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Harmonic analysis of network systems via kernels and their boundary realizations

  • * Corresponding author: James Tian

    * Corresponding author: James Tian
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  • 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.

    Mathematics Subject Classification: Primary: 46N30, 46N50, 42C15; Secondary: 46N20, 31A15, 81S25.


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  • Figure 1.  Current flows in a connected resistance network

    Figure 2.  Transition probabilities $ p_{xy} $ at a vertex $ x $ $ \left(\mbox{in }V\right) $

    Figure 3.  $ v_{x}\left(\cdot\right) = \cdot\wedge x $

    Figure 4.  ${}^{1}\!\!\diagup\!\!{}_{4}\; $-Cantor set

    Figure 5.  A Swiss role

    Figure 6.  SVM using Gaussian kernel

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