Abstract
In this paper, we present a perturbation analysis for the matrices in the multiway normalized cut spectral clustering method based on the matrix perturbation theory. The analytical results show that the eigenvalues and the eigenspaces of the normalized Laplacian matrices are continuous. Therefore, clustering algorithms can be designed according to the special properties of the normalized Laplacian matrices in the ideal case and the method can be extended to the general case based on the continuity of the eigenvalues and the eigenspaces of the normalized Laplacian matrices. The numerical results are consistent with the theoretical results.
摘要
创新点: 本文利用矩阵扰动理论对多路归一化割谱聚类方法用到的矩阵进行了扰动分析. 分析结果表明多路归一化割谱聚类方法用到的矩阵的特征值和特征子空间具有连续性. 因此, 在理想情形下根据多路归一化割谱聚类方法用到的矩阵的特殊性质划分数据, 再将结果推广到一般情形是合理的. 值得一提的是, 本文对非对称规范Laplace矩阵进行了扰动分析. 另外, 本文给出了谱聚类的划分结果保持不变的扰动上限.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Meilă M, Shi J B. Learning segmentation by random walks. In: Advances in Neural Information Processing Systems. Cambridge: MIT Press, 2001. 873–879
Shi J B, Malik J. Normalized cuts and image segmentation. IEEE Trans Pattern Anal Mach Intell, 2000, 22: 888–905
Wu Z Y, Leahy R. A optimal graph theoretic approach to data clustering: theory and its application to image segmentation. IEEE Trans Pattern Anal Mach Intell, 1993, 15: 1101–1113
Hagen L, Kahng A. New spectral methods for ratio cut partitioning and clustering. IEEE Trans Comput Aided Des, 1992, 11: 1074–1085
Sarkar S, Soundararajan P. Supervised learning of large perceptual organization: graph spectral partitioning and learning automata. IEEE Trans Pattern Anal Mach Intell, 2000, 22: 504–525
Gu M, Zha H Y, Ding C, et al. Spectral Relaxation Models and Structure Analysis for K-way Graph Clustering and Bi-clustering. Pennsylvania State University Technical Report. 2001
Meilă M, Xu L. Multiway Cuts and Spectral Clustering. University of Washington Technical Report. 2003
Ng A, Jordan M, Weiss Y. On spectral clustering: analysis and an Algorithm. In: Advances in Neural Information Processing Systems. Cambridge: MIT Press, 2002. 849–856
Verma D, Meilă M. A Comparison of Spectral Clustering Algorithms. University of Washington Technical Report. 2003
Luxburg U. A tutorial on spectral clustering. Stat Comput, 2007, 17: 395–416
Tian Z, Li X, Ju Y. Spectral clustering based on matrix perturbation theory. Sci China Ser-F: Inf Sci, 2007, 50: 63–81
Huang L, Yan D H, Jordan M, et al. Spectral clustering with perturbed data. In: Advances in Neural Information Processing Systems. New York: Curran Associates, 2009. 705–712
Rohe K, Chatterjee S, Yu B. Spectral clustering and the high-dimensional stochastic block model. Ann Stat, 2011, 39: 1878–1915
Balakrishnan S, Xu M, Krishnamurthy A, et al. Noise thresholds for spectral clustering. In: Advances in Neural Information Processing Systems. New York: Curran Associates, 2011. 954–962
Stewart W, Sun J G. Matrix Perturbation Theory. Boston: Academic Press, 1990. 189–283
Bhatia B. Matrix Analysis. New York: Springer, 1997. 212–214
Golub G, Loan C. Matrix Computations. 3rd ed. Baltimore and London: Johns Hopkins University Press, 1996. 250–253
Author information
Authors and Affiliations
Corresponding authors
Electronic supplementary material
Rights and permissions
About this article
Cite this article
Borjigin, S., Guo, C. Perturbation analysis for the normalized Laplacian matrices in the multiway spectral clustering method. Sci. China Inf. Sci. 57, 1–17 (2014). https://doi.org/10.1007/s11432-014-5156-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11432-014-5156-y
Keywords
- spectral clustering method
- normalized Laplacian matrices
- eigenvalue
- eigenspace
- matrix perturbation theory