Opuscula Math. 39, no. 4 (2019), 497-541
https://doi.org/10.7494/OpMath.2019.39.4.497

Opuscula Mathematica

# Decomposition of Gaussian processes, and factorization of positive definite kernels

Palle Jorgensen
Feng Tian

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.

Keywords: reproducing kernel Hilbert space, frames, generalized Ito-integration, the measurable category, analysis/synthesis, interpolation, Gaussian free fields, non-uniform sampling, optimization, transform, covariance, feature space.

Mathematics Subject Classification: 47L60, 46N30, 46N50, 42C15, 65R10, 05C50, 05C75, 31C20, 60J20.

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• Communicated by P.A. Cojuhari.
• Revised: 2019-02-24.
• Accepted: 2019-02-25.
• Published online: 2019-05-23.