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

## TL;DR

This paper establishes a duality between factorizations of positive definite kernels and Gaussian processes, providing explicit correspondences and applications in various fields like point processes and graph Laplacians.

## Contribution

It introduces a novel duality framework linking kernel factorizations with Gaussian process factorizations, addressing measure-theoretic challenges in infinite dimensions.

## Key findings

- Explicit duality between kernel and Gaussian process factorizations
- Measure-theoretic methods for infinite-dimensional factorizations
- Applications to point processes, graph Laplacians, and boundary-value problems

## 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.