Journal of Fourier Analysis and Applications

, Volume 19, Issue 6, pp 1123–1149 | Cite as

The Road to Deterministic Matrices with the Restricted Isometry Property

  • Afonso S. Bandeira
  • Matthew Fickus
  • Dustin G. Mixon
  • Percy Wong
Article

Abstract

The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.

Keywords

Restricted isometry property Compressed sensing Equiangular tight frames 

Mathematics Subject Classification

15A42 05E30 15B52 60F10 94A12 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Afonso S. Bandeira
    • 1
  • Matthew Fickus
    • 2
  • Dustin G. Mixon
    • 1
  • Percy Wong
    • 1
  1. 1.Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of Mathematics and StatisticsAir Force Institute of TechnologyWright-Patterson Air Force BaseUSA

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