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Posterior Contraction Rates for Matérn Gaussian Processes on Riemannian Manifolds
Paul Rosa · Slava Borovitskiy · Alexander Terenin · Judith Rousseau

Wed Dec 13 03:00 PM -- 05:00 PM (PST) @ Great Hall & Hall B1+B2 #1301
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 Matérn 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.

Author Information

Paul Rosa (University of Oxford)
Paul Rosa

I am currently engaged in research focused on the theory and computation of Bayesian nonparametric methods applied to data generating distributions supported on non-Euclidean spaces, specifically graphs or manifolds.

Slava Borovitskiy (ETH Zürich)
Alexander Terenin (Cornell University)
Alexander Terenin

Alexander Terenin is an Assistant Research Professor at Cornell. He is interested in machine learning, particularly for problems where the data is not fixed, but is gathered interactively by the learning machine. His work focuses on data-efficient interactive decision-making algorithms such as Bayesian optimization, and uncertainty-aware probabilistic models that power such algorithms, including Gaussian processes. His technical contributions to this area have won multiple best-paper-type awards at top machine learning conferences.

Judith Rousseau (University of Oxford)

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