On concentration of the empirical measure for radial transport costs

TL;DR

This paper establishes a concentration inequality for the empirical measure's optimal transport cost with radial costs, extending previous results to more general growth conditions.

Contribution

It generalizes and improves existing concentration bounds for empirical measures under radial transport costs with polynomial and superpolynomial growth.

Findings

Provides a new concentration inequality for radial transport costs.

Extends previous bounds to superpolynomial growth functions.

Uses a novel partitioning approach to derive global estimates.

Abstract

Let $\mu$ be a probability measure on $\mathbb{R}^d$ and $\mu_N$ its empirical measure with sample size $N$. We prove a concentration inequality for the optimal transport cost between $\mu$ and $\mu_N$ for radial cost functions with polynomial local growth, that can have superpolynomial global growth. This result generalizes and improves upon estimates of Fournier and Guillin. The proof combines ideas from empirical process theory with known concentration rates for compactly supported $\mu$. By partitioning $\mathbb{R}^d$ into annuli, we infer a global estimate from local estimates on the annuli and conclude that the global estimate can be expressed as a sum of the local estimate and a mean-deviation probability for which efficient bounds are known.