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Latent Low-Rank Representation

  • Guangcan Liu
  • Shuicheng Yan
Chapter

Abstract

As mentioned at the end of previous chapter, a key aspect of LRR is about the configuration of its dictionary matrix. Usually, the observed data matrix itself is chosen as the dictionary, resulting in a powerful method that is useful for both subspace clustering and error correction. However, such a strategy may depress the performance of LRR, especially when the observations are insufficient and/or grossly corrupted. In this chapter we therefore propose to construct the dictionary by using both observed and unobserved, hidden data. We show that the effects of the hidden data can be approximately recovered by solving a nuclear norm minimization problem, which is convex and can be solved efficiently. The formulation of the proposed method, called Latent Low-Rank Representation (LatLRR), seamlessly integrates subspace clustering and feature extraction into a unified framework, and thus provides us with a solution for both subspace clustering and feature extraction. As a subspace clustering algorithm, LatLRR is an enhanced version of LRR and outperforms the state-of-the-art algorithms. Being an unsupervised feature extraction algorithm, LatLRR is able to robustly extract salient features from corrupted data, and thus can work much better than the benchmark that utilizes the original data vectors as features for classification. Compared to dimension reduction based methods, LatLRR is more robust to noise.

Keywords

Low-rank representation Latent variables Subspace clustering  Feature extraction Face recognition 

References

  1. 1.
    G. Liu, S. Yan, Latent low-rank representation for subspace segmentation and feature extraction, in ICCV, pp. 1615–1622 (2011)Google Scholar
  2. 2.
    M. Fazel, Matrix rank minimization with applications, PhD thesis (2002)Google Scholar
  3. 3.
    G. Liu, Z. Lin, S. Yan, J. Sun, Y. Yu, Y. Ma, Robust recovery of subspace structures by low-rank representation, CoRR (2010)Google Scholar
  4. 4.
    G. Liu, Z. Lin, Y. Yu, Robust subspace segmentation by low-rank representation, in ICML, pp. 663–670 (2010)Google Scholar
  5. 5.
    J. Sun, Y. Ni, X. Yuan, S. Yan, L.-F, Cheong, Robust low-rank subspace segmentation with semidefinite guarantees, in ICDM Workshop on Optimization Based Methods for Emerging Data Mining Problems (2010)Google Scholar
  6. 6.
    A.P. Costeira, Jo, T. Kanade, A multibody factorization method for independently moving objects. IJCV 29(3), 159–179 (1998)Google Scholar
  7. 7.
    Y. Zhang, Z. Jiang, L.S. Davis, Learning structured low-rank representations for image classification, in CVPR (2013)Google Scholar
  8. 8.
    E. Elhamifar, R. Vidal, Sparse subspace clustering, in CVPR, vol. 2, pp. 2790–2797 (2009)Google Scholar
  9. 9.
    E. Micheli-Tzanakou (ed.), Supervised and Unsupervised Pattern Recognition: Feature Extraction and Computational Intelligence (CRC Press Inc, Boca Raton, 2000)Google Scholar
  10. 10.
    Z. Lin, M. Chen, L. Wu, Y. Ma, The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices. Technical report, UILU-ENG-09-2215, (2009)Google Scholar
  11. 11.
    J.-F. Cai, E.J. Candès, Z. Shen, A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    J. Shi, J. Malik, Normalized cuts and image segmentation. TPAMI 22, 888–905 (2000)CrossRefGoogle Scholar
  13. 13.
    R. Tron, R. Vidal, A benchmark for the comparison of 3-d motion segmentation algorithms, in CVPR, pp. 1–8 (2007)Google Scholar
  14. 14.
    J. Yan, M. Pollefeys, A general framework for motion segmentation: Independent, articulated, rigid, non-rigid, degenerate and non-degenerate, in ECCV, vol. 4, pp. 94–106 (2006)Google Scholar
  15. 15.
    M.A. Fischler, R.C. Bolles, Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Commun. ACM 24(6), 381–395 (1981)MathSciNetCrossRefGoogle Scholar
  16. 16.
    S. Rao, R. Tron, R. Vidal, Y. Ma, Motion segmentation in the presence of outlying, incomplete, or corrupted trajectories. TPAMI 32(10), 1832–1845 (2010)CrossRefGoogle Scholar
  17. 17.
    F. Lauer, C. Schnòrr, Spectral clustering of linear subspaces for motion segmentation, in ICCV (2009)Google Scholar
  18. 18.
    K. Lee, J. Ho, D. Kriegman, Acquiring linear subspaces for face recognition under variable lighting. TPAMI 27(5), 684–698 (2005)CrossRefGoogle Scholar
  19. 19.
    X. He, D. Cai, H. Liu, W.-Y. Ma, Locality preserving indexing for document representation, in SIGIR, pp. 96–103 (2004)Google Scholar
  20. 20.
    X. He, D. Cai, S. Yan, H.-J. Zhang, Neighborhood preserving embedding, in ICCV, pp. 1208–1213 (2005)Google Scholar
  21. 21.
    D. Guillamet, J. Vitrià, Non-negative matrix factorization for face recognition, in CCIA, pp. 336–344 (2002)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Cornell UniversityIthacaUSA
  2. 2.National University of SingaporeSingaporeSingapore

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