Discrete Reproducing Kernel Hilbert Spaces: Sampling and Distribution of Dirac-masses

Palle Jorgensen; Feng Tian

We study reproducing kernels, and associated reproducing kernel Hilbert spaces (RKHSs)

$\mathscr{H}$ over infinite, discrete and countable sets

$V$ . In this setting we analyze in detail the distributions of the corresponding Dirac point-masses of

$V$ . Illustrations include certain models from neural networks: An Extreme Learning Machine (ELM) is a neural network-configuration in which a hidden layer of weights are randomly sampled, and where the object is then to compute resulting output. For RKHSs

$\mathscr{H}$ of functions defined on a prescribed countable infinite discrete set

$V$ , we characterize those which contain the Dirac masses

$\delta_{x}$ for all points

$x$ in

$V$ . Further examples and applications where this question plays an important role are: (i) discrete Brownian motion-Hilbert spaces, i.e., discrete versions of the Cameron-Martin Hilbert space; (ii) energy-Hilbert spaces corresponding to graph-Laplacians where the set

$V$ of vertices is then equipped with a resistance metric; and finally (iii) the study of Gaussian free fields.

Discrete Reproducing Kernel Hilbert Spaces: Sampling and Distribution of Dirac-masses

Abstract