Aveiro Method in Reproducing Kernel Hilbert Spaces Under Complete Dictionary

TL;DR

The paper introduces the Aveiro Method Under Complete Dictionary (AMUCD), a novel approach that enhances sparse representation in RKHS by utilizing a complete dictionary of derivatives, improving convergence and applicability.

Contribution

It proposes a new Aveiro Method based on a complete dictionary and matching pursuit, addressing challenges in identifying uniqueness sets and improving convergence in RKHS.

Findings

Efficient expansion of elements in Hardy and Paley-Wiener spaces.

Enables better sparse representations using derivatives of kernels.

Reveals new aspects of Aveiro Method and greedy algorithms.

Abstract

Aveiro Method is a sparse representation method in reproducing kernel Hilbert spaces (RKHS) that gives orthogonal projections in linear combinations of reproducing kernels over uniqueness sets. It, however, suffers from determination of uniqueness sets in the underlying RKHS. In fact, in general spaces, uniqueness sets are not easy to be identified, let alone the convergence speed aspect with Aveiro Method. To avoid those difficulties we propose an anew Aveiro Method based on a dictionary and the matching pursuit idea. What we do, in fact, are more: The new Aveiro method will be in relation to the recently proposed, the so called Pre-Orthogonal Greedy Algorithm (P-OGA) involving completion of a given dictionary. The new method is called Aveiro Method Under Complete Dictionary (AMUCD). The complete dictionary consists of all directional derivatives of the underlying reproducing kernels. We show that, under the boundary vanishing condition, bring available for the classical Hardy and Paley-Wiener spaces, the complete dictionary enables an efficient expansion of any given element in the Hilbert space. The proposed method reveals new and advanced aspects in both the Aveiro Method and the greedy algorithm.