Screened optimal transport (Screenkhorn)

This example illustrates the computation of Screenkhorn [26].

[26] Alaya M. Z., Bérar M., Gasso G., Rakotomamonjy A. (2019). Screening Sinkhorn Algorithm for Regularized Optimal Transport, Advances in Neural Information Processing Systems 33 (NeurIPS).

# Author: Mokhtar Z. Alaya <mokhtarzahdi.alaya@gmail.com>
#
# License: MIT License

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

Generate data

n = 100  # nb bins

# bin positions
x = np.arange(n, dtype=np.float64)

# Gaussian distributions
a = gauss(n, m=20, s=5)  # m= mean, s= std
b = gauss(n, m=60, s=10)

# loss matrix
M = ot.dist(x.reshape((n, 1)), x.reshape((n, 1)))
M /= M.max()

Plot distributions and loss matrix

pl.figure(1, figsize=(6.4, 3))
pl.plot(x, a, "b", label="Source distribution")
pl.plot(x, b, "r", label="Target distribution")
pl.legend()

# plot distributions and loss matrix

pl.figure(2, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, M, "Cost matrix M")

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(<Axes: >, <Axes: >, <Axes: >)

Solve Screenkhorn

# Screenkhorn
lambd = 2e-03  # entropy parameter
ns_budget = 30  # budget number of points to be kept in the source distribution
nt_budget = 30  # budget number of points to be kept in the target distribution

G_screen = screenkhorn(
    a, b, M, lambd, ns_budget, nt_budget, uniform=False, restricted=True, verbose=True
)
pl.figure(4, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, G_screen, "OT matrix Screenkhorn")
pl.show()

/home/circleci/project/ot/bregman/_screenkhorn.py:132: UserWarning: Bottleneck module is not installed. Install it from https://pypi.org/project/Bottleneck/ for better performance.
  warnings.warn(
epsilon = 0.020986042861303855

kappa = 3.7476531411890917

Cardinality of selected points: |Isel| = 30      |Jsel| = 30