Abstract
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.
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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)
Bradley, P., Mangasarian, O.: k-Plane clustering. J. Glob. Optim. 16(1), 23–32 (2000). https://doi.org/10.1023/A:1008324625522
Butenko, S., Chaovalitwongse, W., Pardalos, P.: Clustering Challenges in Biological Networks. World Scientific, Singapore (2009)
Cai, J., Candès, E., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)
Candès, E., Plan, Y.: Matrix completion with noise. Proc. IEEE 98(6), 925–936 (2010)
Candès, E., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005)
Chen, G., Lerman, G.: Spectral curvature clustering (SCC). Int. J. Comput. Vis. 81(3), 317–330 (2009)
Elhamifar, E., Vidal, R.: Sparse subspace clustering: algorithm, theory, and applications. IEEE Trans. Pattern Anal. Mach. Intell. 35(11), 2765–2781 (2013). https://doi.org/10.1109/TPAMI.2013.57
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)
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). https://doi.org/10.1109/CVPR.2004.1315101
Han, L., Bi, S.: Two-stage convex relaxation approach to low-rank and sparsity regularized least squares loss. J. Glob. Optim. (2017). https://doi.org/10.1007/s10898-017-0573-2
He, B., Tao, M., Yuan, X.: A splitting method for separable convex programming. IMA J. Numer. Anal. 35(1), 394–426 (2015)
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)
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)
Kanatani, K.: Motion segmentation by subspace separation: model selection and reliability evaluation. Int. J. Image Graph. 2(2), 179–197 (2002)
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)
Lin, Z., Chen, M., Ma, Y.: The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices (2010). Eprint arXiv:1009.5055
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., Yan, S.: Latent low-rank representation for subspace segmentation and feature extraction. In: 2011 International Conference on Computer Vision, pp. 1615–1622 (2011). https://doi.org/10.1109/ICCV.2011.6126422
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)
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)
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)
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)
Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)
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)
Tipping, M., Bishop, C.: Mixtures of probabilistic principal component analyzers. Neural Comput. 11(2), 443–482 (1999)
Tseng, P.: Nearest q-flat to m points. J. Optim. Theory Appl. 105(1), 249–252 (2000)
Vidal, R., Ma, Y., Sastry, S.: Generalized principal component analysis (GPCA). IEEE Trans. Pattern Anal. Mach. Intell. 27(12), 1945–1959 (2005)
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)
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)
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)
Zhang, C., Bitmead, R.: Subspace system identification for training-based MIMO channel estimation. Automatica 41(9), 1623–1632 (2005)
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)
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)
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The authors would like to thank the financial support from the China Scholarship Council.
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This research was supported by a grant from the National Natural Science Foundation of China (No. 11771275).
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Liang, R., Bai, Y. & Lin, H.X. An inexact splitting method for the subspace segmentation from incomplete and noisy observations. J Glob Optim 73, 411–429 (2019). https://doi.org/10.1007/s10898-018-0684-4
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DOI: https://doi.org/10.1007/s10898-018-0684-4