Abstract
In this paper, we develop a new graph kernel by using the quantum Jensen-Shannon divergence and the discrete-time quantum walk. To this end, we commence by performing a discrete-time quantum walk to compute a density matrix over each graph being compared. For a pair of graphs, we compare the mixed quantum states represented by their density matrices using the quantum Jensen-Shannon divergence. With the density matrices for a pair of graphs to hand, the quantum graph kernel between the pair of graphs is defined by exponentiating the negative quantum Jensen-Shannon divergence between the graph density matrices. We evaluate the performance of our kernel on several standard graph datasets, and demonstrate the effectiveness of the new kernel.
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References
Schölkopf, B., Smola, A.: Learning with Kernels. MIT Press (2002)
Haussler, D.: Convolution kernels on discrete structures. In: Technical Report UCS-CRL-99-10, Santa Cruz, CA, USA (1999)
Kashima, H., Tsuda, K., Inokuchi, A.: Marginalized kernels between labeled graphs. In: Proceedings of ICML, pp. 321–328 (2003)
Borgwardt, K.M., Kriegel, H.-P.: Shortest-path kernels on graphs. In: Proceedings of the IEEE International Conference on Data Mining, pp. 74–81 (2005)
Aziz, F., Wilson, R.C., Hancock, E.R.: Backtrackless walks on a graph. IEEE Transactions on Neural Networks and Learning Systems 24, 977–989 (2013)
Ren, P., Wilson, R.C., Hancock, E.R.: Graph characterization via ihara coefficients. IEEE Transactions on Neural Networks 22, 233–245 (2011)
Shervashidze, N., Schweitzer, P., van Leeuwen, E.J., Mehlhorn, K., Borgwardt, K.M.: Weisfeiler-lehman graph kernels. Journal of Machine Learning Research 1, 1–48 (2010)
Harchaoui, Z., Bach, F.: Image classification with segmentation graph kernels. In: Proceedings of CVPR (2007)
Bach, F.R.: Graph kernels between point clouds. In: Proceedings of ICML, pp. 25–32 (2008)
Bai, L., Ren, P., Hancock, E.R.: A hypergraph kernel from isomorphism tests. In: Proceedings of ICPR, pp. 3880–3885 (2014)
Bai, L., Hancock, E.R.: Graph kernels from the jensen-shannon divergence. Journal of Mathematical Imaging and Vision 47, 60–69 (2013)
Bai, L., Hancock, E.R.: Graph clustering using the jensen-shannon kernel. In: Real, P., Diaz-Pernil, D., Molina-Abril, H., Berciano, A., Kropatsch, W. (eds.) CAIP 2011, Part I. LNCS, vol. 6854, pp. 394–401. Springer, Heidelberg (2011)
Bai, L., Hancock, E.R., Ren, P.: Jensen-shannon graph kernel using information functionals. In: Proceedings of ICPR, pp. 2877–2880 (2012)
Bai, L., Rossi, L., Torsello, A., Hancock, E.R.: A quantum jensen-shannon graph kernel for unattributed graphs. Pattern Recognition 48(2), 344–355 (2015)
Bai, L., Hancock, E.R., Torsello, A., Rossi, L.: A quantum jensen-shannon graph kernel using the continuous-time quantum walk. In: Kropatsch, W.G., Artner, N.M., Haxhimusa, Y., Jiang, X. (eds.) GbRPR 2013. LNCS, vol. 7877, pp. 121–131. Springer, Heidelberg (2013)
Rossi, L., Torsello, A., Hancock, E.R.: A continuous-time quantum walk kernel for unattributed graphs. In: Kropatsch, W.G., Artner, N.M., Haxhimusa, Y., Jiang, X. (eds.) GbRPR 2013. LNCS, vol. 7877, pp. 101–110. Springer, Heidelberg (2013)
Lamberti, P., Majtey, A., Borras, A., Casas, M., Plastino, A.: Metric character of the quantum jensen-shannon divergence. Physical Review AÂ 77, 052311 (2008)
Majtey, A., Lamberti, P., Prato, D.: Jensen-shannon divergence as a measure of distinguishability between mixed quantum states. Physical Review AÂ 72, 052310 (2005)
Farhi, E., Gutmann, S.: Quantum computation and decision trees. Physical Review AÂ 58, 915 (1998)
Ren, P., Aleksic, T., Emms, D., Wilson, R.C., Hancock, E.R.: Quantum walks, ihara zeta functions and cospectrality in regular graphs. Quantum Information Process 10, 405–417 (2011)
Emms, D., Severini, S., Wilson, R.C., Hancock, E.R.: Coined quantum walks lift the cospectrality of graphs and trees. Pattern Recognition 42, 1988–2002 (2009)
L.G.: A fast quantum mechanical algorithm for database search. In: Proceedings of ACM Symposium on the Theory of Computation, pp. 212–219 (1996)
Nielsen, M., Chuang, I.: Quantum computation and quantum information. Cambridge university press (2010)
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Bai, L., Rossi, L., Ren, P., Zhang, Z., Hancock, E.R. (2015). A Quantum Jensen-Shannon Graph Kernel Using Discrete-Time Quantum Walks. In: Liu, CL., Luo, B., Kropatsch, W., Cheng, J. (eds) Graph-Based Representations in Pattern Recognition. GbRPR 2015. Lecture Notes in Computer Science(), vol 9069. Springer, Cham. https://doi.org/10.1007/978-3-319-18224-7_25
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DOI: https://doi.org/10.1007/978-3-319-18224-7_25
Publisher Name: Springer, Cham
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