, <Axes: >, <Axes: >)

Sinkhorn vs Low Rank Sinkhorn

Compare Sinkhorn and Low rank sinkhorn with different regularizations and ranks.

/home/circleci/project/ot/lowrank.py:310: UserWarning: Dykstra did not converge. You might want to increase the number of iterations `numItermax`
  warnings.warn(

# Plot sinkhorn vs low rank sinkhorn
pl.figure(1, figsize=(10, 8))

pl.subplot(2, 3, 1)
pl.imshow(list_P_Sin[0], interpolation="nearest", cmap="gray_r")
pl.axis("off")
pl.title("Sinkhorn (reg=0.05)")

pl.subplot(2, 3, 2)
pl.imshow(list_P_Sin[1], interpolation="nearest", cmap="gray_r")
pl.axis("off")
pl.title("Sinkhorn (reg=0.005)")

pl.subplot(2, 3, 3)
pl.imshow(list_P_Sin[2], interpolation="nearest", cmap="gray_r")
pl.axis("off")
pl.title("Sinkhorn (reg=0.001)")
pl.show()

pl.subplot(2, 3, 4)
pl.imshow(list_P_LR[0], interpolation="nearest", cmap="gray_r")
pl.axis("off")
pl.title("Low rank (rank=3)")

pl.subplot(2, 3, 5)
pl.imshow(list_P_LR[1], interpolation="nearest", cmap="gray_r")
pl.axis("off")
pl.title("Low rank (rank=10)")

pl.subplot(2, 3, 6)
pl.imshow(list_P_LR[2], interpolation="nearest", cmap="gray_r")
pl.axis("off")
pl.title("Low rank (rank=50)")

pl.tight_layout()
Sinkhorn (reg=0.05), Sinkhorn (reg=0.005), Sinkhorn (reg=0.001), Low rank (rank=3), Low rank (rank=10), Low rank (rank=50)

Total running time of the script: (0 minutes 19.648 seconds)

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