Performance guarantees of regularized ℓ1−2-minimization for robust sparse recovery

Based on the powerful restricted isometry property (RIP) and the coherence tools, this paper develops two types of robust recovery results for a (non-convex) regularized -minimization model itself, which include some (sufficient) recovery conditions and their resultant recovery error estimates. All these theoretical results show that under some conditions this model is also able to ensure the robust recovery of any signal that is not necessary to be sparse. On the one hand, our RIP-based results not only provide some insights into the regularized -minimization model, but also well complement the previous ones established for the constrained -minimization model. On the other hand, our coherence-based recovery condition is demonstrated to be much better than the state-of-the-art one except for a very small number of cases, which to some degree can be neglected in practice. Furthermore, we also discuss the approximate error resulting from the scenario when the regularized -minimization model is used to replace the classical constrained -minimization model.