Sparse Solutions for Inverse Problems in Reproducing Kernel Hilbert Spaces

TL;DR

This paper explores the H-HK formulation in reproducing kernel Hilbert spaces, connecting it with sparse series representations like POAFD, and demonstrates its theoretical guarantees and practical efficiency for inverse problems.

Contribution

It establishes a connection between the H-HK formulation and sparse series representations, particularly POAFD, with theoretical guarantees and practical benefits.

Findings

The H-HK formulation facilitates solving inverse problems using basis methods.

POAFD provides fast, convergent numerical solutions with guaranteed optimality.

The approach links traditional analysis with modern sparse representation techniques.

Abstract

A linear operator in a Hilbert space defined through the form of Riesz representation naturally introduces a reproducing kernel Hilbert space structure over the range space. Such formulation, called H-HK formulation in this paper, possesses a built-in mechanism to solve some basic type problems in the formulation by using the basis method, that include identification of the range space, the inversion problem, and the Moore-Penrose pseudo- (generalized) inversion problem. After a quick survey of the existing theory, the aim of the article is to establish connection between this formulation with sparse series representation, and in particular with one called pre-orthogonal adaptive Fourier decomposition (POAFD), the latter being one, most recent and well developed, with great efficiency and wide and deep connections with traditional analysis. Within the matching pursuit methodology the optimality of POAFD is theoretically guaranteed. In practice POAFD offers fast converging numerical solutions.