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The restricted isometry property for time–frequency structured random matrices
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  • Published: 26 June 2012

The restricted isometry property for time–frequency structured random matrices

  • Götz E. Pfander1,
  • Holger Rauhut2 &
  • Joel A. Tropp3 

Probability Theory and Related Fields volume 156, pages 707–737 (2013)Cite this article

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Abstract

This paper establishes the restricted isometry property for a Gabor system generated by n 2 time–frequency shifts of a random window function in n dimensions. The sth order restricted isometry constant of the associated n × n 2 Gabor synthesis matrix is small provided that s ≤ c n 2/3 / log2 n. This bound provides a qualitative improvement over previous estimates, which achieve only quadratic scaling of the sparsity s with respect to n. The proof depends on an estimate for the expected supremum of a second-order chaos.

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Author information

Authors and Affiliations

  1. School of Engineering and Science, Jacobs University Bremen, 28759, Bremen, Germany

    Götz E. Pfander

  2. Hausdorff Center for Mathematics and Institute for Numerical Simulation, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Holger Rauhut

  3. California Institute of Technology, Pasadena, CA, 91125, USA

    Joel A. Tropp

Authors
  1. Götz E. Pfander
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  2. Holger Rauhut
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  3. Joel A. Tropp
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Corresponding author

Correspondence to Holger Rauhut.

Additional information

Dedicated to Hans G. Feichtinger on occassion of his 60th birthday.

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Cite this article

Pfander, G.E., Rauhut, H. & Tropp, J.A. The restricted isometry property for time–frequency structured random matrices. Probab. Theory Relat. Fields 156, 707–737 (2013). https://doi.org/10.1007/s00440-012-0441-4

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  • Received: 16 June 2011

  • Revised: 21 May 2012

  • Accepted: 05 June 2012

  • Published: 26 June 2012

  • Issue Date: August 2013

  • DOI: https://doi.org/10.1007/s00440-012-0441-4

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Keywords

  • Compressed sensing
  • Restricted isometry property
  • Gabor system
  • Time–frequency analysis
  • Random matrix
  • Chaos process

Mathematics Subject Classification (2000)

  • 60B20
  • 42C40
  • 94A12
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