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A generalized elastic net regularization with smoothed \(\ell _{q}\) penalty for sparse vector recovery

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Abstract

In this paper, we propose an iterative algorithm for solving the generalized elastic net regularization problem with smoothed \(\ell _{q} (0<q \le 1)\) penalty for recovering sparse vectors. We prove the convergence result of the algorithm based on the algebraic method. Under certain conditions, we show that the iterative solutions converge to a local minimizer of the generalized elastic net regularization problem and we also present an error bound. Theoretical analysis and numerical results show that the proposed algorithm is promising.

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References

  1. Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52, 1289–1306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Lustig, M., Donoho, D.L., Santos, J.M., Pauly, J.M.: Compressed sensing MRI. IEEE Signal Process. Mag. 25, 72–82 (2008)

    Article  Google Scholar 

  3. Duarte, M.F., Eldar, Y.C.: Structured compressed sensing: from theory to applications. IEEE Trans. Signal Process. 59, 4053–4085 (2011)

    Article  MathSciNet  Google Scholar 

  4. Natraajan, B.K.: Sparse approximation to linear systems. SIAM J. Comput. 24(2), 227–234 (1995)

    Article  MathSciNet  Google Scholar 

  5. Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Daubechies, I., Defries, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57, 1413–1457 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B 58(1), 267–288 (1996)

    MathSciNet  MATH  Google Scholar 

  8. Zou, H.: The adaptive lasso and its oracle properties. J. Am. Stat. Assoc. 101(467), 1418–1429 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Meinshausen, N., Yu, B.: Lasso-type recovery of sparse representations for high-dimensional data. Ann. Stat. 37(1), 246–270 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. J. R. Stat. Soc. B 67(2), 301–320 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, X., Xu, F., Ye, Y.: Lower bound theory of nonzero entries in solutions of \(\ell _{2}\)-\(\ell _{p}\) minimization. SIAM J. Sci. Comput. 32, 2832–2852 (2010)

    Article  MathSciNet  Google Scholar 

  12. Chartrand, R.: Exact reconstructions of sparse signals via nonconvex minimization. IEEE Signal Process. Lett. 14(10), 707–710 (2007)

    Article  Google Scholar 

  13. Chartrand, R., Staneva, V.: Restricted isometry properties and nonconvex compressive sensing. Inverse Probl. 24, 1–14 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lai, M.J., Wang, J.Y.: An unconstrained \(\ell _{q}\) minimization with \(0<q\le 1\) for sparse solution of underdetermined linear systems. SIAM J. Optim. 21(1), 82–101 (2011)

    Article  MathSciNet  Google Scholar 

  15. Lai, M.J., Xu, Y.Y., Yin, W.T.: Improved iteratively reweighted least squares for unconstrained smoothed \(\ell _{q}\) minimization. SIAM J. Numer. Anal. 51(2), 927–957 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Efron, B., Hastie, T., Johnstone, I., Tibshirani, R.: Least angle regression. Ann. Stat. 32(2), 407–499 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bian, W., Chen, X.: Worst-case complexity of smoothing quadratic regularization methods for nonLipschitzian optimization. SIAM J. Optim. 23(3), 1718–1741 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bian, W., Chen, X.: Linearly Constrained Non-Lipschitz Optimization for Image Restoration. SIAM J. Imaging Sci. 8(4), 2294–2322 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, Y.F., Ma, S.Q., Dai, Y.H., Zhang, S.Z.: A smoothing SQP framework for a class of composite \(L_{q}\) minimization over polyhedron. Math. Program. Ser. A (2015). doi:10.1007/s10107-015-0939-5

    MATH  Google Scholar 

  20. Candès, E.J., Wakin, M.B., Boyd, S.P.: Enchaning sparsity by reweighted \(\ell _{1}\) minimization. J. Fourier Anal. Appl. 14, 877–905 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chartrand, R., Yin, W.: Iteratively reweighted algorithms for compressive sensing. In: Proceeding of International Conference on Acoustics, Speech, Signal Processing (ICASSP’08), PP. 3869–3872 (2008)

  22. Daubechies, I., Devore, R., Fornasier, M., Güntük, C.S.: Iteratively reweighted least squares minimization for sparse recovery. Commun. Pure Appl. Math. 63, 1–38 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Garcia, C.B., Li, T.Y.: On the number of solutions to polynomial systems of equations. SIAM J. Numer. Anal. 17(4), 540–546 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sun, Q.: Sparse approximation property and stable recovery of sparse singals from noisy measurements. IEEE Trans. Signal Process. 59, 5086–5090 (2011)

    Article  MathSciNet  Google Scholar 

  25. Marjanovic, G., Solo, V.: On \(\ell _{q}\) optimization and matrix completion. IEEE Trans. Signal Process. 60, 5714–5724 (2012)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank deeply the referees and the editor for their helpful suggestions and comments on the paper, which have improved the presentation. This paper is partially supported by the Natural Science Foundation of Shanghai under grant (15ZR1416300) and the National Science Foundation of China under grants (61071186, 11171205).

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Authors and Affiliations

  1. Department of Mathematics, College of Science, Shanghai University, Shanghai, 200444, China

    Yong Zhang, Wanzhou Ye & Jianjun Zhang

  2. School of Science, Jiangsu University of Science and Technology, Zhenjiang, 212003, China

    Yong Zhang

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  1. Yong Zhang
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  2. Wanzhou Ye
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  3. Jianjun Zhang
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Correspondence to Wanzhou Ye.

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Zhang, Y., Ye, W. & Zhang, J. A generalized elastic net regularization with smoothed \(\ell _{q}\) penalty for sparse vector recovery. Comput Optim Appl 68, 437–454 (2017). https://doi.org/10.1007/s10589-017-9916-7

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  • DOI: https://doi.org/10.1007/s10589-017-9916-7

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