Using Sinkhorn in the JKO scheme adds linear diffusion
Aymeric Baradat, Anastasiia Hraivoronska, Filippo Santambrogio
TL;DR
The paper investigates the effect of replacing the quadratic optimal transport cost in the JKO scheme by its entropic regularization $D_ε$ (computed via the Sinkhorn algorithm) at each step. In the joint limit where the time step $τ$ and regularization parameter $ε$ vanish with $ε/τ→α$, the authors prove that the limiting evolution acquires a linear diffusion term, yielding the PDE $\partial_t ρ - \nabla\cdot( ρ(∇V+∇W*ρ) ) = Δ g(ρ) + (α/2) Δ ρ$, with $g(s)=s f'(s)-f(s)$ arising from the internal energy functional. The result unifies the entropic and classical JKO frameworks by showing how entropic regularization induces diffusion in the gradient-flow limit, and it strengthens existing α=0 results by relaxing a prior logarithmic-entropy constraint. The analysis leverages the Benamou–Brenier dynamic formulation of $D_ε$, duality for Schrödinger problems, and novel compactness estimates that control both the density and its entropy, enabling convergence to a well-posed weak (distributional) solution of the limit PDE. Practically, the work explains how entropic regularization used for numerical efficiency (via Sinkhorn) directly translates into a mathematically meaningful diffusion term in the continuum limit, strengthening the link between approximate OT and Wasserstein gradient flows.
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 Schrödinger 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 $ε$ is the regularization parameter of the Schrödinger problem, and $τ$ is the time step parameter, considering the limit $τ,ε\to 0$ with $\fracετ \to α\in \mathbb{R}_+$ results in adding the term $\fracα{2} Δρ$ on the right-hand side of the limiting PDE. In the case $α= 0$ we improve a previous result by Carlier, Duval, Peyr{é} and Schmitzer (2017).