Using Sinkhorn in the JKO scheme adds linear diffusion

TL;DR

This paper investigates how replacing the classical optimal transport problem with the Schrödinger problem in the JKO scheme introduces a linear diffusion term in the limit, connecting entropic regularization to PDE behavior.

Contribution

It demonstrates that using Sinkhorn-regularized optimal transport in the JKO scheme adds a linear diffusion term in the limit, extending previous results and clarifying the effect of entropic regularization.

Findings

Adding Sinkhorn regularization introduces a diffusion term in the PDE limit.

The limit PDE includes a Laplacian term scaled by the regularization parameter ratio.

For zero regularization, the result aligns with previous PDE formulations.

Abstract

The JKO scheme is a time-discrete scheme of implicit Euler type that allows to construct weak solutions of evolution PDEs which have a Wasserstein gradient structure. The purpose of this work is to study the effect of replacing the classical quadratic optimal transport problem by the Schr\"odinger problem (\emph{a.k.a.}\ the entropic regularization of optimal transport, efficiently computed by the Sinkhorn algorithm) at each step of this scheme. We find that if $\epsilon$ is the regularization parameter of the Schr\"odinger problem, and $\tau$ is the time step parameter, considering the limit $\tau,\epsilon \to 0$ with $\frac{\epsilon}{\tau} \to \alpha \in \mathbb{R}_+$ results in adding the term $\frac{\alpha}{2} \Delta \rho$ on the right-hand side of the limiting PDE. In the case $\alpha = 0$ we improve a previous result by Carlier, Duval, Peyr{\'e} and Schmitzer (2017).