A unified approach to convergence rates for $\ell^1$-regularization and lacking sparsity

TL;DR

This paper introduces a unified condition that ensures convergence rates for -regularization, even when the solutions lack sparsity, extending previous results and including the restricted isometry property.

Contribution

It proposes a new sufficient condition that unifies existing conditions for convergence rates in -regularization with non-sparse solutions, incorporating the restricted isometry property.

Findings

Proposes a third sufficient condition for convergence rates.

Unifies previous conditions for non-sparse solutions.

Includes the restricted isometry property in the analysis.

Abstract

In $\ell^1$-regularization, which is an important tool in signal and image processing, one usually is concerned with signals and images having a sparse representation in some suitable basis, e.g. in a wavelet basis. Many results on convergence and convergence rates of sparse approximate solutions to linear ill-posed problems are known, but rate results for the $\ell^1$-regularization in case of lacking sparsity had not been published until 2013. In the last two years, however, two articles appeared providing sufficient conditions for convergence rates in case of non-sparse but almost sparse solutions. In the present paper, we suggest a third sufficient condition, which unifies the existing two and, by the way, also incorporates the well-known restricted isometry property.