1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
|
# -*- coding: utf-8 -*-
"""
==============================
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
# -------------
# %% parameters
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
# ----------------------
# %% 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
# -------------------------
# %% barycenter 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)
# %% plot interpolation
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()
|
