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# -*- coding: utf-8 -*-
"""
================================================================
Regularization path of l2-penalized unbalanced optimal transport
================================================================
This example illustrate the regularization path for 2D unbalanced
optimal transport. We present here both the fully relaxed case
and the semi-relaxed case.
[Chapel et al., 2021] Chapel, L., Flamary, R., Wu, H., Févotte, C.,
and Gasso, G. (2021). Unbalanced optimal transport through non-negative
penalized linear regression.
"""
# Author: Haoran Wu <haoran.wu@univ-ubs.fr>
# License: MIT License
# sphinx_gallery_thumbnail_number = 2
import numpy as np
import matplotlib.pylab as pl
import ot
import matplotlib.animation as animation
##############################################################################
# Generate data
# -------------
#%% parameters and data generation
n = 50 # 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)
a, b = np.ones((n,)) / n, np.ones((n,)) / n # uniform distribution on samples
# loss matrix
M = ot.dist(xs, xt)
M /= M.max()
##############################################################################
# Plot data
# ---------
#%% plot 2 distribution samples
pl.figure(1)
pl.scatter(xs[:, 0], xs[:, 1], c='C0', label='Source')
pl.scatter(xt[:, 0], xt[:, 1], c='C1', label='Target')
pl.legend(loc=2)
pl.title('Source and target distributions')
pl.show()
##############################################################################
# Compute semi-relaxed and fully relaxed regularization paths
# -----------
#%%
final_gamma = 1e-8
t, t_list, g_list = ot.regpath.regularization_path(a, b, M, reg=final_gamma,
semi_relaxed=False)
t2, t_list2, g_list2 = ot.regpath.regularization_path(a, b, M, reg=final_gamma,
semi_relaxed=True)
##############################################################################
# Plot the regularization path
# ----------------
#
# The OT plan is ploted as a function of $\gamma$ that is the inverse of the
# weight on the marginal relaxations.
#%% fully relaxed l2-penalized UOT
pl.figure(2)
selected_gamma = [2e-1, 1e-1, 5e-2, 1e-3]
for p in range(4):
tp = ot.regpath.compute_transport_plan(selected_gamma[p], g_list,
t_list)
P = tp.reshape((n, n))
pl.subplot(2, 2, p + 1)
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,
label='Re-weighted source', alpha=1)
pl.scatter(xt[:, 0], xt[:, 1], c='C1', s=P.sum(0).ravel() * (1 + p) * 2,
label='Re-weighted target', alpha=1)
pl.plot([], [], color='C2', alpha=0.8, label='OT plan')
pl.title(r'$\ell_2$ UOT $\gamma$={}'.format(selected_gamma[p]),
fontsize=11)
if p < 2:
pl.xticks(())
pl.show()
# %%
# Animation of the regpath for UOT l2
# ------------------------
nv = 100
g_list_v = np.logspace(-.5, -2.5, nv)
pl.figure(3)
def _update_plot(iv):
pl.clf()
tp = ot.regpath.compute_transport_plan(g_list_v[iv], g_list,
t_list)
P = tp.reshape((n, n))
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] * 0.5)
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) * 4,
label='Re-weighted source', alpha=1)
pl.scatter(xt[:, 0], xt[:, 1], c='C1', s=P.sum(0).ravel() * (1 + p) * 4,
label='Re-weighted target', alpha=1)
pl.plot([], [], color='C2', alpha=0.8, label='OT plan')
pl.title(r'$\ell_2$ UOT $\gamma$={:1.3f}'.format(g_list_v[iv]),
fontsize=11)
return 1
i = 0
_update_plot(i)
ani = animation.FuncAnimation(pl.gcf(), _update_plot, nv, interval=50, repeat_delay=2000)
##############################################################################
# Plot the semi-relaxed regularization path
# -------------------
#%% semi-relaxed l2-penalized UOT
pl.figure(4)
selected_gamma = [10, 1, 1e-1, 1e-2]
for p in range(4):
tp = ot.regpath.compute_transport_plan(selected_gamma[p], g_list2,
t_list2)
P = tp.reshape((n, n))
pl.subplot(2, 2, p + 1)
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=1, label='Target marginal')
pl.scatter(xs[:, 0], xs[:, 1], c='C0', s=P.sum(1).ravel() * 2 * (1 + p),
label='Source marginal', alpha=1)
pl.plot([], [], color='C2', alpha=0.8, label='OT plan')
pl.title(r'Semi-relaxed $l_2$ UOT $\gamma$={}'.format(selected_gamma[p]),
fontsize=11)
if p < 2:
pl.xticks(())
pl.show()
# %%
# Animation of the regpath for semi-relaxed UOT l2
# ------------------------
nv = 100
g_list_v = np.logspace(2.5, -2, nv)
pl.figure(5)
def _update_plot(iv):
pl.clf()
tp = ot.regpath.compute_transport_plan(g_list_v[iv], g_list2,
t_list2)
P = tp.reshape((n, n))
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] * 0.5)
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) * 4,
label='Re-weighted source', alpha=1)
pl.scatter(xt[:, 0], xt[:, 1], c='C1', s=P.sum(0).ravel() * (1 + p) * 4,
label='Re-weighted target', alpha=1)
pl.plot([], [], color='C2', alpha=0.8, label='OT plan')
pl.title(r'Semi-relaxed $\ell_2$ UOT $\gamma$={:1.3f}'.format(g_list_v[iv]),
fontsize=11)
return 1
i = 0
_update_plot(i)
ani = animation.FuncAnimation(pl.gcf(), _update_plot, nv, interval=50, repeat_delay=2000)
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