2011/08/01 by Arian Maleki, Maleki, Arian, Laura Anitori +5 · 2 citations
Engineering · Medicine · #Advanced MRI Techniques and Applications #Electrical and Bioimpedance Tomography #FOS: Computer and information sciences #Information Theory (cs.IT) #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1108.0477
openalex publication_date 2011/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recovering a sparse signal from an undersampled set of random linear measurements is the main problem of interest in compressed sensing. In this paper, we consider the case where both the signal and the measurements are complex. We study the popular reconstruction method of ℓ1-regularized least squares or LASSO. While several studies have shown that the LASSO algorithm offers desirable solutions under certain conditions, the precise asymptotic performance of this algorithm in the complex setting is not yet known. In this paper, we extend the approximate message passing (AMP) algorithm to the complex signals and measurements and obtain the complex approximate message passing algorithm (CAMP). We then generalize the state evolution framework recently introduced for the analysis of AMP, to the complex setting. Using the state evolution, we derive accurate formulas for the phase transition and noise sensitivity of both LASSO and CAMP.