On ℓ 1 -Regularization in Light of Nashed's Ill-Posedness Concept

Source abstract

Abstract Based on the powerful tool of variational inequalities, in recent papers convergence rates results on ℓ 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 ℓ 1 . In the present paper, we improve those convergence rates results and apply them to the Cesáro operator equation in ℓ 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.