A fast algorithm for nonconvex approaches to sparse recovery problems

  • Authors:
  • Laura B. Montefusco;Damiana Lazzaro;Serena Papi

  • Affiliations:
  • Department of Mathematics, University of Bologna, Piazza di Porta S. Donato, 40123 Bologna, Italy;Department of Mathematics, University of Bologna, Piazza di Porta S. Donato, 40123 Bologna, Italy;CIRI ICT, University of Bologna, Via Rasi e Spinelli 176, 47521 Cesena(FC), Italy

  • Venue:
  • Signal Processing
  • Year:
  • 2013

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Abstract

This paper addresses the problem of sparse signal recovery from a lower number of measurements than those requested by the classical compressed sensing theory. This problem is formalized as a constrained minimization problem, where the objective function is nonconvex and singular at the origin. Several algorithms have been recently proposed, which rely on iterative reweighting schemes, that produce better estimates at each new minimization step. Two such methods are iterative reweighted l"2 and l"1 minimization that have been shown to be effective and general, but very computationally demanding. The main contribution of this paper is the proposal of the algorithm WNFCS, where the reweighted schemes represent the core of a penalized approach to the solution of the constrained nonconvex minimization problem. The algorithm is fast, and succeeds in exactly recovering a sparse signal from a smaller number of measurements than the l"1 minimization and in a shorter time. WNFCS is very general, since it represents an algorithmic framework that can easily be adapted to different reweighting strategies and nonconvex objective functions. Several numerical experiments and comparisons with some of the most recent nonconvex minimization algorithms confirm the capabilities of the proposed algorithm.