Gaussian Processes and Statistical Decision-making in Non-Euclidean Spaces

TL;DR

This paper advances Gaussian process methods by developing pathwise conditioning and extending models to non-Euclidean spaces, enabling more efficient and broader applications in decision-making contexts.

Contribution

It introduces pathwise conditioning techniques and constructs Gaussian processes on Riemannian manifolds and graphs, expanding their applicability and computational efficiency.

Findings

Pathwise conditioning improves efficiency and deployment.

Constructive covariance kernels for non-Euclidean spaces.

Framework for training Gaussian processes on complex domains.

Abstract

Bayesian learning using Gaussian processes provides a foundational framework for making decisions in a manner that balances what is known with what could be learned by gathering data. In this dissertation, we develop techniques for broadening the applicability of Gaussian processes. This is done in two ways. Firstly, we develop pathwise conditioning techniques for Gaussian processes, which allow one to express posterior random functions as prior random functions plus a dependent update term. We introduce a wide class of efficient approximations built from this viewpoint, which can be randomly sampled once in advance, and evaluated at arbitrary locations without any subsequent stochasticity. This key property improves efficiency and makes it simpler to deploy Gaussian process models in decision-making settings. Secondly, we develop a collection of Gaussian process models over non-Euclidean spaces, including Riemannian manifolds and graphs. We derive fully constructive expressions for the covariance kernels of scalar-valued Gaussian processes on Riemannian manifolds and graphs. Building on these ideas, we describe a formalism for defining vector-valued Gaussian processes on Riemannian manifolds. The introduced techniques allow all of these models to be trained using standard computational methods. In total, these contributions make Gaussian processes easier to work with and allow them to be used within a wider class of domains in an effective and principled manner. This, in turn, makes it possible to potentially apply Gaussian processes to novel decision-making settings.