Beyond convergence rates: Exact recovery with Tikhonov regularization with sparsity constraints

TL;DR

This paper investigates conditions under which Tikhonov regularization with an  penalty can exactly recover sparse solutions in ill-posed problems, providing new theoretical insights and practical criteria for support recovery.

Contribution

It derives novel conditions for exact support recovery applicable to ill-posed problems, extending understanding beyond traditional coherence or RIP-based criteria.

Findings

Conditions for exact support recovery are established for ill-posed problems.

Regularized solutions converge in  and  norms, including as a strict inductive limit.

Application demonstrated with digital holography example.

Abstract

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.