Large-Scale Low-Rank Gaussian Process Prediction with Support Points

TL;DR

This paper investigates the theoretical and practical benefits of using support points as knots in low-rank Gaussian process prediction, demonstrating improved accuracy and efficiency through simulations and real data applications.

Contribution

It provides the first theoretical analysis of how knot selection and covariance estimation affect predictive processes, advocating support points for optimal data representation.

Findings

Support points outperform other knot selection methods in simulations.

Theoretical results confirm the asymptotic optimality of support points.

Real data applications show improved prediction accuracy and efficiency.

Abstract

Low-rank approximation is a popular strategy to tackle the "big n problem" associated with large-scale Gaussian process regressions. Basis functions for developing low-rank structures are crucial and should be carefully specified. Predictive processes simplify the problem by inducing basis functions with a covariance function and a set of knots. The existing literature suggests certain practical implementations of knot selection and covariance estimation; however, theoretical foundations explaining the influence of these two factors on predictive processes are lacking. In this paper, the asymptotic prediction performance of the predictive process and Gaussian process predictions is derived and the impacts of the selected knots and estimated covariance are studied. We suggest the use of support points as knots, which best represent data locations. Extensive simulation studies demonstrate the superiority of support points and verify our theoretical results. Real data of precipitation and ozone are used as examples, and the efficiency of our method over other widely used low-rank approximation methods is verified.