Supervised Learning of Graph Structure

  • Andrea Torsello
  • Luca Rossi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7005)


Graph-based representations have been used with considerable success in computer vision in the abstraction and recognition of object shape and scene structure. Despite this, the methodology available for learning structural representations from sets of training examples is relatively limited. In this paper we take a simple yet effective Bayesian approach to attributed graph learning. We present a naïve node-observation model, where we make the important assumption that the observation of each node and each edge is independent of the others, then we propose an EM-like approach to learn a mixture of these models and a Minimum Message Length criterion for components selection. Moreover, in order to avoid the bias that could arise with a single estimation of the node correspondences, we decide to estimate the sampling probability over all the possible matches. Finally we show the utility of the proposed approach on popular computer vision tasks such as 2D and 3D shape recognition.


Message Length Model Node External Node Observation Probability Minimum Message Length 
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  1. 1.
    Babai, L., Erdös, P., Selkow, S.M.: Random Graph Isomorphism. SIAM J. Comput. 9(3), 635–638 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beichl, I., Sullivan, F.: Approximating the permanent via importance sampling with application to the dimer covering problem. J. Comput. Phys. 149(1), 128–147 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bonev, B., et al.: Constellations and the Unsupervised Learning of Graphs. In: Escolano, F., Vento, M. (eds.) GbRPR. LNCS, vol. 4538, pp. 340–350. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Bunke, H., et al.: Graph Clustering Using the Weighted Minimum Common Supergraph. In: Hancock, E.R., Vento, M. (eds.) GbRPR 2003. LNCS, vol. 2726, pp. 235–246. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Cour, T., Srinivasan, P., Shi, J.: Balanced graph matching. In: Advances in NIPS (2006)Google Scholar
  6. 6.
    Friedman, N., Koller, D.: Being Bayesian about Network Structure. Machine Learning 50(1-2), 95–125 (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    He, X.C., Yung, N.H.C.: Curvature scale space corner detector with adaptive threshold and dynamic region of support. In: Proceedings of the Pattern Recognition, 17th International Conference on (ICPR 2004), vol. 2, pp. 791–794. IEEE Computer Society, Washington, DC, USA (2004)Google Scholar
  8. 8.
    Luo, B., Hancock, E.R.: A spectral approach to learning structural variations in graphs. Pattern Recognition 39, 1188–1198 (2006)CrossRefzbMATHGoogle Scholar
  9. 9.
    Nene, S.A., Nayar, S.K., Murase, H.: Columbia Object Image Library (COIL-20). Technical report (February 1996)Google Scholar
  10. 10.
    Pelillo, M.: Replicator equations, maximal cliques, and graph isomorphism. em Neural Computation 11(8), 1933–1955 (1999)CrossRefGoogle Scholar
  11. 11.
    Rabbat, M.G., Figueiredo, M.A.T., Nowak, R.D.: Network Inference From Co-Occurrences. IEEE Trans. Information Theory 54(9), 4053–4068 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Siddiqi, K., et al.: Retrieving Articulated 3D Models Using Medial Surfaces. Machine Vision and Applications 19(4), 261–274 (2008)CrossRefzbMATHGoogle Scholar
  13. 13.
    Sinkhorn, R.: A relationship between arbitrary positive matrices and double stochastic matrices. Ann. Math. Stat. 35, 876–879 (1964)CrossRefzbMATHGoogle Scholar
  14. 14.
    Torsello, A., Hancock, E.R.: Learning Shape-Classes Using a Mixture of Tree-Unions. IEEE Trans. Pattern Anal. Machine Intell. 28(6), 954–967 (2006)CrossRefGoogle Scholar
  15. 15.
    Torsello, A.: An Importance Sampling Approach to Learning Structural Representations of Shape. In: IEEE CVPR (2008)Google Scholar
  16. 16.
    Torsello, A., Dowe, D.: Learning a generative model for structural representations. In: Wobcke, W., Zhang, M. (eds.) AI 2008. LNCS (LNAI), vol. 5360, pp. 573–583. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  17. 17.
    Torsello, A., Hidovic-Rowe, D., Pelillo, M.: Polynomial-time metrics for attributed trees. IEEE Trans. Pattern Anal. Mach. Intell. 27, 1087–1099 (2005)CrossRefGoogle Scholar
  18. 18.
    White, D., Wilson, R.C.: Spectral Generative Models for Graphs. In: Int. Conf. Image Analysis and Processing, pp. 35–42. IEEE Computer Society, Los Alamitos (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Andrea Torsello
    • 1
  • Luca Rossi
    • 1
  1. 1.Dipartimento di Scienze Ambientali, Informatica e StatisticaUniversità Ca’ FoscariVeneziaItaly

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