[MRG] Regularization path for l2 UOT (#274)

* add reg path

* debug examples and verify pep8

* pep8 and move the reg path examples in unbalanced folder

Co-authored-by: haoran010 <haoran.wu@insa-rennes.fr>
Co-authored-by: Rémi Flamary <remi.flamary@gmail.com>

Author: 3 people

examples/unbalanced-partial/plot_regpath.py

@@ -0,0 +1,135 @@
+# -*- 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
+
+
+import numpy as np
+import matplotlib.pylab as pl
+import ot
+
+##############################################################################
+# 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
+# ----------------
+
+#%% 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()
+
+
+##############################################################################
+# Plot the semi-relaxed regularization path
+# -------------------
+
+#%% semi-relaxed l2-penalized UOT
+
+pl.figure(3)
+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()

ot/__init__.py

@@ -34,6 +34,7 @@
 from . import unbalanced
 from . import partial
 from . import backend
+from . import regpath
 
 # OT functions
 from .lp import emd, emd2, emd_1d, emd2_1d, wasserstein_1d
@@ -54,4 +55,4 @@
            'dist', 'unif', 'barycenter', 'sinkhorn_lpl1_mm', 'da', 'optim',
            'sinkhorn_unbalanced', 'barycenter_unbalanced',
            'sinkhorn_unbalanced2', 'sliced_wasserstein_distance',
-           'smooth', 'stochastic', 'unbalanced', 'partial']
+           'smooth', 'stochastic', 'unbalanced', 'partial', 'regpath']