A uniform rate of convergence for the entropic potentials in the quadratic Euclidean setting

TL;DR

This paper establishes a uniform convergence rate for entropic potentials and their gradients towards Brenier potentials in the quadratic Euclidean setting, under certain convexity assumptions on measures.

Contribution

It provides the first explicit uniform convergence rate bounds for entropic potentials and their gradients in the quadratic Euclidean setting.

Findings

Uniform convergence rate bounds derived for entropic potentials.

Results applicable to absolutely continuous measures with convexity conditions.

Convergence established for both potentials and their gradients.

Abstract

We bound the rate of uniform convergence in compact sets for both entropic potentials and their gradients towards the Brenier potential and its gradient, respectively. Both results hold in the quadratic Euclidean setting for absolutely continuous measures satisfying some convexity assumptions.