Mat\'ern Gaussian processes on Riemannian manifolds

TL;DR

This paper develops methods to compute and train Matérn Gaussian processes on Riemannian manifolds, enabling their application in scalable and practical machine learning settings.

Contribution

It introduces spectral techniques for kernel computation and extends to squared exponential processes, making Riemannian Gaussian processes more accessible and scalable.

Findings

Kernel computation via Laplace-Beltrami spectral theory

Extension to squared exponential Gaussian processes

Enabling training with standard scalable methods

Abstract

Gaussian processes are an effective model class for learning unknown functions, particularly in settings where accurately representing predictive uncertainty is of key importance. Motivated by applications in the physical sciences, the widely-used Mat\'ern class of Gaussian processes has recently been generalized to model functions whose domains are Riemannian manifolds, by re-expressing said processes as solutions of stochastic partial differential equations. In this work, we propose techniques for computing the kernels of these processes on compact Riemannian manifolds via spectral theory of the Laplace-Beltrami operator in a fully constructive manner, thereby allowing them to be trained via standard scalable techniques such as inducing point methods. We also extend the generalization from the Mat\'ern to the widely-used squared exponential Gaussian process. By allowing Riemannian Mat\'ern Gaussian processes to be trained using well-understood techniques, our work enables their use in mini-batch, online, and non-conjugate settings, and makes them more accessible to machine learning practitioners.