Sufficient Conditions for Parameter Convergence over Embedded Manifolds using Kernel Techniques

TL;DR

This paper establishes sufficient conditions for parameter convergence in adaptive estimation within embedded RKHS on manifolds, extending traditional PE conditions to more complex geometric settings.

Contribution

It introduces PE conditions for embedded RKHS on manifolds and analyzes their implications for finite and infinite-dimensional cases.

Findings

Finite-dimensional RKHS ensures parameter convergence.

Infinite-dimensional RKHS bounds the estimation error.

Practical example demonstrates the effectiveness of the conditions.

Abstract

The persistence of excitation (PE) condition is sufficient to ensure parameter convergence in adaptive estimation problems. Recent results on adaptive estimation in reproducing kernel Hilbert spaces (RKHS) introduce PE conditions for RKHS. This paper presents sufficient conditions for PE for the particular class of uniformly embedded reproducing kernel Hilbert spaces (RKHS) defined over smooth Riemannian manifolds. This paper also studies the implications of the sufficient condition in the case when the RKHS is finite or infinite-dimensional. When the RKHS is finite-dimensional, the sufficient condition implies parameter convergence as in the conventional analysis. On the other hand, when the RKHS is infinite-dimensional, the same condition implies that the function estimate error is ultimately bounded by a constant that depends on the approximation error in the infinite-dimensional RKHS. We illustrate the effectiveness of the sufficient condition in a practical example.