Discrete reproducing kernel Hilbert spaces: Sampling and distribution of\n Dirac-masses

Abstract

We study reproducing kernels, and associated reproducing kernel Hilbert\nspaces (RKHSs) $\\mathscr{H}$ over infinite, discrete and countable sets $V$. In\nthis setting we analyze in detail the distributions of the corresponding Dirac\npoint-masses of $V$. Illustrations include certain models from neural networks:\nAn Extreme Learning Machine (ELM) is a neural network-configuration in which a\nhidden layer of weights are randomly sampled, and where the object is then to\ncompute resulting output. For RKHSs $\\mathscr{H}$ of functions defined on a\nprescribed countable infinite discrete set $V$, we characterize those which\ncontain the Dirac masses $\\delta_{x}$ for all points $x$ in $V$. Further\nexamples and applications where this question plays an important role are: (i)\ndiscrete Brownian motion-Hilbert spaces, i.e., discrete versions of the\nCameron-Martin Hilbert space; (ii) energy-Hilbert spaces corresponding to\ngraph-Laplacians where the set $V$ of vertices is then equipped with a\nresistance metric; and finally (iii) the study of Gaussian free fields.\n