Sharp local sparsity of regularized optimal transport

TL;DR

This paper investigates the local sparsity and convergence rates of entropy-regularized optimal transport solutions, providing sharp geometric and potential convergence results in multivariate settings.

Contribution

It establishes sharp local support size estimates and convergence rates for regularized optimal transport potentials, extending previous results to multivariate and non-self transport cases.

Findings

Supports behave like balls of radius ε^{1/(d(p-1)+2)}

Regularized potentials are uniformly strongly convex

Derived convergence rates of potentials to unregularized limits

Abstract

In recent years, the use of entropy-regularized optimal transport with $L^p$-type entropies has become increasingly popular. In this setting, the solutions are sparse, in the sense that the support of the regularized optimal coupling, $\mathrm{supp}(\pi_\varepsilon)$, shrinks to the support of the original optimal transport problem as $\varepsilon \to 0$. The main open question concerns the rate of this convergence. In this paper, we obtain sharp local results away from the boundary. We prove that the supports $\mathrm{supp}(\pi_\varepsilon(\cdot \mid x))$ of the conditional measures, $\pi_\varepsilon(\cdot \mid x)$, behave like balls of radius $\varepsilon^\frac 1 {d(p-1)+2}$. This allows us to show that the regularized potentials are uniformly strongly convex and to derive the rate of convergence of these potentials toward their unregularized limit. Our results generalize the results of (Gonz\'alez-Sanz and Nutz, SIAM J.~Math.~Anal.) and (Wiesel and Xu, Ibid.) to the multivariate case and beyond the case of self-transport.