Beyond convergence rates: Exact recovery with Tikhonov regularization with sparsity constraints
Provides theoretical guarantees for exact recovery in ill-posed inverse problems with sparsity constraints, benefiting practitioners in imaging and signal processing.
The paper derives conditions for exact support recovery in Tikhonov regularization with ℓ¹ penalty, applicable to ill-posed problems where standard conditions like coherence or RIP fail. It shows that regularized solutions converge in ℓ⁰ and demonstrates applicability via a digital holography example.
The Tikhonov regularization of linear ill-posed problems with an $\ell^1$ 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 $\ell^1$-norm but also in the vector space $\ell^0$ (when considered as the strict inductive limit of the spaces $\R^n$ 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 Tikhonov regularization with sparsity constraints.