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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

An adaptive inverse scale space method for compressed sensing


Authors: Martin Burger, Michael Möller, Martin Benning and Stanley Osher
Journal: Math. Comp. 82 (2013), 269-299
MSC (2010): Primary 49M29, 90C25, 65F20, 65F22
Published electronically: June 7, 2012
MathSciNet review: 2983025
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Abstract: In this paper we introduce a novel adaptive approach for solving $ \ell ^1$-minimization problems as frequently arising in compressed sensing, which is based on the recently introduced inverse scale space method. The scheme allows to efficiently compute minimizers by solving a sequence of low-dimensional nonnegative least-squares problems.

We provide a detailed convergence analysis in a general setup as well as refined results under special conditions. In addition, we discuss experimental observations in several numerical examples.


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Additional Information

Martin Burger
Affiliation: Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte Mathematik, Einsteinstr. 62, D 48149 Münster, Germany
Email: martin.burger@wwu.de

Michael Möller
Affiliation: Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte Mathematik, Einsteinstr. 62, D 48149 Münster, Germany
Email: m.moeller@gmx.net

Martin Benning
Affiliation: Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte Mathematik, Einsteinstr. 62, D 48149 Münster, Germany
Email: martin.benning@wwu.de

Stanley Osher
Affiliation: Department of Mathematics, University of California Los Angeles. Portola Plaza, Los Angeles, California 90095
Email: sjo@math.ucla.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02599-3
PII: S 0025-5718(2012)02599-3
Keywords: Compressed sensing, inverse scale space, sparsity, adaptivity, greedy methods
Received by editor(s): February 23, 2011
Received by editor(s) in revised form: July 11, 2011
Published electronically: June 7, 2012
Additional Notes: The work of MB and MB has been supported by the German Research Foundation DFG through the project Regularization with Singular Energies. M.M. and S.O. were supported by NSF grants DMS-0835863, DMS-0914561, DMS-0914856 and ONR grant N00014-08-1119. M.M. also acknowledges the support of the German Academic Exchange Service (DAAD)
Article copyright: © Copyright 2012 American Mathematical Society