Journal of Scientific Computing

, Volume 74, Issue 2, pp 767–785 | Cite as

Fast L1–L2 Minimization via a Proximal Operator



This paper aims to develop new and fast algorithms for recovering a sparse vector from a small number of measurements, which is a fundamental problem in the field of compressive sensing (CS). Currently, CS favors incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent, and conventional methods such as \(L_1\) minimization do not work well. Recently, the difference of the \(L_1\) and \(L_2\) norms, denoted as \(L_1\)\(L_2\), is shown to have superior performance over the classic \(L_1\) method, but it is computationally expensive. We derive an analytical solution for the proximal operator of the \(L_1\)\(L_2\) metric, and it makes some fast \(L_1\) solvers such as forward–backward splitting (FBS) and alternating direction method of multipliers (ADMM) applicable for \(L_1\)\(L_2\). We describe in details how to incorporate the proximal operator into FBS and ADMM and show that the resulting algorithms are convergent under mild conditions. Both algorithms are shown to be much more efficient than the original implementation of \(L_1\)\(L_2\) based on a difference-of-convex approach in the numerical experiments.


Compressive sensing Proximal operator Forward–backward splitting Alternating direction method of multipliers Difference-of-convex 

Mathematics Subject Classification

90C26 65K10 49M29 



The authors would like to thank Zhi Li and the anonymous reviewers for valuable comments.


