Kernel Methods for Regression in Continuous Time over Subsets and Manifolds

TL;DR

This paper develops error bounds and convergence analysis for kernel-based regression in continuous time on Riemannian manifolds, introducing a new PE notion and illustrating results with simulations and motion capture data.

Contribution

It introduces a novel PE concept for manifold regression and derives convergence rates, expanding kernel methods to nonlinear manifold-valued data in continuous time.

Findings

Derived error bounds for manifold regression.

Established convergence rates using PE condition.

Validated methods with Lorenz system and motion capture data.

Abstract

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 (RKHS). 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. Numerical results from function estimates generated over a trajectory of the Lorenz system are presented. Additionally, we analyze an implementation of the two approximation methods using motion capture data.