Vector-Valued Gaussian Processes and their Kernels on a Class of Metric Graphs

TL;DR

This paper develops a theoretical framework for constructing vector-valued Gaussian process kernels on metric graphs, especially Euclidean trees, ensuring positive semidefiniteness and introducing new classes of covariance functions.

Contribution

It introduces conditions for matrix-valued functions to form valid kernels on metric graphs, focusing on Euclidean trees and the $ ext{l}_1$ metric, filling a gap in multivariate Gaussian process literature.

Findings

Characterization of kernels on Euclidean trees with finite leaves

Conditions for matrix-valued functions to be positive semidefinite

Introduction of compactly supported covariance functions

Abstract

Despite the increasing importance of stochastic processes on linear networks and graphs, current literature on multivariate (vector-valued) Gaussian random fields on metric graphs is elusive. This paper challenges several aspects related to the construction of proper matrix-valued kernels structures. We start by considering matrix-valued metrics that can be composed with scalar- or matrix-valued functions to implement valid kernels associated with vector-valued Gaussian fields. We then provide conditions for certain classes of matrix-valued functions to be composed with the univariate resistance metric and ensure positive semidefiniteness. Special attention is then devoted to Euclidean trees, where a substantial effort is required given the absence of literature related to multivariate kernels depending on the $\ell_1$ metric. Hence, we provide a foundational contribution to certain classes of matrix-valued positive semidefinite functions depending on the $\ell_1$ metric. This fact is then used to characterise kernels on Euclidean trees with a finite number of leaves. Amongst those, we provide classes of matrix-valued covariance functions that are compactly supported.