Advertisement

Robust Affine Subspace Clustering via Smoothed \(\ell _{0}\)-Norm

  • Wenhua Dong
  • Xiao-jun Wu
Article
  • 53 Downloads

Abstract

In the past few years, sparse representation based method has been used in many fields with breathtaking speed due to its superior sparse recovery performance. Sparse subspace clustering (SSC), as one of its application hot-spots, has attracted considerable attention. Traditional sparse subspace clustering methods employ the \(\ell _{1}\)-norm to induce sparse representation of data points. Typically, the use of \(\ell _{1}\)-regularization instead of the \(\ell _{0}\) one can make the objective function convex while it also causes large errors on large coefficients in some cases. In this work, we propose using the non-convex smoothed \(\ell _{0}\)-norm to replace the \(\ell _{0}\) one for affine subspace clustering. This leads to a non-convex minimization problem. We then propose an effective method to solve the problem which minimizes the objective function by using the gradient method and proximal projection. In addition, the proposed algorithm is robust to noise and can provide a fast solution. Extensive experiments on real datasets demonstrate the effectiveness of our proposed method.

Keywords

Affine subspace clustering Smoothed \(\ell _{0}\)-norm Gradient method and Proximal projection 

Notes

Acknowledgements

This paper is jointly supported by the 111 Project of Chinese Ministry of Education under Grant B12018 and the National Natural Science Foundation of China under Grant 61373055; 61672265.

