Dynamic Gaussian Processes and the Vanilla-SPDE Exchange

Gaussian process inference is often limited by cubic computational costs, a challenge that becomes more pronounced in spatio-temporal settings where posterior inference is required over dense grids. While state-space SPDE formulations enable linear complexity in time, exact inference remains cubic in space and deteriorates further when observation locations are disjoint from the prediction locations, which inflates the number of considered spatial points. To address this, we propose the Vanilla-SPDE Exchange, which exploits an equivalence between the standard and SPDE formulations of GP inference to construct a hybrid scheme with improved computational cost. We demonstrate these gains through complexity analysis and numerical experiments.

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

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, \[ (κ^2 - Δ)^{α/2} u = \mathcal{W}, \;\; κ\in \mathbb{R}, \; α\in \mathbb{N}. \] 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 Matérn fields over complete, boundaryless Riemannian manifolds discretized as well-centered simplicial complexes. This convergent method (i) is agnostic to $α, κ$ and thus allows a universal approximation scheme for the precision and covariance matrices of the entire $(α, κ)$-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 Matérn GMRFs and mention a use case: reducing the number of measurements needed to model the GMRF by compressed-sensing.