Nystr\"om Kernel Stein Discrepancy

TL;DR

This paper introduces a Nyström-based acceleration method for Kernel Stein Discrepancy (KSD) that significantly reduces computational complexity, enabling large-scale goodness-of-fit testing with theoretical guarantees.

Contribution

It proposes a Nyström approximation for KSD, achieving faster runtime while maintaining statistical consistency, and demonstrates its effectiveness in large-scale testing scenarios.

Findings

Nyström-based KSD has runtime $ ext{O}(mn + m^3)$, much faster than quadratic methods.

The proposed method is $ ext{O}(rac{1}{ oot n})$-consistent under sub-Gaussian assumptions.

Empirical tests show the method's applicability for large-scale goodness-of-fit testing.

Abstract

Kernel methods underpin many of the most successful approaches in data science and statistics, and they allow representing probability measures as elements of a reproducing kernel Hilbert space without loss of information. Recently, the kernel Stein discrepancy (KSD), which combines Stein's method with the flexibility of kernel techniques, gained considerable attention. Through the Stein operator, KSD allows the construction of powerful goodness-of-fit tests where it is sufficient to know the target distribution up to a multiplicative constant. However, the typical U- and V-statistic-based KSD estimators suffer from a quadratic runtime complexity, which hinders their application in large-scale settings. In this work, we propose a Nystr\"om-based KSD acceleration -- with runtime $\mathcal O\left(mn+m^3\right)$ for $n$ samples and $m\ll n$ Nystr\"om points -- , show its $\sqrt{n}$-consistency with a classical sub-Gaussian assumption, and demonstrate its applicability for goodness-of-fit testing on a suite of benchmarks. We also show the $\sqrt n$-consistency of the quadratic-time KSD estimator.