# 2D examples of exact and entropic unbalanced optimal transport

This example is designed to show how to compute unbalanced and partial OT in POT.

UOT aims at solving the following optimization problem:

\begin{align}\begin{aligned}W = \min_{\gamma} <\gamma, \mathbf{M}>_F + \mathrm{reg}\cdot\Omega(\gamma) + \mathrm{reg_m} \cdot \mathrm{div}(\gamma \mathbf{1}, \mathbf{a}) + \mathrm{reg_m} \cdot \mathrm{div}(\gamma^T \mathbf{1}, \mathbf{b})\\s.t. \gamma \geq 0\end{aligned}\end{align}

where $$\mathrm{div}$$ is a divergence. When using the entropic UOT, $$\mathrm{reg}>0$$ and $$\mathrm{div}$$ should be the Kullback-Leibler divergence. When solving exact UOT, $$\mathrm{reg}=0$$ and $$\mathrm{div}$$ can be either the Kullback-Leibler or the quadratic divergence. Using $$\ell_1$$ norm gives the so-called partial OT.

# Author: Laetitia Chapel <laetitia.chapel@univ-ubs.fr>

import numpy as np
import matplotlib.pylab as pl
import ot


## Generate data

n = 40  # nb samples

mu_s = np.array([-1, -1])
cov_s = np.array([[1, 0], [0, 1]])

mu_t = np.array([4, 4])
cov_t = np.array([[1, -.8], [-.8, 1]])

np.random.seed(0)
xs = ot.datasets.make_2D_samples_gauss(n, mu_s, cov_s)
xt = ot.datasets.make_2D_samples_gauss(n, mu_t, cov_t)

n_noise = 10

xs = np.concatenate((xs, ((np.random.rand(n_noise, 2) - 4))), axis=0)
xt = np.concatenate((xt, ((np.random.rand(n_noise, 2) + 6))), axis=0)

n = n + n_noise

a, b = np.ones((n,)) / n, np.ones((n,)) / n  # uniform distribution on samples

# loss matrix
M = ot.dist(xs, xt)
M /= M.max()


## Compute entropic kl-regularized UOT, kl- and l2-regularized UOT

reg = 0.005
reg_m_kl = 0.05
reg_m_l2 = 5
mass = 0.7

entropic_kl_uot = ot.unbalanced.sinkhorn_unbalanced(a, b, M, reg, reg_m_kl)
kl_uot = ot.unbalanced.mm_unbalanced(a, b, M, reg_m_kl, div='kl')
l2_uot = ot.unbalanced.mm_unbalanced(a, b, M, reg_m_l2, div='l2')
partial_ot = ot.partial.partial_wasserstein(a, b, M, m=mass)


## Plot the results

pl.figure(2)
transp = [partial_ot, l2_uot, kl_uot, entropic_kl_uot]
title = ["partial OT \n m=" + str(mass), "$\ell_2$-UOT \n $\mathrm{reg_m}$=" +
str(reg_m_l2), "kl-UOT \n $\mathrm{reg_m}$=" + str(reg_m_kl),
"entropic kl-UOT \n $\mathrm{reg_m}$=" + str(reg_m_kl)]

for p in range(4):
pl.subplot(2, 4, p + 1)
P = transp[p]
if P.sum() > 0:
P = P / P.max()
for i in range(n):
for j in range(n):
if P[i, j] > 0:
pl.plot([xs[i, 0], xt[j, 0]], [xs[i, 1], xt[j, 1]], color='C2',
alpha=P[i, j] * 0.3)
pl.scatter(xs[:, 0], xs[:, 1], c='C0', alpha=0.2)
pl.scatter(xt[:, 0], xt[:, 1], c='C1', alpha=0.2)
pl.scatter(xs[:, 0], xs[:, 1], c='C0', s=P.sum(1).ravel() * (1 + p) * 2)
pl.scatter(xt[:, 0], xt[:, 1], c='C1', s=P.sum(0).ravel() * (1 + p) * 2)
pl.title(title[p])
pl.yticks(())
pl.xticks(())
if p < 1:
pl.ylabel("mappings")
pl.subplot(2, 4, p + 5)
pl.imshow(P, cmap='jet')
pl.yticks(())
pl.xticks(())
if p < 1:
pl.ylabel("transport plans")
pl.show()


Total running time of the script: ( 0 minutes 4.119 seconds)

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