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

摘要

The stochastic partial differential equation approach to Gaussian processes (GPs) represents Matern 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.