Stationary MMD Points

TL;DR

This paper studies stationary points of the maximum mean discrepancy (MMD) for distribution approximation, showing they can be computed accurately and have super-convergence properties, with practical methods based on MMD gradient flows.

Contribution

It proves stationary MMD points can be computed efficiently and have faster convergence than MMD, introducing gradient flow methods with finite-particle error bounds.

Findings

Stationary MMD points enable accurate distribution approximation.

Super-convergence: integration error vanishes faster than MMD.

Gradient flow methods effectively compute stationary MMD points.

Abstract

Approximation of a target probability distribution using a finite set of points is a problem of fundamental importance in numerical integration. Several authors have proposed to select points by minimising a maximum mean discrepancy (MMD), but the non-convexity of this objective typically precludes global minimisation. Instead, we consider the concept of \emph{stationary points of the MMD} which, in contrast to points globally minimising the MMD, can be accurately computed. Our main contributions are two-fold and theoretical in nature. We first prove the (perhaps surprising) result that, for integrands in the associated reproducing kernel Hilbert space, the numerical integration error of stationary MMD points vanishes \emph{faster} than the MMD. Motivated by this \emph{super-convergence} property, we consider MMD gradient flows as a practical strategy for computing stationary points of the MMD. We then prove that MMD gradient flow can indeed compute stationary MMD points, based on a refined convergence analysis that establishes a novel non-asymptotic finite-particle error bound.