Stein's method for steady-state diffusion approximation in Wasserstein distance

TL;DR

This paper develops a Stein's method-based approach to quantify the Wasserstein distance between the invariant measures of diffusion processes and Markov chain approximations, with applications in machine learning.

Contribution

It introduces a generalized Stein's method framework for steady-state diffusion approximation in Wasserstein distance, applicable to complex stochastic models.

Findings

Provides bounds on Wasserstein distance between measures

Applies method to random walk on k-nearest neighbors graph

Offers quantitative insights for machine learning models

Abstract

We provide a general steady-state diffusion approximation result which bounds the Wasserstein distance between the reversible measure $\mu$ of a diffusion process and the measure $\nu$ of an approximating Markov chain. Our result is obtained thanks to a generalization of a new approach to Stein's method which may be of independent interest. As an application, we study the invariant measure of a random walk on a $k$-nearest neighbors graph, providing a quantitative answer to a problem of interest to the machine learning community.