With view to applications to harmonic and stochastic analysis of infinite network/graph models, we introduce new tools for realizations and transforms of positive definite kernels (p.d.) $ K $ and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of factorizations: (i) Probabilistic: Starting with a positive definite kernel $ K $ we analyze associated Gaussian processes $ V $. Properties of the Gaussian processes will be derived from certain factorizations of $ K $, arising as a covariance kernel of $ V $. (ii) Geometric analysis: We discuss families of measure spaces arising as boundaries for $ K $. Our results entail an analysis of a partial order on families of p.d. kernels, a duality for operators and frames, optimization, Karhunen–Loève expansions, and factorizations. Applications include a new boundary analysis for the Drury-Arveson kernel, and for certain fractals arising as iterated function systems; and an identification of optimal feature spaces in machine learning models.
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