Abstract
In this paper, we describe the use of concepts from structural and statistical pattern recognition for recovering a mapping which can be viewed as an operator on the graph attribute-set. This mapping can be used to embed graphs into spaces where tasks such as categorisation and relational matching can be effected. We depart from concepts in graph theory to introduce mappings as operators over graph spaces. This treatment leads to the recovery of a mapping based upon the graph attributes which is related to the edge-space of the graphs under study. As a result, this mapping is a linear operator over the attribute set which is associated with the graph topology. Here, we employ an optimisation approach whose cost function is related to the target function used in discrete Markov Random Field approaches. Thus, the proposed method provides a link between concepts in graph theory, statistical inference and linear operators. We illustrate the utility of the recovered embedding for shape matching and categorisation on MPEG7 CE-Shape-1 dataset. We also compare our results to those yielded by alternatives.
Chapter PDF
References
Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)
Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000)
Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. In: NIPS. Number, vol. 14, pp. 634–640 (2002)
Chung, F.R.K.: Spectral Graph Theory. American Mathematical Society, Providence (1997)
Sebastian, T.B., Klein, P.N., Kimia, B.B.: Shock-based indexing into large shape databases. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2352, pp. 731–746. Springer, Heidelberg (2002)
Wong, A.K.C., You, M.: Entropy and distance of random graphs with application to structural pattern recognition. IEEE TPAMI 7, 599–609 (1985)
Christmas, W.J., Kittler, J., Petrou, M.: Structural matching in computer vision using probabilistic relaxation. IEEE TPAMI 17(8), 749–764 (1995)
Wilson, R., Hancock, E.R.: Structural matching by discrete relaxation. IEEE TPAMI 19(6), 634–648 (1997)
Caetano, T., Cheng, L., Le, Q., Smola, A.: Learning graph matching. In: ICCV, pp. 14–21 (2007)
Biggs, N.L.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1993)
Bremaud, P.: Markov Chains, Gibbs Fields, Monte Carlo Simulation and Queues. Springer, Heidelberg (2001)
Keuchel, J.: Multiclass image labeling with semidefinite programming. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3952, pp. 454–467. Springer, Heidelberg (2006)
Kumar, M., Torr, P., Zisserman, A.: Solving markov random fields using second order cone programming relaxations. In: CVPR, pp. 1045–1052 (2006)
Cour, T., Shi, J.: Solving markov random fields with spectral relaxation. In: Intl. Conf. on Artificial Intelligence and Statistics (2007)
Young, G., Householder, A.S.: Discussion of a set of points in terms of their mutual distances. Psychometrika 3, 19–22 (1938)
Ding, C., He, X.: K-means clustering via principal component analysis. In: ICML, pp. 225–232 (2004)
Gold, S., Rangarajan, A.: A graduated assignment algorithm for graph matching. IEEE TPAMI 18(4), 377–388 (1996)
Demirci, M.F., Shokoufandeh, A., Dickinson, S.J.: Skeletal shape abstraction from examples. IEEE TPAMI 31(5), 944–952 (2009)
Belongie, S., Malik, J., Puzicha, J.: Shape matching and object recognition using shape contexts. IEEE TPAMI 24(24), 509–522 (2002)
Chen, L., McAuley, J.J., Feris, R.S., Caetano, T.S., Turk, M.: Shape classification through structured learning of matching measures. In: CVPR (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zhao, H., Zhou, J., Robles-Kelly, A. (2010). A Structured Learning Approach to Attributed Graph Embedding. In: Hancock, E.R., Wilson, R.C., Windeatt, T., Ulusoy, I., Escolano, F. (eds) Structural, Syntactic, and Statistical Pattern Recognition. SSPR /SPR 2010. Lecture Notes in Computer Science, vol 6218. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14980-1_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-14980-1_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14979-5
Online ISBN: 978-3-642-14980-1
eBook Packages: Computer ScienceComputer Science (R0)