Finite Element Representations of Gaussian Processes: Balancing Numerical and Statistical Accuracy

The stochastic partial differential equation approach to Gaussian processes (GPs) represents Mate ' rn GP priors in terms of n 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 N by setting n N and exploiting sparsity. In this paper we reconsider the standard choice n N 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 n << N in the large N asymptotics. Numerical experiments illustrate the applicability of our theory and the effect of the prior lengthscale in the preasymptotic regime.