New bounds for restricted isometry constants

  • Authors:
  • T. Tony Cai;Lie Wang;Guangwu Xu

  • Affiliations:
  • Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA;Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA;Department of Electrical Engineering and Computer Science, University of Wisconsin, Milwaukee, WI

  • Venue:
  • IEEE Transactions on Information Theory
  • Year:
  • 2010

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Abstract

This paper discusses new bounds for restricted isometry constants in compressed sensing. Let φ be an n×p real matrix and k be a positive integer with k ≤ n. One of the main results of this paper shows that if the restricted isometry constant δk of φ satisfies δk k-sparse signals are guaranteed to be recovered exactly via l1 minimization when no noise is present and k-sparse signals can be estimated stably in the noisy case. It is also shown that the bound cannot be substantially improved. An explicit example is constructed in which δk = k-1/2k-1 k-sparse signals.