Targeted Separation and Convergence with Kernel Discrepancies

TL;DR

This paper develops new theoretical conditions for kernel-based discrepancy measures like MMDs and KSDs, enabling better separation of distributions and control of convergence, with applications in hypothesis testing and sampling.

Contribution

It introduces necessary and sufficient conditions for kernel-based measures to separate distributions and control weak convergence, including the first KSDs that exactly metrize weak convergence.

Findings

Characterized kernels that separate Bochner embeddable measures.

Established conditions for unbounded and bounded kernels to separate measures and control convergence.

Developed KSDs that exactly metrize weak convergence to the target distribution.

Abstract

Maximum mean discrepancies (MMDs) like the kernel Stein discrepancy (KSD) have grown central to a wide range of applications, including hypothesis testing, sampler selection, distribution approximation, and variational inference. In each setting, these kernel-based discrepancy measures are required to (i) separate a target P from other probability measures or even (ii) control weak convergence to P. In this article we derive new sufficient and necessary conditions to ensure (i) and (ii). For MMDs on separable metric spaces, we characterize those kernels that separate Bochner embeddable measures and introduce simple conditions for separating all measures with unbounded kernels and for controlling convergence with bounded kernels. We use these results on $\mathbb{R}^d$ to substantially broaden the known conditions for KSD separation and convergence control and to develop the first KSDs known to exactly metrize weak convergence to P. Along the way, we highlight the implications of our results for hypothesis testing, measuring and improving sample quality, and sampling with Stein variational gradient descent.