Stein kernels and moment maps

TL;DR

This paper introduces a novel method for constructing Stein kernels via moment maps, linking geometric solutions to the Monge-Ampère equation with convergence rates in the central limit theorem, especially for log-concave measures.

Contribution

It presents a new approach to Stein kernels using moment maps and establishes explicit convergence rates in Wasserstein distance for log-concave distributions.

Findings

Regularity bounds on moment maps control convergence rates.

Derived new polynomial dimension-dependent rates in Wasserstein distance.

Applied to log-concave measures for improved CLT convergence estimates.

Abstract

We describe a construction of Stein kernels using moment maps, which are solutions to a variant of the Monge-Amp\`ere equation. As a consequence, we show how regularity bounds on these maps control the rate of convergence in the classical central limit theorem, and derive new rates in Kantorovitch-Wasserstein distance in the log-concave situation, with explicit polynomial dependence on the dimension.