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.. _sphx_glr_auto_examples_plot_barycenter_lp_vs_entropic.py:
=================================================================================
1D Wasserstein barycenter comparison between exact LP and entropic regularization
=================================================================================
This example illustrates the computation of regularized Wasserstein Barycenter
as proposed in [3] and exact LP barycenters using standard LP solver.
It reproduces approximately Figure 3.1 and 3.2 from the following paper:
Cuturi, M., & Peyré, G. (2016). A smoothed dual approach for variational
Wasserstein problems. SIAM Journal on Imaging Sciences, 9(1), 320-343.
[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.
.. code-block:: python
# Author: Remi Flamary <remi.flamary@unice.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 # noqa
#import ot.lp.cvx as cvx
Gaussian Data
-------------
.. code-block:: python
#%% parameters
problems = []
n = 100 # nb bins
# bin positions
x = np.arange(n, dtype=np.float64)
# Gaussian distributions
# 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()
#%% 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()
#%% barycenter computation
alpha = 0.5 # 0<=alpha<=1
weights = np.array([1 - alpha, alpha])
# l2bary
bary_l2 = A.dot(weights)
# wasserstein
reg = 1e-3
ot.tic()
bary_wass = ot.bregman.barycenter(A, M, reg, weights)
ot.toc()
ot.tic()
bary_wass2 = ot.lp.barycenter(A, M, weights, solver='interior-point', verbose=True)
ot.toc()
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='Reg Wasserstein')
pl.plot(x, bary_wass2, 'b', label='LP Wasserstein')
pl.legend()
pl.title('Barycenters')
pl.tight_layout()
problems.append([A, [bary_l2, bary_wass, bary_wass2]])
.. rst-class:: sphx-glr-horizontal
*
.. image:: /auto_examples/images/sphx_glr_plot_barycenter_lp_vs_entropic_001.png
:scale: 47
*
.. image:: /auto_examples/images/sphx_glr_plot_barycenter_lp_vs_entropic_002.png
:scale: 47
.. rst-class:: sphx-glr-script-out
Out::
Elapsed time : 0.010712385177612305 s
Primal Feasibility Dual Feasibility Duality Gap Step Path Parameter Objective
1.0 1.0 1.0 - 1.0 1700.336700337
0.006776453137632 0.006776453137633 0.006776453137633 0.9932238647293 0.006776453137633 125.6700527543
0.004018712867874 0.004018712867874 0.004018712867874 0.4301142633 0.004018712867874 12.26594150093
0.001172775061627 0.001172775061627 0.001172775061627 0.7599932455029 0.001172775061627 0.3378536968897
0.0004375137005385 0.0004375137005385 0.0004375137005385 0.6422331807989 0.0004375137005385 0.1468420566358
0.000232669046734 0.0002326690467341 0.000232669046734 0.5016999460893 0.000232669046734 0.09381703231432
7.430121674303e-05 7.430121674303e-05 7.430121674303e-05 0.7035962305812 7.430121674303e-05 0.0577787025717
5.321227838876e-05 5.321227838875e-05 5.321227838876e-05 0.308784186441 5.321227838876e-05 0.05266249477203
1.990900379199e-05 1.990900379196e-05 1.990900379199e-05 0.6520472013244 1.990900379199e-05 0.04526054405519
6.305442046799e-06 6.30544204682e-06 6.3054420468e-06 0.7073953304075 6.305442046798e-06 0.04237597591383
2.290148391577e-06 2.290148391582e-06 2.290148391578e-06 0.6941812711492 2.29014839159e-06 0.041522849321
1.182864875387e-06 1.182864875406e-06 1.182864875427e-06 0.508455204675 1.182864875445e-06 0.04129461872827
3.626786381529e-07 3.626786382468e-07 3.626786382923e-07 0.7101651572101 3.62678638267e-07 0.04113032448923
1.539754244902e-07 1.539754249276e-07 1.539754249356e-07 0.6279322066282 1.539754253892e-07 0.04108867636379
5.193221323143e-08 5.193221463044e-08 5.193221462729e-08 0.6843453436759 5.193221708199e-08 0.04106859618414
1.888205054507e-08 1.888204779723e-08 1.88820477688e-08 0.6673444085651 1.888205650952e-08 0.041062141752
5.676855206925e-09 5.676854518888e-09 5.676854517651e-09 0.7281705804232 5.676885442702e-09 0.04105958648713
3.501157668218e-09 3.501150243546e-09 3.501150216347e-09 0.414020345194 3.501164437194e-09 0.04105916265261
1.110594251499e-09 1.110590786827e-09 1.11059083379e-09 0.6998954759911 1.110636623476e-09 0.04105870073485
5.770971626386e-10 5.772456113791e-10 5.772456200156e-10 0.4999769658132 5.77013379477e-10 0.04105859769135
1.535218204536e-10 1.536993317032e-10 1.536992771966e-10 0.7516471627141 1.536205005991e-10 0.04105851679958
6.724209350756e-11 6.739211232927e-11 6.739210470901e-11 0.5944802416166 6.735465384341e-11 0.04105850033766
1.743382199199e-11 1.736445896691e-11 1.736448490761e-11 0.7573407808104 1.734254328931e-11 0.04105849088824
Optimization terminated successfully.
