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

*Palle Jorgensen, Feng Tian*; 16(Dec):3079−3114, 2015.

### Abstract

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.

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