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Compressed sensing using sparse-graph codes for the continuous-alphabet setting

In this paper, we consider the compressive sensing (CS) problem in the presence of noise. The problem is to recover a K-sparse signal s ∈ ℝn from noisy linear measurements y = As + w. We propose a fast recovery algorithm that can reconstruct any K-sparse signal s with time complexity that grows linearly in K and sublinearly in n. Specifically, with high probability, our algorithm is able to recover an arbitrarily large fraction of the support of the sparse signal using Θ(K log(n) log log(n)) samples and Θ(K log1+r(n)) computational cost, where r > 0 is an arbitrarily small constant. The sample and time complexities are near order-optimal. Further, our algorithm is able to recover the exact support with Θ(K log(K) log(n) log log(n)) measurements and time complexity of Θ(K log(K) log1+r(n)). With a mild technical assumption on the existence of a code with universal decoding algorithm and small decoding complexity, our algorithm can achieve Θ(K log(n)) sample and time complexities for the large fraction recovery, and furthermore, Θ(K log(K) log(n)) sample and time complexities for the full support recovery. The design of measurements and the recovery algorithm are based on sparse graph codes. We also justify our theoretical results with numerical experiments.