Accuracy guaranties and optimal l1 recovery of sparse signals
Résumé
We discuss new methods for recovery of sparse signals which are based on l1 minimization. Our emphasis is on verifiable conditions on the problem parameters (sensing matrix and sparsity structure) for accurate signal recovery from noisy observations. In particular, we show how the certificates underlying sufficient conditions of exact recovery in the case without noise are used to provide efficiently computable bounds for the recovery error in different models of imperfect observation. These bounds are then optimized with respect to the parameters of the recovery procedures to construct the estimators with improved statistical properties. To justify the proposed approach we provide oracle inequalities which link the properties of the recovery algorithms to the best estimation performance in the Gaussian observation model.