Approximations of the Reproducing Kernel Hilbert Space (RKHS) Embedding Method over Manifolds

Abstract

The reproducing kernel Hilbert space (RKHS) embedding method is a recently introduced estimation approach that seeks to identify the unknown or uncertain function in the governing equations of a nonlinear set of ordinary differential equations (ODEs). While the original state estimate evolves in Euclidean space, the function estimate is constructed in an infinite dimensional RKHS and must be approximated in practice. When a finite dimensional approximation is constructed using a basis defined in terms of shifted kernel functions centered at the observations along a trajectory, the RKHS embedding method can be understood as a data-driven approach. This paper derives sufficient conditions that ensure that approximations of the unknown function converge in a Sobolev norm over a submanifold that supports the dynamics. Moreover, the rate of convergence for the finite dimensional approximations is derived in terms of the fill distance of the samples in the embedded manifold. A numerical simulation of an example problem is carried out to illustrate the qualitative nature of convergence results derived in the paper.