[MRG] Add weak OT solver (#341)

* add info in release file

* update tests

* pep8

* add weak OT example

* update plot in doc

* correction ewample with empirical sinkhorn

* better thumbnail

* comment from review

* update documenation

2 files changed, 100 insertions, 3 deletions

diff --git a/examples/others/plot_WeakOT_VS_OT.py b/examples/others/plot_WeakOT_VS_OT.py
new file mode 100644
index 0000000..a29c875
--- /dev/null
+++ b/examples/others/plot_WeakOT_VS_OT.py
@@ -0,0 +1,98 @@
+# -*- coding: utf-8 -*-
+"""
+====================================================
+Weak Optimal Transport VS exact Optimal Transport
+====================================================
+
+Illustration of 2D optimal transport between distributions that are weighted
+sum of diracs. The OT matrix is plotted with the samples.
+
+"""
+
+# Author: Remi Flamary <remi.flamary@polytechnique.edu>
+#
+# License: MIT License
+
+# sphinx_gallery_thumbnail_number = 4
+
+import numpy as np
+import matplotlib.pylab as pl
+import ot
+import ot.plot
+
+##############################################################################
+# Generate data an plot it
+# ------------------------
+
+#%% parameters and data generation
+
+n = 50  # nb samples
+
+mu_s = np.array([0, 0])
+cov_s = np.array([[1, 0], [0, 1]])
+
+mu_t = np.array([4, 4])
+cov_t = np.array([[1, -.8], [-.8, 1]])
+
+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 = ot.unif(n), ot.unif(n)  # uniform distribution on samples
+
+# loss matrix
+M = ot.dist(xs, xt)
+M /= M.max()
+
+#%% plot samples
+
+pl.figure(1)
+pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+pl.legend(loc=0)
+pl.title('Source and target distributions')
+
+pl.figure(2)
+pl.imshow(M, interpolation='nearest')
+pl.title('Cost matrix M')
+
+
+##############################################################################
+# Compute Weak OT and exact OT solutions
+# --------------------------------------
+
+#%% EMD
+
+G0 = ot.emd(a, b, M)
+
+#%% Weak OT
+
+Gweak = ot.weak_optimal_transport(xs, xt, a, b)
+
+
+##############################################################################
+# Plot weak OT and exact OT solutions
+# --------------------------------------
+
+pl.figure(3, (8, 5))
+
+pl.subplot(1, 2, 1)
+pl.imshow(G0, interpolation='nearest')
+pl.title('OT matrix')
+
+pl.subplot(1, 2, 2)
+pl.imshow(Gweak, interpolation='nearest')
+pl.title('Weak OT matrix')
+
+pl.figure(4, (8, 5))
+
+pl.subplot(1, 2, 1)
+ot.plot.plot2D_samples_mat(xs, xt, G0, c=[.5, .5, 1])
+pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+pl.title('OT matrix with samples')
+
+pl.subplot(1, 2, 2)
+ot.plot.plot2D_samples_mat(xs, xt, Gweak, c=[.5, .5, 1])
+pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
+pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
+pl.title('Weak OT matrix with samples')
diff --git a/examples/plot_OT_2D_samples.py b/examples/plot_OT_2D_samples.py
index af1bc12..c3a7cd8 100644
--- a/examples/plot_OT_2D_samples.py
+++ b/examples/plot_OT_2D_samples.py
@@ -42,7 +42,6 @@ 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
@@ -87,7 +86,7 @@ pl.title('OT matrix with samples')
 #%% sinkhorn
 
 # reg term
-lambd = 1e-3
+lambd = 1e-1
 
 Gs = ot.sinkhorn(a, b, M, lambd)
 
@@ -112,7 +111,7 @@ pl.show()
 #%% sinkhorn
 
 # reg term
-lambd = 1e-3
+lambd = 1e-1
 
 Ges = ot.bregman.empirical_sinkhorn(xs, xt, lambd)