Approximating Gaussian Whittle-Matérn Fields over Riemannian Manifolds

One low-dimensional subspace contains optimal approximations to an entire two-parameter family of higher order Gaussian Processes over some Riemannian manifolds. Makes an existing fast approach for simulating scalar GMRFs even faster.

Available at arXiv:2606.13827 and cited thus¹.

Overview

Markovian Whittle-Matérn fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family,

of SPDEs^{2 3}. Using recent developements in the analysis of Discrete Exterior Calculus (DEC)^{4 5}, we present a different, yet closely related, convergent GMRF approximation to these Matérn fields over complete boundaryless Riemannian manifolds of any dimension discretized as well-centered simplicial complexes. This convergent method

1. is agnostic to , and thus allows a universal approximation scheme for the precision and covariance matrices of the entire -family of GMRFs, so they may be inferred rather than guessed,

2. inherently models pointwise and piecewise-smoothed measurements of a random field and approximates both equally well,

3. is computationally independent of the interpolants used - it suffers no overhead if one convergent interpolant were replaced with another suitable interpolant over the same mesh.

Furthermore, on discretizations that are well-connected in a precise sense, and volume-concentrated, the precision matrices are approximately spectral functions of a single graph-laplacian. The eigenspaces of such precisions and covariances are thus invariants. By the Eckart-Young theorem, the top -dimensional subspace of the covariance “contains” the best -rank approximation to it. By the polynomial decay of the spectrum, the approximation is poorest when . So guaranteeing an error for guarantees it for all .

One use-case is reducing the number of measurements needed to model the GMRF - compressed sensing.

References

1.

Nambirajan, S. Approximating gaussian whittle-matern fields over well-centered triangulations of riemannian manifolds. (2026).

2.

Lindgren, F., Rue, H. & Lindström, J. An explicit link between gaussian fields and gaussian markov random fields: The stochastic partial differential equation approach. Journal of the Royal Statistical Society Series B: Statistical Methodology 73, 423–498 (2011).

3.

Lindgren, F., Bolin, D. & Rue, H. The SPDE approach for gaussian and non-gaussian fields: 10 years and still running. Spatial Statistics 50, 100599 (2022).

4.

Hirani, A. N. Discrete exterior calculus. (California Institute of Technology, Pasadena, CA, USA, 2003).

5.

Desbrun, M., Hirani, A. N., Leok, M. & Marsden, J. E. Discrete exterior calculus. arXiv preprint cs/0310014, (2003).