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Signal reconstruction by conjugate gradient algorithm based on smoothing \(l_1\)-norm

  • Caiying Wu
  • Jiaming Zhan
  • Yue Lu
  • Jein-Shan ChenEmail author
Article
  • 69 Downloads

Abstract

The \(l_1\)-norm regularized minimization problem is a non-differentiable problem and has a wide range of applications in the field of compressive sensing. Many approaches have been proposed in the literature. Among them, smoothing \(l_1\)-norm is one of the effective approaches. This paper follows this path, in which we adopt six smoothing functions to approximate the \(l_1\)-norm. Then, we recast the signal recovery problem as a smoothing penalized least squares optimization problem, and apply the nonlinear conjugate gradient method to solve the smoothing model. The algorithm is shown globally convergent. In addition, the simulation results not only suggest some nice smoothing functions, but also show that the proposed algorithm is competitive in view of relative error.

Keywords

\(l_1\)-norm regularization Compressive sensing Conjugate gradient algorithm Smoothing function 

Mathematics Subject Classification

90C33 

Notes

Acknowledgements

The authors would like to thank the anonymous referee and the editor for their valuable comments, viewpoints, and suggestions, which help improve the manuscript a lot.

References

  1. 1.
    Beck, A., Teboulle, M.: A fast iterative shrinkage–thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Becker, S., Bobin, J., Candes, E.: NESTA: a fast and accurate first-order method for sparse recovery. SIAM J. Imaging Sci. 4, 1–39 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Berg, E., Friedlander, M.P.: Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31, 890–912 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bioucas-Dias, J.M., Figueiredo, M.: A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration. IEEE Trans. Image Process. 16, 2992–3004 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Broughton, R.L., Coope, I.D., Renaud, P.F., Tappenden, R.E.H.: A box constrained gradient projection algorithm for compressed sensing. Signal Process. 91, 1985–1992 (2011)CrossRefGoogle Scholar
  6. 6.
    Candes, E.J.: The restricted isometry property and its implications for compressed sensing. Comptes Rendus Mathematique 346(9–10), 589–592 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Candes, E.J., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Candes, E.J., Tao, T.: Near optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Candes, E.J., Romberg, J.K., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Candes, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, C., Mangasarian, O.L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comput. Optim. Appl. 5, 97–138 (1996)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen, X.: Smoothing methods for nonsmooth, nonconvex minimization. Math. Program. 134, 71–99 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen, X., Womersley, R.S., Ye, J.: Minimizing the condition number of Gram matrix. SIAM J. Optim. 21(1), 127–148 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chen, X., Zhou, W.: Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization. SIAM J. Imaging Sci. 3(4), 765–790 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    De Mol, C., Defrise, M.: A note on wavelet-based inversion algorithms. Contemp. Math. 313, 85–96 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Elad, M.: Why simple shrinkage is still relevant for redundant representations? IEEE Trans. Inf. Theory 52(12), 5559–5569 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Figueiredo, M., Nowak, R., Wright, S.J.: Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 1, 586–597 (2007)CrossRefGoogle Scholar
  19. 19.
    Hale, E.T., Yin, W., Zhang, Y.: Fixed-point continuation for \(l_1\)-minimization: methodology and convergence. SIAM J. Optim. 19, 1107–1130 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hale, E.T., Yin, W., Zhang, Y.: Fixed-point continuation applied to compressed sensing: implementation and numerical experiments. J. Comput. Math. 28, 170–194 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Nowak, R., Figueiredo, M.: Fast wavelet-based image deconvolution using the EM algorithm. In: Proceedings of the 35th Asilomar Conference on Signals, Systems and Computers, vol. 1, pp. 371–375 (2001)Google Scholar
  22. 22.
    Qi, L., Sun, D.: Smoothing functions and smoothing Newton method for complementarity and variational inequality problems. J. Optim. Theory Appl. 113, 121–147 (2001)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Saheya, B., Yu, C.H., Chen, J.-S.: Numerical comparisons based on four smoothing functions for absolute value equation. J. Appl. Math. Comput. 56(1–2), 131–149 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Starck, J.L., Candes, E., Donoho, D.: Astronomical image representation by the curvelet transform. Astron. Astrophys. 398, 785–800 (2003)CrossRefGoogle Scholar
  25. 25.
    Starck, J.L., Nguyen, M., Murtagh, F.: Wavelets and curvelets for image deconvolution: a combined approach. Signal Process. 83, 2279–2283 (2003)CrossRefGoogle Scholar
  26. 26.
    Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. 58, 267–268 (1996)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Voronin, S., Ozkaya, G., Yoshida, D.: Convolution based smooth approximations to the absolute value function with application to non-smooth regularization. arXiv:1408.6795v2 [math.NA] 1, (July  2015)
  28. 28.
    Wang, X., Liu, F., Jiao, L.C., Wu, J., Chen, J.: Incomplete variables truncated conjugate gradient method for signal reconstruction in compressed sensing. Inf. Sci. 288, 387–411 (2014)CrossRefGoogle Scholar
  29. 29.
    Wright, S., Nowak, R., Figueiredo, M.: Sparse reconstruction by separable approximation. In: Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (October 2008)Google Scholar
  30. 30.
    Yang, J., Zhang, Y.: Alternating direction algorithms for \(l_1\)-problems in compressive sensing. SIAM J. Sci. Comput. 33, 250–278 (2011)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Yin, K., Xiao, Y.H., Zhang, M.L.: Nonlinear conjugate gradient method for \(l_1\)-norm regularization problems in compressive sensing. J. Comput. Inf. Syst. 7, 880–885 (2011)Google Scholar
  32. 32.
    Zhang, L., Zhou, W., Li, D.: A descent modified Polak–Ribiere–Polyak conjugate gradient method and its global convergence. IMA J. Numer. Anal. 26, 629–640 (2006)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zhu, H., Xiao, Y.H., Wu, S.Y.: Large sparse signal recovery by conjugate gradient algorithm based on smoothing technique. Comput. Math. Appl. 66, 24–32 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica (IIT) 2019

Authors and Affiliations

  • Caiying Wu
    • 1
  • Jiaming Zhan
    • 1
  • Yue Lu
    • 2
  • Jein-Shan Chen
    • 3
    Email author
  1. 1.College of Mathematics ScienceInner Mongolia UniversityHohhotChina
  2. 2.School of Mathematical SciencesTianjin Normal UniversityTianjinChina
  3. 3.Department of MathematicsNational Taiwan Normal UniversityTaipeiTaiwan

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