Derandomizing Restricted Isometries via the Legendre Symbol

Abstract

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.

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Acknowledgments

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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Correspondence to Joel Moreira.

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Communicated by Joel A. Tropp.

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Bandeira, A.S., Fickus, M., Mixon, D.G. et al. Derandomizing Restricted Isometries via the Legendre Symbol. Constr Approx 43, 409–424 (2016). https://doi.org/10.1007/s00365-015-9310-6

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Keywords

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

Mathematics Subject Classification

  • Primary 94A20
  • Secondary 05D40
  • 94B05