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
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
|
.. _sphx_glr_auto_examples_plot_UOT_barycenter_1D.py:
===========================================================
1D Wasserstein barycenter demo for Unbalanced distributions
===========================================================
This example illustrates the computation of regularized Wassersyein Barycenter
as proposed in [10] for Unbalanced inputs.
[10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.
.. code-block:: python
# Author: Hicham Janati <hicham.janati@inria.fr>
#
# License: MIT License
import numpy as np
import matplotlib.pylab as pl
import ot
# necessary for 3d plot even if not used
from mpl_toolkits.mplot3d import Axes3D # noqa
from matplotlib.collections import PolyCollection
Generate data
-------------
.. code-block:: python
# 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)
# make unbalanced dists
a2 *= 3.
# 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()
Plot data
---------
.. code-block:: python
# plot the distributions
pl.figure(1, figsize=(6.4, 3))
for i in range(n_distributions):
pl.plot(x, A[:, i])
pl.title('Distributions')
pl.tight_layout()
.. image:: /auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_001.png
:align: center
Barycenter computation
----------------------
.. code-block:: python
# non weighted barycenter computation
weight = 0.5 # 0<=weight<=1
weights = np.array([1 - weight, weight])
# l2bary
bary_l2 = A.dot(weights)
# wasserstein
reg = 1e-3
alpha = 1.
bary_wass = ot.unbalanced.barycenter_unbalanced(A, M, reg, alpha, weights)
pl.figure(2)
pl.clf()
pl.subplot(2, 1, 1)
for i in range(n_distributions):
pl.plot(x, A[:, i])
pl.title('Distributions')
pl.subplot(2, 1, 2)
pl.plot(x, bary_l2, 'r', label='l2')
pl.plot(x, bary_wass, 'g', label='Wasserstein')
pl.legend()
pl.title('Barycenters')
pl.tight_layout()
.. image:: /auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_003.png
:align: center
Barycentric interpolation
-------------------------
.. code-block:: python
# barycenter interpolation
n_weight = 11
weight_list = np.linspace(0, 1, n_weight)
B_l2 = np.zeros((n, n_weight))
B_wass = np.copy(B_l2)
for i in range(0, n_weight):
weight = weight_list[i]
weights = np.array([1 - weight, weight])
B_l2[:, i] = A.dot(weights)
B_wass[:, i] = ot.unbalanced.barycenter_unbalanced(A, M, reg, alpha, weights)
# plot interpolation
pl.figure(3)
cmap = pl.cm.get_cmap('viridis')
verts = []
zs = weight_list
for i, z in enumerate(zs):
ys = B_l2[:, i]
verts.append(list(zip(x, ys)))
ax = pl.gcf().gca(projection='3d')
poly = PolyCollection(verts, facecolors=[cmap(a) for a in weight_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(r'$\alpha$')
ax.set_ylim3d(0, 1)
ax.set_zlabel('')
ax.set_zlim3d(0, B_l2.max() * 1.01)
pl.title('Barycenter interpolation with l2')
pl.tight_layout()
pl.figure(4)
cmap = pl.cm.get_cmap('viridis')
verts = []
zs = weight_list
for i, z in enumerate(zs):
ys = B_wass[:, i]
verts.append(list(zip(x, ys)))
ax = pl.gcf().gca(projection='3d')
poly = PolyCollection(verts, facecolors=[cmap(a) for a in weight_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(r'$\alpha$')
ax.set_ylim3d(0, 1)
ax.set_zlabel('')
ax.set_zlim3d(0, B_l2.max() * 1.01)
pl.title('Barycenter interpolation with Wasserstein')
pl.tight_layout()
pl.show()
.. rst-class:: sphx-glr-horizontal
*
.. image:: /auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_005.png
:scale: 47
*
.. image:: /auto_examples/images/sphx_glr_plot_UOT_barycenter_1D_006.png
:scale: 47
**Total running time of the script:** ( 0 minutes 0.344 seconds)
.. only :: html
.. container:: sphx-glr-footer
.. container:: sphx-glr-download
:download:`Download Python source code: plot_UOT_barycenter_1D.py <plot_UOT_barycenter_1D.py>`
.. container:: sphx-glr-download
:download:`Download Jupyter notebook: plot_UOT_barycenter_1D.ipynb <plot_UOT_barycenter_1D.ipynb>`
.. only:: html
.. rst-class:: sphx-glr-signature
`Gallery generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_
|
