Multiway spectral clustering: A maragin-based persoective

Statistical Science

Volumn 23, Issue 3, 2008, Pages 383-403

  Zhang, Zhihua ^a   Jordan, Michael I ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Gaussian intrinsic autoregression; Graph partitioning; Large margin classification; Reproducing kernel Hilbert space; Spectral clustering; Spectral relaxation

Indexed keywords

EID: 59349084916     PISSN: 08834237     EISSN: None     Source Type: Journal    
DOI: 10.1214/08-STS266     Document Type: Article

References (41)

1. BACH, F. R. and JORDAN, M. I. (2006). Learning spectral clustering, with application to speech separation. J.Mach. Learn. Res. 7 1963-2001. MR2274430
2. BARTLETT, P. L., JORDAN, M. I. andMCAULIFFE, J. D. (2006). Convexity, classification, and risk bounds. J. Amer. Statist. Assoc. 101 138-156. MR2268032
3. BELKIN, M. and NIYOGI, P. (2002). Laplacian eigenmaps and spectral techniques for embedding and clustering. In Advances in Neural Information Processing Systems 14 585-592. MIT Press, Cambridge, MA.
4. BESAG, J. (1974). Spatial interaction and statistical analysis of lattice systems (with discussion). J. Roy. Statist. Soc. Ser. B 36 192-236. MR0373208
5. BESAG, J. and KOOPERBERG, C. (1995). On conditional and intrinsic autoregression. Biometrika 82 733-746. MR1380811
6. CARLIN, B. P. and BANERJEE, S. (2003). Hierarchical multivariate CAR models for spatio-temporally correlated survival data (with discussion). In Bayesian Statistics 7 45-63. Oxford Univ. Press. MR2003166
7. CHAN, P. K., SCHLAG, M. D. F. and ZIEN, J. Y. (1994). Spectral K-way ratio-cut partitioning and clustering. IEEE Trans. Computer-Aided Design Integrated Circuits Syst. 13 1088-1096.
8. CHUNG, F. R. (1997). Spectral Graph Theory. Amer.Math. Soc., Providence, RI. MR1421568
9. DHILLON, I. S., GUAN, Y. andKULIS, B. (2007).Weighted graph cuts without eigenvectors: A multilevel approach. IEEE Trans. Pattern Anal. Mach. Intell. 29 1944-1957.
10. DING, C., HE, X. and SIMON, H. D. (2005). On the equivalence of nonnegative matrix factorization and spectral clustering. In SIAM Conference on Data Mining (SDM). Newport Beach, CA.
11. DONATH, W. E. and HOFMANN, A. J. (1973). Lower bounds for the partitioning of graphs. IBM J. Res. Develop. 17 420-425. MR0329965
12. EDELMAN, A., ARIAS, T. A. and SMITH, S. T. (1999). The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20 303-353. MR1646856
13. FIEDLER, M. (1973). Algebraic connectivity of graphs. Czechoslovak Math. J. 23 298-305. MR0318007
14. GOWER, J. C. and DIJKSTERHUIS, G. B. (2004). Procrustes Problems. Oxford Univ. Press. MR2051013
15. GUATTERY, S. andMILLER, G. L. (2000). Graph embeddings and Laplacian eigenvalues. SIAM J. Matrix Anal. Appl. 21 703-723. MR1740868
16. GUPTA, A. K. and NAGAR, D. K. (2000). Matrix Variate Distributions. Chapman and Hall, London. MR1738933
17. HORN, R. A. and JOHNSON, C. R. (1985). Matrix Analysis. Cambridge Univ. Press. MR0832183
18. JUHÁSZ, F. and MÁLYUSZ, K. (1977). Problems of cluster analysis from the viewpoint of numerical analysis. In Numerical Methods, Colloquia Mathematica Societatis Janos Bolyai (P. Rózsa, ed.) 22 405-415. North-Holland, Amsterdam. MR0590916