References

  1. 1.
    Basri R, Jacobs D (2003) Lambertian reflectance and linear subspaces. IEEE Trans Pattern Anal Mach Intell 25(2):218–233CrossRefGoogle Scholar
  2. 2.
    Chen G, Lerman G (2009) Spectral curvature clustering (SCC). Int J Comput Vis 81(3):317–330CrossRefGoogle Scholar
  3. 3.
    David A, Jean P (2002) Computer vision: a modern approach. Prentice Hall, Upper Saddle River, pp 654–659Google Scholar
  4. 4.
    Eftekhari A, Babaie-Zadeh M, Jutten C, Moghaddam HA (2009) Robust-SL0 for stable sparse representation in noisy settings. In: IEEE international conference on acoustics, speech and signal processing, ICASSP 2009. IEEE, pp 3433–3436Google Scholar
  5. 5.
    Elhamifar E, Vidal R (2013) Sparse subspace clustering: algorithm, theory, and applications. IEEE Trans Pattern Anal Mach Intell 35(11):2765–2781CrossRefGoogle Scholar
  6. 6.
    Flusser J, Suk T (1993) Pattern recognition by affine moment invariants. Pattern Recognit 26(1):167–174MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gabay D, Mercier B (1976) A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput Math Appl 2(1):17–40CrossRefGoogle Scholar
  8. 8.
    Georghiades AS, Belhumeur PN (1998) Illumination cone models for faces recognition under variable lighting. In: Proceedings of CVPR 1998Google Scholar
  9. 9.
    Hong C, Yu J, Wan J, Tao D, Wang M (2015) Multimodal deep autoencoder for human pose recovery. IEEE Trans Image Process 24(12):5659–5670MathSciNetCrossRefGoogle Scholar
  10. 10.
    Huang S, Yeh Y, Eguchi S (2009) Robust kernel principal component analysis. Neural Comput 21(11):3179–3213MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hull JJ (1994) A database for handwritten text recognition research. IEEE Trans Pattern Anal Mach Intell 16(5):550–554CrossRefGoogle Scholar
  12. 12.
    Jolliffe I (2011) Principal component analysis. In: International encyclopedia of statistical science. Springer, Berlin, pp 1094–1096Google Scholar
  13. 13.
    Lee K, Ho J, Kriegman DJ (2005) Acquiring linear subspaces for face recognition under variable lighting. IEEE Trans Pattern Anal Mach Intell 5:684–698Google Scholar
  14. 14.
    Li C, Vidal R (2015) Structured sparse subspace clustering: a unified optimization framework. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 277–286Google Scholar
  15. 15.
    Li J, Liang X, Shen S, Xu T, Feng J, Yan S (2018) Scale-aware fast r-cnn for pedestrian detection. IEEE Trans Multimed 20(4):985–996Google Scholar
  16. 16.
    Liu G, Lin Z, Yan S, Sun J, Yu Y, Ma Y (2013) Robust recovery of subspace structures by low-rank representation. IEEE Trans Pattern Anal Mach Intell 35(1):171–184CrossRefGoogle Scholar
  17. 17.
    Lowe DG (1999) Object recognition from local scale-invariant features. In: The proceedings of the 7th IEEE international conference on Computer vision, 1999, 2:1150–1157. IEEEGoogle Scholar
  18. 18.
    Lu C, Min H, Zhao Z, Zhu L, Huang D, Yan S (2012) Robust and efficient subspace segmentation via least squares regression. In: European conference on computer vision. Springer, Berlin, pp 347–360Google Scholar
  19. 19.
    Martinez AM (1998) The AR face database. CVC Technical Report 24Google Scholar
  20. 20.
    Mohimani H, Babaie-Zadeh M, Jutten C (2009) A fast approach for overcomplete sparse decomposition based on smoothed \(\ell _{0}\) norm. IEEE Trans Signal Process 57(1):289–301MathSciNetCrossRefGoogle Scholar
  21. 21.
    Nene SA, Nayar SK, Murase H et al (1996) Columbia object image library (coil-20)Google Scholar
  22. 22.
    Ng AY, Jordan MI, Weiss Y (2002) On spectral clustering: Analysis and an algorithm. In: Advances in neural information processing systems, pp 849–856Google Scholar
  23. 23.
    Siddiquie B, Feris RS, Davis LS (2011) Image ranking and retrieval based on multi-attribute queries. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 801–808Google Scholar
  24. 24.
    Sonka M, Hlavac V, Boyle R (2014) Image processing, analysis, and machine vision. Cengage LearningGoogle Scholar
  25. 25.
    Tron R, Vidal R (2007) A benchmark for the comparison of 3-D motion segmentation algorithms. In: IEEE conference on computer vision and pattern recognition, 2007. IEEE, pp 1–8Google Scholar
  26. 26.
    Vidal R, Favaro P (2014) Low rank subspace clustering (LRSC). Pattern Recognit Lett 43:47–61CrossRefGoogle Scholar
  27. 27.
    Wang J, Shi D, Cheng D, Zhang Y, Gao J (2016) LRSR: low-rank-sparse representation for subspace clustering. Neurocomputing 214:1026–1037CrossRefGoogle Scholar
  28. 28.
    Wang Y, Xu H, Leng C (2013) Provable subspace clustering: When LRR meets SSC. In: Advances in Neural Information Processing Systems, pp 64–72Google Scholar
  29. 29.
    Wei L, Wang X, Wu A, Zhou R, Zhu C (2018) Robust subspace segmentation by self-representation constrained low-rank representation. Neural Process Lett 48(3):1671–1691CrossRefGoogle Scholar
  30. 30.
    Wu Z, Yin M, Zhou Y, Fang X, Xie S (2017) Robust spectral subspace clustering based on least square regression. Neural Process Lett 48:1359–1372CrossRefGoogle Scholar
  31. 31.
    Yan J, Pollefeys M (2006) A general framework for motion segmentation: independent, articulated, rigid, non-rigid, degenerate and non-degenerate. In: European conference on computer vision. Springer, Berlin, pp 94–106Google Scholar
  32. 32.
    Yang Y, Feng J, Jojic N, Yang J, Huang TS (2016) \(\ell _{0}\)-sparse subspace clustering. In: European conference on computer vision. Springer, Berlin, pp 731–747Google Scholar
  33. 33.
    You C, Robinson D, Vidal R (2016) Scalable sparse subspace clustering by orthogonal matching pursuit. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 3918–3927Google Scholar
  34. 34.
    Yu J, Yang X, Gao F, Tao D (2017) Deep multimodal distance metric learning using click constraints for image ranking. IEEE Trans Cybern 47(12):4014–4024CrossRefGoogle Scholar
  35. 35.
    Zhang J, Yu J, Tao D (2018) Local deep-feature alignment for unsupervised dimension reduction. IEEE Trans Image Process 27(5):2420–2432MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zhang Y (2010) Recent advances in alternating direction methods: Practice and theory. In: IPAM workshop on continuous optimizationGoogle Scholar
  37. 37.
    Zheng X, Cai D, He X, Ma WY, Lin X (2004) Locality preserving clustering for image database. In: Proceedings of the 12th annual ACM international conference on Multimedia. ACM, pp 885–891Google Scholar
  38. 38.
    Zou H, Li R (2008) One-step sparse estimates in nonconcave penalized likelihood models. Ann Stat 36(4):1509MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Jiangnan UniversityWuxiChina

Personalised recommendations