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Fast sparse reconstruction: Greedy inverse scale space flows


Authors: Michael Moeller and Xiaoqun Zhang
Journal: Math. Comp. 85 (2016), 179-208
MSC (2010): Primary 65K10, 90C59, 90C26; Secondary 92C55
DOI: https://doi.org/10.1090/mcom/3004
Published electronically: July 29, 2015
MathSciNet review: 3404447
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Abstract: In this paper we analyze the connection between the recently proposed adaptive inverse scale space methods for basis pursuit and the well-known orthogonal matching pursuit method for the recovery of sparse solutions to underdetermined linear systems. Furthermore, we propose a new greedy sparse recovery method, which approximates $ \ell ^1$ minimization more closely. A variant of our new approach can increase the support of the current iterate by many indices at once, resulting in an extremely efficient algorithm. Our new method has the advantage that there is a simple criterion to determine a posteriori if an $ \ell ^1$ minimizer was found. Numerical comparisons with orthogonal matching pursuit, weak orthogonal matching pursuit, hard thresholding pursuit and compressive sampling matching pursuit underline that our methods indeed inherit some advantageous properties from the inverse scale space flow.


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Additional Information

Michael Moeller
Affiliation: Institue for Computational and Applied Mathematics, University of Münster, Germany
Address at time of publication: Department of Mathematics, Technische Universität München, Boltzmannstrasse 3, 85748 Garching, Germany
Email: m.moeller@gmx.net

Xiaoqun Zhang
Affiliation: Department of Mathematics, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University, No. 800, Dongchuan Road, Shanghai 200240, People’s Republic of China
Email: xqzhang@sjtu.edu.cn

DOI: https://doi.org/10.1090/mcom/3004
Received by editor(s): January 30, 2013
Received by editor(s) in revised form: January 26, 2014
Published electronically: July 29, 2015
Additional Notes: This work was supported by the DFG grant “Sparsity constrained inversion with Tomographic Applications”
The second author was additionally supported by the National Science Foundation of China (grant numbers NSFC91330102, NSFC11101277 and NSFC11161130004) and by the Shanghai Pujiang Talent program (grant number 11PJ1405900)
Article copyright: © Copyright 2015 American Mathematical Society