The Restricted Isometry Property of Subsampled Fourier Matrices

  • Ishay Haviv
  • Oded Regev
Part of the Lecture Notes in Mathematics book series (LNM, volume 2169)


A matrix \(A \in \mathbb{C}^{q\times N}\) satisfies the restricted isometry property of order k with constant ε if it preserves the 2 norm of all k-sparse vectors up to a factor of 1 ±ε. We prove that a matrix A obtained by randomly sampling q = O(k ⋅ log2k ⋅ logN) rows from an N × N Fourier matrix satisfies the restricted isometry property of order k with a fixed ε with high probability. This improves on Rudelson and Vershynin (Comm Pure Appl Math, 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).



We thank Afonso S. Bandeira, Mahdi Cheraghchi, Michael Kapralov, Jelani Nelson, and Eric Price for useful discussions, and anonymous reviewers for useful comments.

Oded Regev was supported by the Simons Collaboration on Algorithms and Geometry and by the National Science Foundation (NSF) under Grant No. CCF-1320188. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.


  1. 1.
    N. Ailon, B. Chazelle, The fast Johnson–Lindenstrauss transform and approximate nearest neighbors. SIAM J. Comput. 39 (1), 302–322 (2009). Preliminary version in STOC’06Google Scholar
  2. 2.
    N. Ailon, E. Liberty, Fast dimension reduction using Rademacher series on dual BCH codes. Discrete Comput. Geom. 42 (4), 615–630 (2009). Preliminary version in SODA’08Google Scholar
  3. 3.
    N. Ailon, E. Liberty, An almost optimal unrestricted fast Johnson–Lindenstrauss transform. ACM Trans. Algorithms 9 (3), 21 (2013). Preliminary version in SODA’11Google Scholar
  4. 4.
    A.S. Bandeira, E. Dobriban, D.G. Mixon, W.F. Sawin, Certifying the restricted isometry property is hard. IEEE Trans. Inform. Theory 59 (6), 3448–3450 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    A.S. Bandeira, M.E. Lewis, D.G. Mixon, Discrete uncertainty principles and sparse signal processing. CoRR abs/1504.01014 (2015)Google Scholar
  6. 6.
    R. Baraniuk, M. Davenport, R. DeVore, M. Wakin, A simple proof of the restricted isometry property for random matrices. Constr. Approx. 28 (3), 253–263 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J. Bourgain, An improved estimate in the restricted isometry problem, in Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 2116, pp. 65–70 (Springer, Berlin, 2014)Google Scholar
  8. 8.
    E.J. Candès, The restricted isometry property and its implications for compressed sensing. C. R. Math. 346 (9–10), 589–592 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    E.J. Candès, T. Tao, Decoding by linear programming. IEEE Trans. Inform. Theory 51 (12), 4203–4215 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    E.J. Candès, T. Tao, Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. on Inform. Theory 52 (12), 5406–5425 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    E.J. Candès, M. Rudelson, T. Tao, R. Vershynin, Error correction via linear programming, in 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 295–308 (2005)Google Scholar
  12. 12.
    E.J. Candès, J.K. Romberg, T. Tao, Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59 (8), 1207–1223 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by basis pursuit. SIAM J. Comput. 20 (1), 33–61 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    M. Cheraghchi, V. Guruswami, A. Velingker, Restricted isometry of Fourier matrices and list decodability of random linear codes. SIAM J. Comput. 42 (5), 1888–1914 (2013). Preliminary version in SODA’13Google Scholar
  15. 15.
    S. Dirksen, Tail bounds via generic chaining. Electron. J. Prob. 20 (53), 1–29 (2015)MathSciNetzbMATHGoogle Scholar
  16. 16.
    D.L. Donoho, M. Elad, V.N. Temlyakov, Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inform. Theory 52 (1), 6–18 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    S. Foucart, A. Pajor, H. Rauhut, T. Ullrich, The Gelfand widths of p-balls for 0 < p ≤ 1. J. Complex. 26 (6), 629–640 (2010)Google Scholar
  18. 18.
    A.Y. Garnaev, E.D. Gluskin, On the widths of Euclidean balls. Sov. Math. Dokl. 30, 200–203 (1984)zbMATHGoogle Scholar
  19. 19.
    I. Haviv, O. Regev, The list-decoding size of Fourier-sparse boolean functions, in Proceedings of the 30th Conference on Computational Complexity, CCC, pp. 58–71 (2015)Google Scholar
  20. 20.
    P. Indyk, I. Razenshteyn, On model-based RIP-1 matrices, in Automata, Languages, and Programming - 40th International Colloquium, ICALP, pp. 564–575 (2013)Google Scholar
  21. 21.
    A.C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Society of Industrial and Applied Mathematics, Philadelphia, 2001)CrossRefzbMATHGoogle Scholar
  22. 22.
    F. Krahmer, R. Ward, New and improved Johnson-Lindenstrauss embeddings via the restricted isometry property. SIAM J. Math. Anal. 43 (3), 1269–1281 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    F. Krahmer, S. Mendelson, H. Rauhut, Suprema of chaos processes and the restricted isometry property. CoRR abs/1207.0235 (2012)Google Scholar
  24. 24.
    C. McDiarmid, Concentration, in Probabilistic Methods for Algorithmic Discrete Mathematics. Algorithms Combination, vol. 16 (Springer, Berlin, 1998), pp. 195–248Google Scholar
  25. 25.
    A. Natarajan, Y. Wu, Computational complexity of certifying restricted isometry property, in Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX, pp. 371–380 (2014)Google Scholar
  26. 26.
    D. Needell, J.A. Tropp, CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Commun. ACM 53 (12), 93–100 (2010)CrossRefzbMATHGoogle Scholar
  27. 27.
    J. Nelson, E. Price, M. Wootters, New constructions of RIP matrices with fast multiplication and fewer rows, in Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 1515–1528 (2014)Google Scholar
  28. 28.
    D.G. Nishimura, Principles of Magnetic Resonance Imaging (Stanford University, Stanford, CA, 2010)Google Scholar
  29. 29.
    H. Rauhut, Compressive sensing and structured random matrices, in Theoretical Foundations and Numerical Methods for Sparse Recovery, vol. 9, ed. by M. Fornasier (De Gruyter, Berlin, 2010), pp. 1–92Google Scholar
  30. 30.
    M. Rudelson, R. Vershynin, On sparse reconstruction from Fourier and Gaussian measurements. Commun. Pure Appl. Math. 61 (8), 1025–1045 (2008). Preliminary version in CISS’06Google Scholar
  31. 31.
    A.M. Tillmann, M.E. Pfetsch, The computational complexity of the restricted isometry property, the nullspace property, and related concepts in compressed sensing. IEEE Trans. Inform. Theory 60 (2), 1248–1259 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    S. Wenger, S. Darabi, P. Sen, K. Glassmeier, M.A. Magnor, Compressed sensing for aperture synthesis imaging, in Proceedings of the International Conference on Image Processing, ICIP, pp. 1381–1384 (2010)Google Scholar
  33. 33.
    M. Wootters, On the list decodability of random linear codes with large error rates, in Proceedings of the 45th Annual ACM Symposium on Theory of Computing, STOC, pp. 853–860 (2013)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Ishay Haviv
    • 1
  • Oded Regev
    • 2
  1. 1.School of Computer ScienceThe Academic College of Tel Aviv-YaffoTel AvivIsrael
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNYUSA

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