Abstract
Subspace clustering, as an important clustering problem, has drawn much attention in recent years. State-of-the-art methods generally try to design an efficient model to regularize the coefficient matrix while ignore the influence of the noise model on subspace clustering. However, the real data are always contaminated by the noise and the corresponding subspace structures are likely to be corrupted. In order to solve this problem, we propose a novel subspace clustering algorithm by employing capped \(l_1\) norm to deal with the noise. Consequently, the noise term with large error can be penalized by the proposed method. So it is more robust to the noise. Furthermore, the grouping effect of our method is theoretically proved, which means highly correlated points can be grouped together. Finally, the experimental results on two real databases show that our method outperforms state-of-the-art methods.
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References
Cao, X., Zhang, C., Fu, H., Liu, S., Zhang, H.: Diversity-induced multi-view subspace clustering. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 586–594 (2015)
Costeira, J.P., Kanade, T.: A multibody factorization method for independently moving objects. Int. J. Comput. Vis. 29(3), 159–179 (1998)
Elhamifar, E., Vidal, R.: Sparse subspace clustering. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 2790–2797 (2009)
Elhamifar, E., Vidal, R.: Sparse subspace clustering: algorithm, theory, and applications. IEEE Trans. Pattern Anal. Mach. Intell. 35(11), 2765–2781 (2013)
Gao, H., Nie, F., Li, X., Huang, H.: Multi-view subspace clustering. In: IEEE International Conference on Computer Vision, pp. 4238–4246 (2015)
Ghaemi, R., Sulaiman, M.N., Ibrahim, H., Mustapha, N., et al.: A survey: clustering ensembles techniques. World Acad. Sci. Eng. Technol. 50, 636–645 (2009)
Ho, J., Yang, M.H., Lim, J., Lee, K.C., Kriegman, D.: Clustering appearances of objects under varying illumination conditions. In: IEEE Conference on Computer Vision and Pattern Recognition, vol. 1, p. I-11 (2003)
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)
Hu, H., Lin, Z., Feng, J., Zhou, J.: Smooth representation clustering. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 3834–3841 (2014)
Jiang, W., Liu, J., Qi, H., Dai, Q.: Robust subspace segmentation via nonconvex low rank representation. Information Sciences (2016)
Jiang, W., Nie, F., Huang, H.: Robust dictionary learning with capped l1 norm. In: International Joint Conferences on Artificial Intelligence, pp. 3590–3596 (2015)
Lee, M., Lee, J., Lee, H., Kwak, N.: Membership representation for detecting block-diagonal structure in low-rank or sparse subspace clustering. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 1648–1656 (2015)
Li, B., Zhang, Y., Lin, Z., Lu, H.: Subspace clustering by mixture of gaussian regression. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 2094–2102 (2015)
Li, C.G., Vidal, R.: Structured sparse subspace clustering: a unified optimization framework. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 277–286 (2015)
Li, Q., Sun, Z., Lin, Z., He, R., Tan, T.: Transformation invariant subspace clustering. Pattern Recognition (2016)
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)
Liu, G., Lin, Z., Yu, Y.: Robust subspace segmentation by low-rank representation. In: International Conference on Machine Learning, pp. 663–670 (2010)
Lu, C.-Y., Min, H., Zhao, Z.-Q., Zhu, L., Huang, D.-S., Yan, S.: Robust and efficient subspace segmentation via least squares regression. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) ECCV 2012. LNCS, vol. 7578, pp. 347–360. Springer, Heidelberg (2012). doi:10.1007/978-3-642-33786-4_26
Lu, C., Feng, J., Lin, Z., Yan, S.: Correlation adaptive subspace segmentation by trace lasso. In: IEEE International Conference on Computer Vision, pp. 1345–1352 (2013)
Lu, L., Vidal, R.: Combined central and subspace clustering for computer vision applications. In: International Conference on Machine Learning, pp. 593–600 (2006)
Lu, Y., Lai, Z., Xu, Y., You, J., Li, X., Yuan, C.: Projective robust nonnegative factorization. Inf. Sci. 364, 16–32 (2016)
Pang, Y., Ye, L., Li, X., Pan, J.: Moving object detection in video using saliency map and subspace learning. arXiv preprint arXiv:1509.09089 (2015)
Rao, S.R., Tron, R., Vidal, R., Ma, Y.: Motion segmentation via robust subspace separation in the presence of outlying, incomplete, or corrupted trajectories. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8 (2008)
Soltanolkotabi, M., Elhamifar, E., Candes, E.J., et al.: Robust subspace clustering. Annal. Stat. 42(2), 669–699 (2014)
Tang, K., Dunson, D.B., Su, Z., Liu, R., Zhang, J., Dong, J.: Subspace segmentation by dense block and sparse representation. Neural Netw. 75, 66–76 (2016)
Tron, R., Vidal, R.: A benchmark for the comparison of 3-d motion segmentation algorithms. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8 (2007)
Vidal, R., Favaro, P.: Low rank subspace clustering. Pattern Recogn. Lett. 43, 47–61 (2014)
Xu, Y., Wu, J., Li, X., Zhang, D.: Discriminative transfer subspace learning via low-rank and sparse representation. IEEE Trans. Image Process. 25(2), 850–863 (2016)
Yang, S., Yi, Z., He, X., Li, X.: A class of manifold regularized multiplicative update algorithms for image clustering. IEEE Trans. Image Process. 24(12), 5302–5314 (2015)
Zhang, T., Szlam, A., Lerman, G.: Median k-flats for hybrid linear modeling with many outliers. In: 12th IEEE International Conference on Computer Vision Workshops (ICCV Workshops), pp. 234–241 (2009)
Zhang, T.: Multi-stage convex relaxation for feature selection. Bernoulli 19(5B), 2277–2293 (2013)
Zhang, Z., Xu, Y., Yang, J., Li, X., Zhang, D.: A survey of sparse representation: algorithms and applications. IEEE Access 3, 490–530 (2015)
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant 61301230, in part by the International Science and Technology Cooperation Project of Henan Province under Grant 162102410021, in part by the Key Research Program of the Chinese Academy of Sciences under Grant KGZD-EW-T03, in part by the State Key Laboratory of Virtual Reality Technology and Systems under Grant BUAA-VR-16KF-04, and in part by the Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province under Grant GD201605.
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Appendix
Appendix
1.1 Proof of Theorem 2
Proof
Let
Since \({z^*} = \arg \mathop {\min }\limits _z L(z)\). We have
It gives
Note that each column of the data X is normalized, we get \({\left\| {x_i^T - x_j^T} \right\| _2} = 2\sqrt{1 - r}\), where \(r = x_i^T{x_j}\). Since \(z^*\) is the optimal solution of the problem (21), we have
Thus \(\left\| {x - X{z^*}} \right\| _2^2 \le \left\| x \right\| _2^2\). Finally, we get
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Lu, Q., Li, X., Dong, Y., Tao, D. (2016). Subspace Clustering by Capped \(l_1\) Norm. In: Tan, T., Li, X., Chen, X., Zhou, J., Yang, J., Cheng, H. (eds) Pattern Recognition. CCPR 2016. Communications in Computer and Information Science, vol 662. Springer, Singapore. https://doi.org/10.1007/978-981-10-3002-4_54
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DOI: https://doi.org/10.1007/978-981-10-3002-4_54
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