🤖 AI Summary
Kernelized Stein discrepancy (KSD) suffers from theoretical limitations in controlling weak convergence and precisely separating target distributions. Method: We integrate Bochner embedding theory, Stein’s method, kernel analysis, and weak topology theory to systematically address these limitations. Contribution/Results: First, we establish the necessary and sufficient conditions for KSD to metrize weak convergence—its first rigorous characterization. Second, we construct a novel class of unbounded kernels that are universally discriminative—capable of separating all Borel probability measures—overcoming the inherent discriminability constraints of bounded kernels. Third, we propose the first KSD variant provably equivalent to weak convergence. Our framework significantly enhances KSD’s separation power and convergence control: on ℝᵈ, it enables precise quantitative characterization of weak convergence toward any target distribution P. This advancement strengthens theoretical guarantees and empirical performance in statistical hypothesis testing, sample quality assessment, and Stein variational gradient descent (SVGD) sampling.
📝 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.
