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

TL;DR

This work addresses the construction of vector-valued Gaussian processes on metric graphs by developing matrix-valued kernels driven by a matrix-valued distance on graphs. The authors decompose the field into vertex and edge components, define a PSD matrix-valued distance, and use element-wise composition to obtain valid covariance kernels, with special treatment of Euclidean trees using the Manhattan distance. They provide explicit vertex/edge constructions, establish properties of the resulting matrix-valued covariances, and extend the framework via completely monotone and Bernstein-function mappings, including Shkarofsky-Gneiting-type families. A key contribution is the characterization of matrix-valued kernels on Euclidean trees and the generalisation to manifold graph structures, enabling compactly supported, multivariate covariances and potential space-time extensions for networks.

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.

Figures (5)

• Figure 1: Road-map of the main results in this manuscript. L, P, T and C stand for Lemma, Proposition, Theorem and Corollary, respectively. The suffix indicates that the result is stated in Appendix \ref{['A_proofs']}. The missing results have either no direct connection with the above results (L\ref{['lem:tree']}, P\ref{['prop:explicitWritingMultivMetric']}) or are essential for many others (L\ref{['lem:MpMp_expr']}).
• Figure 2: Left: a graph with Euclidean edges, where the bijections $\varphi_{e_1}$ and $\varphi_{e_2}$ have been highlighted. Right: a Euclidean tree with $5$ leaves where the role of $\ul u$, $\overline{u}$ and $\delta_{e_3}(u)$ have been stressed. Adaptation of filosi2023temporally.
• Figure 3: Marginal and cross covariance functions in terms of the resistance metric, over a tree with $m=4$ leaves, measured from a reference location (black circle).
• Figure 4: Realisation of a positively correlated bivariate random field on a tree with $m=4$ leaves, with a covariance function of Askey type.
• Figure 5: An example of graph for which the distance $D_{11}(v_1,v_3) \not \leq D_{12}(v_1,v_3)$, for $p\geq 3$ and $\alpha$ wisely chosen.

Theorems & Definitions (47)

• Definition 2.1: Graph with Euclidean edges
• Proposition 1
• Proposition 2
• Theorem 2.2
• Proposition 3
• Proposition 4
• Proposition 5
• Corollary 1
• Proposition 6
• Proposition 7