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An Implementable Splitting Algorithm for the \(\ell _1\)-norm Regularized Split Feasibility Problem

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Abstract

The split feasibility problem (SFP) captures a wide range of inverse problems, such as signal processing, image reconstruction, and so on. Recently, applications of \(\ell _1\)-norm regularization to linear inverse problems, a special case of SFP, have been received a considerable amount of attention in the signal/image processing and statistical learning communities. However, the study of the \(\ell _1\)-norm regularized SFP still deserves attention, especially in terms of algorithmic issues. In this paper, we shall propose an algorithm for solving the \(\ell _1\)-norm regularized SFP. More specifically, we first formulate the \(\ell _1\)-norm regularized SFP as a separable convex minimization problem with linear constraints, and then introduce our splitting method, which takes advantage of the separable structure and gives rise to subproblems with closed-form solutions. We prove global convergence of the proposed algorithm under certain mild conditions. Moreover, numerical experiments on an image deblurring problem verify the efficiency of our algorithm.

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References

  1. Afonso, M., Bioucas-Dias, J., Figueiredo, M.: Fast image recovery using variable splitting and constrained optimization. IEEE Trans. Image Process. 19, 2345–2356 (2010)

    Article  MathSciNet  Google Scholar 

  2. Bai, Z., Chen, M., Yuan, X.: Applications of the alternating direction method of multipliers to the semidefinite inverse quadratic eigenvalue problems. Inverse Probl. 29, 075,011 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Beck, A., Teboulle, M.: Gradient-based algorithms with applications to signal recovery problems. In: Palomar, D.P., Eldar, Y.C. (eds.) Convex Optimization in Signal Processing and Communications, pp. 33–88. Cambridge University Press, New York (2010)

    Google Scholar 

  5. Bertsekas, D., Tsitsiklis, J.: Parallel and Distributed Computation, Numerical Methods. Prentice-Hall, Englewood Cliffs, NJ (1989)

    MATH  Google Scholar 

  6. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3, 1–122 (2010)

    Article  MATH  Google Scholar 

  7. Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Ceng, L., Ansari, Q., Yao, J.: Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem. Nonlinear Anal. 75, 2115–2116 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)

    Article  Google Scholar 

  12. Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in product space. Numer. Algorithms 8, 221–239 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Censor, Y., Jiang, M., Wang, G.: Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning, and Inverse Problems. Medical Physics Publishing Madison, Wisconsin (2010)

    Google Scholar 

  16. Censor, Y., Motova, A., Segal, A.: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl. 327, 1244–1256 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Censor, Y., Segal, A.: Iterative projection methods in biomedical inverse problems. In: Censor, Y., Jiang, M., Louis, A. (eds.) Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), pp. 65–96. Edizioni della Normale, Pisa (2008)

    Google Scholar 

  18. Chan, R., Tao, M., Yuan, X.: Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers. SIAM J. Imaging Sci. 6, 680–697 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chan, R., Yang, J., Yuan, X.: Alternating direction method for image inpainting in wavelet domain. SIAM J. Imaging Sci. 4, 807–826 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chan, R.H., Tao, M., Yuan, X.: Linearized alternating direction method of multipliers for constrained linear least squares problems. East Asian J. Appl. Math. 2, 326–341 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Chen, C., He, B., Yuan, X.: Matrix completion via alternating direction method. IMA J. Numer. Anal. 32, 227–245 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, S., Donoho, D., Saunders, M.: Automatic decomposition by basis pursuit. SIAM Rev. 43, 129–159 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Combettes, P., Pesquet, J.: Proximal splitting methods in signal processing. In: Bauschke, H., Burachik, R., Combettes, P., Elser, V., Luke, D., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optimization and Its Applications, vol. 49, pp. 185–212. Springer, New York (2011)

    Chapter  Google Scholar 

  24. Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dai, Y., Fletcher, R.: Projected Barzilai–Borwein methods for large-scale box-constrained quadratic programming. Numerische Mathematik 100, 21–47 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dang, Y., Gao, Y.: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 27, 015007 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Donoho, D.: Compressed sensing. IEEE Trans. Inform. Theory 52, 1289–1306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Eckstein, J., Bertsekas, D.: On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  29. Eicke, B.: Iteration methods for convexly constrained ill-posed problems in Hilbert space. Numer. Funct. Anal. Optim. 13, 413–429 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  30. Engl, H., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publishers Group, Dordrecht, The Netherlands (1996)

    Book  MATH  Google Scholar 

  31. Figueiredo, M., Nowak, R., Wright, S.: Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 1, 586–598 (2007)

    Article  Google Scholar 

  32. Frick, K., Grasmair, M.: Regularization of linear ill-posed problems by the augmented Lagrangian method and variational inequalities. Inverse Probl. 28, 104,005 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximations. Comput. Math. Appl. 2, 16–40 (1976)

