Skip to main content
Springer Nature Link
Log in

Spectral Clustering by Recursive Partitioning

  • Conference paper
Algorithms – ESA 2006 (ESA 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4168))

Included in the following conference series:

Abstract

In this paper, we analyze the second eigenvector technique of spectral partitioning on the planted partition random graph model, by constructing a recursive algorithm using the second eigenvectors in order to learn the planted partitions. The correctness of our algorithm is not based on the ratio-cut interpretation of the second eigenvector, but exploits instead the stability of the eigenvector subspace. As a result, we get an improved cluster separation bound in terms of dependence on the maximum variance. We also extend our results for a clustering problem in the case of sparse graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Achlioptas, D., McSherry, F.: On spectral learning of mixtures of distributions. In: Auer, P., Meir, R. (eds.) COLT 2005. LNCS, vol. 3559, pp. 458–469. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  2. Alon, N.: Eigenvalues and expanders. Combinatorica 6(2), 83–96 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alon, N., Kahale, N.: A spectral technique for coloring random 3-colorable graphs. SIAM Journal on Computing 26(6), 1733–1748 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Azar, Y., Fiat, A., Karlin, A.R., McSherry, F., Saia, J.: Spectral analysis of data. In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pp. 619–626 (2001)

    Google Scholar 

  5. Boppana, R.: Eigenvalues and graph bisection: an average case analysis. In: Proceedings of the 28th IEEE Symposium on Foundations of Computer Science (1987)

    Google Scholar 

  6. Bui, T., Chaudhuri, S., Leighton, T., Sipser, M.: Graph bisection algorithms with good average case behavior. Combinatorica 7, 171–191 (1987)

    Article  MathSciNet  Google Scholar 

  7. Cheng, D., Kannan, R., Vempala, S., Wang, G.: A Divide-and- Merge methodology for Clustering. In: Proc. of the 24th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PODS), pp. 196–205

    Google Scholar 

  8. Coja-Oghlan, A.: A spectral heuristic for bisecting random graphs. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (2005)

    Google Scholar 

  9. Coja-Oghlan, A.: An adaptive spectral heuristic for partitioning random graphs. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 691–702. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  10. Dasgupta, A., Hopcroft, J., McSherry, F.: Spectral analysis of random Graphs with skewed degree distributions. In: Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, pp. 602–610 (2004)

    Google Scholar 

  11. Dyer, M., Frieze, A.: Fast Solution of Some Random NP-Hard Problems. In: Proceedings of the 27th IEEE Symposium on Foundations of Computer Science, pp. 331–336 (1986)

    Google Scholar 

  12. Feige, U., Ofek, E.: Spectral techniques applied to sparse random graphs. Random Structures and Algorithms 27(2), 251–275 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fiedler, M.: Algebraic connectibility of graphs. Czechoslovak Mathematical Journal 23(98), 298–305 (1973)

    MathSciNet  Google Scholar 

  14. Friedman, J., Kahn, J., Szemeredi, E.: On the second eigenvalue of random regular graphs. In: Proceedings of the 21st annual ACM Symposium on Theory of computing, pp. 587–598 (1989)

    Google Scholar 

  15. Furedi, Z., Komlos, J.: The eigenvalues of random symmetric matrices. Combinatorica 1(3), 233–241 (1981)

    Article  MathSciNet  Google Scholar 

  16. Golub, G., Van Loan, C.: Matrix computations, 3rd edn. Johns Hopkins University Press, London (1996)

    MATH  Google Scholar 

  17. Kannan, R., Vempala, S., Vetta, A.: On Clusterings: Good, bad and spectral. In: Proceedings of the Symposium on Foundations of Computer Science, pp. 497–515 (2000)

    Google Scholar 

  18. McSherry, F.: Spectral partitioning of random graphs. In: Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, pp. 529–537 (2001)

    Google Scholar 

  19. Sinclair, A., Jerrum, M.: Conductance and the mixing property of markov chains, the approximation of the permenant resolved. In: Proc. of the 20th Annual ACM Symposium on Theory of Computing, pp. 235–244 (1988)

    Google Scholar 

  20. Spielman, D., Teng, S.-h.: Spectral Partitioning Works: Planar graphs and finite element meshes. In: Proc. of the 37th Annual Symposium on Foundations of Computer Science (FOCS 1996), pp. 96–105 (1996)

    Google Scholar 

  21. Verma, D., Meila, M.: A comparison of spectral clustering algorithms, TR UW-CSE-03-05-01, Department of Computer Science and Engineering, University of Washington (2005)

    Google Scholar 

  22. Van Vu: Spectral norm of random matrices. In: Proc. of the 36th annual ACM Symposium on Theory of computing, pp. 619–626 (2005)

    Google Scholar 

  23. Zhao, Y., Karypis, G.: Evaluation of hierarchical clustering algorithms for document datasets. In: Proc. of the 11 International Conference on Information and Knowledge Management, pp. 515–524 (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Yahoo! Research Labs,  

    Anirban Dasgupta

  2. Department of Computer Science, Cornell University,  

    John Hopcroft

  3. Department of Computer Science, Yale University,  

    Ravi Kannan & Pradipta Mitra

Authors
  1. Anirban Dasgupta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John Hopcroft
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ravi Kannan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Pradipta Mitra
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA and School of Computer Science, Tel-Aviv University, 69978, Tel-Aviv, Israel

    Yossi Azar

  2. Department of Computer Science, University of Leicester, University Road, LE1 7RH, Leicester, UK

    Thomas Erlebach

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Dasgupta, A., Hopcroft, J., Kannan, R., Mitra, P. (2006). Spectral Clustering by Recursive Partitioning. In: Azar, Y., Erlebach, T. (eds) Algorithms – ESA 2006. ESA 2006. Lecture Notes in Computer Science, vol 4168. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11841036_25

Download citation

  • DOI: https://doi.org/10.1007/11841036_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-38875-3

  • Online ISBN: 978-3-540-38876-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics