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

Abstract

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.

References

  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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bourgain, J., Glibichuk, A.: Exponential sum estimate over subgroup in an arbitrary finite field. http://www.math.ias.edu/files/avi/Bourgain_Glibichuk.pdf (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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fickus, M., Mixon, D.G., Tremain, J.C.: Steiner equiangular tight frames. Linear Algebra Appl. 436, 1014–1027 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Springer, Berlin (2013)CrossRefMATHGoogle 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). http://www.dustingmixon.wordpress.com/2013/12/02/deterministic-rip-matrices-breaking-the-square-root-bottleneck/ (2013)
  16. 16.
    Mixon, D.G.: Deterministic RIP matrices: breaking the square-root bottleneck, II, short, fat matrices (weblog). http://www.dustingmixon.wordpress.com/2013/12/11/deterministic-rip-matrices-breaking-the-square-root-bottleneck-ii/ (2013)
  17. 17.
    Mixon, D.G.: Deterministic RIP matrices: breaking the square-root bottleneck, III, short, fat matrices (weblog). http://www.dustingmixon.wordpress.com/2014/01/14/deterministic-rip-matrices-breaking-the-square-root-bottleneck-iii/ (2013)
  18. 18.
    Tao, T.: Open question: deterministic UUP matrices. What’s new (weblog). http://www.terrytao.wordpress.com/2007/07/02/open-question-deterministic-uup-matrices/ (2007)
  19. 19.
    Tao, T., Vu, V.H.: Additive Combinatorics. Cambridge University Press, Cambridge (2006)CrossRefMATHGoogle Scholar
  20. 20.
    Welch, L.R.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inform. Theory 20, 397–399 (1974)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

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

Personalised recommendations