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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Fast sparse reconstruction: Greedy inverse scale space flows
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by Michael Moeller and Xiaoqun Zhang PDF
Math. Comp. 85 (2016), 179-208 Request permission

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
  • MR Author ID: 974311
  • 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
  • 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)
  • © Copyright 2015 American Mathematical Society
  • Journal: Math. Comp. 85 (2016), 179-208
  • MSC (2010): Primary 65K10, 90C59, 90C26; Secondary 92C55
  • DOI: https://doi.org/10.1090/mcom/3004
  • MathSciNet review: 3404447