Sampling with positive definite kernels and an associated dichotomy

TL;DR

This paper characterizes classes of positive definite kernels on general domains that allow for countable discrete sample-sets, with implications for machine learning models based on reproducing kernel Hilbert spaces.

Contribution

It provides a characterization of kernels admitting countable discrete sample-sets, expanding understanding of sampling in reproducing kernel Hilbert spaces.

Findings

Characterization of kernels with countable discrete sample-sets

Applications to specific kernels in machine learning

Conditions for kernels to admit such sample-sets

Abstract

We study classes of reproducing kernels $K$ on general domains; these are kernels which arise commonly in machine learning models; models based on certain families of reproducing kernel Hilbert spaces. They are the positive definite kernels $K$ with the property that there are countable discrete sample-subsets $S$; i.e., proper subsets $S$ having the property that every function in $\mathscr{H}\left(K\right)$ admits an $S$-sample representation. We give a characterizations of kernels which admit such non-trivial countable discrete sample-sets. A number of applications and concrete kernels are given in the second half of the paper.