Journal of Global Optimization

, Volume 73, Issue 2, pp 411–429 | Cite as

An inexact splitting method for the subspace segmentation from incomplete and noisy observations

  • Renli Liang
  • Yanqin BaiEmail author
  • Hai Xiang Lin


Subspace segmentation is a fundamental issue in computer vision and machine learning, which segments a collection of high-dimensional data points into their respective low-dimensional subspaces. In this paper, we first propose a model for segmenting the data points from incomplete and noisy observations. Then, we develop an inexact splitting method for solving the resulted model. Moreover, we prove the global convergence of the proposed method. Finally, the inexact splitting method is implemented on the clustering problems in synthetic and benchmark data, respectively. Numerical results demonstrate that the proposed method is computationally efficient, robust as well as more accurate compared with the state-of-the-art algorithms.


Subspace segmentation Low rank representation Inexact augmented Lagrange multiplier method 

Mathematics Subject Classification

65K05 90C25 90C30 94A08 



The authors would like to thank the financial support from the China Scholarship Council.


  1. 1.
    Bauschke, H., Combettes, P.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 26(2), 248–264 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bradley, P., Mangasarian, O.: k-Plane clustering. J. Glob. Optim. 16(1), 23–32 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Butenko, S., Chaovalitwongse, W., Pardalos, P.: Clustering Challenges in Biological Networks. World Scientific, Singapore (2009)CrossRefGoogle Scholar
  4. 4.
    Cai, J., Candès, E., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Candès, E., Plan, Y.: Matrix completion with noise. Proc. IEEE 98(6), 925–936 (2010)CrossRefGoogle Scholar
  6. 6.
    Candès, E., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, G., Lerman, G.: Spectral curvature clustering (SCC). Int. J. Comput. Vis. 81(3), 317–330 (2009)CrossRefGoogle Scholar
  8. 8.
    Elhamifar, E., Vidal, R.: Sparse subspace clustering: algorithm, theory, and applications. IEEE Trans. Pattern Anal. Mach. Intell. 35(11), 2765–2781 (2013). CrossRefGoogle Scholar
  9. 9.
    Goh, A., Vidal, R.: Segmenting motions of different types by unsupervised manifold clustering. In: 2007 IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–6 (2007)Google Scholar
  10. 10.
    Gruber, A., Weiss, Y.: Multibody factorization with uncertainty and missing data using the EM algorithm. In: Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 707–714 (2004).
  11. 11.
    Han, L., Bi, S.: Two-stage convex relaxation approach to low-rank and sparsity regularized least squares loss. J. Glob. Optim. (2017). zbMATHGoogle Scholar
  12. 12.
    He, B., Tao, M., Yuan, X.: A splitting method for separable convex programming. IMA J. Numer. Anal. 35(1), 394–426 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ho, J., Yang, M., Lim, J., Lee, K., Kriegman, D.: Clustering appearances of objects under varying illumination conditions. In: 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 11–18 (2003)Google Scholar
  14. 14.
    Hong, W., Wright, J., Huang, K., Ma, Y.: Multiscale hybrid linear models for lossy image representation. IEEE Trans. Image Process. 15(12), 3655–3671 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kanatani, K.: Motion segmentation by subspace separation: model selection and reliability evaluation. Int. J. Image Graph. 2(2), 179–197 (2002)CrossRefGoogle Scholar
  16. 16.
    Lee, K., Ho, J., Kriegman, D.: Acquiring linear subspaces for face recognition under variable lighting. IEEE Trans. Pattern Anal. Mach. Intell. 27(5), 684–698 (2005)CrossRefGoogle Scholar
  17. 17.
    Lin, Z., Chen, M., Ma, Y.: The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices (2010). Eprint arXiv:1009.5055
  18. 18.
    Liu, G., Lin, Z., Yan, S., Sun, J., Yu, Y., Ma, Y.: Robust recovery of subspace structures by low-rank representation. IEEE Trans. Pattern Anal. Mach. Intell. 35(1), 171–184 (2013)CrossRefGoogle Scholar
  19. 19.
    Liu, G., Yan, S.: Latent low-rank representation for subspace segmentation and feature extraction. In: 2011 International Conference on Computer Vision, pp. 1615–1622 (2011).
  20. 20.
    Liu, Y., Jiao, L., Shang, F.: A fast tri-factorization method for low-rank matrix recovery and completion. Pattern Recognit. 46(1), 163–173 (2013)CrossRefzbMATHGoogle Scholar
  21. 21.
    Lu, C., Min, H., Zhao, Z., Zhu, L., Huang, D., Yan, S.: Robust and Efficient Subspace Segmentation via Least Squares Regression, pp. 347–360. Springer, Berlin (2012)Google Scholar
  22. 22.
    Ma, Y., Yang, A., Derksen, H., Fossum, R.: Estimation of subspace arrangements with applications in modeling and segmenting mixed data. SIAM Rev. 50(3), 413–458 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rao, S., Tron, R., Vidal, R., Ma, Y.: Motion segmentation in the presence of outlying, incomplete, or corrupted trajectories. IEEE Trans. Pattern Anal. Mach. Intell. 32(10), 1832–1845 (2010)CrossRefGoogle Scholar
  24. 24.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)CrossRefGoogle Scholar
  25. 25.
    Tao, M., Yuan, X.: Recovering low-rank and sparse components of matrices from incomplete and noisy observations. SIAM J. Optim. 21(1), 57–81 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tipping, M., Bishop, C.: Mixtures of probabilistic principal component analyzers. Neural Comput. 11(2), 443–482 (1999)CrossRefGoogle Scholar
  27. 27.
    Tseng, P.: Nearest q-flat to m points. J. Optim. Theory Appl. 105(1), 249–252 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Vidal, R., Ma, Y., Sastry, S.: Generalized principal component analysis (GPCA). IEEE Trans. Pattern Anal. Mach. Intell. 27(12), 1945–1959 (2005)CrossRefGoogle Scholar
  29. 29.
    Xiao, Y., Wu, S., Li, D.: Splitting and linearizing augmented Lagrangian algorithm for subspace recovery from corrupted observations. Adv. Comput. Math. 38(4), 837–858 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Yan, J., Pollefeys, M.: A General Framework for Motion Segmentation: Independent, Articulated, Rigid, Non-rigid, Degenerate and Non-degenerate, pp. 94–106. Springer, Berlin (2006)Google Scholar
  31. 31.
    Yang, J., Yin, W., Zhang, Y., Wang, Y.: A fast algorithm for edge-preserving variational multichannel image restoration. SIAM J. Imaging Sci. 2(2), 569–592 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Zhang, C., Bitmead, R.: Subspace system identification for training-based MIMO channel estimation. Automatica 41(9), 1623–1632 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zhang, T., Szlam, A., Lerman, G.: Median k-flats for hybrid linear modeling with many outliers. In: 2009 IEEE 12th International Conference on Computer Vision Workshops, ICCV Workshops, pp. 234–241 (2009)Google Scholar
  34. 34.
    Zhang, T., Szlam, A., Wang, Y., Lerman, G.: Hybrid linear modeling via local best-fit flats. Int. J. Comput. Vis. 100(3), 217–240 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiChina
  2. 2.Department of Applied MathematicsDelft University of TechnologyDelftThe Netherlands

Personalised recommendations