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Spectral clustering based on matrix perturbation theory

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Abstract

This paper exposes some intrinsic characteristics of the spectral clustering method by using the tools from the matrix perturbation theory. We construct a weight matrix of a graph and study its eigenvalues and eigenvectors. It shows that the number of clusters is equal to the number of eigenvalues that are larger than 1, and the number of points in each of the clusters can be approximated by the associated eigenvalue. It also shows that the eigenvector of the weight matrix can be used directly to perform clustering; that is, the directional angle between the two-row vectors of the matrix derived from the eigenvectors is a suitable distance measure for clustering. As a result, an unsupervised spectral clustering algorithm based on weight matrix (USCAWM) is developed. The experimental results on a number of artificial and real-world data sets show the correctness of the theoretical analysis.

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Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, China

    Tian Zheng, Li XiaoBin & Ju YanWei

  2. National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Science, Beijing, 100080, China

    Tian Zheng

Authors
  1. Tian Zheng
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  2. Li XiaoBin
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  3. Ju YanWei
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Corresponding author

Correspondence to Tian Zheng.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 60375003) and the Aeronatical Science Foundation of China (Grant No. 03I53059)

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Tian, Z., Li, X. & Ju, Y. Spectral clustering based on matrix perturbation theory. SCI CHINA SER F 50, 63–81 (2007). https://doi.org/10.1007/s11432-007-0007-8

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  • DOI: https://doi.org/10.1007/s11432-007-0007-8

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