1D Wasserstein barycenter demo

This example illustrates the computation of regularized Wasserstein Barycenter as proposed in [3].

[3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. (2015). Iterative Bregman projections for regularized transportation problems SIAM Journal on Scientific Computing, 37(2), A1111-A1138.

# Author: Remi Flamary <remi.flamary@unice.fr>
#
# License: MIT License

# sphinx_gallery_thumbnail_number = 1

import numpy as np
import matplotlib.pyplot as plt
import ot

# necessary for 3d plot even if not used
from mpl_toolkits.mplot3d import Axes3D  # noqa
from matplotlib.collections import PolyCollection

Generate data

n = 100  # nb bins

# bin positions
x = np.arange(n, dtype=np.float64)

# Gaussian distributions
a1 = ot.datasets.make_1D_gauss(n, m=20, s=5)  # m= mean, s= std
a2 = ot.datasets.make_1D_gauss(n, m=60, s=8)

# creating matrix A containing all distributions
A = np.vstack((a1, a2)).T
n_distributions = A.shape[1]

# loss matrix + normalization
M = ot.utils.dist0(n)
M /= M.max()

Barycenter computation

alpha = 0.2  # 0<=alpha<=1
weights = np.array([1 - alpha, alpha])

# l2bary
bary_l2 = A.dot(weights)

# wasserstein
reg = 1e-3
bary_wass = ot.bregman.barycenter(A, M, reg, weights)

f, (ax1, ax2) = plt.subplots(2, 1, tight_layout=True, num=1)
ax1.plot(x, A, color="black")
ax1.set_title("Distributions")

ax2.plot(x, bary_l2, "r", label="l2")
ax2.plot(x, bary_wass, "g", label="Wasserstein")
ax2.set_title("Barycenters")

plt.legend()
plt.show()

Barycentric interpolation

n_alpha = 11
alpha_list = np.linspace(0, 1, n_alpha)


B_l2 = np.zeros((n, n_alpha))

B_wass = np.copy(B_l2)

for i in range(n_alpha):
    alpha = alpha_list[i]
    weights = np.array([1 - alpha, alpha])
    B_l2[:, i] = A.dot(weights)
    B_wass[:, i] = ot.bregman.barycenter(A, M, reg, weights)

plt.figure(2)

cmap = plt.get_cmap("viridis")
verts = []
zs = alpha_list
for i, z in enumerate(zs):
    ys = B_l2[:, i]
    verts.append(list(zip(x, ys)))

ax = plt.gcf().add_subplot(projection="3d")

poly = PolyCollection(verts, facecolors=[cmap(a) for a in alpha_list])
poly.set_alpha(0.7)
ax.add_collection3d(poly, zs=zs, zdir="y")
ax.set_xlabel("x")
ax.set_xlim3d(0, n)
ax.set_ylabel("$\\alpha$")
ax.set_ylim3d(0, 1)
ax.set_zlabel("")
ax.set_zlim3d(0, B_l2.max() * 1.01)
plt.title("Barycenter interpolation with l2")
plt.tight_layout()

plt.figure(3)
cmap = plt.get_cmap("viridis")
verts = []
zs = alpha_list
for i, z in enumerate(zs):
    ys = B_wass[:, i]
    verts.append(list(zip(x, ys)))

ax = plt.gcf().add_subplot(projection="3d")

poly = PolyCollection(verts, facecolors=[cmap(a) for a in alpha_list])
poly.set_alpha(0.7)
ax.add_collection3d(poly, zs=zs, zdir="y")
ax.set_xlabel("x")
ax.set_xlim3d(0, n)
ax.set_ylabel("$\\alpha$")
ax.set_ylim3d(0, 1)
ax.set_zlabel("")
ax.set_zlim3d(0, B_l2.max() * 1.01)
plt.title("Barycenter interpolation with Wasserstein")
plt.tight_layout()

plt.show()

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