Injectivity and weak*-to-weak continuity suffice for convergence rates in $\ell^1$-regularization

TL;DR

This paper demonstrates that injectivity and weak*-to-weak continuity of operators guarantee linear convergence rates in -regularization for both sparse and non-sparse solutions of ill-posed linear equations.

Contribution

It establishes that injectivity and weak*-to-weak continuity are sufficient conditions for convergence rates in -regularization, simplifying previous source condition requirements.

Findings

Convergence rate is O(elta) for sparse solutions under given conditions.

Convergence rates are proven for non-sparse solutions as well.

Source-type conditions are automatically satisfied under the assumptions.

Abstract

We show that the convergence rate of $\ell^1$-regularization for linear ill-posed equations is always $O(\delta)$ if the exact solution is sparse and if the considered operator is injective and weak*-to-weak continuous. Under the same assumptions convergence rates in case of non-sparse solutions are proven. The results base on the fact that certain source-type conditions used in the literature for proving convergence rates are automatically satisfied.