Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions

TL;DR

This paper investigates the properties of kernel mean embeddings, focusing on conditions for injectivity, the types of measures that can be embedded, and how the induced metrics relate to other topologies, unifying and extending existing results.

Contribution

It provides a unified framework for understanding when kernel embeddings are injective, how they can be extended to distributions, and characterizes kernels that induce metrics compatible with weak convergence.

Findings

Characterizes when kernel embeddings are injective.

Extends kernel methods to distributions beyond measures.

Shows that characteristic kernels metrize weak convergence under certain conditions.

Abstract

Kernel mean embeddings have recently attracted the attention of the machine learning community. They map measures $\mu$ from some set $M$ to functions in a reproducing kernel Hilbert space (RKHS) with kernel $k$. The RKHS distance of two mapped measures is a semi-metric $d_k$ over $M$. We study three questions. (I) For a given kernel, what sets $M$ can be embedded? (II) When is the embedding injective over $M$ (in which case $d_k$ is a metric)? (III) How does the $d_k$-induced topology compare to other topologies on $M$? The existing machine learning literature has addressed these questions in cases where $M$ is (a subset of) the finite regular Borel measures. We unify, improve and generalise those results. Our approach naturally leads to continuous and possibly even injective embeddings of (Schwartz-) distributions, i.e., generalised measures, but the reader is free to focus on measures only. In particular, we systemise and extend various (partly known) equivalences between different notions of universal, characteristic and strictly positive definite kernels, and show that on an underlying locally compact Hausdorff space, $d_k$ metrises the weak convergence of probability measures if and only if $k$ is continuous and characteristic.