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Lossless Expanders in Compressive Sensing

  • Simon Foucart
  • Holger Rauhut
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

A new class of measurement matrices is introduced in this chapter. It consists of adjacency matrices of certain left regular bipartite graphs called lossless expanders. After pointing out basic properties of lossless expanders, their existence is established using probabilistic methods. The adjacency matrices are then proved to satisfy the null space property, hence enable sparse recovery via basis pursuit. Sparse recovery is also shown to be achieved via a variation of iterative hard thresholding. Finally, a rudimentary example of another type of algorithms that run in sublinear time is presented.

Keywords

bipartite graph sdjacency matrix 1-minimization thresholding-based algorithm sublinear-time algorithm 

References

  1. 19.
    S. Arora, B. Barak, Computational Complexity: A Modern Approach (Cambridge University Press, Cambridge, 2009)CrossRefGoogle Scholar
  2. 41.
    R. Berinde, A. Gilbert, P. Indyk, H. Karloff, M. Strauss, Combining geometry and combinatorics: A unified approach to sparse signal recovery. In Proc. of 46th Annual Allerton Conference on Communication, Control, and Computing, pp. 798–805, 2008Google Scholar
  3. 42.
    R. Berinde, P. Indyk, M. Rz̆ić, Practical near-optimal sparse recovery in the L1 norm. In Proc. Allerton, 2008Google Scholar
  4. 101.
    M. Capalbo, O. Reingold, S. Vadhan, A. Wigderson, Randomness conductors and constant-degree lossless expanders. In Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing (electronic) (ACM, New York, 2002), pp. 659–668Google Scholar
  5. 225.
    A.C. Gilbert, M. Strauss, J.A. Tropp, R. Vershynin, One sketch for all: fast algorithms for compressed sensing. In Proceedings of the Thirty-ninth Annual ACM Symposium on Theory of Computing, STOC ’07, pp. 237–246, ACM, New York, NY, USA, 2007Google Scholar
  6. 251.
    V. Guruswani, C. Umans, S. Vadhan, Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes. In IEEE Conference on Computational Complexity, pp. 237–246, 2007Google Scholar
  7. 261.
    H. Hassanieh, P. Indyk, D. Katabi, E. Price, Nearly optimal sparse Fourier transform. In Proceedings of the 44th Symposium on Theory of Computing, STOC ’12, pp. 563–578, ACM, New York, NY, USA, 2012Google Scholar
  8. 262.
    H. Hassanieh, P. Indyk, D. Katabi, E. Price, Simple and practical algorithm for sparse Fourier transform. In Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’12, pp. 1183–1194. SIAM, 2012Google Scholar
  9. 279.
    S. Hoory, N. Linial, A. Wigderson, Expander graphs and their applications. Bull. Am. Math. Soc. (N.S.) 43(4), 439–561 (electronic) (2006)Google Scholar
  10. 285.
    P. Indyk, A. Gilbert, Sparse recovery using sparse matrices. Proc. IEEE 98(6), 937–947 (2010)CrossRefGoogle Scholar
  11. 286.
    P. Indyk, M. Ruz̆ić, Near-optimal sparse recovery in the L1 norm. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’08, pp. 199–207 (2008)Google Scholar
  12. 287.
    M. Iwen, Combinatorial sublinear-time Fourier algorithms. Found. Comput. Math. 10(3), 303–338 (2010)MathSciNetMATHCrossRefGoogle Scholar
  13. 291.
    S. Jafarpour, W. Xu, B. Hassibi, R. Calderbank, Efficient and robust compressed sensing using optimized expander graphs. IEEE Trans. Inform. Theor. 55(9), 4299–4308 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Simon Foucart
    • 1
  • Holger Rauhut
    • 2
  1. 1.Department of MathematicsDrexel UniversityPhiladelphiaUSA
  2. 2.Lehrstuhl C für Mathematik (Analysis)RWTH Aachen UniversityAachenGermany

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