Support Recovery in Compressed Sensing: An Estimation Theoretic Approach

Compressed sensing (CS) deals with the reconstruction of sparse signals from a small number of linear measurements. One of the main challenges in CS is to find the support of a sparse signal from a set of noisy observations. In the CS literature, several information-theoretic bounds on the scaling law of the required number of measurements for exact support recovery have been derived, where the focus is mainly on random measurement matrices. In this paper, we investigate the support recovery problem from an estimation theory point of view, where no specific assumption is made on the underlying measurement matrix. By using the Hammersley-Chapman-Robbins (HCR) bound, we derive a fundamental lower bound on the performance of any unbiased estimator which provides necessary conditions for reliable $\ell_2$-norm support recovery. We then analyze the optimal decoder to provide conditions under which the HCR bound is achievable. This leads to a set of sufficient conditions for reliable $\ell_2$-norm support recovery.