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# Compressed Counting Meets Compressed Sensing

*Proceedings of The 27th Conference on Learning Theory*, PMLR 35:1058-1077, 2014.

#### Abstract

Compressed sensing (sparse signal recovery) has been a popular and important research topic in recent years. By observing that natural signals (e.g., images or network data) are often nonnegative, we propose a framework for nonnegative signal recovery using \em Compressed Counting (CC). CC is a technique built on \em maximally-skewed α-stable random projections originally developed for data stream computations (e.g., entropy estimations). Our recovery procedure is computationally efficient in that it requires only one linear scan of the coordinates. In our settings, the signal \mathbfx∈\mathbbR^N is assumed to be nonnegative, i.e., x_i≥0, ∀i. We prove that, when α∈(0, 0.5], it suffices to use M=(C_α+o(1)) ε^-α \left(\sum_i=1^N x_i^α\right)\log N/δmeasurements so that, with probability 1-δ, all coordinates will be recovered within εadditive precision, in one scan of the coordinates. The constant C_α=1 when α\rightarrow0 and C_α=\pi/2 when α=0.5. In particular, when α\rightarrow0, the required number of measurements is essentially M=K\log N/δ, where K = \sum_i=1^N 1{x_i≠0} is the number of nonzero coordinates of the signal.