Skip to main content
SpringerLink
Log in

Properties and iterative methods for the lasso and its variants

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

The lasso of Tibshirani (1996) is a least-squares problem regularized by the ℓ 1 norm. Due to the sparseness promoting property of the ℓ 1 norm, the lasso has been received much attention in recent years. In this paper some basic properties of the lasso and two variants of it are exploited. Moreover, the proximal method and its variants such as the relaxed proximal algorithm and a dual method for solving the lasso by iterative algorithms are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Candes, E. J., Romberg, J. and Tao, T., Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52(2), 2006, 489–509.

    Article  MATH  MathSciNet  Google Scholar 

  2. Candes, E. J., Romberg, J. and Tao, T., Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Applied Math., 59(2), 2006, 1207–1223.

    MATH  MathSciNet  Google Scholar 

  3. Candes, E. J. and Tao, T., Near-optimal signal recovery from random projections: Universal encoding strategies? IEEE Trans. Inform. Theory, 52(12), 2006, 5406–5425.

    Article  MathSciNet  Google Scholar 

  4. Candes, E. J. and Wakin, M. B., An introduction to compressive sampling, IEEE Signal Processing Magazine, 2008, 21–30.

    Google Scholar 

  5. Cipra, B. A., ℓ 1-magic, SIAM News, 39(9), 2006.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Donoho, D., Compressed sensing, IEEE Trans. Inform. Theory, 52(4), 2006, 1289–1306.

    Article  MathSciNet  Google Scholar 

  8. Friedman, J., Hastie, T. and Tibshirani, R., A note on the group lasso and a sparse group lasso, arXiv:1001.0736V1.

  9. Geobel, K. and Kirk, W. A., Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, Vol. 28, Cambridge University Press, 1990.

    Google Scholar 

  10. Hebiri, M. and van de Geer, S., The smooth-lasso and other ℓ 1+ℓ 2-penalized methods, Electron. J. Statist., 5, 2011, 1184–1226.

    Article  MATH  Google Scholar 

  11. Marino, G. and Xu, H. K., Convergence of generalized proximal point algorithms, Comm. Pure Appl. Anal., 3(3), 2004, 791–808.

    MATH  MathSciNet  Google Scholar 

  12. Micchelli, C. A., Shen, L. and Xu, Y., Proximity algorithms for image models: Denoising, Inverse Problems, 27, 2011, 045009, 30pp.

    Article  MathSciNet  Google Scholar 

  13. Moreau, J. J., Proprietes des applications “prox”, C. R. Acad. Sci. Paris Ser. A Math., 256, 1963, 1069–1071.

    MATH  MathSciNet  Google Scholar 

  14. Moreau, J. J., Proximite et dualite dans un espace hilbertien, Bull. Soc. Math. France, 93, 1965, 272–299.

    MathSciNet  Google Scholar 

  15. Raasch, T., On the L-curve criterion in ℓ 1 regularization of linear discrete ill-posed problems, International Conference on Inverse Problems and Related Topics, Nanjing, 2012.

    Google Scholar 

  16. Tibshirani, R., Regression shrinkage and selection via the lasso, J. Royal Statist. Soc. Ser. B, 58, 1996, 267–288.

    MATH  MathSciNet  Google Scholar 

  17. Tibshirani, R., Saunders, M., Rosset, R., et al., Sparsity and smoothness via the fused lasso, J. Royal Statist. Soc., Ser. B, 67, 2005, 91–108.

    Article  MATH  MathSciNet  Google Scholar 

  18. Xu, H. K., Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl., 150, 2011, 360–378.

    Article  MATH  MathSciNet  Google Scholar 

  19. Yuan, M. and Lin, Y., Model selection and estimation in regression with grouped variables, J. Royal Statist. Soc., Ser. B, 68, 2006, 49–67.

    Article  MATH  MathSciNet  Google Scholar 

  20. Zou, H. and Hastie, T., Regularization and variable selection via the elastic net, J. Royal Statist. Soc., Ser. B, 67, 2005, 301–320.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, 80424, Taiwan

    Hong-Kun Xu

  2. Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah, 21589, Saudi Arabia

    Hong-Kun Xu

Authors
  1. Hong-Kun Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hong-Kun Xu.

Additional information

This work was supported by NSC 102-2115-M-110-001-MY3.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, HK. Properties and iterative methods for the lasso and its variants. Chin. Ann. Math. Ser. B 35, 501–518 (2014). https://doi.org/10.1007/s11401-014-0829-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-014-0829-9

Keywords

2010 MR Subject Classification

Search

Navigation