[MRG] Add Bures-Wasserstein arycenetrs example (and debug the solver) (#584)

* add exmaple and debug barycenters

* debug barycenter again

Author: rflamary

RELEASES.md

@@ -21,7 +21,7 @@
 + Add `stop_criterion` feature to (un)regularized (f)gw barycenter solvers (PR #578)
 + Add `fixed_structure` and `fixed_features` to entropic fgw barycenter solver (PR #578)
 + Add new BAPG solvers with KL projections for GW and FGW (PR #581)
-+ Add Bures-Wasserstein barycenter in `ot.gaussian` (PR #582)
++ Add Bures-Wasserstein barycenter in `ot.gaussian` and example (PR #582, PR #584)
 
 
 #### Closed issues

examples/barycenters/plot_gaussian_barycenter.py

@@ -0,0 +1,127 @@
+# -*- coding: utf-8 -*-
+"""
+========================================================
+Gaussian Bures-Wasserstein barycenters
+========================================================
+
+Illustration of Gaussian Bures-Wasserstein barycenters.
+
+"""
+
+# Authors: Rémi Flamary <remi.flamary@polytechnique.edu>
+#
+# License: MIT License
+
+# sphinx_gallery_thumbnail_number = 2
+# %%
+from matplotlib import colors
+from matplotlib.patches import Ellipse
+import numpy as np
+import matplotlib.pylab as pl
+import ot
+
+
+# %%
+# Define Gaussian Covariances and distributions
+# ---------------------------------------------
+
+C1 = np.array([[0.5, -0.4], [-0.4, 0.5]])
+C2 = np.array([[1, 0.3], [0.3, 1]])
+C3 = np.array([[1.5, 0], [0, 0.5]])
+C4 = np.array([[0.5, 0], [0, 1.5]])
+
+C = np.stack((C1, C2, C3, C4))
+
+m1 = np.array([0, 0])
+m2 = np.array([0, 4])
+m3 = np.array([4, 0])
+m4 = np.array([4, 4])
+
+m = np.stack((m1, m2, m3, m4))
+
+# %%
+# Plot the distributions
+# ----------------------
+
+
+def draw_cov(mu, C, color=None, label=None, nstd=1):
+
+    def eigsorted(cov):
+        vals, vecs = np.linalg.eigh(cov)
+        order = vals.argsort()[::-1]
+        return vals[order], vecs[:, order]
+
+    vals, vecs = eigsorted(C)
+    theta = np.degrees(np.arctan2(*vecs[:, 0][::-1]))
+    w, h = 2 * nstd * np.sqrt(vals)
+    ell = Ellipse(xy=(mu[0], mu[1]),
+                  width=w, height=h, alpha=0.5,
+                  angle=theta, facecolor=color, edgecolor=color, label=label, fill=True)
+    pl.gca().add_artist(ell)
+    #pl.scatter(mu[0],mu[1],color=color, marker='x')
+
+
+axis = [-1.5, 5.5, -1.5, 5.5]
+
+pl.figure(1, (8, 2))
+pl.clf()
+
+pl.subplot(1, 4, 1)
+draw_cov(m1, C1, color='C0')
+pl.axis(axis)
+pl.title('$\mathcal{N}(m_1,\Sigma_1)$')
+
+pl.subplot(1, 4, 2)
+draw_cov(m2, C2, color='C1')
+pl.axis(axis)
+pl.title('$\mathcal{N}(m_2,\Sigma_2)$')
+
+pl.subplot(1, 4, 3)
+draw_cov(m3, C3, color='C2')
+pl.axis(axis)
+pl.title('$\mathcal{N}(m_3,\Sigma_3)$')
+
+pl.subplot(1, 4, 4)
+draw_cov(m4, C4, color='C3')
+pl.axis(axis)
+pl.title('$\mathcal{N}(m_4,\Sigma_4)$')
+
+# %%
+# Compute Bures-Wasserstein barycenters and plot them
+# -------------------------------------------
+
+# basis for bilinear interpolation
+v1 = np.array((1, 0, 0, 0))
+v2 = np.array((0, 1, 0, 0))
+v3 = np.array((0, 0, 1, 0))
+v4 = np.array((0, 0, 0, 1))
+
+
+colors = np.stack((colors.to_rgb('C0'),
+                   colors.to_rgb('C1'),
+                   colors.to_rgb('C2'),
+                   colors.to_rgb('C3')))
+
+pl.figure(2, (8, 8))
+
+nb_interp = 6
+
+for i in range(nb_interp):
+    for j in range(nb_interp):
+        tx = float(i) / (nb_interp - 1)
+        ty = float(j) / (nb_interp - 1)
+
+        # weights are constructed by bilinear interpolation
+        tmp1 = (1 - tx) * v1 + tx * v2
+        tmp2 = (1 - tx) * v3 + tx * v4
+        weights = (1 - ty) * tmp1 + ty * tmp2
+
+        color = np.dot(colors.T, weights)
+
+        mb, Cb = ot.gaussian.bures_wasserstein_barycenter(m, C, weights)
+
+        draw_cov(mb, Cb, color=color, label=None, nstd=0.3)
+
+pl.axis(axis)
+pl.axis('off')
+pl.tight_layout()

ot/gaussian.py

@@ -399,14 +399,14 @@ def bures_wasserstein_barycenter(m, C, weights=None, num_iter=1000, eps=1e-7, lo
     """
     nx = get_backend(*C, *m,)
 
+    if weights is None:
+        weights = nx.ones(C.shape[0], type_as=C[0]) / C.shape[0]
+
     # Compute the mean barycenter
-    mb = nx.mean(m)
+    mb = nx.sum(m * weights[:, None], axis=0)
 
     # Init the covariance barycenter
-    Cb = nx.mean(C, axis=0)
-
-    if weights is None:
-        weights = nx.ones(len(C), type_as=C[0]) / len(C)
+    Cb = nx.mean(C * weights[:, None, None], axis=0)
 
     for it in range(num_iter):
         # fixed point update