Permutation and Grouping Methods for Sharpening Gaussian Process Approximations

TL;DR

This paper demonstrates that optimizing the ordering and grouping of observations in Gaussian process approximations significantly improves accuracy and reduces computational costs, challenging the assumption that default orderings are optimal.

Contribution

It introduces automatic grouping methods and shows that random orderings can outperform default coordinate-based orderings in Gaussian process likelihood approximations.

Findings

Reordering observations can drastically improve approximation accuracy.

Grouping calculations reduces computational burden and enhances quality.

Random orderings often outperform default coordinate-based orderings.

Abstract

Vecchia's approximate likelihood for Gaussian process parameters depends on how the observations are ordered, which can be viewed as a deficiency because the exact likelihood is permutation-invariant. This article takes the alternative standpoint that the ordering of the observations can be tuned to sharpen the approximations. Advantageously chosen orderings can drastically improve the approximations, and in fact, completely random orderings often produce far more accurate approximations than default coordinate-based orderings do. In addition to the permutation results, automatic methods for grouping calculations of components of the approximation are introduced, having the result of simultaneously improving the quality of the approximation and reducing its computational burden. In common settings, reordering combined with grouping reduces Kullback-Leibler divergence from the target model by a factor of 80 and computation time by a factor of 2 compared to ungrouped approximations with default ordering. The claims are supported by theory and numerical results with comparisons to other approximations, including tapered covariances and stochastic partial differential equation approximations. Computational details are provided, including efficiently finding the orderings and ordered nearest neighbors, and profiling out linear mean parameters and using the approximations for prediction and conditional simulation. An application to space-time satellite data is presented.