Finite Element Representations of Gaussian Processes: Balancing Numerical and Statistical Accuracy
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The stochastic partial differential equation approach to Gaussian processes (GPs) represents Matérn GP priors in terms of n finite element basis functions and Gaussian coefficients with 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 pre-asymptotic regime.
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