# Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds

arXiv: https://arxiv.org/abs/2606.13827 · HTML: https://mlsanity.com/paper/2606.13827/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 54 references that are in the map.
- First posted on arXiv 11 Jun 2026.

## Builds on (most cited first, 8 of 8)

- [Simplicial Closure and higher-order link prediction](https://mlsanity.com/paper/1802.06916/) (2018)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [LazySVD: Even Faster SVD Decomposition Yet Without Agonizing Pain](https://mlsanity.com/paper/1607.03463/) (2016)
- [The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running](https://mlsanity.com/paper/2111.01084/) (2021)
- [Intrinsic Gaussian Vector Fields on Manifolds](https://mlsanity.com/paper/2310.18824/) (2023)
- [Exploring a New Class of Non-stationary Spatial Gaussian Random Fields with Varying Local Anisotropy](https://mlsanity.com/paper/1304.6949/) (2013)
- [Accelerated graph-based spectral polynomial filters](https://mlsanity.com/paper/1509.02468/) (2015)
- [Structure-Aware Spectral Sparsification via Uniform Edge Sampling](https://mlsanity.com/paper/2510.12669/) (2025)

## Abstract (arXiv)

Markovian Whittle-Mat'ern 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, [ (kappa2 - Delta)α/2 u = W, ;; kappa ∈ ℝ, ; α ∈ mathbbN. ] of SPDEs. Using recent developements in the analysis of Discrete Exterior Calculus (DEC), we present a different, yet closely related, convergent GMRF approximation to these Mat'ern fields over complete, boundaryless Riemannian manifolds discretized as well-centered simplicial complexes. This convergent method (i) is agnostic to α, kappa and thus allows a universal approximation scheme for the precision and covariance matrices of the entire (α, kappa)-family of GMRFs, so they may be inferred rather than guessed. (ii) inherently models pointwise and piecewise-smoothed measurements of a random field and approximates both equally well (iii) 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, we show that, on discretizations that are well-connected in a precise sense, and volume-concentrated, the precision matrices are spectral functions of a graph-laplacian. We provide a low rank approximator to the family of such Mat'ern GMRFs and mention a use case: reducing the number of measurements needed to model the GMRF by compressed-sensing.