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Decomposition of Gaussian processes, and factorization of positive definite kernels
Journal article   Open access   Peer reviewed

Decomposition of Gaussian processes, and factorization of positive definite kernels

Palle Jorgensen and Feng Tian
Rocznik Akademii Górniczo-Hutniczej im. Stanisława Staszica. Opuscula Mathematica, Vol.39(4), pp.497-541
01/01/2019
DOI: 10.7494/OpMath.2019.39.4.497
url
https://doi.org/10.7494/OpMath.2019.39.4.497View
Published (Version of record) Open Access

Abstract

We establish a duality for two factorization questions, one for general positive definite (p.d.) kernels \(K\), and the other for Gaussian processes, say \(V\). The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel \(K\), presented as a covariance kernel for a Gaussian process \(V\). We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel \(K\), vs for Gaussian process \(V\). Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for \(K\) is the exact same as that which yield factorizations for \(V\). Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems.
 analysis/synthesis covariance feature space frames gaussian free fields generalized ito-integration interpolation non-uniform sampling optimization reproducing kernel hilbert space the measurable category transform

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