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 = 2(p)) using a linear scaling of observations (n = 2(p)). Our second contribution is to prove necessary conditions on the number of observations n required for asymptotical...

• Type: Journal
• Year: 2008
• Where: CORR
• Authors: Wei Wang, Martin J. Wainwright, Kannan Ramchandran
• Keywords: CORR 2008, Education, Measurement Matrices, Signal Sparsity, Sparsity