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Successive concave sparsity approximation for compressed sensing

Malek Mohammadi, M ; Sharif University of Technology | 2016

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  1. Type of Document: Article
  2. DOI: 10.1109/TSP.2016.2585096
  3. Publisher: Institute of Electrical and Electronics Engineers Inc , 2016
  4. Abstract:
  5. In this paper, based on a successively accuracy-increasing approximation of the ℓ0 norm, we propose a new algorithm for recovery of sparse vectors from underdetermined measurements. The approximations are realized with a certain class of concave functions that aggressively induce sparsity and their closeness to the ℓ0 norm can be controlled. We prove that the series of the approximations asymptotically coincides with the ℓ1 and ℓ0 norms when the approximation accuracy changes from the worst fitting to the best fitting. When measurements are noise-free, an optimization scheme is proposed that leads to a number of weighted ℓ1 minimization programs, whereas, in the presence of noise, we propose two iterative thresholding methods that are computationally appealing. A convergence guarantee for the iterative thresholding method is provided, and, for a particular function in the class of the approximating functions, we derive the closed-form thresholding operator. We further present some theoretical analyses via the restricted isometry, null space, and spherical section properties. Our extensive numerical simulations indicate that the proposed algorithm closely follows the performance of the oracle estimator for a range of sparsity levels wider than those of the state-of-the-art algorithms
  6. Keywords:
  7. Compressed sensing (CS) ; Compressed sensing ; Iterative methods ; Optimization ; Signal reconstruction ; Compressive sensing ; Iterative thresholding ; Nonconvex optimization ; Oracle estimator ; The LASSO estimator ; Approximation algorithms
  8. Source: IEEE Transactions on Signal Processing ; Volume 64, Issue 21 , 2016 , Pages 5657-5671 ; 1053587X (ISSN)
  9. URL: http://ieeexplore.ieee.org/document/7500117/?reload=true