Constructive Approximation

, Volume 43, Issue 3, pp 409–424 | Cite as

Derandomizing Restricted Isometries via the Legendre Symbol

  • Afonso S. Bandeira
  • Matthew Fickus
  • Dustin G. Mixon
  • Joel MoreiraEmail author


The restricted isometry property (RIP) is an important matrix condition in compressed sensing, but the best matrix constructions to date use randomness. This paper leverages pseudorandom properties of the Legendre symbol to reduce the number of random bits in an RIP matrix with Bernoulli entries. In this regard, the Legendre symbol is not special—our main result naturally generalizes to any small-bias sample space. We also conjecture that no random bits are necessary for our Legendre symbol-based construction.


Derandomization Legendre symbol Small-bias sample space Restricted isometry property Compressed sensing 

Mathematics Subject Classification

Primary 94A20 Secondary 05D40 94B05 



The authors thank Prof. Peter Sarnak for insightful discussions. A. S. Bandeira was supported by AFOSR Grant No. FA9550-12-1-0317, and M. Fickus and D. G. Mixon were supported by NSF Grant No. DMS-1321779. The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Afonso S. Bandeira
    • 1
  • Matthew Fickus
    • 2
  • Dustin G. Mixon
    • 2
  • Joel Moreira
    • 3
    Email author
  1. 1.Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of Mathematics and StatisticsAir Force Institute of TechnologyWright-Patterson AFBUSA
  3. 3.Department of MathematicsOhio State UniversityColumbusUSA

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