From ac830dd2b85cfd39f4fadd879a721b36ded033ea Mon Sep 17 00:00:00 2001
From: Nathan Cassereau <84033440+ncassereau-idris@users.noreply.github.com>
Date: Sat, 3 Dec 2022 22:44:41 +0100
Subject: [MRG] Wrong documentation in weak OT solver (#410)

* Docstrings of weak.py updated

* releases.md
---
 RELEASES.md | 1 +
 ot/weak.py  | 6 ++++--
 2 files changed, 5 insertions(+), 2 deletions(-)

diff --git a/RELEASES.md b/RELEASES.md
index 564fd4a..68487e8 100644
--- a/RELEASES.md
+++ b/RELEASES.md
@@ -26,6 +26,7 @@ roughly 2^31) (PR #381)
 - Added a work-around for scipy's bug, where you cannot compute the Hamming distance with a "None" weight attribute. (Issue #400, PR #402)
 - Fixed an issue where the doc could not be built due to some changes in matplotlib's API (Issue #403, PR #402)
 - Replaced Numpy C Compiler with Setuptools C Compiler due to deprecation issues (Issue #408, PR #409)
+- Fixed weak optimal transport docstring (Issue #404, PR #410)
 
 
 ## 0.8.2
diff --git a/ot/weak.py b/ot/weak.py
index f7d5b23..7364e68 100644
--- a/ot/weak.py
+++ b/ot/weak.py
@@ -18,7 +18,7 @@ def weak_optimal_transport(Xa, Xb, a=None, b=None, verbose=False, log=False, G0=
 
 
     .. math::
-        \gamma = \mathop{\arg \min}_\gamma \quad \|X_a-diag(1/a)\gammaX_b\|_F^2
+        \gamma = \mathop{\arg \min}_\gamma \quad \sum_i \mathbf{a}_i \left(\mathbf{X^a}_i - \frac{1}{\mathbf{a}_i} \sum_j \gamma_{ij} \mathbf{X^b}_j \right)^2
 
         s.t. \ \gamma \mathbf{1} = \mathbf{a}
 
@@ -28,7 +28,7 @@ def weak_optimal_transport(Xa, Xb, a=None, b=None, verbose=False, log=False, G0=
 
     where :
 
-    - :math:`X_a` :math:`X_b`  are the sample matrices.
+    - :math:`X^a` and  :math:`X^b`  are the sample matrices.
     - :math:`\mathbf{a}` and :math:`\mathbf{b}` are the sample weights
 
 
@@ -49,6 +49,8 @@ def weak_optimal_transport(Xa, Xb, a=None, b=None, verbose=False, log=False, G0=
         Source histogram (uniform weight if empty list)
     b : (nt,) array-like, float
         Target histogram (uniform weight if empty list))
+    G0 : (ns,nt) array-like, float
+        initial guess (default is indep joint density)
     numItermax : int, optional
         Max number of iterations
     numItermaxEmd : int, optional
-- 
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