The Restricted Isometry Property of Subsampled Fourier Matrices

  • Ishay Haviv
  • Oded Regev
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2169)

Abstract

A matrix \(A \in \mathbb{C}^{q\times N}\) satisfies the restricted isometry property of order k with constant ε if it preserves the 2 norm of all k-sparse vectors up to a factor of 1 ±ε. We prove that a matrix A obtained by randomly sampling q = O(k ⋅ log2k ⋅ logN) rows from an N × N Fourier matrix satisfies the restricted isometry property of order k with a fixed ε with high probability. This improves on Rudelson and Vershynin (Comm Pure Appl Math, 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).

Notes

Acknowledgements

We thank Afonso S. Bandeira, Mahdi Cheraghchi, Michael Kapralov, Jelani Nelson, and Eric Price for useful discussions, and anonymous reviewers for useful comments.

Oded Regev was supported by the Simons Collaboration on Algorithms and Geometry and by the National Science Foundation (NSF) under Grant No. CCF-1320188. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Ishay Haviv
    • 1
  • Oded Regev
    • 2
  1. 1.School of Computer ScienceThe Academic College of Tel Aviv-YaffoTel AvivIsrael
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNYUSA

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