Asymptotic analysis of complex LASSO via complex approximate message passing (CAMP)

Arian Maleki, Laura Anitori, Zai Yang, Richard G. Baraniuk

Research output: Contribution to journalArticle

129 Scopus citations

Abstract

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 l1- 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.

Original languageEnglish (US)
Article number6478821
Pages (from-to)4290-4308
Number of pages19
JournalIEEE Transactions on Information Theory
Volume59
Issue number7
DOIs
StatePublished - Jan 1 2013

Keywords

  • Approximate message passing (AMP)
  • Complexvalued LASSO
  • Compressed sensing (CS)
  • Minimax analysis

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences

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