[MRG] Sliced wasserstein (#203)

* example for log treatment in bregman.py

* Improve doc

* Revert "example for log treatment in bregman.py"

This reverts commit 9f51c14e

* Add comments by Flamary

* Delete repetitive description

* Added raw string to avoid pbs with backslashes

* Implements sliced wasserstein

* Changed formatting of string for py3.5 support

* Docstest, expected 0.0 and not 0.

* Adressed comments by @rflamary

* No 3d plot here

* add sliced to the docs

* Incorporate comments by @rflamary

* add link to pdf

Co-authored-by: Rémi Flamary <remi.flamary@gmail.com>

2 files changed, 88 insertions, 0 deletions

diff --git a/examples/sliced-wasserstein/README.txt b/examples/sliced-wasserstein/README.txt
new file mode 100644
index 0000000..a575345
--- /dev/null
+++ b/examples/sliced-wasserstein/README.txt
@@ -0,0 +1,4 @@
+
+
+Sliced Wasserstein Distance
+---------------------------
\ No newline at end of file
diff --git a/examples/sliced-wasserstein/plot_variance.py b/examples/sliced-wasserstein/plot_variance.py
new file mode 100644
index 0000000..f3deeff
--- /dev/null
+++ b/examples/sliced-wasserstein/plot_variance.py
@@ -0,0 +1,84 @@
+# -*- coding: utf-8 -*-
+"""
+==============================
+2D Sliced Wasserstein Distance
+==============================
+
+This example illustrates the computation of the sliced Wasserstein Distance as proposed in [31].
+
+[31] Bonneel, Nicolas, et al. "Sliced and radon wasserstein barycenters of measures." Journal of Mathematical Imaging and Vision 51.1 (2015): 22-45
+
+"""
+
+# Author: Adrien Corenflos <adrien.corenflos@aalto.fi>
+#
+# License: MIT License
+
+import matplotlib.pylab as pl
+import numpy as np
+
+import ot
+
+##############################################################################
+# Generate data
+# -------------
+
+# %% parameters and data generation
+
+n = 500  # 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 = np.ones((n,)) / n, np.ones((n,)) / n  # uniform distribution on samples
+
+##############################################################################
+# Plot data
+# ---------
+
+# %% 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')
+
+###################################################################################
+# Compute Sliced Wasserstein distance for different seeds and number of projections
+# -----------
+
+n_seed = 50
+n_projections_arr = np.logspace(0, 3, 25, dtype=int)
+res = np.empty((n_seed, 25))
+
+# %% Compute statistics
+for seed in range(n_seed):
+    for i, n_projections in enumerate(n_projections_arr):
+        res[seed, i] = ot.sliced_wasserstein_distance(xs, xt, a, b, n_projections, seed)
+
+res_mean = np.mean(res, axis=0)
+res_std = np.std(res, axis=0)
+
+###################################################################################
+# Plot Sliced Wasserstein Distance
+# -----------
+
+pl.figure(2)
+pl.plot(n_projections_arr, res_mean, label="SWD")
+pl.fill_between(n_projections_arr, res_mean - 2 * res_std, res_mean + 2 * res_std, alpha=0.5)
+
+pl.legend()
+pl.xscale('log')
+
+pl.xlabel("Number of projections")
+pl.ylabel("Distance")
+pl.title('Sliced Wasserstein Distance with 95% confidence inverval')
+
+pl.show()