Optimal Mass Transport (OMT) distance
Wasserstein distance is a metric for distributions derived from optimal mass transport. The formulation of OMT due to Monge and Kantorovich [3,4] may be expressed as follows:
where
denotes the set of all the couplings between
and
(joint distributions whose two marginal distributions are
and
;
is the cost of moving unit mass from x to y. The optimal
gives a transport plan from
to
. And the optimal value of the object function is called Wasserstein distance.
See Jupyter notebook demonstrating OMT distance between two dose distributions, computed using CERRx.
- Jean-David Benamou and Yann Brenier. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numerische Mathematik, 84(3):375{393, 2000.
- Lenaic Chizat, Gabriel Peyre, Bernhard Schmitzer, and Francois-Xavier Vialard. An interpolating distance between optimal transport and Fisher-Rao metrics. Foundations of Computational Mathematics, 10:1{44, 2016.
- Cedric Villani. Topics in Optimal Transportation. American Mathematical Soc., 2003.
- Cedric Villani. Optimal Transport: Old and New, volume 338. Springer Science & Business Media, 2008.