  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.
