Advertisement

Representable Hierarchical Clustering Methods for Asymmetric Networks

  • Gunnar Carlsson
  • Facundo MémoliEmail author
  • Alejandro Ribeiro
  • Santiago Segarra
Conference paper
  • 2k Downloads
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)

Abstract

This paper introduces the generative model of representability for hierarchical clustering methods in asymmetric networks, i.e., the possibility to describe a method through its action on a collection of networks called representers. We characterize the necessary and sufficient structural conditions needed on these representers in order to generate a method which is scale preserving and admissible with respect to two known axioms and, based on this result, we construct the family of cyclic clustering methods.

References

  1. 1.
    Boyd, J.: Asymmetric clusters of internal migration regions of France. IEEE Trans. Syst. Man Cybern. 10(2), 101–104 (1980)CrossRefGoogle Scholar
  2. 2.
    Carlsson, G., Mémoli, F.: Characterization, stability and convergence of hierarchical clustering methods. J. Mach. Learn. Res. 11, 1425–1470 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Carlsson, G., Mémoli, F.: Classifying clustering schemes. Found. Comput. Math. 13(2), 221–252 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Carlsson, G., Memoli, F., Ribeiro, A., Segarra, S.: Axiomatic construction of hierarchical clustering in asymmetric networks. In: IEEE International Conference on Acoustics, Speech and Signal Process (ICASSP), pp. 5219–5223 (2013)Google Scholar
  5. 5.
    Carlsson, G., Memoli, F., Ribeiro, A., Segarra, S.: Axiomatic construction of hierarchical clustering in asymmetric networks (2014). arXiv:1301.7724v2Google Scholar
  6. 6.
    Guyon, I., von Luxburg, U., Williamson, R.: Clustering: science or art? Tech. rep. Paper presented at the NIPS 2009 Workshop Clustering: Science or Art? (2009)Google Scholar
  7. 7.
    Hubert, L.: Min and max hierarchical clustering using asymmetric similarity measures. Psychometrika 38(1), 63–72 (1973)CrossRefzbMATHGoogle Scholar
  8. 8.
    Jain, A.K., Dubes, R.C.: Algorithms for Clustering Data. Prentice-Hall, Upper Saddle River, NJ (1988)zbMATHGoogle Scholar
  9. 9.
    Jardine, N., Sibson, R.: Mathematical Taxonomy. Wiley Series in Probability and Mathematical Statistics. Wiley, London (1971)zbMATHGoogle Scholar
  10. 10.
    Kleinberg, J.M.: An impossibility theorem for clustering. In: NIPS, pp. 446–453 (2002)Google Scholar
  11. 11.
    Lance, G.N., Williams, W.T.: A general theory of classificatory sorting strategies 1. Hierarchical systems. Comput. J. 9(4), 373–380 (1967)Google Scholar
  12. 12.
    Meila, M.: Comparing clusterings: an axiomatic view. In: Proceedings of the 22nd International Conference on Machine Learning, pp. 577–584. ACM, New York (2005)Google Scholar
  13. 13.
    Meila, M.: Comparing clusterings – an information based distance. J. Multivar. Anal. 98(5), 873–895 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Murtagh, F.: Multidimensional clustering algorithms. In: Compstat Lectures, vol. 1. Physika Verlag, Vienna (1985)Google Scholar
  15. 15.
    Pentney, W., Meila, M.: Spectral clustering of biological sequence data. In: Proceedings of National Conference on Artificial Intelligence (2005)Google Scholar
  16. 16.
    Punj, G., Stewart, D.W.: Cluster analysis in marketing research: review and suggestions for application. J. Market. Res. 20(2), 134–148 (1983)CrossRefGoogle Scholar
  17. 17.
    Rui, X., Wunsch-II, D.: Survey of clustering algorithms. IEEE Trans. Neural Netw. 16(3), 645–678 (2005)CrossRefGoogle Scholar
  18. 18.
    Saito, T., Yadohisa, H.: Data Analysis of Asymmetric Structures: Advanced Approaches in Computational Statistics. CRC Press, Boca Raton, FL (2004)CrossRefzbMATHGoogle Scholar
  19. 19.
    Slater, P.: Hierarchical internal migration regions of France. IEEE Trans. Syst. Man Cybern. 6(4), 321–324 (1976)CrossRefGoogle Scholar
  20. 20.
    Tarjan, R.E.: An improved algorithm for hierarchical clustering using strong components. Inf. Process. Lett. 17(1), 37–41 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Van Laarhoven, T., Marchiori, E.: Axioms for graph clustering quality functions. J. Mach. Learn. Res. 15(1), 193–215 (2014)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Walsh, D., Rybicki, L.: Symptom clustering in advanced cancer. Support. Care Cancer 14(8), 831–836 (2006)CrossRefGoogle Scholar
  23. 23.
    West, D.B., et al.: Introduction to Graph Theory, vol. 2. Prentice Hall, Upper Saddle River (2001)Google Scholar
  24. 24.
    Zadeh, R., Ben-David, S.: A uniqueness theorem for clustering. In: Proceedings of Uncertainty in Artificial Intelligence (2009)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Gunnar Carlsson
    • 1
  • Facundo Mémoli
    • 2
    • 3
    Email author
  • Alejandro Ribeiro
    • 4
  • Santiago Segarra
    • 4
  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusUSA
  3. 3.Department of Computer ScienceThe Ohio State UniversityColumbusUSA
  4. 4.Department of Electrical and Systems EngineeringUniversity of PennsylvaniaPhiladelphiaUSA

Personalised recommendations