Add comments by Flamary

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:
@@ -438,12 +438,7 @@ def free_support_barycenter(measures_locations, measures_weights, X_init, b=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)
 
- 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.
-
- Formally we solve the following problem:
+ The function solves the Wasserstein barycenter problem when the barycenter measure is constrained to be supported on k atoms, formally:
 
  .. math::
  \min_X \sum_{i=1}^N w_i W_2^2(b, X, a_i, X_i)
@@ -455,6 +450,11 @@ def free_support_barycenter(measures_locations, measures_weights, X_init, b=None
  - 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
 
+ 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 N (k_i,d) numpy.ndarray