Learning Mixtures of Weighted Tree-Unions by Minimizing Description Length

  • Andrea Torsello
  • Edwin R. Hancock
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3023)


This paper focuses on how to perform the unsupervised clustering of tree structures in an information theoretic setting. We pose the problem of clustering as that of locating a series of archetypes that can be used to represent the variations in tree structure present in the training sample. The archetypes are tree-unions that are formed by merging sets of sample trees, and are attributed with probabilities that measure the node frequency or weight in the training sample. The approach is designed to operate when the correspondences between nodes are unknown and must be inferred as part of the learning process. We show how the tree merging process can be posed as the minimisation of an information theoretic minimum descriptor length criterion. We illustrate the utility of the resulting algorithm on the problem of classifying 2D shapes using a shock graph representation.


Sample Tree Tree Union Edit Distance Shape Class Node Probability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Cyr, C., Kimia, B.: 3D Object Recognition Using Shape Similarity-Based Aspect Graph. In: ICCV (2001)Google Scholar
  2. 2.
    Dickinson, S.J., Pentland, A.P., Rosenfeld, A.: 3-D shape recovery using distributed aspect matching. PAMI 14(2), 174–198 (1992)Google Scholar
  3. 3.
    Friedman, N., Koller, D.: Being Bayesian about Network Structure. Machine Learning (2002) (to appear)Google Scholar
  4. 4.
    Getoor, L., et al.: Learning Probabilistic models of relational structure. In: 8th Int. Conf. on Machine Learning (2001)Google Scholar
  5. 5.
    Heckerman, D., Geiger, D., Chickering, D.M.: Learning Bayesian networks: the combination of knowledge and statistical data. Machine Learning 20(3), 197–243 (1995)zbMATHGoogle Scholar
  6. 6.
    Heap, T., Hogg, D.: Wormholes in shape space: tracking through discontinuous changes in shape. In: ICCV, pp. 344-349 (1998)Google Scholar
  7. 7.
    Hjaltason, G.R., Samet, H.: Properties of embedding methods for similarity searching in metric spaces. PAMI (25), 530–549 (2003)Google Scholar
  8. 8.
    Ioffe, S., Forsyth, D.A.: Human Tracking with Mixtures of Trees. In: ICCV, vol. I, pp. 690–695 (2001)Google Scholar
  9. 9.
    Keselman, Y., Shokoufandeh, A., Demirci, M.F., Dickinson, S.: Many-to-many graph matching via metric embedding. In: CVPR 2003, vol. I, pp. 850–857 (2003)Google Scholar
  10. 10.
    Kimia, B.B., Tannenbaum, A.R., Zucker, S.W.: Shapes, shocks, and deformations I. International Journal of Computer Vision 15, 189–224 (1995)CrossRefGoogle Scholar
  11. 11.
    Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some of its applications. In: 35th Anual Symposium on Foundations of Computer Science, pp. 169–175 (1994)Google Scholar
  12. 12.
    Meilă, M.: Learning with Mixtures of Trees. PhD thesis, MIT (1999)Google Scholar
  13. 13.
    Rissanen, J.: Stochastic complexity and modeling. Annals of Statistics 14, 1080–1100 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Robles-Kelly, A., Hancock, E.R.: A maximum likelihood framework for iterative eigendecomposition. In: ICCV, vol. I, pp. 654–661 (2001)Google Scholar
  15. 15.
    Shokoufandeh, A., Dickinson, S.J., Siddiqi, K., Zucker, S.W.: Indexing using a spectral encoding of topological structure. In: CVPR (1999)Google Scholar
  16. 16.
    Sebastian, T., Klein, P., Kimia, B.: Recognition of shapes by editing shock graphs. In: ICCV, vol. I, pp. 755–762 (2001)Google Scholar
  17. 17.
    Torsello, A., Hancock, E.R.: Efficiently computing weighted tree edit distance using relaxation labeling. In: Figueiredo, M., Zerubia, J., Jain, A.K. (eds.) EMMCVPR 2001. LNCS, vol. 2134, pp. 438–453. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. 18.
    Torsello, A., Hancock, E.R.: Matching and embedding through edit-union of trees. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2352, pp. 822–836. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  19. 19.
    Zhu, S.C., Yuille, A.L.: FORMS: A Flexible Object Recognition and Modelling System. IJCV 20(3), 187–212 (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Andrea Torsello
    • 1
  • Edwin R. Hancock
    • 2
  1. 1.Dipartimento di InformaticaUniversita’ Ca’ Foscari di VeneziaVenezia MestreItaly
  2. 2.Department of Computer ScienceUniversity of YorkYorkEngland

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