Finite Element Representations of Gaussian Processes: Balancing Numerical and Statistical Accuracy
Daniel Sanz-Alonso, Ruiyi Yang · SIAM/ASA Journal on Uncertainty Quantification · 2022
Abstract. The stochastic partial differential equation approach to Gaussian processes (GPs) represents Matérn GP priors in terms of [Formula: see text] finite element basis functions and Gaussian coefficients with a sparse precision matrix. Such representations enhance the scalability of GP regression and classification to datasets of large size [Formula: see text] by setting [Formula: see text] and exploiting sparsity. In this paper we reconsider the standard choice [Formula: see text] through an analysis of the estimation performance. Our theory implies that, under certain smoothness assumptions, one can reduce the computation and memory cost without hindering the estimation accuracy by setting [Formula: see text] in the large [Formula: see text] asymptotics. Numerical experiments illustrate the applicability of our theory and the effect of the prior lengthscale in the preasymptotic regime.