Nonuniform sampling, reproducing kernels, and the associated Hilbert spaces

TL;DR

This paper develops tools and algorithms for sampling and reconstructing functions in reproducing kernel Hilbert spaces, with applications to networks, graph Laplacians, and Gaussian fields, based on nonuniform sampling strategies.

Contribution

It introduces a framework for sampling in RKHSs, providing conditions for finite norm configurations and analyzing induced kernels with applications to various systems.

Findings

Established necessary and sufficient conditions for sampling configurations.

Analyzed the properties of induced positive definite kernels.

Applied the theory to networks, graph Laplacians, and Gaussian fields.

Abstract

In a general context of positive definite kernels $k$, we develop tools and algorithms for sampling in reproducing kernel Hilbert space $\mathscr{H}$ (RKHS). With reference to these RKHSs, our results allow inference from samples; more precisely, reconstruction of an "entire" (or global) signal, a function $f$ from $\mathscr{H}$, via generalized interpolation of $f$ from partial information obtained from carefully chosen distributions of sample points. We give necessary and sufficient conditions for configurations of point-masses $\delta_{x}$ of sample-points $x$ to have finite norm relative to the particular RKHS $\mathscr{H}$ considered. When this is the case, and the kernel $k$ is given, we obtain an induced positive definite kernel $\left\langle \delta_{x},\delta_{y}\right\rangle _{\mathscr{H}}$. We perform a comparison, and we study when this induced positive definite kernel has $l^{2}$ rows and columns. The latter task is accomplished with the use of certain symmetric pairs of operators in the two Hilbert spaces, $l^{2}$ on one side, and the RKHS $\mathscr{H}$ on the other. A number of applications are given, including to infinite network systems, to graph Laplacians, to resistance metrics, and to sampling of Gaussian fields.