The SPDE Approach to Mat\'ern Fields: Graph Representations

TL;DR

This paper introduces a graph-based SPDE approach for approximating nonstationary Gaussian fields, enabling efficient inference and unifying various models across spatial statistics, machine learning, and inverse problems.

Contribution

It generalizes the Matérn model to unstructured point clouds using graph representations, with theoretical error guarantees and broad interdisciplinary applications.

Findings

Provides approximation error bounds based on spectral convergence

Enables inference and sampling with sparse matrix techniques

Unifies models across spatial statistics, machine learning, and inverse problems

Abstract

This paper investigates Gaussian Markov random field approximations to nonstationary Gaussian fields using graph representations of stochastic partial differential equations. We establish approximation error guarantees building on the theory of spectral convergence of graph Laplacians. The proposed graph representations provide a generalization of the Mat\'ern model to unstructured point clouds, and facilitate inference and sampling using linear algebra methods for sparse matrices. In addition, they bridge and unify several models in Bayesian inverse problems, spatial statistics and graph-based machine learning. We demonstrate through examples in these three disciplines that the unity revealed by graph representations facilitates the exchange of ideas across them.