Information-Theoretic Limits on Sparse Signal Recovery: Dense versus Sparse Measurement Matrices

We study the information-theoretic limits of exactly recovering the support set of a sparse signal, using noisy projections defined by various classes of measurement matrices. Our analysis is high-dimensional in nature, in which the number of observations n, the ambient signal dimension p, and the signal sparsity k are all allowed to tend to infinity in a general manner. This paper makes two novel contributions. First, we provide sharper necessary conditions for exact support recovery using general (including non-Gaussian) dense measurement matrices. Combined with previously known sufficient conditions, this result yields sharp characterizations of when the optimal decoder can recover a signal for various scalings of the signal sparsity k and sample size n, including the important special case of linear sparsity (k = ¿(p)) using a linear scaling of observations (n = ¿(p)). Our second contribution is to prove necessary conditions on the number of observations n required for asymptotically reliable recovery using a class of ¿-sparsified measurement matrices, where the measurement sparsity parameter ¿(n, p, k) ¿ (0,1] corresponds to the fraction of nonzero entries per row. Our analysis allows general scaling of the quadruplet (n, p, k, ¿) , and reveals three different regimes, corresponding to whether measurement sparsity has no asymptotic effect, a minor effect, or a dramatic effect on the information-theoretic limits of the subset recovery problem.