add free support barycenter algorithm

Author: Vivien Seguy

README.md

@@ -17,6 +17,7 @@ It provides the following solvers:
 * Entropic regularization OT solver with Sinkhorn Knopp Algorithm [2] and stabilized version [9][10] with optional GPU implementation (requires cudamat).
 * Smooth optimal transport solvers (dual and semi-dual) for KL and squared L2 regularizations [17].
 * Non regularized Wasserstein barycenters [16] with LP solver (only small scale).
+* Non regularized free support Wasserstein barycenters [20].
 * 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].
@@ -225,3 +226,5 @@ You can also post bug reports and feature requests in Github issues. Make sure t
 [18] Genevay, A., Cuturi, M., Peyré, G. & Bach, F. (2016) [Stochastic Optimization for Large-scale Optimal Transport](arXiv preprint arxiv:1605.08527). Advances in Neural Information Processing Systems (2016).
 
 [19] Seguy, V., Bhushan Damodaran, B., Flamary, R., Courty, N., Rolet, A.& Blondel, M. [Large-scale Optimal Transport and Mapping Estimation](https://arxiv.org/pdf/1711.02283.pdf). International Conference on Learning Representation (2018)
+
+[20] Cuturi, M. and Doucet, A. (2014) [Fast Computation of Wasserstein Barycenters](http://proceedings.mlr.press/v32/cuturi14.html). International Conference in Machine Learning

examples/plot_free_support_barycenter.py

@@ -21,7 +21,7 @@
 # Generate data
 # -------------
 #%% parameters and data generation
-N = 6
+N = 3
 d = 2
 measures_locations = []
 measures_weights = []
@@ -33,24 +33,25 @@
     mu = np.random.normal(0., 4., (d,))
 
     A = np.random.rand(d, d)
-    cov = np.dot(A,A.transpose())
+    cov = np.dot(A, A.transpose())
 
-    xs = ot.datasets.make_2D_samples_gauss(n, mu, cov)
-    b = np.random.uniform(0., 1., (n,))
-    b = b/np.sum(b)
+    x_i = ot.datasets.make_2D_samples_gauss(n, mu, cov)
+    b_i = np.random.uniform(0., 1., (n,))
+    b_i = b_i / np.sum(b_i)
 
-    measures_locations.append(xs)
-    measures_weights.append(b)
-
-k = 10
-X_init = np.random.normal(0., 1., (k,d))
-b_init = np.ones((k,)) / k
+    measures_locations.append(x_i)
+    measures_weights.append(b_i)
 
 
 ##############################################################################
 # Compute free support barycenter
 # -------------
-X = ot.lp.cvx.free_support_barycenter(measures_locations, measures_weights, X_init, b_init)
+
+k = 10
+X_init = np.random.normal(0., 1., (k, d))
+b = np.ones((k,)) / k
+
+X = ot.lp.cvx.free_support_barycenter(measures_locations, measures_weights, X_init, b)
 
 
 ##############################################################################
@@ -60,10 +61,10 @@
 #%% plot samples
 
 pl.figure(1)
-for (xs, b) in zip(measures_locations, measures_weights):
-    color = np.random.randint(low=1, high=10*N)
-    pl.scatter(xs[:, 0], xs[:, 1], s=b*1000, label='input measure')
-pl.scatter(X[:, 0], X[:, 1], s=b_init*1000, c='black' , marker='^', label='2-Wasserstein barycenter')
+for (x_i, b_i) in zip(measures_locations, measures_weights):
+    color = np.random.randint(low=1, high=10 * N)
+    pl.scatter(x_i[:, 0], x_i[:, 1], s=b * 1000, label='input measure')
+pl.scatter(X[:, 0], X[:, 1], s=b * 1000, c='black', marker='^', label='2-Wasserstein barycenter')
 pl.title('Data measures and their barycenter')
 pl.legend(loc=0)
-pl.show()
+pl.show()

ot/lp/cvx.py

@@ -147,10 +147,7 @@ def barycenter(A, M, weights=None, verbose=False, log=False, solver='interior-po
         return b
 
 
-
-
-def free_support_barycenter(measures_locations, measures_weights, X_init, b_init, weights=None, numItermax=100, stopThr=1e-6, verbose=False):
-
+def free_support_barycenter(measures_locations, measures_weights, X_init, b, weights=None, numItermax=100, stopThr=1e-6, verbose=False):
     """
     Solves the free support (locations of the barycenters are optimized, not the weights) Wasserstein barycenter problem (i.e. the weighted Frechet mean for the 2-Wasserstein distance)
 
@@ -168,7 +165,7 @@ def free_support_barycenter(measures_locations, measures_weights, X_init, b_init
 
     X_init : (k,d) np.ndarray
         Initialization of the support locations (on k atoms) of the barycenter
-    b_init : (k,) np.ndarray
+    b : (k,) np.ndarray
         Initialization of the weights of the barycenter (non-negatives, sum to 1)
     weights : (k,) np.ndarray
         Initialization of the coefficients of the barycenter (non-negatives, sum to 1)
@@ -199,27 +196,27 @@ def free_support_barycenter(measures_locations, measures_weights, X_init, b_init
     iter_count = 0
 
     d = X_init.shape[1]
-    k = b_init.size
+    k = b.size
     N = len(measures_locations)
 
     if not weights:
-        weights = np.ones((N,))/N
+        weights = np.ones((N,)) / N
 
     X = X_init
 
-    displacement_square_norm = stopThr+1.
+    displacement_square_norm = stopThr + 1.
 
-    while ( displacement_square_norm > stopThr and iter_count < numItermax ):
+    while (displacement_square_norm > stopThr and iter_count < numItermax):
 
         T_sum = np.zeros((k, d))
 
         for (measure_locations_i, measure_weights_i, weight_i) in zip(measures_locations, measures_weights, weights.tolist()):
 
             M_i = ot.dist(X, measure_locations_i)
-            T_i = ot.emd(b_init, measure_weights_i, M_i)
-            T_sum = T_sum + weight_i*np.reshape(1. / b_init, (-1, 1)) * np.matmul(T_i, measure_locations_i)
+            T_i = ot.emd(b, measure_weights_i, M_i)
+            T_sum = T_sum + weight_i * np.reshape(1. / b, (-1, 1)) * np.matmul(T_i, measure_locations_i)
 
-        displacement_square_norm = np.sum(np.square(X-T_sum))
+        displacement_square_norm = np.sum(np.square(X - T_sum))
         X = T_sum
 
         if verbose:
@@ -228,4 +225,3 @@ def free_support_barycenter(measures_locations, measures_weights, X_init, b_init
         iter_count += 1
 
     return X
-