$n$ data points into a number of clusters, with each cluster being a (linear) subspace. The recently established algorithms such as Sparse Subspace Clustering (SSC), Low-Rank Representation (LRR) and Low-Rank Subspace Segmentation (LRSS) are effective in terms of segmentation accuracy, but computationally inefficient as they possess a complexity of $O(n^{3})$ , which is too high to afford for the case where $n$ is very large. In this paper we devise a fast subspace segmentation algorithm with complexity of $O(n\log (n))$ . This is achieved by firstly using partial Singular Value Decomposition (SVD) to approximate the solution of LRSS, secondly utilizing Locality Sensitive Hashing (LSH) to build a sparse affinity graph that encodes the subspace memberships, and finally adopting a fast Normalized Cut (NCut) algorithm to produce the final segmentation results. Besides of high efficiency, our algorithm also has comparable effectiveness as the original LRSS method." /> $n$ data points into a number of clusters, with each cluster being a (linear) subspace. The recently established algorithms such as Sparse Subspace Clustering (SSC), Low-Rank Representation (LRR) and Low-Rank Subspace Segmentation (LRSS) are effective in terms of segmentation accuracy, but computationally inefficient as they possess a complexity of $O(n^{3})$ , which is too high to afford for the case where $n$ is very large. In this paper we devise a fast subspace segmentation algorithm with complexity of $O(n\log (n))$ . This is achieved by firstly using partial Singular Value Decomposition (SVD) to approximate the solution of LRSS, secondly utilizing Locality Sensitive Hashing (LSH) to build a sparse affinity graph that encodes the subspace memberships, and finally adopting a fast Normalized Cut (NCut) algorithm to produce the final segmentation results. Besides of high efficiency, our algorithm also has comparable effectiveness as the original LRSS method." /> $n$ data points into a number of clusters, with each cluster being a (linear) subspace. The recently established algorithms such as Sparse Subspace Clustering (SSC), Low-Rank Representation (LRR) and Low-Rank Subspace Segmentation (LRSS) are effective in terms of segmentation accuracy, but computationally inefficient as they possess a complexity of $O(n^{3})$ , which is too high to afford for the case where $n$ is very large. In this paper we devise a fast subspace segmentation algorithm with complexity of $O(n\log (n))$ . This is achieved by firstly using partial Singular Value Decomposition (SVD) to approximate the solution of LRSS, secondly utilizing Locality Sensitive Hashing (LSH) to build a sparse affinity graph that encodes the subspace memberships, and finally adopting a fast Normalized Cut (NCut) algorithm to produce the final segmentation results. Besides of high efficiency, our algorithm also has comparable effectiveness as the original LRSS method." /> Fast Low-Rank Subspace Segmentation
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Issue No.05 - May (2014 vol.26)
pp: 1293-1297
Xin Zhang , Dept. of Comput. Sci. & Technol., Tsinghua Univ., Beijing, China
Fuchun Sun , Dept. of Comput. Sci. & Technol., Tsinghua Univ., Beijing, China
Guangcan Liu , Univ. of Illinois at Urbana-Champaign, Champaign, IL, USA
Yi Ma , Microsoft Res. Asia, Beijing, China
ABSTRACT
Subspace segmentation is the problem of segmenting (or grouping) a set of n data points into a number of clusters, with each cluster being a (linear) subspace. The recently established algorithms such as Sparse Subspace Clustering (SSC), Low-Rank Representation (LRR) and Low-Rank Subspace Segmentation (LRSS) are effective in terms of segmentation accuracy, but computationally inefficient as they possess a complexity of O(n3), which is too high to afford for the case where n is very large. In this paper we devise a fast subspace segmentation algorithm with complexity of O(n log (n)). This is achieved by firstly using partial Singular Value Decomposition (SVD) to approximate the solution of LRSS, secondly utilizing Locality Sensitive Hashing (LSH) to build a sparse affinity graph that encodes the subspace memberships, and finally adopting a fast Normalized Cut (NCut) algorithm to produce the final segmentation results. Besides of high efficiency, our algorithm also has comparable effectiveness as the original LRSS method.
INDEX TERMS
singular value decomposition, approximation theory, computational complexity, file organisation, graph theory, image representation, image segmentation, pattern clustering,NCut, fast low-rank subspace segmentation, data points, sparse subspace clustering, SSC, low-rank representation, LRR, LRSS, segmentation accuracy, partial singular value decomposition, SVD, locality sensitive hashing, LSH, sparse affinity graph, subspace memberships, fast normalized cut algorithm,Clustering algorithms, Complexity theory, Accuracy, Sparse matrices, Algorithm design and analysis, Vectors, Buildings,Algorithms, Clustering,singular value decomposition, Locality sensitive hashing, low-rank subspace segmentation
CITATION
Xin Zhang, Fuchun Sun, Guangcan Liu, Yi Ma, "Fast Low-Rank Subspace Segmentation", IEEE Transactions on Knowledge & Data Engineering, vol.26, no. 5, pp. 1293-1297, May 2014, doi:10.1109/TKDE.2013.114