Here is a new interesting study by Wei Wang, Martin Wainwright, and Kannan Ramchandran that is trying to answer the generic question on the theoretical limits entailed by using either dense or sparse measurement matrices. The technical report is entitled: Information-theoretic limits on sparse signal recovery: Dense versus sparse measurement matrices. The abstract reads: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 sharp characterizations of when the optimal decoder can recover a signal for various scalings of the sparsity k and sample size n, including the important special case of linear sparsity (k =) using a linear scaling of observations (n =
). 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
element of (0, 1] corresponds to the fraction of non-zero entries per row. Our analysis allows general scaling of the quadruplet
, 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.
The conference paper version is entitled Information-theoretic limits on sparse signal recovery: Dense versus sparse measurement matrices.
Image Credit: NASA/JPL/Space Science Institute, Thieving Moon.
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