Fast and Robust Support Recovery for Compressive Sensing with Continuous Alphabet

In this paper, we consider the compressive sensing (CS) problem in the presence of noise. Theproblem is to recover a K-sparse signal s ∈ R from noisy linear measurements y = As+w. Wepropose the Fast-Continuous algorithm with large fraction recovery guarantee and full recoveryguarantee. Specifically, with high probability, to recover an arbitrarily large fraction of the sup-port of the sparse signal, the algorithm uses Θ(K log(n) log log(n)) samples and Θ(K log(n))computational cost, where r > 0 is an arbitrarily small constant. The sample and time com-plexities are almost order optimal. To recover the full support, the algorithm uses sample andtime complexities Θ(K log(K) log(n) log log(n)) and Θ(K log(K) log(n)), respectively. Witha mild technical assumption, our algorithm can have Θ(K log(n)) sample and time complexitiesfor the large fraction recovery and Θ(K log(K) log(n)) sample and time complexities for the fullsupport recovery. The design of measurements and the recovery algorithm are based on sparsegraph codes. The main contribution of our algorithm compared with previous CS algorithmsbased on sparse graph codes is that our algorithm relax the assumption that the non-zero ele-ments of the sparse signal lie in finite constellation points, i.e., we allow the non-zero elementstaking continuous-valued real numbers. The algorithm that we propose also provides a generalframework for the sparse-graph-code based algorithms in other problems such as compressivephase retrieval and sparse covariance estimation to work with continuous-valued signals.