  1. 1.
    Beck, A., Teboulle, M.: A fast iterative shrinkage–thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bredies, K., Lorenz, D.A., Reiterer, S.: Minimization of non-smooth, non-convex functionals by iterative thresholding. J. Optim. Theory Appl. 165(1), 78–112 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Candès, E.J., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59, 1207–1223 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chartrand, R.: Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Process. Lett. 10(14), 707–710 (2007)CrossRefGoogle Scholar
  5. 5.
    Chartrand, R., Yin, W.: Iteratively reweighted algorithms for compressive sensing. In: International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 3869–3872 (2008)Google Scholar
  6. 6.
    Cheney, W., Goldstein, A.A.: Proximity maps for convex sets. Proc. Am. Math. Soc. 10(3), 448–450 (1959)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Donoho, D., Elad, M.: Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization. Proc. Natl. Acad. Sci. U.S.A. 100, 2197–2202 (2003)CrossRefMATHGoogle Scholar
  8. 8.
    Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Esser, E., Lou, Y., Xin, J.: A method for finding structured sparse solutions to non-negative least squares problems with applications. SIAM J. Imaging Sci. 6(4), 2010–2046 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fannjiang, A., Liao, W.: Coherence pattern-guided compressive sensing with unresolved grids. SIAM J. Imaging Sci. 5(1), 179–202 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gribonval, R., Nielsen, M.: Sparse representations in unions of bases. IEEE Trans. Inf. Theory 49(12), 3320–3325 (2003)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Huang, X., Shi, L., Yan, M.: Nonconvex sorted l1 minimization for sparse approximation. J. Oper. Res. Soc. China 3, 207–229 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Krishnan, D., Fergus, R.: Fast image deconvolution using hyper-Laplacian priors. In: Advances in Neural Information Processing Systems (NIPS), pp. 1033–1041 (2009)Google Scholar
  14. 14.
    Lai, M.J., Xu, Y., Yin, W.: Improved iteratively reweighted least squares for unconstrained smoothed lq minimization. SIAM J. Numer. Anal. 5(2), 927–957 (2013)CrossRefMATHGoogle Scholar
  15. 15.
    Li, G., Pong, T.K.: Global convergence of splitting methods for nonconvex composite optimization. SIAM J. Optim. 25, 2434–2460 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Li, H., Lin, Z.: Accelerated proximal gradient methods for nonconvex programming. In: Advances in Neural Information Processing Systems, pp. 379–387 (2015)Google Scholar
  17. 17.
    Liu, T., Pong, T.K.: Further properties of the forward-backward envelope with applications to difference-of-convex programming. Comput. Optim. Appl. 67(3), 489–520 (2017)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lorenz, D.A.: Constructing test instances for basis pursuit denoising. Trans. Signal Process. 61(5), 1210–1214 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lou, Y., Osher, S., Xin, J.: Computational aspects of l1-l2 minimization for compressive sensing. In: Le Thi, H., Pham Dinh, T., Nguyen, N. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences. Advances in Intelligent Systems and Computing, vol. 359, pp. 169–180. Springer, Cham (2015)Google Scholar
  20. 20.
    Lou, Y., Yin, P., He, Q., Xin, J.: Computing sparse representation in a highly coherent dictionary based on difference of l1 and l2. J. Sci. Comput. 64(1), 178–196 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lou, Y., Yin, P., Xin, J.: Point source super-resolution via non-convex l1 based methods. J. Sci. Comput. 68(3), 1082–1100 (2016)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Mammone, R.J.: Spectral extrapolation of constrained signals. J. Opt. Soc. Am. 73(11), 1476–1480 (1983)CrossRefGoogle Scholar
  23. 23.
    Natarajan, B.K.: Sparse approximate solutions to linear systems. SIAM J. Comput. 24, 227–234 (1995)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Papoulis, A., Chamzas, C.: Improvement of range resolution by spectral extrapolation. Ultrason. Imaging 1(2), 121–135 (1979)CrossRefGoogle Scholar
  25. 25.
    Pham-Dinh, T., Le-Thi, H.A.: A DC optimization algorithm for solving the trust-region subproblem. SIAM J. Optim. 8(2), 476–505 (1998)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Repetti, A., Pham, M.Q., Duval, L., Chouzenoux, E., Pesquet, J.C.: Euclid in a taxicab: sparse blind deconvolution with smoothed regularization. IEEE Signal Process. Lett. 22(5), 539–543 (2015)CrossRefGoogle Scholar
  27. 27.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1997)MATHGoogle Scholar
  28. 28.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Dordrecht (2009)MATHGoogle Scholar
  29. 29.
    Santosa, F., Symes, W.W.: Linear inversion of band-limited reflection seismograms. SIAM J. Sci. Stat. Comput. 7(4), 1307–1330 (1986)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Wang, Y., Yin, W., Zeng, J.: Global convergence of ADMM in nonconvex nonsmooth optimization. arXiv:1511.06324 [cs, math] (2015)
  31. 31.
    Woodworth, J., Chartrand, R.: Compressed sensing recovery via nonconvex shrinkage penalties. Inverse Probl. 32(7), 075,004 (2016)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Wu, L., Sun, Z., Li, D.H.: A Barzilai–Borwein-like iterative half thresholding algorithm for the \(l_{1/2}\) regularized problem. J. Sci. Comput. 67, 581–601 (2016)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Xu, Z., Chang, X., Xu, F., Zhang, H.: \(l_{1/2}\) regularization: a thresholding representation theory and a fast solver. IEEE Trans. Neural Netw. Learn. Syst. 23, 1013–1027 (2012)CrossRefGoogle Scholar
  34. 34.
    Yin, P., Lou, Y., He, Q., Xin, J.: Minimization of \(l_1\)\(l_2\) for compressed sensing. SIAM J. Sci. Comput. 37, A536–A563 (2015)CrossRefMATHGoogle Scholar
  35. 35.
    Zhang, S., Xin, J.: Minimization of transformed \(l_1\) penalty: Theory, difference of convex function algorithm, and robust application in compressed sensing. arXiv preprint arXiv:1411.5735 (2014)

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of Texas at DallasRichardsonUSA
  2. 2.Department of Computational Mathematics, Science and Engineering (CMSE)Michigan State UniversityEast LansingUSA
  3. 3.Department of MathematicsMichigan State UniversityEast LansingUSA

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