Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels

TL;DR

This paper introduces gauge independent kernels for vector-valued Gaussian processes on Riemannian manifolds, enabling their use in physical sciences and extending Gaussian process models to non-Euclidean domains.

Contribution

It presents a general method to construct gauge independent kernels for vector fields on Riemannian manifolds, facilitating their training with standard Gaussian process techniques.

Findings

Developed gauge independent kernels for vector-valued Gaussian processes.

Extended variational inference to Riemannian vector fields.

Enabled practical training of Gaussian vector fields on manifolds.

Abstract

Gaussian processes are machine learning models capable of learning unknown functions in a way that represents uncertainty, thereby facilitating construction of optimal decision-making systems. Motivated by a desire to deploy Gaussian processes in novel areas of science, a rapidly-growing line of research has focused on constructively extending these models to handle non-Euclidean domains, including Riemannian manifolds, such as spheres and tori. We propose techniques that generalize this class to model vector fields on Riemannian manifolds, which are important in a number of application areas in the physical sciences. To do so, we present a general recipe for constructing gauge independent kernels, which induce Gaussian vector fields, i.e. vector-valued Gaussian processes coherent with geometry, from scalar-valued Riemannian kernels. We extend standard Gaussian process training methods, such as variational inference, to this setting. This enables vector-valued Gaussian processes on Riemannian manifolds to be trained using standard methods and makes them accessible to machine learning practitioners.