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

2021/09/06 by Sanz-Alonso, Daniel, Yang, Ruiyi · 2 citations
#Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2109.02777

Abstract

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.

Cited by

• Bayesian Optimization on Networks
• Bayesian Inverse Problems on Metric Graphs

Related

• Adaptive Resolution for Finite-Rank Gaussian Processes
• GP-Recipe: Gaussian Process approximation to linear operations in numerical methods
• Gaussian Process Regression under Computational and Epistemic Misspecification
• BdryGP: a new Gaussian process model for incorporating boundary information
• A Bulirsch-Stoer algorithm using Gaussian processes