19. KANNAN, R., VEMPALA, S. and VETTA, A. (2000). On clusterings: Good, bad, and spectral. In Proceedings of the 41st Annual Symposium on the Foundation of Computer Science 367-380. MR1931834
20. MARDIA, K. V. (1988). Multi-dimensional multivariate Gaussian Markov random fields with application to image processing. J. Multivariate Anal. 24 265-284. MR0926357
21. MARDIA, K. V., KENT, J. T. and BIBBY, J. M. (1979). Multivariate Analysis. Academic Press, New York. MR0560319
22. MEILǍ, M. and PENTNEY, W. (2007). Clustering by weighted cuts in directed graphs. In SIAM Conference on Data Mining (SDM). Minneapolis, MN.
23. MEILǍ, M. and SHI, J. (2000). Learning segmentation by random walks. In Advances in Neural Information Processing 12 470-477. MIT Press, Cambridge, MA.
24. MOHAR, B. (1991). The Laplacian Spectrum of Graphs 871-898. Wiley, New York. MR1170831
25. NG, A. Y., JORDAN, M. I. and WEISS, Y. (2002). On spectral clustering: Analysis and an algorithm. In Advances in Neural Information Processing Systems 14 849-856. MIT Press, Cambridge, MA.
26. RAHIMI, A. and RECHT, B. (2004). Clustering with normalized cuts is clustering with a hyperplane. In Workshop on Statistical Learning in Computer Vision. Prague, Czech Republic.
27. RAND, W. M. (1971). Objective criteria for the evaluation of clustering methods. J. Amer. Statist. Assoc. 66 846-850.
28. RIPLEY, B. D. (1981). Spatial Statistics. Wiley, New York. MR0624436
29. SAERENS, M., FOUSS, F., YEN, L. and DUPONT, P. (2004). The principal components analysis of a graph, and its relationships to spectral clustering. In The 15th European Conference on Machine Learning (ECML) 371-383.
30. SHEN, X. and WANG, L. (2007). Generalization error for multiclass margin classification. Electron. J. Statist. 1 307-330. MR2336036
31. SHI, J. andMALIK, J. (2000). Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22 888-905.
32. SHORTREED, S. and MEILǍ , M. (2005). Unsupervised spectral learning. In Proceedings of the Twenty-First Conference on Uncertainty in Artificial Intelligence 534-541. AUAI Press.
33. VON LUXBURG, U., BELKIN, M. and BOUSQUET, O. (2008). Consistency of spectral clustering. Ann. Statist. 36 555-586. MR2396807
34. VON LUXBURG, U. (2007). A tutorial on spectral clustering. Statist. Comput. 17 395-416. MR2409803
35. WAHBA, G. (1990). Spline Models for Observational Data. SIAM, Philadelphia. MR1045442
36. WEBB, A. R. (2002). Statistical Pattern Recognition, 2nd ed. Wiley, Hoboken. MR2191640
37. WEISS, Y. (1999). Segmentation using eigenvectors: A unifying view. In IEEE International Conference on Computer Vision 975-982. IEEE Computer Society.
38. YEUNG, K. Y., FRALEY, C., MURUA, A., RAFTERY, A. E. and RUZZO, W. L. (2001). Model-based clustering and data transformations for gene expression data. Bioinform. 17 977-987.
39. YU, S. X. and SHI, J. (2003). Multiclass spectral clustering. In IEEE International Conference on Computer Vision 313-319. IEEE Computer Society.
40. ZHA, H., DING, C., GU, M., HE, X. and SIMON, H. (2002). Spectral relaxation for k-means clustering. In Advances in Neural Information Processing Systems 14 1057-1064. MIT Press, Cambridge, MA.
41. ZOU, H., ZHU, J. and HASTIE, T. (2006). The margin vector, admissible loss and multi-class margin-based classifiers. Technical report, Dept. Statistics, Stanford Univ.