Gaussian processes are an effective model class for learning unknown functions, particularly in settings where accurately representing predictive uncertainty is of key importance. Motivated by applications in the physical sciences, the widely-used Mat\'ern class of Gaussian processes has recently been generalized to model functions whose domains are Riemannian manifolds, by re-expressing said processes as solutions of stochastic partial differential equations. In this work, we propose techniques for computing the kernels of these processes on compact Riemannian manifolds via spectral theory of the Laplace-Beltrami operator in a fully constructive manner, thereby allowing them to be trained via standard scalable techniques such as inducing point methods. We also extend the generalization from the Mat\'ern to the widely-used squared exponential Gaussian process. By allowing Riemannian Mat\'ern Gaussian processes to be trained using well-understood techniques, our work enables their use in mini-batch, online, and non-conjugate settings, and makes them more accessible to machine learning practitioners.

翻译：高斯过程是学习未知函数的有效模型类，尤其在需要精确表征预测不确定性的场景中具有关键作用。受物理科学应用驱动，通过将广泛使用的Matérn类高斯过程重新表述为随机偏微分方程的解，学界近期已将其推广至定义域为黎曼流形的函数建模。本文提出一种通过拉普拉斯-贝尔特拉米算子的谱理论，以完全构造性方式计算紧致黎曼流形上这些过程核函数的方法，从而允许通过标准可扩展技术（如诱导点方法）对其进行训练。我们还将该推广从Matérn过程扩展至广泛使用的平方指数高斯过程。通过使黎曼Matérn高斯过程能够采用成熟技术进行训练，本工作为小批量、在线及非共轭设置下的应用铺平道路，并显著提升了机器学习从业者对该类方法的可及性。