Beyond convergence rates: exact recovery with the Tikhonov regularization with sparsity constraints

Abstract

The Tikhonov regularization of linear ill-posed problems with an ℓ1 penalty is considered. We recall results for linear convergence rates and results on exact recovery of the support. Moreover, we derive conditions for exact support recovery which are especially applicable in the case of ill-posed problems, where other conditions, e.g., based on the so-called coherence or the restricted isometry property are usually not applicable. The obtained results also show that the regularized solutions do not only converge in the ℓ1-norm but also in the vector space ℓ0 (when considered as the strict inductive limit of the spaces as n tends to infinity). Additionally, the relations between different conditions for exact support recovery and linear convergence rates are investigated. With an imaging example from digital holography the applicability of the obtained results is illustrated, i.e. that one may check a priori if the experimental setup guarantees exact recovery with the Tikhonov regularization with sparsity constraints.