On $\ell^1$-regularization in light of Nashed's ill-posedness concept

TL;DR

This paper advances convergence rate results for $\,\ell^1$-regularization in ill-posed inverse problems, applying variational inequalities and exploring connections with Nashed's ill-posedness types in specific operator contexts.

Contribution

It improves existing convergence rate results for $\,\ell^1$-regularization and relates ill-posedness concepts to operator properties like compactness and strict singularity.

Findings

Enhanced convergence rates for $\,\ell^1$-regularization.

Application to Cesàro operator equation in $\,\ell^2$.

Relationships between ill-posedness types and operator properties.

Abstract

Based on the powerful tool of variational inequalities, in recent papers convergence rates results on $\ell^1$-regularization for ill-posed inverse problems have been formulated in infinite dimensional spaces under the condition that the sparsity assumption slightly fails, but the solution is still in $\ell^1$. In the present paper we improve those convergence rates results and apply them to the Ces\'aro operator equation in $\ell^2$ and to specific denoising problems. Moreover, we formulate in this context relationships between Nashed's types of ill-posedness and mapping properties like compactness and strict singularity.