Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels

Gaussian processes are machine learning models capable of learning unknown\nfunctions in a way that represents uncertainty, thereby facilitating\nconstruction of optimal decision-making systems. Motivated by a desire to\ndeploy Gaussian processes in novel areas of science, a rapidly-growing line of\nresearch has focused on constructively extending these models to handle\nnon-Euclidean domains, including Riemannian manifolds, such as spheres and\ntori. We propose techniques that generalize this class to model vector fields\non Riemannian manifolds, which are important in a number of application areas\nin the physical sciences. To do so, we present a general recipe for\nconstructing gauge independent kernels, which induce Gaussian vector fields,\ni.e. vector-valued Gaussian processes coherent with geometry, from\nscalar-valued Riemannian kernels. We extend standard Gaussian process training\nmethods, such as variational inference, to this setting. This enables\nvector-valued Gaussian processes on Riemannian manifolds to be trained using\nstandard methods and makes them accessible to machine learning practitioners.\n