Random Field Representations of Kernel Distances

TL;DR

This paper introduces a novel perspective on kernel-induced distances between measures using random fields, connecting them to well-known processes like fractional Brownian motion and the Gaussian free field, with practical implications.

Contribution

It provides a new random field framework for understanding kernel distances, extending classical metrics with connections to stochastic processes like fractional Brownian motion.

Findings

Energy distance is induced by fields with dense support and scale invariance.

Fractional Brownian motion induces a generalized metric preserving continuity.

Gaussian free field extends the Cramér-von Mises distance with practical signal-to-noise insights.

Abstract

Positive semi-definite kernels are used to induce pseudo-metrics, or ``distances'', between measures. We write these as an expected quadratic variation of, or expected inner product between, a random field and the difference of measures. This alternate viewpoint offers important intuition and interesting connections to existing forms. Metric distances leading to convenient finite sample estimates are shown to be induced by fields with dense support, stationary increments, and scale invariance. The main example of this is energy distance. We show that the common generalization preserving continuity is induced by fractional Brownian motion. We induce an alternate generalization with the Gaussian free field, formally extending the Cram\'er-von Mises distance. Pathwise properties give intuition about practical aspects of each. This is demonstrated through signal to noise ratio studies.