Radial Neighbors for Provably Accurate Scalable Approximations of Gaussian Processes

TL;DR

This paper introduces Radial Neighbors Gaussian Processes (RadGP), a scalable approximation method for Gaussian processes that uses a radial neighborhood structure, with theoretical error bounds and empirical validation on real and simulated data.

Contribution

The paper proposes RadGP, a new class of Gaussian processes with provable approximation guarantees based on radial neighborhood graphs, improving scalability and theoretical understanding.

Findings

RadGP achieves accurate approximation in Wasserstein-2 distance.

Empirical results show excellent performance on real and simulated data.

Theoretical bounds relate approximation error to radius and covariance properties.

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 which usually include a subset of the nearest neighbors. 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 neighbors Gaussian processes (RadGP), a class of Gaussian processes based on directed acyclic graphs in which directed edges connect every location to all of its neighbors within a predetermined radius. We prove that any radial neighbors Gaussian process can accurately approximate the corresponding unrestricted Gaussian process in 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.