Optimal sampling points in reproducing kernel Hilbert spaces

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-Loève 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.

Sampling based approximation of linear functionals in reproducing kernel Hilbert spaces

“…This implies for example that one could use (24) with γ i > c ′ i −p for some c ′ , p > 0 (as done e.g. in [21]), and the bound of Corollary 12 would become min…”