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Compressed sensing (sparse signal recovery) often encounters nonnegative data(e.g., images). Recently we developed the methodology of using (dense)Compressed Counting for recovering nonnegative K-sparse signals. In this paper,we adopt very sparse Compressed Counting for nonnegative signal recovery. Ourdesign matrix is sampled from a maximally-skewed p-stable distribution (0<p<1),and we sparsify the design matrix so that on average (1-g)-fraction of theentries become zero. The idea is related to very sparse stable randomprojections (Li et al 2006 and Li 2007), the prior work for estimating summarystatistics of the data.  In our theoretical analysis, we show that, when p->0, it suffices to use M=K/(1-exp(-gK) log N measurements, so that all coordinates can be recovered inone scan of the coordinates. If g = 1 (i.e., dense design), then M = K log N.If g= 1/K or 2/K (i.e., very sparse design), then M = 1.58K log N or M = 1.16Klog N. This means the design matrix can be indeed very sparse at only a minorinflation of the sample complexity.  Interestingly, as p->1, the required number of measurements is essentially M= 2.7K log N, provided g= 1/K. It turns out that this result is a generalworst-case bound.

Sparse Recovery with Very Sparse Compressed Counting