free support barycenter

Author: vivienseguy

ot/lp/cvx.py

@@ -10,6 +10,7 @@
 import numpy as np
 import scipy as sp
 import scipy.sparse as sps
+import ot
 
 try:
     import cvxopt
@@ -144,3 +145,82 @@ def barycenter(A, M, weights=None, verbose=False, log=False, solver='interior-po
         return b, sol
     else:
         return b
+
+
+
+
+def free_support_barycenter(data_positions, data_weights, X_init, b_init, lamda, numItermax=100, stopThr=1e-5, verbose=False, log=False, **kwargs):
+
+    """
+    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)
+
+    The function solves the Wasserstein barycenter problem when the barycenter measure is constrained to be supported on k atoms.
+    This problem is considered in [1] (Algorithm 2). There are two differences with the following codes:
+    - we do not optimize over the weights
+    - we do not do line search for the locations updates, we use i.e. theta = 1 in [1] (Algorithm 2). This can be seen as a discrete implementation of the fixed-point algorithm of [2] proposed in the continuous setting.
+
+    Parameters
+    ----------
+    data_positions : list of (k_i,d) np.ndarray
+        The discrete support of a measure supported on k_i locations of a d-dimensional space (k_i can be different for each element of the list)
+    data_weights : list of (k_i,) np.ndarray
+        Numpy arrays where each numpy array has k_i non-negatives values summing to one representing the weights of each discrete input measure
+
+    X_init : (k,d) np.ndarray
+        Initialization of the support locations (on k atoms) of the barycenter
+    b_init : (k,) np.ndarray
+        Initialization of the weights of the barycenter (non-negatives, sum to 1)
+    lambda : (k,) np.ndarray
+        Initialization of the coefficients of the barycenter (non-negatives, sum to 1)
+
+    numItermax : int, optional
+        Max number of iterations
+    stopThr : float, optional
+        Stop threshol on error (>0)
+    verbose : bool, optional
+        Print information along iterations
+    log : bool, optional
+        record log if True
+
+    Returns
+    -------
+    X : (k,d) np.ndarray
+        Support locations (on k atoms) of the barycenter
+
+    References
+    ----------
+
+    .. [1] Cuturi, Marco, and Arnaud Doucet. "Fast computation of Wasserstein barycenters." International Conference on Machine Learning. 2014.
+
+    .. [2]  Álvarez-Esteban, Pedro C., et al. "A fixed-point approach to barycenters in Wasserstein space." Journal of Mathematical Analysis and Applications 441.2 (2016): 744-762.
+
+    """
+
+    iter_count = 0
+
+    d = X_init.shape[1]
+    k = b_init.size
+    N = len(data_positions)
+
+    X = X_init
+
+    displacement_square_norm = 1e3
+
+    while ( displacement_square_norm > stopThr and iter_count < numItermax ):
+
+        T_sum = np.zeros((k, d))
+
+        for (data_positions_i, data_weights_i) in zip(data_positions, data_weights):
+            M_i = ot.dist(X, data_positions_i)
+            T_i = ot.emd(b_init, data_weights_i, M_i)
+            T_sum += np.reshape(1. / b_init, (-1, 1)) * np.matmul(T_i, data_positions_i)
+
+        X_previous = X
+        X = T_sum / N
+
+        displacement_square_norm = np.sum(np.square(X-X_previous))
+
+        iter_count += 1
+
+    return X
+