Radial neighbours for provably accurate scalable approximations of Gaussian processes

Abstract

In geostatistical problems with massive sample size, Gaussian processes can be approximated using sparse directed acyclic graphs to achieve scalable O(n) computational complexity. In these models, data at each location are typically assumed conditionally dependent on a small set of parents that usually include a subset of the nearest neighbours. These methodologies often exhibit excellent empirical performance, but the lack of theoretical validation leads to unclear guidance in specifying the underlying graphical model and sensitivity to graph choice. We address these issues by introducing radial-neighbour Gaussian processes, a class of Gaussian processes based on directed acyclic graphs in which directed edges connect every location to all of its neighbours within a predetermined radius. We prove that any radial-neighbour Gaussian process can accurately approximate the corresponding unrestricted Gaussian process in the Wasserstein-2 distance, with an error rate determined by the approximation radius, the spatial covariance function and the spatial dispersion of samples. We offer further empirical validation of our approach via applications on simulated and real-world data, showing excellent performance in both prior and posterior approximations to the original Gaussian process.