On Verifiable Sufficient Conditions for Sparse Signal Recovery via l_1 Minimization
A. Juditsky and A. Nemirovski
We propose novel necessary and sufficient conditions for a sensing matrix to be "s-good'' -- to allow for exact l_1recovery of sparse signals with s nonzero entries when no measurement noise is present. Then we express the error bounds for imperfect l_1-recovery (nonzero measurement noise, nearly s-sparse signal, near-optimal solution of the optimization problem yielding the l_1-recovery) in terms of the characteristics underlying these conditions. Further, we demonstrate (and this is the principal result of the paper) that these characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse l_1-recovery and to efficiently computable upper bounds on those s for which a given sensing matrix is s-good. We establish also instructive links between our approach and the basic concepts of the Compressed Sensing theory, like Restricted Isometry or Restricted Eigenvalue properties.