Elapsed time : 2.883899211883545 s
Dirac Data
----------
.. code-block:: python
#%% parameters
a1 = 1.0 * (x > 10) * (x < 50)
a2 = 1.0 * (x > 60) * (x < 80)
a1 /= a1.sum()
a2 /= a2.sum()
# 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 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()
#%% barycenter computation
alpha = 0.5 # 0<=alpha<=1
weights = np.array([1 - alpha, alpha])
# l2bary
bary_l2 = A.dot(weights)
# wasserstein
reg = 1e-3
ot.tic()
bary_wass = ot.bregman.barycenter(A, M, reg, weights)
ot.toc()
ot.tic()
bary_wass2 = ot.lp.barycenter(A, M, weights, solver='interior-point', verbose=True)
ot.toc()
problems.append([A, [bary_l2, bary_wass, bary_wass2]])
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='Reg Wasserstein')
pl.plot(x, bary_wass2, 'b', label='LP Wasserstein')
pl.legend()
pl.title('Barycenters')
pl.tight_layout()
#%% parameters
a1 = np.zeros(n)
a2 = np.zeros(n)
a1[10] = .25
a1[20] = .5
a1[30] = .25
a2[80] = 1
a1 /= a1.sum()
a2 /= a2.sum()
# 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 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()
#%% barycenter computation
alpha = 0.5 # 0<=alpha<=1
weights = np.array([1 - alpha, alpha])
# l2bary
bary_l2 = A.dot(weights)
# wasserstein
reg = 1e-3
ot.tic()
bary_wass = ot.bregman.barycenter(A, M, reg, weights)
ot.toc()
ot.tic()
bary_wass2 = ot.lp.barycenter(A, M, weights, solver='interior-point', verbose=True)
ot.toc()
problems.append([A, [bary_l2, bary_wass, bary_wass2]])
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='Reg Wasserstein')
pl.plot(x, bary_wass2, 'b', label='LP Wasserstein')
pl.legend()
pl.title('Barycenters')
pl.tight_layout()
.. rst-class:: sphx-glr-horizontal
*
.. image:: /auto_examples/images/sphx_glr_plot_barycenter_lp_vs_entropic_003.png
:scale: 47
*
.. image:: /auto_examples/images/sphx_glr_plot_barycenter_lp_vs_entropic_004.png
:scale: 47
.. rst-class:: sphx-glr-script-out
Out::
Elapsed time : 0.014938592910766602 s
Primal Feasibility Dual Feasibility Duality Gap Step Path Parameter Objective
1.0 1.0 1.0 - 1.0 1700.336700337
0.006776466288966 0.006776466288966 0.006776466288966 0.9932238515788 0.006776466288966 125.6649255808
0.004036918865495 0.004036918865495 0.004036918865495 0.4272973099316 0.004036918865495 12.3471617011
0.00121923268707 0.00121923268707 0.00121923268707 0.749698685599 0.00121923268707 0.3243835647408
0.0003837422984432 0.0003837422984432 0.0003837422984432 0.6926882608284 0.0003837422984432 0.1361719397493
0.0001070128410183 0.0001070128410183 0.0001070128410183 0.7643889137854 0.0001070128410183 0.07581952832518
0.0001001275033711 0.0001001275033711 0.0001001275033711 0.07058704837812 0.0001001275033712 0.0734739493635
4.550897507844e-05 4.550897507841e-05 4.550897507844e-05 0.5761172484828 4.550897507845e-05 0.05555077655047
8.557124125522e-06 8.5571241255e-06 8.557124125522e-06 0.8535925441152 8.557124125522e-06 0.04439814660221
3.611995628407e-06 3.61199562841e-06 3.611995628414e-06 0.6002277331554 3.611995628415e-06 0.04283007762152
7.590393750365e-07 7.590393750491e-07 7.590393750378e-07 0.8221486533416 7.590393750381e-07 0.04192322976248
8.299929287441e-08 8.299929286079e-08 8.299929287532e-08 0.9017467938799 8.29992928758e-08 0.04170825633295
3.117560203449e-10 3.117560130137e-10 3.11756019954e-10 0.997039969226 3.11756019952e-10 0.04168179329766
1.559749653711e-14 1.558073160926e-14 1.559756940692e-14 0.9999499686183 1.559750643989e-14 0.04168169240444
Optimization terminated successfully.
