doi: 10.3934/dcdss.2021105
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Harmonic analysis of network systems via kernels and their boundary realizations

1. 

Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, USA

2. 

Mathematical Reviews, 416 4th Street, Ann Arbor, MI 48103-4816, USA

* Corresponding author: James Tian

Received  March 2021 Revised  June 2021 Early access September 2021

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.

Citation: Palle Jorgensen, James Tian. Harmonic analysis of network systems via kernels and their boundary realizations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021105
References:
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References:
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D. Alpay and V. Bolotnikov, On tangential interpolation in reproducing kernel Hilbert modules and applications, in Topics in Interpolation Theory (Leipzig, 1994), vol. 95 of Oper. Theory Adv. Appl., Birkhäuser, Basel, 1997, 37–68.  Google Scholar

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D. Alpay and P. Jorgensen, Spectral theory for Gaussian processes: Reproducing kernels, boundaries, and $L^2$-wavelet generators with fractional scales, Numer. Funct. Anal. Optim., 36 (2015), 1239-1285.  doi: 10.1080/01630563.2015.1062777.  Google Scholar

[10]

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D. Alpay and P. E. T. Jorgensen, Stochastic processes induced by singular operators, Numer. Funct. Anal. Optim., 33 (2012), 708-735.  doi: 10.1080/01630563.2012.682132.  Google Scholar

[12]

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[13]

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[14]

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G. ChengX. Hou and C. Liu, The singular integral operator induced by Drury-Arveson kernel, Complex Anal. Oper. Theory, 12 (2018), 917-929.  doi: 10.1007/s11785-016-0537-4.  Google Scholar

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[22]

D. E. Dutkay and P. E. T. Jorgensen, Affine fractals as boundaries and their harmonic analysis, Proc. Amer. Math. Soc., 139 (2011), 3291-3305.  doi: 10.1090/S0002-9939-2011-10752-4.  Google Scholar

[23]

D. E. Dutkay and P. E. T. Jorgensen, Unitary groups and spectral sets, J. Funct. Anal., 268 (2015), 2102-2141.  doi: 10.1016/j.jfa.2015.01.018.  Google Scholar

[24]

W. E and S. Wojtowytsch, Kolmogorov width decay and poor approximators in machine learning: Shallow neural networks, random feature models and neural tangent kernels, Res. Math. Sci., 8 (2021), Paper No. 5, 28 pp. doi: 10.1007/s40687-020-00233-4.  Google Scholar

[25]

M. Geiger, A. Jacot, S. Spigler, F. Gabriel, L. Sagun, S. d'Ascoli, G. Biroli, C. Hongler and M. Wyart, Scaling description of generalization with number of parameters in deep learning, J. Stat. Mech. Theory Exp., (2020), 023401, 23 pp. doi: 10.1088/1742-5468/ab633c.  Google Scholar

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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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