**Data-driven optimal transport**

Esteban Tabak, CIMS

Abstract:

Monge's optimal transport problem seeks a function y(x) which maps a given probability distribution into another while minimizing a cost function.

In Kantorovich’s formulation, the problem is relaxed so that, instead of a map, a joint probability distribution or “plan" pi(x,y) is sought that has the given distributions as marginals.
This talk will discuss the frequently occurring situation in which the two distributions are only known through samples {x_i}, {y_j}.

In addition, we will consider more general scenarios where a plan is sought coupling two or more distributions, not necessarily over spaces of equal dimension, with marginals only known through samples.
Applications include density estimation, classification, regression from doubly unlabeled data, resource allocation, risk estimation, aggregation of data from diverse sources and fluid-flow reconstruction from tracers.