Smoothing operators in multi-marginal Optimal Transport

TL;DR

This paper introduces a universal smoothing technique for multi-marginal optimal transport plans that preserves marginals and provides control over energy and continuity, enhancing regularity properties.

Contribution

The authors develop a method to approximate transport plans with Sobolev regular plans while maintaining marginals and controlling energy and continuity.

Findings

Approximation of transport plans with Sobolev regularity

Sharp energy control of approximations

Continuity properties of the smoothing family

Abstract

Given $N$ absolutely continuous probabilities $\rho_1, \dotsc, \rho_N$ over $\mathbb{R}^d$ which have Sobolev regularity, and given a transport plan $P$ with marginals $\rho_1, \dotsc, \rho_N$, we provide a universal technique to approximate $P$ with Sobolev regular transport plans with the same marginals. Moreover, we prove a sharp control of the energy and some continuity property of the approximating family.