  We discuss two new methods of recovery of sparse signals from noisy
observation based on $\ell_1$- minimization. They are closely related to the
well-known techniques such as Lasso and Dantzig Selector. However, 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 also show how these techniques allow to provide
efficiently computable accuracy bounds for Lasso and Dantzig Selector. 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 demonstrate how the estimates can be computed
using the Non-Euclidean Basis Pursuit algorithm.
