From d20d471a1806bde43c23e67c1f805aa3c8908ec3 Mon Sep 17 00:00:00 2001
From: Rémi Flamary <remi.flamary@gmail.com>
Date: Thu, 27 Jun 2019 14:34:23 +0200
Subject: update part 1

---
 docs/source/quickstart.rst | 23 +++++++++++++++++++++++
 1 file changed, 23 insertions(+)

(limited to 'docs/source/quickstart.rst')

diff --git a/docs/source/quickstart.rst b/docs/source/quickstart.rst
index d8d4838..a14358c 100644
--- a/docs/source/quickstart.rst
+++ b/docs/source/quickstart.rst
@@ -83,6 +83,29 @@ properties. It can computed from an already estimated OT matrix with
 Regularized Optimal Transport
 -----------------------------
 
+Recent developments have shown the interest of regularized OT both in terms of
+computational and statistical properties.
+
+We address in this section the regularized OT problem that can be expressed as
+
+.. math::
+    \gamma^* = arg\min_\gamma <\gamma,M>_F + reg*\Omega(\gamma)
+
+    s.t. \gamma 1 = a
+
+            \gamma^T 1= b
+
+            \gamma\geq 0
+where :
+
+- :math:`M\in\mathbb{R}_+^{m\times n}` is the metric cost matrix defining the cost to move mass from bin :math:`a_i` to bin :math:`b_j`.
+- :math:`a` and :math:`b` are histograms (positive, sum to 1) that represent the weights of each samples in the source an target distributions.
+- :math:`\Omega` is the regularization term.
+
+We disvuss in the following specific algorithms 
+
+
+
 Entropic regularized OT
 ^^^^^^^^^^^^^^^^^^^^^^^
 
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