  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.
