Year: 2017
Author: Penghang Yin, Jack Xin
Journal of Computational Mathematics, Vol. 35 (2017), Iss. 4 : pp. 439–451
Abstract
An algorithmic framework, based on the difference of convex functions algorithm (DCA), is proposed for minimizing a class of concave sparse metrics for compressed sensing problems. The resulting algorithm iterates a sequence of $ℓ_1$ minimization problems. An exact sparse recovery theory is established to show that the proposed framework always improves on the basis pursuit ($ℓ_1$ minimization) and inherits robustness from it. Numerical examples on success rates of sparse solution recovery illustrate further that, unlike most existing non-convex compressed sensing solvers in the literature, our method always out-performs basis pursuit, no matter how ill-conditioned the measurement matrix is. Moreover, the iterative $ℓ_1$ (IL1) algorithm lead by a wide margin the state-of-the-art algorithms on $ℓ_{1/2}$ and logarithimic minimizations in the strongly coherent (highly ill-conditioned) regime, despite the same objective functions. Last but not least, in the application of magnetic resonance imaging (MRI), IL1 algorithm easily recovers the phantom image with just 7 line projections.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1610-m2016-0620
Journal of Computational Mathematics, Vol. 35 (2017), Iss. 4 : pp. 439–451
Published online: 2017-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 13
Keywords: Compressed sensing Non-convexity Difference of convex functions algorithm Iterative $ℓ_1$ minimization.