Kernel Center Adaptation in the Reproducing Kernel Hilbert Space Embedding Method

TL;DR

This paper develops a theoretical framework and algorithms for optimally positioning kernel centers in RKHS embedding methods, improving adaptive estimator performance for systems with specific state-space visitation patterns.

Contribution

It introduces criteria and algorithms for selecting kernel centers in RKHS, enhancing convergence and approximation in adaptive estimation.

Findings

Algorithms based on Voronoi tessellations and Kohonen maps improve kernel center placement.

The methods enhance estimator accuracy in practical system examples.

Theoretical links between kernel center location and estimation performance are established.

Abstract

The performance of adaptive estimators that employ embedding in reproducing kernel Hilbert spaces (RKHS) depends on the choice of the location of basis kernel centers. Parameter convergence and error approximation rates depend on where and how the kernel centers are distributed in the state-space. In this paper, we develop the theory that relates parameter convergence and approximation rates to the position of kernel centers. We develop criteria for choosing kernel centers in a specific class of systems - ones in which the state trajectory regularly visits the neighborhood of the positive limit set. Two algorithms, based on centroidal Voronoi tessellations and Kohonen self-organizing maps, are derived to choose kernel centers in the RKHS embedding method. Finally, we implement these methods on two practical examples and test their effectiveness.