Information-theoretic limits on sparse support recovery: Dense versus sparse measurements

We study the information-theoretic limits of exactly recovering the support 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 (non-Gaussian) dense measurement matrices. Combined with previously known sufficient conditions, this result yields a sharp characterization of when the optimal decoder can recover a signal with linear sparsity (k = Theta(p)) using a linear scaling of observations (n = Theta(p)) in the presence of noise. Our second contribution is to prove necessary conditions on the number of observations n required for asymptotically reliable recovery using a class of gamma-sparsified measurement matrices, where the measurement sparsity gamma(n, p, k) isin (0,1] corresponds to the fraction of non-zero entries per row. Our analysis allows general scaling of the quadruplet (n, p, k, gamma), and reveals three different regimes, corresponding to whether measurement sparsity has no effect, a minor effect, or a dramatic effect on the information theoretic limits of the subset recovery problem.