CS: On Verifiable Sufficient Conditions for Sparse Signal Recovery via $\ell_1$ Minimization

If you recall, it looked like the ability to check whether a measurement matrix is acceptable or notfor the purpose of recovering a sparse signal through an l_1 minimization method, was pretty bleak. Anatoli Juditsky and Arkadii Nemirovski seem to have found a way out of this conundrum with a new preprint entitled: On Verifiable Sufficient Conditions for Sparse Signal Recovery via $\ell_1$ Minimization. The abstract reads:

We propose novel necessary and sufficient conditions for a sensing matrix to be "$s$-good" -- to allow for exact $\ell_1$-recovery of sparse signals with $s$ nonzero entries when no measurement noise is present. Then we express the error bounds for imperfect $\ell_1$-recovery (nonzero measurement noise, nearly $s$-sparse signal, near-optimal solution of the optimization problem yielding the $\ell_1$-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 $\ell_1$-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.

wow.

Credit: NASA/JPL-Caltech/University of Arizona/Texas A&M, clouds on Mars as seen from Phoenix on sol 111.