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. arXiv:2606.13827.

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^{1 2}. Using recent developements in the analysis of Discrete Exterior Calculus (DEC)^{3 4}, 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 spectral functions of a graph-laplacian. So the eigenspaces of such precision matrices and covariance matrices are 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 error is worst when . So guaranteeing a good error for suffices. One use-case is reducing the number of measurements needed to model the GMRF - compressed sensing.

References

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