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Iterative Hard Thresholding Based on Randomized Kaczmarz Method

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Abstract

The iterative hard thresholding methods have been recently developed to deal with the sparse regularization problems arising in compressed sensing and other sparse signal processing methods. The methods are attractive due to their simplicity. In this paper, we propose acceleration schemes for iterative hard thresholding methods based on randomized Kaczmarz method. The speedup is achieved by replacing the gradient step in the iterative hard thresholding methods with a randomized Kaczmarz algorithm. Numerical experiments with the sparse signal recovery show the simplicity and effectiveness of the proposed algorithms.

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References

  1. T. Blumensath, M.E. Davies, Iterative thresholding for sparse approximations. J. Fourier Anal. Appl. 14(5), 629–654 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. T. Blumensath, M.E. Davies, Iterative hard thresholding for compressed sensing. Appl. Comput. Harmon. Anal. 27(3), 265–274 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. T. Blumensath, M.E. Davies, Normalised iterative hard thresholding: guaranteed stability and performance. IEEE J. Select. Topics Signal Process. 4(2), 298–309 (2010)

    Article  Google Scholar 

  4. A.M. Bruckstein, D.L. Donoho, M. Elad, From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51(1), 34–81 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, New York, 2010)

    Book  Google Scholar 

  6. M. Elad, Sparse and redundant representation modeling–what next ? IEEE Signal Process. Lett. 19(12), 922–928 (2012)

    Article  Google Scholar 

  7. Y.C. Eldar, G. Kutyniok, Compressed Sensing: Theory and Applications (Cambridge University Press, Cambridge, 2012)

    Book  Google Scholar 

  8. S. Foucart, Hard thresholding pursuit: an algorithm for compressive sensing. SIAM J. Numer. Anal. 49(6), 2543–2563 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. G.T. Herman, L.B. Meyer, Algebraic reconstruction techniques can be made computationally efficient. IEEE Trans. Med. Imaging 12(3), 600–609 (1993)

    Article  Google Scholar 

  10. F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986)

    MATH  Google Scholar 

  11. K. Qiu, A. Dogandzic, Sparse signal reconstruction via ECME hard thresholding. IEEE Trans. Signal Process. 60(9), 5451–4569 (2012)

    Article  MathSciNet  Google Scholar 

  12. T. Strohmer, R. Vershynin, A randomized Kaczmarz algorithm with exponential convergence. J. Fourier Anal. Appl. 15(3), 262–278 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Z. Zhang, C. Chui, K. Liu, Iterative-hard threshold algorithm with momentum. Electronics Lett. 47(4), 257–259 (2011)

    Article  Google Scholar 

  14. Z.S. Zhang, H.X. Lv, Q.B. Zhou, H.Q. Zhang, Reconstructing sparse signals from dyadic wavelet transform modulus maxima. Circuits Syst. Signal Process. 33(8), 2667–2674 (2014)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

This work is supported by the ‘973’ National Basic Research Program of China (2010CB731900). The authors would like to thank the reviewers for their comments and suggestions that helped improve the quality of this paper.

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Correspondence to Zhuosheng Zhang.

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Zhang, Z., Yu, Y. & Zhao, S. Iterative Hard Thresholding Based on Randomized Kaczmarz Method. Circuits Syst Signal Process 34, 2065–2075 (2015). https://doi.org/10.1007/s00034-014-9934-y

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  • DOI: https://doi.org/10.1007/s00034-014-9934-y

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