Convergence of Empirical Optimal Transport in Unbounded Settings

TL;DR

This paper extends the understanding of how quickly empirical optimal transport costs converge to their true values from compact to unbounded spaces, using a decomposition approach under broad moment conditions.

Contribution

It introduces a decomposition-based method to generalize convergence rates of empirical optimal transport to unbounded settings with minimal assumptions.

Findings

Convergence rates similar to compact spaces are achieved in unbounded settings.

Properties like adaptation to lower complexity carry over to unbounded cases.

Results hold under generic, sharp moment assumptions.

Abstract

In compact settings, the convergence rate of the empirical optimal transport cost to its population value is well understood for a wide class of spaces and cost functions. In unbounded settings, however, hitherto available results require strong assumptions on the ground costs and the concentration of the involved measures. In this work, we pursue a decomposition-based approach to generalize the convergence rates found in compact spaces to unbounded settings under generic moment assumptions that are sharp up to an arbitrarily small $\epsilon > 0$. Hallmark properties of empirical optimal transport on compact spaces, like the recently established adaptation to lower complexity, are shown to carry over to the unbounded case.