SpringerLink
Log in

Improved bounds for the RIP of Subsampled Circulant Matrices

  • Published:
Sampling Theory in Signal and Image Processing Aims and scope Submit manuscript

Abstract

In this paper, we study the restricted isometry property of partial random circulant matrices. For a bounded subgaussian generator with independent entries, we prove that the partial random circulant matrices satisfy s-order RIP with high probability if one chooses m ≳ s log2 s log n rows randomly where n is the vector length. This improves the previously known bound m ≳ s log2 s log2 n.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W. U. Bajwa, J. Haupt, G. Raz and R. Nowak, Compressed channel sensing, In Information Sciences and Systems, 42nd Annual Conference on Information Sciences and Systems, IEEE, 5–10, 2008.

  2. W. U. Bajwa, J. D. Haupt, G.M. Raz, S. J. Wright and R. D. Nowak, Toeplitz-structured compressed sensing matrices, 14th Workshop on Statistical Signal Processing, IEEE, 294–298, IEEE, 2007.

  3. T. Blumensath and M. E. Davies, Iterative hard thresholding for compressed sensing, Applied and Computational Harmonic Analysis, 27, 265–274, 2009.

    Article  MathSciNet  Google Scholar 

  4. J. Bourgain, An improved estimate in the restricted isometry problem, In Geometric Aspects of Functional Analysis, 65–70, Springer, 2014.

  5. T. Cai and A. Zhang, Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-rank Matrices, IEEE Transactions on Information Theory, 60, 122–132, 2014.

    Article  MathSciNet  Google Scholar 

  6. E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Transactions on Information Theory, 52, 489–509, 2006.

    Article  MathSciNet  Google Scholar 

  7. E. J. Candès, J. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math., 59, 1207–1223, 2006.

    Article  MathSciNet  Google Scholar 

  8. S. S. Chen, D. L. Donoho and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM Review, 43,129–159, 2001.

  9. D. L. Donoho, Compressed sensing, IEEE Transactions on Information Theory, 52, 1289–1306, 2006.

    Article  MathSciNet  Google Scholar 

  10. S. Foucart, Hard thresholding pursuit: an algorithm for compressive sensing, SIAM Journal on Numerical Analysis, 49, 2543–2563, 2011.

    Article  MathSciNet  Google Scholar 

  11. S. Foucart, Sparse recovery algorithms: sufficient conditions in terms of restricted isometry constants, In Approximation Theory XIII: San Antonio 2010, 65–77, Springer, 2012.

    MATH  Google Scholar 

  12. S. Foucart, A. Pajor, H. Rauhut and T. Ullrich, The gelfand widths of ℓp-balls for 0 < p ≤ 1, Journal of Complexity, 26, 629–640, 2010.

    Article  Google Scholar 

  13. S. Foucart and H. Rauhut, A mathematical introduction to compressive sensing, Applied and Numerical Harmonic Analysis, Birkhauser Basel, 2013.

  14. J. Haupt, W.U. Bajwa, G. Raz and R. Nowak, Toeplitz compressed sensing matrices with applications to sparse channel estimation, IEEE Transactions on Information Theory, 56, 5862–5875, 2010.

    Article  MathSciNet  Google Scholar 

  15. I. Haviv and O. Regev, The restricted isometry property of subsampled fourier matrices, In Geometric Aspects of Functional Analysis, 163–179, Springer, 2017.

    Book  Google Scholar 

  16. F. Krahmer, S. Mendelson and H. Rauhut, Suprema of chaos processes and the restricted isometry property, Communications on Pure and Applied Mathematics, 67, 1877–1904, 2014.

    Article  MathSciNet  Google Scholar 

  17. D. Liang, G. Xu, H. Wang, K. F. King, D. Xu and L. Ying, Toeplitz random encoding mr imaging using compressed sensing, IEEE International Symposium on Biomedical Imaging: From Nano to Macro, IEEE, 270–273, 2009.

    Book  Google Scholar 

  18. S. Mendelson, A. Pajor and N. T. Jaegermann, Uniform uncertainty principle for bernoulli and subgaussian ensembles, Constructive Approximation, 28, 277–289, 2008.

    Article  MathSciNet  Google Scholar 

  19. S. Mendelson, H. Rauhut and R. Ward, Improved bounds for sparse recovery from sub-sampled random convolutions, The Annals of Applied Probability, 28, 3491–3527, 2018.

    Article  MathSciNet  Google Scholar 

  20. B. K. Natarajan, Sparse approximate solutions to linear systems, SIAM Journal on Computing, 24, 227–234, 1995.

    Article  MathSciNet  Google Scholar 

  21. D. Needell and J. A. Tropp, CoSaMP: Iterative signal recovery from incomplete and inaccurate samples, Applied and Computational Harmonic Analysis, 26, 301–321, 2009.

    Article  MathSciNet  Google Scholar 

  22. H. Rauhut, Compressive sensing and structured random matrices, Theoretical Foundations and Numerical Methods for Sparse Recovery, 9, 1–92, 2010.

    MathSciNet  MATH  Google Scholar 

  23. J. Romberg, Compressive sensing by random convolution, SIAM Journal on Imaging Sciences, 2, 1098–1128, 2009.

    Article  MathSciNet  Google Scholar 

  24. M. Rudelson and R. Vershynin, On sparse reconstruction from Fourier and Gaussian measurements, Communications on Pure and Applied Mathematics, 61, 1025–1045, 2008.

    Article  MathSciNet  Google Scholar 

  25. Z. Xu, The performance of orthogonal multi-matching pursuit under RIP, J. Comp. Math., 33, 495–516, 2015.

    Article  Google Scholar 

  26. T. Zhang, Sparse recovery with orthogonal matching pursuit under RIP, IEEE Transactions on Information Theory, 57, 6215–6221, 2011.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Academy of Mathematics and System Science, Chinese Academy of Sciences, 100091, Bei**g, China

    Meng Huang, Yuxuan Pang & Zhiqiang Xu

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, 100049, Bei**g, China

    Zhiqiang Xu

Authors
  1. Meng Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuxuan Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhiqiang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Meng Huang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, M., Pang, Y. & Xu, Z. Improved bounds for the RIP of Subsampled Circulant Matrices. STSIP 18, 1–8 (2019). https://doi.org/10.1007/BF03549617

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03549617

Key words and phrases

2010 AMS Mathematics Subject Classficatio

Search

Navigation