Explicit Matrices with the Restricted Isometry Property: Breaking the Square-Root Bottleneck

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


Matrices with the restricted isometry property (RIP) are of particular interest in compressed sensing. To date, the best known RIP matrices are constructed using random processes, while explicit constructions are notorious for performing at the “square-root bottleneck,” i.e., they only accept sparsity levels on the order of the square root of the number of measurements. The only known explicit matrix which surpasses this bottleneck was constructed by Bourgain, Dilworth, Ford, Konyagin, and Kutzarova in Bourgain et al. (Duke Math. J. 159:145–185, 2011). This chapter provides three contributions to advance the groundbreaking work of Bourgain et al.: (i) we develop an intuition for their matrix construction and underlying proof techniques; (ii) we prove a generalized version of their main result; and (iii) we apply this more general result to maximize the extent to which their matrix construction surpasses the square-root bottleneck.



The author thanks the anonymous referees for their helpful suggestions. This work was supported by NSF Grant No. DMS-1321779. The views expressed in this chapter are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.


  1. 1.
    Applebaum, L., Howard, S.D., Searle, S., Calderbank, R.: Chirp sensing codes: deterministic compressed sensing measurements for fast recovery. Appl. Comput. Harmon. Anal. 26, 283–290 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bandeira, A.S., Dobriban, E., Mixon, D.G., Sawin, W.F.: Certifying the restricted isometry property is hard. IEEE Trans. Inf. Theory 59, 3448–3450 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bandeira, A.S., Fickus, M., Mixon, D.G., Wong, P.: The road to deterministic matrices with the restricted isometry property. J. Fourier Anal. Appl. 19, 1123–1149 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baraniuk, R., Davenport, M., DeVore, R., Wakin, M.: A simple proof of the restricted isometry property for random matrices. Constr. Approx. 28, 253–263 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bourgain, J., Garaev, M.Z.: On a variant of sum-product estimates and explicit exponential sum bounds in prime fields. Math. Proc. Camb. Philos. Soc. 146, 1–21 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bourgain, J., Glibichuk, A.: Exponential sum estimate over subgroup in an arbitrary finite field. (2011)
  7. 7.
    Bourgain, J., Dilworth, S.J., Ford, K., Konyagin, S.V., Kutzarova, D.: Breaking the k 2 barrier for explicit RIP matrices. In: STOC 2011, pp. 637–644 (2011)MathSciNetGoogle Scholar
  8. 8.
    Bourgain, J., Dilworth, S.J., Ford, K., Konyagin, S., Kutzarova, D.: Explicit constructions of RIP matrices and related problems. Duke Math. J. 159, 145–185 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cai, T.T., Zhang, A.: Sharp RIP bound for sparse signal and low-rank matrix recovery. Appl. Comput. Harmon. Anal. 35, 74–93 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Casazza, P.G., Fickus, M.: Fourier transforms of finite chirps. EURASIP J. Appl. Signal Process. 2006, 7 p (2006)Google Scholar
  11. 11.
    DeVore, R.A.: Deterministic constructions of compressed sensing matrices. J. Complexity 23, 918–925 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fickus, M., Mixon, D.G., Tremain, J.C.: Steiner equiangular tight frames. Linear Algebra Appl. 436, 1014–1027 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  14. 14.
    Koiran, P., Zouzias, A.: Hidden cliques and the certification of the restricted isometry property. arXiv:1211.0665 (2012)Google Scholar
  15. 15.
    Mixon, D.G.: Deterministic RIP matrices: breaking the square-root bottleneck, short, fat matrices (weblog). (2013)
  16. 16.
    Mixon, D.G.: Deterministic RIP matrices: breaking the square-root bottleneck, II, short, fat matrices (weblog). (2013)
  17. 17.
    Mixon, D.G.: Deterministic RIP matrices: breaking the square-root bottleneck, III, short, fat matrices (weblog). (2013)
  18. 18.
    Tao, T.: Open question: deterministic UUP matrices. What’s new (weblog). (2007)
  19. 19.
    Tao, T., Vu, V.H.: Additive Combinatorics. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  20. 20.
    Welch, L.R.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inform. Theory 20, 397–399 (1974)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Air Force Institute of TechnologyWright-Patterson Air Force BaseUSA

Personalised recommendations