A Scalable Method to Exploit Screening in Gaussian Process Models with Noise

TL;DR

This paper introduces a scalable EM-based method to better exploit the screening effect in Gaussian process models with noise, improving approximation accuracy and applicability across various methods.

Contribution

The paper presents a novel EM algorithm approach that enhances the exploitation of the screening effect in noisy Gaussian process models, applicable to multiple approximation techniques.

Findings

Improved Gaussian likelihood approximation in noisy settings.

Demonstrates second-order optimization of EM steps in Vecchia's framework.

Method applicable to a wide class of precision matrix-based methods.

Abstract

A common approach to approximating Gaussian log-likelihoods at scale exploits the fact that precision matrices can be well-approximated by sparse matrices in some circumstances. This strategy is motivated by the \emph{screening effect}, which refers to the phenomenon in which the linear prediction of a process $Z$ at a point $\mathbf{x}_0$ depends primarily on measurements nearest to $\mathbf{x}_0$. But simple perturbations, such as i.i.d. measurement noise, can significantly reduce the degree to which this exploitable phenomenon occurs. While strategies to cope with this issue already exist and are certainly improvements over ignoring the problem, in this work we present a new one based on the EM algorithm that offers several advantages. While in this work we focus on the application to Vecchia's approximation (1988), a particularly popular and powerful framework in which we can demonstrate true second-order optimization of M steps, the method can also be applied using entirely matrix-vector products, making it applicable to a very wide class of precision matrix-based approximation methods.