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Robust Recovery of Subspace Structures by Low-Rank Representation
Jan. 2013 (vol. 35 no. 1)
pp. 171-184
Guangcan Liu, Dept. of Comput. Sci. & Eng., Shanghai Jiao Tong Univ., Shanghai, China
Zhouchen Lin, Key Lab. of Machine Perception (MOE), Peking Univ., Beijing, China
Shuicheng Yan, Dept. of Electr. & Comput. Eng., Nat. Univ. of Singapore, Singapore, Singapore
Ju Sun, Dept. of Electr. Eng., Columbia Univ., New York, NY, USA
Yong Yu, Dept. of Comput. Sci. & Eng., Shanghai Jiao Tong Univ., Shanghai, China
Yi Ma, Visual Comput. Group, Microsoft Res. Asia, Beijing, China
In this paper, we address the subspace clustering problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to cluster the samples into their respective subspaces and remove possible outliers as well. To this end, we propose a novel objective function named Low-Rank Representation (LRR), which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that the convex program associated with LRR solves the subspace clustering problem in the following sense: When the data is clean, we prove that LRR exactly recovers the true subspace structures; when the data are contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for data corrupted by arbitrary sparse errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace clustering and error correction in an efficient and effective way.
Index Terms:
sparse matrices,convex programming,pattern clustering,subspace membership,robust subspace structure recovery,low-rank representation,subspace clustering problem,data sample vectors,outlier removal,objective function,LRR,dictionary,convex program,row space,sparse error correction,Robustness,Noise,Dictionaries,Optimization,Polynomials,Data models,Vectors,outlier detection,Low-rank representation,subspace clustering,segmentation
Citation:
Guangcan Liu, Zhouchen Lin, Shuicheng Yan, Ju Sun, Yong Yu, Yi Ma, "Robust Recovery of Subspace Structures by Low-Rank Representation," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 35, no. 1, pp. 171-184, Jan. 2013, doi:10.1109/TPAMI.2012.88
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