Advertisement

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 Moreira
Article

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.

Keywords

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

Mathematics Subject Classification

Primary 94A20 Secondary 05D40 94B05 

Notes

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.

References

  1. 1.
    Achlioptas, D.: Database-friendly random projections: Johnson–Lindenstrauss with binary coins. J. Comput. Syst. Sci. 66, 671–687 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Ann. Math. 160, 781–793 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alon, N., Goldreich, O., Håstad, J., Paralta, R.: Simple constructions of almost k-wise independent random variables. Random Struct. Algorithms 3, 289–304 (1992)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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
  5. 5.
    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
  6. 6.
    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
  7. 7.
    Baraniuk, R.G., Davenport, M., DeVore, R.A., Wakin, M.: A simple proof of the restricted isometry property for random matrices. Constr. Approx. 28, 253–263 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ben-Aroya, A., Ta-Shma, A.: Constructing small-bias sets from algebraic-geometric codes. In: FOCS, pp. 191–197 (2009)Google Scholar
  9. 9.
    Bourgain, J., Dilworth, S., Ford, K., Konyagin, S., Kutzarova, D.: Explicit constructions of RIP matrices and related problems. Duke Math. J. 159, 145–185 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bourgain, J., Dilworth, S., Ford, K., Konyagin, S., Kutzarova, D.: Breaking the \(k^2\) barrier for explicit RIP matrices. In: STOC, pp. 637–644 (2011)Google Scholar
  11. 11.
    Calderbank, R., Jafarpour, S., Nastasescu, M.: Covering radius and the restricted isometry property. In: IEEE Information Theory Workshop, pp. 558–562 (2011)Google Scholar
  12. 12.
    Candès, E.J.: The restricted isometry property and its implications for compressed sensing. C. R. Acad. Sci. Paris Ser. I 346, 589–592 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Candès, E.J., Tao, T.: Near-optimal signal recovery from random projections: Universal encoding strategies? IEEE Trans. Inf. Theory 52, 5406–5425 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cheraghchi, M., Guruswami, V., Velingker, A.: Restricted isometry of Fourier matrices and list decodability of random linear codes. SIAM J. Comput. 42, 1888–1914 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chowla, S.: The Riemann Hypothesis and Hilbert’s Tenth Problem. Mathematics and Its Applications, vol. 4, pp. xv+119. Gordon and Breach Science Publishers, New York, London, Paris (1965)Google Scholar
  16. 16.
    Chung, F.R.K.: Several generalizations of Weil sums. J. Number Theory 49, 95–106 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Clarkson, K.L., Woodruff, D.P.: Numerical linear algebra in the streaming model. In: STOC, pp. 205–214 (2009)Google Scholar
  18. 18.
    Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  19. 19.
    DeVore, R.A.: Deterministic constructions of compressed sensing matrices. J. Complex. 23, 918–925 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Erdős, P.: On a problem in graph theory. Math. Gaz. 47, 220–223 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Erdős, P.: Some remarks on the theory of graphs. Bull. Am. Math. Soc. 53, 292–294 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fickus, M., Mixon, D.G., Tremain, J.C.: Steiner equiangular tight frames. Linear Algebra Appl. 436, 1014–1027 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Graham, R.L., Spencer, J.H.: A constructive solution to a tournament problem. Can. Math. Bull. 14, 45–48 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kane, D.M., Nelson, J.: A derandomized sparse Johnson–Lindenstrauss transform. arXiv:1006.3585
  25. 25.
    Krahmer, F., Mendelson, S., Rauhut, H.: Suprema of chaos processes and the restricted isometry property. Commun. Pure Appl. Math. 67, 1877–1904 (2014). doi: 10.1002/cpa.21504
  26. 26.
    Krahmer, F., Ward, R.: New and improved Johnson–Lindenstrauss embeddings via the restricted isometry property. SIAM J. Math. Anal. 43, 1269–1281 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mauduit, C., Sárközy, A.: On finite pseudorandom binary sequences I: measure of pseudorandomness, the Legendre symbol. Acta Arith. 82, 365–377 (1997)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mendelson, S., Pajor, A., Tomczak-Jaegermann, N.: Uniform uncertainty principle for Bernoulli and subgaussian ensembles. Constr. Approx. 28, 277–289 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mixon, D.G.: Explicit matrices with the restricted isometry property: breaking the square-root bottleneck. arXiv:1403.3427
  30. 30.
    Naor, J., Naor, M.: Small-bias probability spaces: efficient constructions and applications. In: STOC, pp. 213–223 (1990)Google Scholar
  31. 31.
    Nelson, J., Price, E., Wootters, M.: New constructions of RIP matrices with fast multiplication and fewer rows. In: SODA, pp. 1515–1528 (2014)Google Scholar
  32. 32.
    Peralta, R.: On the randomness complexity of algorithms, University of Wisconsin, Milwaukee, CS Research Report TR 90-1Google Scholar
  33. 33.
    Rudelson, M., Vershynin, R.: On sparse reconstruction from Fourier and Gaussian measurements. Commun. Pure Appl. Math. 61, 1025–1045 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Schmidt, W.M.: Equations Over Finite Fields: An Elementary Approach. Lecture Notes in Mathematics. Springer Verlag (1976)Google Scholar
  35. 35.
  36. 36.
    Tao, T., Croot III, E., Helfgott, H.: Deterministic methods to find primes. Math. Comput. 81, 1233–1246 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Vazirani, U.: Randomness, adversaries and computation. Ph.D. Thesis, University of California, Berkeley (1986)Google Scholar
  38. 38.
    Welch, L.R.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inf. Theory 20, 397–399 (1974)MathSciNetCrossRefzbMATHGoogle Scholar

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
  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

Personalised recommendations