Intrinsic Gaussian Vector Fields on Manifolds
This work addresses the need for uncertainty quantification in vector-valued signals on non-Euclidean domains, such as in robotics or climate science, representing an incremental advance by extending Gaussian process models from scalar to vector fields with intrinsic geometric considerations.
The authors tackled the problem of modeling vector-valued signals on manifolds, which is crucial for applications like wind speed or force field modeling, by proposing intrinsically defined Gaussian process models that account for the geometry of the space, showing that these fields provide more refined inductive biases than previous extrinsic methods.
Various applications ranging from robotics to climate science require modeling signals on non-Euclidean domains, such as the sphere. Gaussian process models on manifolds have recently been proposed for such tasks, in particular when uncertainty quantification is needed. In the manifold setting, vector-valued signals can behave very differently from scalar-valued ones, with much of the progress so far focused on modeling the latter. The former, however, are crucial for many applications, such as modeling wind speeds or force fields of unknown dynamical systems. In this paper, we propose novel Gaussian process models for vector-valued signals on manifolds that are intrinsically defined and account for the geometry of the space in consideration. We provide computational primitives needed to deploy the resulting Hodge-Matérn Gaussian vector fields on the two-dimensional sphere and the hypertori. Further, we highlight two generalization directions: discrete two-dimensional meshes and "ideal" manifolds like hyperspheres, Lie groups, and homogeneous spaces. Finally, we show that our Gaussian vector fields constitute considerably more refined inductive biases than the extrinsic fields proposed before.