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Accelerated Bregman Method for Linearly Constrained \(\ell _1\)–\(\ell _2\) Minimization

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Abstract

In this paper, we consider the linearly constrained \(\ell _1\)–\(\ell _2\) minimization, and we propose an accelerated Bregman method for solving this minimization problem. The proposed method is based on the extrapolation technique, which is used in accelerated proximal gradient methods proposed by Nesterov and others, and on the equivalence between the Bregman method and the augmented Lagrangian method. A convergence rate of \(\mathcal{O }(\frac{1}{k^2})\) is proved for the proposed method when it is applied to solve a more general linearly constrained nonsmooth convex minimization problem. We numerically test our proposed method on a synthetic problem from compressive sensing. The numerical results confirm that our accelerated Bregman method is faster than the original Bregman method.

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References

  1. Baraniuk, R.: Compressive sensing. IEEE Signal Process. Mag. 24, 118–121 (2007)

    Article  Google Scholar 

  2. Barzilai, J., Borwein, J.: 1988 Two point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beck, A., Teboulle, M.: Fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  5. Bunea, R., She, Y., Ombao, H., Gongvatana, A., Devlin, K., Cohen, R.: Penalized least squares regression methods and applications to neuroimaging. NeuroImage 55, 1519–1527 (2011)

    Article  Google Scholar 

  6. Cai, J., Osher, S., Shen, Z.: Convergence of the linearized Bregman iteration for \(\ell _1\)-norm minimization. Math. Comput. 78, 2127–2136 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cai, J., Osher, S., Shen, Z.: Linearized Bregman iterations for compressed sensing. Math. Comput. 78, 1515–1536 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Candés, E., Romberg, J.: \(\ell _1\)-MAGIC : recovery of sparse signals via convex programming. Technical Report, Caltech (2005)

  9. Candés, E., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52, 489–509 (2006)

    Article  MATH  Google Scholar 

  10. Candés, E., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51, 4203–4215 (2005)

    Article  MATH  Google Scholar 

  11. Candés, E., Tao, T.: Near optimal signal recovery from random projections: Universal encoding strategies? IEEE Trans. Inf. Theory 52, 5406–5425 (2006)

    Article  Google Scholar 

  12. Candés, E., Wakin, M.: An introduction to compressive sampling. IEEE Signal Process. Mag. 21, 21–30 (2008)

    Article  Google Scholar 

  13. Chen, S., Donoho, D., Saunders, M.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20, 33–61 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cho, S., Kim, H., Oh, S., Kim, K., Park, T.: Elastic-net regularization approaches for genome-wide association studies of rheumatoid arthritis. BMC Proc. 3, S25 (2009)

    Article  Google Scholar 

  15. De Mol, C., De Vito, E., Rosasco, L.: Elastic net regularization in learning theory. J. Complex. 25, 201–230 (2009)

    Article  MATH  Google Scholar 

  16. Friedlander, M., Tseng, P.: Exact regularization of convex programs. SIAM J. Opt. 18, 1326–1350 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Friedman, J., Hastie, T., Hofling, H., Tibshirani, R.: Pathwise coordinate optimization. Ann. Appl. Stat. 1, 302–332 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hale, E., Yin, W., Zhang, Y.: Fixed-point continuation applied to compressed sensing: Implementation and numerical experiments. J. Comput. Math. 28, 170–194 (2010)

    MathSciNet  MATH  Google Scholar 

  19. He, B. and Yuan, X. : The acceleration of augmented Lagrangian method for linearly constrained optimization. optimization online. http://www.optimization-online.org/DB_FILE/2010/10/2760.pdf (2010).

  20. Hestenes, M.: Multiplier and gradient methods. J. Opt. Theory Appl. 4, 303–320 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hiriart-Urruty, J.-B., Lemarechal, C.: Convex Analysis and Minimization Algorithms: Part 1: Fundamentals. Springer, New York (1996)

    Google Scholar 

  22. Huang, B., Ma, S., Goldfarb, D.: Accelerated linearized Bregman method. J. Sci. Comput. (2012). doi: 10.1007/s10915-012-9592-9

  23. Li, Q., Lin, N.: The Bayesian elastic net. Bayesian. Analysis 5, 151–70 (2010)

    Google Scholar 

  24. Liu, D., Nocedal, J.: On the limited memory method for large scale optimization. Math. Program. B 45, 503–528 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nesterov, Y.: A method of solving a convex programming problem with convergence rate \({\cal O}(\frac{1}{k^2})\). Sov. Math. Doklady 27, 372–376 (1983)

    Google Scholar 

  26. Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. SIAM J. Multiscale Model. Simul. 4, 460–489 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Osher, S., Mao, Y., Dong, B., Yin, W.: Fast Linearized Bregman iteration for compressive sensing and sparse denoising. Commun. Math. Sci. 8, 93–111 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (1998)

    Book  MATH  Google Scholar 

  29. Saunders, M. : PDCO. Matlab software for convex optimization. Technical report, Stanford University. http://www.stanford.edu/group/SOL/software/pdco.html (2002)

  30. Sra, S., Tropp. J. A.: Row-action methods for compressive sensing, vol 3. In ICASSP, Toulouse, pp. 868–871 (2006).

  31. Van den Breg, E., Friedlander, M.P.: Probing the pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31, 890–912 (2009)

    Article  Google Scholar 

  32. Yang, Y., Moller, M., Osher, S. : A dual split Bregman method for fast L1 minimization. Math. Comput. (to appear)

  33. Yin, W.: Analysis and generalizations of the linearized Bregman method. SIAM J. Imaging Sci. 3, 856–877 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yin, W., Osher, S., Goldfarb, D., Darbon, J.: Bregman iterative algorithms for \(\ell _1\)-minimization with applications to compressive sensing. SIAM J. Imaging Sci. 1, 143–168 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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Acknowledgments

Sangwoon Yun and Hyenkyun Woo were supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2012R1A1A1006406 and 2010-0510-1-3 respectively). Myungjoo Kang was supported by the NRF funded by the Ministry of Education, Science and Technology(2012001766).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Seoul National University, Seoul, Korea

    Myeongmin Kang & Myungjoo Kang

  2. School of Computational Science and Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, United States of America

    Hyenkyun Woo

  3. Department of Mathematics Education, Sungkyunkwan University, Seoul, Korea

    Sangwoon Yun

Authors
  1. Myeongmin Kang
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  2. Sangwoon Yun
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  3. Hyenkyun Woo
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  4. Myungjoo Kang
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Correspondence to Myungjoo Kang.

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Kang, M., Yun, S., Woo, H. et al. Accelerated Bregman Method for Linearly Constrained \(\ell _1\)–\(\ell _2\) Minimization. J Sci Comput 56, 515–534 (2013). https://doi.org/10.1007/s10915-013-9686-z

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  • DOI: https://doi.org/10.1007/s10915-013-9686-z

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