Abstract
We propose a family of attributed graph kernels based on mutual information measures, i.e., the Jensen-Tsallis (JT) q-differences (for q ∈ [1,2]) between probability distributions over the graphs. To this end, we first assign a probability to each vertex of the graph through a continuous-time quantum walk (CTQW). We then adopt the tree-index approach [1] to strengthen the original vertex labels, and we show how the CTQW can induce a probability distribution over these strengthened labels. We show that our JT kernel (for q = 1) overcomes the shortcoming of discarding non-isomorphic substructures arising in the R-convolution kernels. Moreover, we prove that the proposed JT kernels generalize the Jensen-Shannon graph kernel [2] (for q = 1) and the classical subtree kernel [3] (for q = 2), respectively. Experimental evaluations demonstrate the effectiveness and efficiency of the JT kernels.
Chapter PDF
Similar content being viewed by others
References
Dahm, N., Bunke, H., Caelli, T., Gao, Y.: A Unified Framework for Strengthening Topological Node Features and Its Application to Subgraph Isomorphism Detection. In: Kropatsch, W.G., Artner, N.M., Haxhimusa, Y., Jiang, X. (eds.) GbRPR 2013. LNCS, vol. 7877, pp. 11–20. Springer, Heidelberg (2013)
Bai, L., Hancock, E.R.: Graph Kernels from the Jensen-Shannon Divergence. Journal of Mathematical Imaging and Vision 47(1-2), 60–69 (2013)
Gärtner, T., Flach, P.A., Wrobel, S.: On Graph Kernels: Hardness Results and Efficient Alternatives. In: Schölkopf, B., Warmuth, M.K. (eds.) COLT/Kernel 2003. LNCS (LNAI), vol. 2777, pp. 129–143. Springer, Heidelberg (2003)
Bunke, H., Riesen, K.: A Family of Novel Graph Kernels for Structural Pattern Recognition. In: Rueda, L., Mery, D., Kittler, J. (eds.) CIARP 2007. LNCS, vol. 4756, pp. 20–31. Springer, Heidelberg (2007)
Jebara, T., Kondor, R.I., Howard, A.: Probability Product Kernels. Journal of Machine Learning Research 5, 819–844 (2004)
Haussler, D.: Convolution Kernels on Discrete Structures. Technical Report UCS-CRL-99-10 (1999)
Kashima, H., Tsuda, K., Inokuchi, A.: Marginalized Kernels Between Labeled Graphs. In: Proc. ICML, pp. 321–328 (2003)
Borgwardt, K.M., Kriegel, H.P.: Shortest-path Kernels on Graphs. In: Proc. ICDM, pp. 74–81 (2005)
Shervashidze, N., Borgwardt, K.M.: Fast Subtree Kernels on Graphs. In: Proc. NIPS, pp. 1660–1668 (2009)
Costa, F., Grave, K.D.: Fast Neighborhood Subgraph Pairwise Distance Kernel. In: Proc. ICML, pp. 255–262 (2010)
Shervashidze, N., Vishwanathan, S.V.N., Petri, T., Mehlhorn, K., Borgwardt, K.M.: Efficient Graphlet Kernels for Large Graph Comparison. Journal of Machine Learning Research 5, 488–495 (2009)
Martins, A.F.T., Smith, N.A., Xing, E.P., Aguiar, P.M.Q., Figueiredo, M.A.T.: Nonextensive Information Theoretic Kernels on Measures. Journal of Machine Learning Research 10, 935–975 (2009)
Julia, K.: Quantum Random Walks: An Introductory Overview. Contemporary Physics 44(4), 307–327 (2003)
Farhi, E., Gutmann, S.: Quantum Computation and Decision Trees. Physical Review A 58, 915 (1998)
Aubry, M., Schlickewei, U., Cremers, D.: The Wave Kernel Signature: A Quantum Mechanical Approach to Shape Analysis. In: Proc. ICCV Workshops, pp. 1626–1633 (2011)
Rossi, L., Tosello, A., Hancock, E.R., Wilson, R.C.: Characterizing Graph Symmetries through Quantum Jensen-Shannon Divergence. Physical Review E 88(3-1), 032806 (2013)
Suau, P., Hancock, E.R., Escolano, F.: Graph Characteristics from the Schrödinger Operator. In: Kropatsch, W.G., Artner, N.M., Haxhimusa, Y., Jiang, X. (eds.) GbRPR 2013. LNCS, vol. 7877, pp. 172–181. Springer, Heidelberg (2013)
Cottrell, S., Hillery, M.: Finding Structural Anomalies in Star Graphs Using Quantum Walks. Physical Review Letters 112(3), 030501 (2014)
Aziz, F., Wilson, R.C., Hancock, E.R.: Backtrackless Walks on A Graph. IEEE Transactions on Neural Networks and Learning System 24(6), 977–989 (2013)
Mahé, P., Ueda, N., Akutsu, T., Perret, J., Vert, J.: Extensions of marginalized graph kernels. In: Proc. ICML (2004)
Alon, N., Benjamini, I., Lubetzky, E., Sodin, S.: Non-backtracking Random Walks Mix Faster. Communications in Contemporary Mathematics 9(4), 585–603 (2007)
Tsallis, C.: Possible Generalization of Boltzman-Gibbs Statistacs. J. Stats. Physics 52, 479–487 (1988)
Furuichi, S.: Information Theoretical Properities of Tsallis Entropies. Journal of Math. Physics 47, 2 (2006)
Riesen, K., Bunke, H.: Graph Classification and Clustering based on Vector Space Embedding. World Scientific Publishing (2010)
Bai, L., Hancock, E.R., Ren, P.: Jensen-Shannon Graph Kernel using Information Functionals. In: Proc. ICPR, pp. 2877–2880 (2012)
Bai, L., Rossi, L., Torsello, A., Hancock, E.R.: A Quantum Jensen-Shannon Kernel for Unattributed Graphs. To appear in Pattern Recognition (2014)
Kriege, N., Mutzel, P.: Subgraph Matching Kernels for Attributed Graphs. In: Proc. ICML (2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bai, L., Rossi, L., Bunke, H., Hancock, E.R. (2014). Attributed Graph Kernels Using the Jensen-Tsallis q-Differences. In: Calders, T., Esposito, F., Hüllermeier, E., Meo, R. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2014. Lecture Notes in Computer Science(), vol 8724. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44848-9_7
Download citation
DOI: https://doi.org/10.1007/978-3-662-44848-9_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-44847-2
Online ISBN: 978-3-662-44848-9
eBook Packages: Computer ScienceComputer Science (R0)