Optimal Sampling Points in Reproducing Kernel Hilbert Spaces

TL;DR

This paper investigates the optimal placement of finite sampling points in Reproducing Kernel Hilbert Spaces to improve information extraction efficiency, proposing algorithms and demonstrating their effectiveness through numerical experiments.

Contribution

It introduces a new approach to determine optimal sampling points in RKHS, including a computationally efficient algorithm and theoretical estimates.

Findings

The proposed algorithms effectively identify optimal sampling points.

Numerical experiments validate the algorithms' performance.

Optimal sampling improves information extraction in compressed sensing applications.

Abstract

The recent developments of basis pursuit and compressed sensing seek to extract information from as few samples as possible. In such applications, since the number of samples is restricted, one should deploy the sampling points wisely. We are motivated to study the optimal distribution of finite sampling points. Formulation under the framework of optimal reconstruction yields a minimization problem. In the discrete case, we estimate the distance between the optimal subspace resulting from a general Karhunen-Loeve transform and the kernel space to obtain another algorithm that is computationally favorable. Numerical experiments are then presented to illustrate the performance of the algorithms for the searching of optimal sampling points.