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author | Rémi Flamary <remi.flamary@gmail.com> | 2019-06-27 14:34:23 +0200 |
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committer | Rémi Flamary <remi.flamary@gmail.com> | 2019-06-27 14:34:23 +0200 |
commit | d20d471a1806bde43c23e67c1f805aa3c8908ec3 (patch) | |
tree | db40ba091643cd6945c087d583bbd8856d8c9767 /docs/source/quickstart.rst | |
parent | 8ae85fd6b3649058da07b16c9ea139864c7f94a1 (diff) |
update part 1
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-rw-r--r-- | docs/source/quickstart.rst | 23 |
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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 ^^^^^^^^^^^^^^^^^^^^^^^ |