Efficient reduced-rank methods for Gaussian processes with eigenfunction expansions

TL;DR

This paper presents a reduced-rank Gaussian process regression method using eigenfunction expansions, enabling efficient computation of the Karhunen-Loève expansion independent of data, with applications demonstrated in 1D and 2D.

Contribution

It introduces a novel numerical scheme for Gaussian processes that computes the Karhunen-Loève expansion once, without requiring translation invariance, and develops fast algorithms for hyperparameter fitting.

Findings

Efficient computation of Gaussian process regression via eigenfunction expansions.

Numerical experiments demonstrate the method's effectiveness in 1D and 2D.

Extensions to higher dimensions are possible but computationally challenging.

Abstract

In this work we introduce a reduced-rank algorithm for Gaussian process regression. Our numerical scheme converts a Gaussian process on a user-specified interval to its Karhunen-Lo\`eve expansion, the $L^2$-optimal reduced-rank representation. Numerical evaluation of the Karhunen-Lo\`eve expansion is performed once during precomputation and involves computing a numerical eigendecomposition of an integral operator whose kernel is the covariance function of the Gaussian process. The Karhunen-Lo\`eve expansion is independent of observed data and depends only on the covariance kernel and the size of the interval on which the Gaussian process is defined. The scheme of this paper does not require translation invariance of the covariance kernel. We also introduce a class of fast algorithms for Bayesian fitting of hyperparameters, and demonstrate the performance of our algorithms with numerical experiments in one and two dimensions. Extensions to higher dimensions are mathematically straightforward but suffer from the standard curses of high dimensions.