Random Field Representations of Kernel Distances

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.