Correct documentation for support barycenter (#201)

* example for log treatment in bregman.py

* Improve doc

* Revert "example for log treatment in bregman.py"

This reverts commit 9f51c14

* Add comments by Flamary

* Delete repetitive description

* Added raw string to avoid pbs with backslashes

Author: AdrienCorenflos

ot/lp/__init__.py

@@ -272,7 +272,7 @@ def emd(a, b, M, numItermax=100000, log=False, center_dual=True):
 
     if np.any(~asel) or np.any(~bsel):
         u, v = estimate_dual_null_weights(u, v, a, b, M)
-    
+
     result_code_string = check_result(result_code)
     if log:
         log = {}
@@ -389,7 +389,7 @@ def emd2(a, b, M, processes=multiprocessing.cpu_count(),
     if log or return_matrix:
         def f(b):
             bsel = b != 0
-            
+
             G, cost, u, v, result_code = emd_c(a, b, M, numItermax)
 
             if center_dual:
@@ -435,26 +435,36 @@ def f(b):
 
 def free_support_barycenter(measures_locations, measures_weights, X_init, b=None, weights=None, numItermax=100,
                             stopThr=1e-7, verbose=False, log=None):
-    """
-    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)
+    r"""
+    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), formally:
+
+    .. math::
+        \min_X \sum_{i=1}^N w_i W_2^2(b, X, a_i, X_i)
+
+    where :
+
+    - :math:`w \in \mathbb{(0, 1)}^{N}`'s are the barycenter weights and sum to one
+    - the :math:`a_i \in \mathbb{R}^{k_i}` are the empirical measures weights and sum to one for each :math:`i`
+    - the :math:`X_i \in \mathbb{R}^{k_i, d}` are the empirical measures atoms locations
+    - :math:`b \in \mathbb{R}^{k}` is the desired weights vector of the barycenter
 
-    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
     ----------
-    measures_locations : list of (k_i,d) numpy.ndarray
+    measures_locations : list of N (k_i,d) numpy.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)
-    measures_weights : list of (k_i,) numpy.ndarray
+    measures_weights : list of N (k_i,) numpy.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 : (k,) np.ndarray
         Initialization of the weights of the barycenter (non-negatives, sum to 1)
-    weights : (k,) np.ndarray
+    weights : (N,) np.ndarray
         Initialization of the coefficients of the barycenter (non-negatives, sum to 1)
 
     numItermax : int, optional