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.

本研究论证了紧致黎曼流形上的内蕴Matérn高斯过程和外蕴过程在收敛速率上相符，通过在多个实例上的实证结果表明内蕴过程在实践中可以获得更好的性能，这加强了对几何高斯过程的数据效率水平进行细粒度分析的必要性。

黎曼流形上 Matérn 高斯过程的后置收缩速率