Entropic Optimal Transport: Geometry and Large Deviations

TL;DR

This paper analyzes how entropically regularized optimal transport solutions converge to classical optimal transport, providing a large deviations principle that quantifies the exponential convergence rate as regularization diminishes.

Contribution

It establishes a large deviations principle for the convergence of entropic optimal transport to classical optimal transport, linking the rate function to the Kantorovich potential.

Findings

Derived the large deviations rate function for entropic optimal transport

Linked the convergence rate to the geometry of optimizers and Kantorovich potential

Applicable to Schrödinger bridge formulations

Abstract

We study the convergence of entropically regularized optimal transport to optimal transport. The main result is concerned with the convergence of the associated optimizers and takes the form of a large deviations principle quantifying the local exponential convergence rate as the regularization parameter vanishes. The exact rate function is determined in a general setting and linked to the Kantorovich potential of optimal transport. Our arguments are based on the geometry of the optimizers and inspired by the use of $c$-cyclical monotonicity in classical transport theory. The results can also be phrased in terms of Schr\"odinger bridges.