fix doc

Author: Kilian Fatras

ot/bregman.py

@@ -1317,9 +1317,9 @@ def empirical_sinkhorn(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', numI
              \gamma\geq 0
     where :
 
-    - M is the (ns,nt) metric cost matrix
+    - :math:`M` is the (ns,nt) metric cost matrix
     - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`
-    - a and b are source and target weights (sum to 1)
+    - :math:`a` and :math:`b` are source and target weights (sum to 1)
 
 
     Parameters
@@ -1399,7 +1399,7 @@ def empirical_sinkhorn2(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', num
     The function solves the following optimization problem:
 
     .. math::
-        W = \min_\gamma_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma)
+        W = \min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma)
 
         s.t. \gamma 1 = a
 
@@ -1408,9 +1408,9 @@ def empirical_sinkhorn2(X_s, X_t, reg, a=None, b=None, metric='sqeuclidean', num
              \gamma\geq 0
     where :
 
-    - M is the (ns,nt) metric cost matrix
+    - :math:`M` is the (ns,nt) metric cost matrix
     - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`
-    - a and b are source and target weights (sum to 1)
+    - :math:`a` and :math:`b` are source and target weights (sum to 1)
 
 
     Parameters
@@ -1484,13 +1484,20 @@ def empirical_sinkhorn_divergence(X_s, X_t, reg, a=None, b=None, metric='sqeucli
     '''
     Compute the sinkhorn divergence loss from empirical data
 
-    The function solves the following optimization problem:
+    The function solves the following optimization problems and return the
+    sinkhorn divergence :math:`S`:
 
     .. math::
-        S = \min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma) -
-        \min_\gamma_a <\gamma_a,M_a>_F + reg\cdot\Omega(\gamma_a) -
-        \min_\gamma_b <\gamma_b,M_b>_F + reg\cdot\Omega(\gamma_b)
 
+        W &= \min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma)
+
+        W_a &= \min_{\gamma_a} <\gamma_a,M_a>_F + reg\cdot\Omega(\gamma_a)
+
+        W_b &= \min_{\gamma_b} <\gamma_b,M_b>_F + reg\cdot\Omega(\gamma_b)
+
+        S &= W - 1/2 * (W_a + W_b)
+
+    .. math::
         s.t. \gamma 1 = a
 
              \gamma^T 1= b
@@ -1510,9 +1517,9 @@ def empirical_sinkhorn_divergence(X_s, X_t, reg, a=None, b=None, metric='sqeucli
              \gamma_b\geq 0
     where :
 
-    - M (resp. :math:`M_a, M_b) is the (ns,nt) metric cost matrix (resp (ns, ns) and (nt, nt))
+    - :math:`M` (resp. :math:`M_a, M_b`) is the (ns,nt) metric cost matrix (resp (ns, ns) and (nt, nt))
     - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`
-    - a and b are source and target weights (sum to 1)
+    - :math:`a` and :math:`b` are source and target weights (sum to 1)
 
 
     Parameters

ot/stochastic.py

@@ -348,8 +348,11 @@ def solve_semi_dual_entropic(a, b, M, reg, method, numItermax=10000, lr=None,
 
     .. math::
         \gamma = arg\min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma)
+
         s.t. \gamma 1 = a
+
              \gamma^T 1= b
+
              \gamma \geq 0
 
     Where :