Tight Sufficient Conditions on Exact Sparsity Pattern Recovery

A noisy underdetermined system of linear equations is considered in which a sparse vector (a vector with a few nonzero elements) is subject to measurement. The measurement matrix elements are drawn from a Gaussian distribution. We study the information-theoretic constraints on exact support recovery of a sparse vector from the measurement vector and matrix. We compute a tight, sufficient condition that is applied to ergodic wide-sense stationary sparse vectors. We compare our results with the existing bounds and recovery conditions. Finally, we extend our results to approximately sparse signals.