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# -*- coding: utf-8 -*-
"""
========================================
Low rank Sinkhorn
========================================
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
# -------------
# %% parameters
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")
##############################################################################
# Sinkhorn vs Low Rank Sinkhorn
# -----------------------
# Compare Sinkhorn and Low rank sinkhorn with different regularizations and ranks.
# %% Sinkhorn
# Compute cost matrix for sinkhorn OT
M = ot.dist(X, Y)
M = M / np.max(M)
# Solve sinkhorn with different regularizations using ot.solve
list_reg = [0.05, 0.005, 0.001]
list_P_Sin = []
for reg in list_reg:
P = ot.solve(M, a, b, reg=reg, max_iter=2000, tol=1e-8).plan
list_P_Sin.append(P)
# %% Low rank sinkhorn
# Solve low rank sinkhorn with different ranks using ot.solve_sample
list_rank = [3, 10, 50]
list_P_LR = []
for rank in list_rank:
P = ot.solve_sample(X, Y, a, b, method="lowrank", rank=rank).plan
P = P[:]
list_P_LR.append(P)
# %%
# 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")
pl.axis("off")
pl.title("Sinkhorn (reg=0.05)")
pl.subplot(2, 3, 2)
pl.imshow(list_P_Sin[1], interpolation="nearest")
pl.axis("off")
pl.title("Sinkhorn (reg=0.005)")
pl.subplot(2, 3, 3)
pl.imshow(list_P_Sin[2], interpolation="nearest")
pl.axis("off")
pl.title("Sinkhorn (reg=0.001)")
pl.show()
pl.subplot(2, 3, 4)
pl.imshow(list_P_LR[0], interpolation="nearest")
pl.axis("off")
pl.title("Low rank (rank=3)")
pl.subplot(2, 3, 5)
pl.imshow(list_P_LR[1], interpolation="nearest")
pl.axis("off")
pl.title("Low rank (rank=10)")
pl.subplot(2, 3, 6)
pl.imshow(list_P_LR[2], interpolation="nearest")
pl.axis("off")
pl.title("Low rank (rank=50)")
pl.tight_layout()
|
