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

arXiv: https://arxiv.org/abs/2004.08000 · HTML: https://mlsanity.com/paper/2004.08000/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Apr 2020.

## Built on by (most cited first, 6 of 6)

- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Non-separable Spatio-temporal Graph Kernels via SPDEs](https://mlsanity.com/paper/2111.08524/) (2021)
- [Optimization on Manifolds via Graph Gaussian Processes](https://mlsanity.com/paper/2210.10962/) (2022)
- [Rates of Convergence for Regression with the Graph Poly-Laplacian](https://mlsanity.com/paper/2209.02305/) (2022)
- [Mathematical Foundations of Graph-Based Bayesian Semi-Supervised Learning](https://mlsanity.com/paper/2207.01093/) (2022)
- [Bayesian Optimization on Networks](https://mlsanity.com/paper/2510.27643/) (2025)

## Abstract (arXiv)

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.