Derivative and curvature processes of Gaussian processes on compact manifolds

Music City Center 

In this talk, we introduce and formalize the concepts of derivative and curvature processes of Gaussian processes on compact Riemannian manifolds, both identified as Gaussian processes themselves. The derivation of covariance functions for these derivative and curvature processes forms a substantial part of our work, together with establishing the joint distribution among the original process, the derivative process, and the curvature process, which allows for principled statistical inference. Our examination finds its practicality in the case study of spheres of arbitrary dimensions. By scrutinizing the smoothness of 13 extant kernels on spheres, we derive necessary and sufficient conditions for a kernel to be continuous or differentiable on this manifold. Additionally, we determine the covariance structure of the process, its derivative, and its curvature in this setting. Empirical evidence obtained through simulations lends substantial weight to our theoretical findings. By extending the understanding of Gaussian processes on Riemannian manifolds, this study unlocks a variety of potential applications in machine learning and statistics where Gaussian processes are used.

Derivative and curvature processes

Gaussian processes

Manifold learning