LP Wasserstein barycenter with scipy linear solver and/or cvxopt

Makefile

@@ -58,9 +58,9 @@ notebook :
 	ipython notebook --matplotlib=inline  --notebook-dir=notebooks/
 
 autopep8 :
-	autopep8 -ir test ot examples
+	autopep8 -ir test ot examples --jobs -1
 
 aautopep8 :
-	autopep8 -air test ot examples
+	autopep8 -air test ot examples --jobs -1
 
 FORCE :

README.md

@@ -14,7 +14,8 @@ This open source Python library provide several solvers for optimization problem
 It provides the following solvers:
 
 * OT Network Flow solver for the linear program/ Earth Movers Distance [1].
-* Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2] and stabilized version [9][10] with optional GPU implementation (required cudamat).
+* Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2] and stabilized version [9][10] with optional GPU implementation (requires cudamat).
+* Non regularized Wasserstein barycenters [16] with LP solver.
 * Bregman projections for Wasserstein barycenter [3] and unmixing [4].
 * Optimal transport for domain adaptation with group lasso regularization [5]
 * Conditional gradient [6] and Generalized conditional gradient for regularized OT [7].
@@ -210,3 +211,5 @@ You can also post bug reports and feature requests in Github issues. Make sure t
 [14] Knott, M. and Smith, C. S. (1984).[On the optimal mapping of distributions](https://link.springer.com/article/10.1007/BF00934745), Journal of Optimization Theory and Applications Vol 43.
 
 [15] Peyré, G., & Cuturi, M. (2018). [Computational Optimal Transport](https://arxiv.org/pdf/1803.00567.pdf) .
+
+[16] Agueh, M., & Carlier, G. (2011). [Barycenters in the Wasserstein space](https://hal.archives-ouvertes.fr/hal-00637399/document). SIAM Journal on Mathematical Analysis, 43(2), 904-924.

examples/plot_barycenter_lp_vs_entropic.py

@@ -0,0 +1,292 @@
+# -*- coding: utf-8 -*-
+"""
+=================================================================================
+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.
+
+
+
+"""
+
+# 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
+
+#
+# Generate data
+# -------------
+
+#%% parameters
+
+problems = []
+
+n = 100  # nb bins
+
+# bin positions
+x = np.arange(n, dtype=np.float64)
+
+# Gaussian distributions
+# Gaussian distributions
+a1 = ot.datasets.get_1D_gauss(n, m=20, s=5)  # m= mean, s= std
+a2 = ot.datasets.get_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 data
+# ---------
+
+#%% 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
+# ----------------------
+
+#%% 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]])
+
+#%% 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
+# ----------------------
+
+#%% 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
+# ----------------------
+
+#%% 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()
+
+
+#
+# Final figure
+# ------------
+#
+
+#%% 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(())

ot/bregman.py

@@ -839,11 +839,13 @@ def barycenter(A, M, reg, weights=None, numItermax=1000,
     Parameters
     ----------
     A : np.ndarray (d,n)
-        n training distributions of size d
+        n training distributions a_i of size d
     M : np.ndarray (d,d)
         loss matrix   for OT
     reg : float
         Regularization term >0
+    weights : np.ndarray (n,)
+        Weights of each histogram a_i on the simplex (barycentric coodinates)
     numItermax : int, optional
         Max number of iterations
     stopThr : float, optional

ot/lp/__init__.py

@@ -11,9 +11,12 @@
 
 import numpy as np
 
+from .import cvx
+
 # import compiled emd
 from .emd_wrap import emd_c, check_result
 from ..utils import parmap
+from .cvx import barycenter
 
 
 def emd(a, b, M, numItermax=100000, log=False):