Exchangeable optimal transportation and log-concavity

TL;DR

This paper explores exchangeable measures in infinite-dimensional optimal transportation, reducing the problem to Hilbert space and establishing conditions for convergence, with implications for log-concave sequences.

Contribution

It introduces a novel approach to infinite-dimensional optimal transport using de Finetti decomposition and provides new convergence criteria under log-concavity assumptions.

Findings

Finite-dimensional approximations converge to the Monge solution.

Uniformly log-concave exchangeable sequences are i.i.d.

Conditions for convergence are established under log-concavity.

Abstract

We study the Monge and Kantorovich transportation problems on $\mathbb{R}^{\infty}$ within the class of exchangeable measures. With the help of the de Finetti decomposition theorem the problem is reduced to an unconstrained optimal transportation problem on the Hilbert space. We find sufficient conditions for convergence of finite-dimensional approximations to the Monge solution. The result holds, in particular, under certain analytical assumptions involving log-concavity of the target measure. As a by-product we obtain the following result: any uniformly log-concave exchangeable sequence of random variables is i.i.d.