Sampling with positive definite kernels and an associated dichotomy

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 H (K) 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.