    Article  MATH  Google Scholar 

  34. Han, D., He, H., Yang, H., Yuan, X.: A customized Douglas--Rachford splitting algorithm for separable convex minimization with linear constraints. Numer. Math. 127, 167–200 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. He, B., Liao, L., Han, D., Yang, H.: A new inexact alternating direction method for monotone variational inequalities. Math. Program. 92, 103–118 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. He, B., Xu, M., Yuan, X.: Solving large-scale least squares covariance matrix problems by alternating direction methods. SIAM J. Matrix Anal. Appl. 32, 136–152 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. He, B., Yuan, X., Zhang, W.: A customized proximal point algorithm for convex minimization with linear constraints. Comput. Optim. Appl. 56, 559–572 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. He, S., Zhu, W.: A note on approximating curve with 1-norm regularization method for the split feasibility problem. J. Appl. Math. 2012, Article ID 683,890, 10 pp (2012)

  39. Hochstenbach, M., Reichel, L.: Fractional Tikhonov regularization for linear discrete ill-posed problems. BIT Numer. Math. 51, 197–215 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ma, S.: Alternating direction method of multipliers for sparse principal component analysis. J. Oper. Res. Soc. China 1, 253–274 (2013)

    Article  MATH  Google Scholar 

  41. Martinet, B.: Régularization d’ inéquations variationelles par approximations sucessives. Rev. Fr. Inform. Rech. Opér. 4, 154–159 (1970)

    MathSciNet  Google Scholar 

  42. Moreau, J.: Fonctions convexe duales et points proximaux dans un espace hilbertien. C. R. Acad. Sci. Paris Ser. A Math. 255, 2897–2899 (1962)

    MathSciNet  MATH  Google Scholar 

  43. Morini, S., Porcelli, M., Chan, R.: A reduced Newton method for constrained linear least squares problems. J. Comput. Appl. Math. 233, 2200–2212 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Parikh, N., Boyd, S.: Proximal algorithms. Found. Trends Optim. 1, 123–231 (2013)

    Google Scholar 

  45. Potter, L., Arun, K.: A dual approach to linear inverse problems with convex constraints. SIAM J. Control Optim. 31, 1080–1092 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  46. Qu, B., Xiu, N.: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 21, 1655–1665 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  47. Rockafellar, R.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  48. Sabharwal, A., Potter, L.: Convexly constrained linear inverse problems: iterative least-squares and regularization. IEEE Trans. Signal Process. 46, 2345–2352 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  49. Schopfer, F., Louis, A., Schuster, T.: Nonlinear iterative methods for linear ill-posed problems in Banach spaces. Inverse Probl. 22, 311–329 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  51. Tikhonov, A., Arsenin, V.: Solutions of Ill-Posed Problems. Winston, New York (1977)

    MATH  Google Scholar 

  52. Wang, X., Yuan, X.: The linearized alternating direction method of multipliers for Dantzig selector. SIAM J. Sci. Comput. 34, A2792–A2811 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  53. Wright, S., Nowak, R., Figueiredo, M.: Sparse reconstruction by separable approximation. IEEE Trans. Signal Process. 57, 2479–2493 (2009)

    Article  MathSciNet  Google Scholar 

  54. Xu, H.: A variable Krasnosel’skii–Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006)

    Article  MATH  Google Scholar 

  55. Xu, H.: Iterative methods for the split feasibility problem in infinite dimensional Hilbert spaces. Inverse Probl. 26, 105,018 (2010)

    Article  MathSciNet  Google Scholar 

  56. Yang, J., Zhang, Y.: Alternating direction algorithms for \(\ell _1\)-problems in compressive sensing. SIAM J. Sci. Comput. 332, 250–278 (2011)

    Article  Google Scholar 

  57. Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20, 1261–1266 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  58. Yao, Y., Wu, J., Liou, Y.: Regularized methods for the split feasibility problem. Abstr. Appl. Anal. 2012, Article ID 140,679, 13 pp (2012)

  59. Yuan, X.: Alternating direction methods for covariance selection models. J. Sci. Comput. 51, 261–273 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  60. Zhang, H., Wang, Y.: A new CQ method for solving split feasibility problem. Front. Math. China 5, 37–46 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  61. Zhang, J., Morini, B.: Solving regularized linear least-squares problems by the alternating direction method with applications to image restoration. Electron. Trans. Numer. Anal. 40, 356–372 (2013)

    MathSciNet  MATH  Google Scholar 

  62. Zhang, W., Han, D., Li, Z.: A self-adaptive projection method for solving the multiple-sets split feasibility problem. Inverse Probl. 25, 115,001 (2009)

    Article  MathSciNet  Google Scholar 

  63. Zhang, W., Han, D., Yuan, X.: An efficient simultaneous method for the constrained multiple-sets split feasibility problem. Comput. Optim. Appl. 52, 825–843 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors would like to thank the two anonymous referees for their constructive comments, which significantly improved the presentation of this paper. The first two authors were supported by National Natural Science Foundation of China (11301123; 11171083) and the Zhejiang Provincial NSFC Grant No. LZ14A010003, and the third author in part by NSC 102-2115-M-110-001-MY3.

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Authors and Affiliations

  1. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, Zhejiang, China

    Hongjin He, Chen Ling & Hong-Kun Xu

  2. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

    Hong-Kun Xu

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  1. Hongjin He
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Correspondence to Hong-Kun Xu.

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He, H., Ling, C. & Xu, HK. An Implementable Splitting Algorithm for the \(\ell _1\)-norm Regularized Split Feasibility Problem. J Sci Comput 67, 281–298 (2016). https://doi.org/10.1007/s10915-015-0078-4

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