Injectivity and weak*-to-weak continuity suffice for convergence rates in $\ell^1$-regularization
Provides theoretical convergence rate guarantees for ℓ1-regularization in inverse problems, simplifying assumptions needed for rate proofs.
The paper proves that ℓ1-regularization for linear ill-posed equations achieves an O(δ) convergence rate for sparse solutions under injectivity and weak*-to-weak continuity, and also provides rates for non-sparse solutions, showing that certain source conditions are automatically satisfied.
We show that the convergence rate of $\ell^1$-regularization for linear ill-posed equations is always $O(δ)$ 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.