Convergence rates in $\mathbf{\ell^1}$-regularization if the sparsity assumption fails

TL;DR

This paper analyzes the convergence rates of $ ext{l}^1$-regularization for ill-posed linear problems when solutions are not fully sparse but have rapidly decaying nonzero components, using variational inequalities.

Contribution

It introduces a novel convergence analysis for solutions with decaying nonzero parts, extending the applicability of variational inequalities beyond standard source conditions.

Findings

Provides the first examples where variational inequalities outperform source conditions.

Shows convergence rates for solutions with fast decaying nonzero components.

Extends regularization theory to less sparse solutions.

Abstract

Variational sparsity regularization based on $\ell^1$-norms and other nonlinear functionals has gained enormous attention recently, both with respect to its applications and its mathematical analysis. A focus in regularization theory has been to develop error estimation in terms of regularization parameter and noise strength. For this sake specific error measures such as Bregman distances and specific conditions on the solution such as source conditions or variational inequalities have been developed and used. In this paper we provide, for a certain class of ill-posed linear operator equations, a convergence analysis that works for solutions that are not completely sparse, but have a fast decaying nonzero part. This case is not covered by standard source conditions, but surprisingly can be treated with an appropriate variational inequality. As a consequence the paper also provides the first examples where the variational inequality approach, which was often believed to be equivalent to appropriate source conditions, can indeed go farther than the latter.