Accuracy Guarantees for $\ell_1$-Recovery

We discuss two new methods of recovery of sparse signals from noisy observation based on
ℓ
₁
-minimization. While they are closely related to the well-known techniques such as Lasso and Dantzig Selector, these estimators come with efficiently verifiable guaranties of performance. By optimizing these bounds with respect to the method parameters we are able to construct the estimators which possess better statistical properties than the commonly used ones. We link our performance estimations to the well known results of Compressive Sensing and justify our proposed approach with an oracle inequality which links the properties of the recovery algorithms and the best estimation performance when the signal support is known. We also show how the estimates can be computed using the Non-Euclidean Basis Pursuit algorithm.