Asymptotic Analysis of Complex LASSO via Complex Approximate Message Passing (CAMP)

• Published in: IEEE transactions on information theory Vol. 59; no. 7; pp. 4290 - 4308
• Main Authors: Maleki, A., Anitori, L., Yang, Z., Baraniuk, R. G.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.07.2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Approximate message passing (AMP); Approximation theory; Asymptotic methods; complex-valued LASSO; Compressed sensing; compressed sensing (CS); Exact sciences and technology; Gaussian processes; Information theory; Information, signal and communications theory; Measurement; Message passing; minimax analysis; Minimax techniques; Normal distribution; Sampling, quantization; Signal and communications theory; Simulation; Telecommunications and information theory; Phase noise; compressed sensing (CS); Asymptotic behavior; Theoretical study; Algorithm; Phase transitions; Approximate message passing (AMP); Complex variable method; complex-valued LASSO; Message passing; Simulation; Least squares method; minimax analysis; Minimax method; Compressed sensing
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2013.2252232

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-valued. We study the popular recovery method of l 1 -regularized least squares or LASSO. While several studies have shown that LASSO provides 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 solve the complex-valued LASSO problem 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. Our theoretical results are concerned with the case of i.i.d. Gaussian sensing matrices. Simulations confirm that our results hold for a larger class of random matrices.