Posterior Contraction Rates for Mat\'ern Gaussian Processes on Riemannian Manifolds

TL;DR

This paper establishes optimal theoretical contraction rates for intrinsic and extrinsic Matérn Gaussian processes on Riemannian manifolds, demonstrating their comparable performance and practical advantages in geometric machine learning.

Contribution

It provides the first theoretical analysis of contraction rates for Gaussian processes on manifolds, showing intrinsic models can outperform extrinsic ones in practice.

Findings

Intrinsic and extrinsic processes have matching contraction rates when smoothness is aligned.

Empirical results show intrinsic processes often perform better with small datasets.

Theoretical results bridge geometric Gaussian process theory with practical applications.

Abstract

Gaussian processes are used in many machine learning applications that rely on uncertainty quantification. Recently, computational tools for working with these models in geometric settings, such as when inputs lie on a Riemannian manifold, have been developed. This raises the question: can these intrinsic models be shown theoretically to lead to better performance, compared to simply embedding all relevant quantities into $\mathbb{R}^d$ and using the restriction of an ordinary Euclidean Gaussian process? To study this, we prove optimal contraction rates for intrinsic Mat\'ern Gaussian processes defined on compact Riemannian manifolds. We also prove analogous rates for extrinsic processes using trace and extension theorems between manifold and ambient Sobolev spaces: somewhat surprisingly, the rates obtained turn out to coincide with those of the intrinsic processes, provided that their smoothness parameters are matched appropriately. We illustrate these rates empirically on a number of examples, which, mirroring prior work, show that intrinsic processes can achieve better performance in practice. Therefore, our work shows that finer-grained analyses are needed to distinguish between different levels of data-efficiency of geometric Gaussian processes, particularly in settings which involve small data set sizes and non-asymptotic behavior.