Elapsed time : 2.642659902572632 s
Elapsed time : 0.002908945083618164 s
Primal Feasibility Dual Feasibility Duality Gap Step Path Parameter Objective
1.0 1.0 1.0 - 1.0 1700.336700337
0.006774675520727 0.006774675520727 0.006774675520727 0.9932256422636 0.006774675520727 125.6956034743
0.002048208707562 0.002048208707562 0.002048208707562 0.7343095368143 0.002048208707562 5.213991622123
0.000269736547478 0.0002697365474781 0.0002697365474781 0.8839403501193 0.000269736547478 0.505938390389
6.832109993943e-05 6.832109993944e-05 6.832109993944e-05 0.7601171075965 6.832109993943e-05 0.2339657807272
2.437682932219e-05 2.43768293222e-05 2.437682932219e-05 0.6663448297475 2.437682932219e-05 0.1471256246325
1.13498321631e-05 1.134983216308e-05 1.13498321631e-05 0.5553643816404 1.13498321631e-05 0.1181584941171
3.342312725885e-06 3.342312725884e-06 3.342312725885e-06 0.7238133571615 3.342312725885e-06 0.1006387519747
7.078561231603e-07 7.078561231509e-07 7.078561231604e-07 0.8033142552512 7.078561231603e-07 0.09474734646269
1.966870956916e-07 1.966870954537e-07 1.966870954468e-07 0.752547917788 1.966870954633e-07 0.09354342735766
4.19989524849e-10 4.199895164852e-10 4.199895238758e-10 0.9984019849375 4.19989523951e-10 0.09310367785861
2.101015938666e-14 2.100625691113e-14 2.101023853438e-14 0.999949974425 2.101023691864e-14 0.09310274466458
Optimization terminated successfully.
Elapsed time : 2.690450668334961 s
Final figure
------------
.. code-block:: python
#%% plot
nbm = len(problems)
nbm2 = (nbm // 2)
pl.figure(2, (20, 6))
pl.clf()
for i in range(nbm):
A = problems[i][0]
bary_l2 = problems[i][1][0]
bary_wass = problems[i][1][1]
bary_wass2 = problems[i][1][2]
pl.subplot(2, nbm, 1 + i)
for j in range(n_distributions):
pl.plot(x, A[:, j])
if i == nbm2:
pl.title('Distributions')
pl.xticks(())
pl.yticks(())
pl.subplot(2, nbm, 1 + i + nbm)
pl.plot(x, bary_l2, 'r', label='L2 (Euclidean)')
pl.plot(x, bary_wass, 'g', label='Reg Wasserstein')
pl.plot(x, bary_wass2, 'b', label='LP Wasserstein')
if i == nbm - 1:
pl.legend()
if i == nbm2:
pl.title('Barycenters')
pl.xticks(())
pl.yticks(())
.. image:: /auto_examples/images/sphx_glr_plot_barycenter_lp_vs_entropic_006.png
:align: center
**Total running time of the script:** ( 0 minutes 8.892 seconds)
.. only :: html
.. container:: sphx-glr-footer
.. container:: sphx-glr-download
:download:`Download Python source code: plot_barycenter_lp_vs_entropic.py <plot_barycenter_lp_vs_entropic.py>`
.. container:: sphx-glr-download
:download:`Download Jupyter notebook: plot_barycenter_lp_vs_entropic.ipynb <plot_barycenter_lp_vs_entropic.ipynb>`
.. only:: html
.. rst-class:: sphx-glr-signature
`Gallery generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_
|
