Kernel methods for regression in continuous time over subsets and manifolds

This paper derives error bounds for regression in continuous time over subsets of certain types of Riemannian manifolds. The regression problem is typically driven by a nonlinear evolution law taking values on the manifold, and it is cast as one of optimal estimation in a reproducing kernel Hilbert space. A new notion of persistency of excitation (PE) is defined for the estimation problem over the manifold, and rates of convergence of the continuous time estimates are derived using the PE condition. We discuss and analyze two approximation methods of the exact regression solution. We then conclude the paper with some numerical simulations that illustrate the qualitative character of the computed function estimates. Examples of function estimates generated over a trajectory of the Lorenz system and based on experimental motion capture data are included.