# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1901.07407