Multivariate Stein Factors for a Class of Strongly Log-concave Distributions

TL;DR

This paper derives uniform bounds on derivatives of Stein equation solutions for strongly log-concave distributions, enabling better analysis of distribution distances and Stein discrepancies using probabilistic coupling methods.

Contribution

It introduces new Stein factor bounds for multivariate strongly log-concave distributions, facilitating improved analysis of distributional distances.

Findings

Established uniform bounds on derivatives of Stein solutions

Provided control over Wasserstein and related distances

Applied probabilistic coupling of Langevin diffusions

Abstract

We establish uniform bounds on the low-order derivatives of Stein equation solutions for a broad class of multivariate, strongly log-concave target distributions. These "Stein factor" bounds deliver control over Wasserstein and related smooth function distances and are well-suited to analyzing the computable Stein discrepancy measures of Gorham and Mackey. Our arguments of proof are probabilistic and feature the synchronous coupling of multiple overdamped Langevin diffusions.