Infinitesimal behavior of Quadratically Regularized Optimal Transport and its relation with the Porous Medium Equation

TL;DR

This paper investigates the mathematical effects of quadratic regularization on optimal transport, revealing a limit behavior related to the porous medium equation and confirming a conjecture about the approximation of transport costs.

Contribution

It provides the first rigorous analysis of quadratic regularization in optimal transport, establishing the limit of the cost difference and linking it to the porous medium equation's solutions.

Findings

The scaled difference of costs converges to a nontrivial limit as regularization vanishes.

The Barenblatt--Pattle solution approximates the regularized transport cost for small regularization parameters.

The work confirms a conjecture connecting quadratic regularization with porous medium equation solutions.

Abstract

The quadratically regularized optimal transport problem has recently been considered in various applications where the coupling needs to be \emph{sparse}, i.e., the density of the coupling needs to be zero for a large subset of the product of the supports of the marginals. However, unlike the acclaimed entropy-regularized optimal transport, the effect of quadratic regularization on the transport problem is not well understood from a mathematical standpoint. In this work, we take a first step towards its understanding. We prove that the difference between the cost of optimal transport and its regularized version multiplied by the ratio $\varepsilon^{-\frac{2}{d+2}}$ converges to a nontrivial limit as the regularization parameter $\varepsilon$ tends to 0. The proof confirms a conjecture from Zhang et al. (2023) where it is claimed that a modification of the self-similar solution of the porous medium equation, the Barenblatt--Pattle solution, can be used as an approximate solution of the regularized transport cost for small values of $\varepsilon$.