Compressed Counting Meets Compressed Sensing

Compressed1 sensing (sparse signal recovery) has been a popular and important research topic in recent years. By observing that natural signals are often nonnegative, we propose a new framework for nonnegative signal recovery using Compressed Counting (CC). CC is a technique built on maximally-skewed α-stable random projections originally developed for data stream computations. Our recovery procedure is computationally very efficient in that it requires only one linear scan of the coordinates. In our settings, the signal x ∈ R is assumed to be nonnegative, i.e., xi ≥ 0, ∀ i. Our analysis demonstrates that, when α ∈ (0, 0.5], it suffices to use M = (Cα + o(1))ε−α (∑N i=1 x α i ) logN/δ measurements so that, with probability 1 − δ, all coordinates will be recovered within ε additive precision, in one scan of the coordinates. The constant Cα = 1 when α → 0 and Cα = π/2 when α = 0.5. In particular, when α → 0, the required number of measurements is essentially M = K logN/δ, where K = ∑N i=1 1{xi ̸= 0} is the number of nonzero coordinates of the signal. The work was presented at Simons Institute Workshop on Succinct Data Representations and Applications in September 2013.