Universality, Characteristic Kernels and RKHS Embedding of Measures

TL;DR

This paper extends the RKHS embedding of probability measures to finite signed measures, establishing that injectivity of the embedding is equivalent to the kernel being universal, thus linking universality and characteristic kernels.

Contribution

It generalizes the RKHS embedding to signed measures and characterizes universal kernels through this embedding, connecting them to characteristic kernels.

Findings

Embedding is injective iff the kernel is universal

Universal kernels are characterized via RKHS embedding of signed measures

Establishes a relation between universal and characteristic kernels

Abstract

A Hilbert space embedding for probability measures has recently been proposed, wherein any probability measure is represented as a mean element in a reproducing kernel Hilbert space (RKHS). Such an embedding has found applications in homogeneity testing, independence testing, dimensionality reduction, etc., with the requirement that the reproducing kernel is characteristic, i.e., the embedding is injective. In this paper, we generalize this embedding to finite signed Borel measures, wherein any finite signed Borel measure is represented as a mean element in an RKHS. We show that the proposed embedding is injective if and only if the kernel is universal. This therefore, provides a novel characterization of universal kernels, which are proposed in the context of achieving the Bayes risk by kernel-based classification/regression algorithms. By exploiting this relation between universality and the embedding of finite signed Borel measures into an RKHS, we establish the relation between universal and characteristic kernels.