This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Sparse Subspace Clustering: Algorithm, Theory, and Applications
Nov. 2013 (vol. 35 no. 11)
pp. 2765-2781
E. Elhamifar, Dept. of Electr. Eng. & Comput. Sci., Univ. of California, Berkeley, Berkeley, CA, USA
R. Vidal, Dept. of Biomed. Eng., Johns Hopkins Univ., Baltimore, MD, USA
Many real-world problems deal with collections of high-dimensional data, such as images, videos, text, and web documents, DNA microarray data, and more. Often, such high-dimensional data lie close to low-dimensional structures corresponding to several classes or categories to which the data belong. In this paper, we propose and study an algorithm, called sparse subspace clustering, to cluster data points that lie in a union of low-dimensional subspaces. The key idea is that, among the infinitely many possible representations of a data point in terms of other points, a sparse representation corresponds to selecting a few points from the same subspace. This motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of the data into subspaces. Since solving the sparse optimization program is in general NP-hard, we consider a convex relaxation and show that, under appropriate conditions on the arrangement of the subspaces and the distribution of the data, the proposed minimization program succeeds in recovering the desired sparse representations. The proposed algorithm is efficient and can handle data points near the intersections of subspaces. Another key advantage of the proposed algorithm with respect to the state of the art is that it can deal directly with data nuisances, such as noise, sparse outlying entries, and missing entries, by incorporating the model of the data into the sparse optimization program. We demonstrate the effectiveness of the proposed algorithm through experiments on synthetic data as well as the two real-world problems of motion segmentation and face clustering.
Index Terms:
pattern clustering,computational complexity,convex programming,data structures,minimisation,synthetic data,sparse subspace clustering algorithm,high-dimensional data collection,data point clustering,data point representation,sparse representation,sparse optimization program,spectral clustering framework,general NP-hard problem,convex relaxation,minimization program,motion segmentation,face clustering,Clustering algorithms,Noise,Optimization,Sparse matrices,Vectors,Computer vision,Face,face clustering,High-dimensional data,intrinsic low-dimensionality,subspaces,clustering,sparse representation,$(\ell_1)$-minimization,convex programming,spectral clustering,principal angles,motion segmentation
Citation:
E. Elhamifar, R. Vidal, "Sparse Subspace Clustering: Algorithm, Theory, and Applications," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 35, no. 11, pp. 2765-2781, Nov. 2013, doi:10.1109/TPAMI.2013.57
Usage of this product signifies your acceptance of the Terms of Use.