Manifold Learning and the Quantum Jensen-Shannon Divergence Kernel

  • Luca Rossi
  • Andrea Torsello
  • Edwin R. Hancock
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8047)


The quantum Jensen-Shannon divergence kernel [1] was recently introduced in the context of unattributed graphs where it was shown to outperform several commonly used alternatives. In this paper, we study the separability properties of this kernel and we propose a way to compute a low-dimensional kernel embedding where the separation of the different classes is enhanced. The idea stems from the observation that the multidimensional scaling embeddings on this kernel show a strong horseshoe shape distribution, a pattern which is known to arise when long range distances are not estimated accurately. Here we propose to use Isomap to embed the graphs using only local distance information onto a new vectorial space with a higher class separability. The experimental evaluation shows the effectiveness of the proposed approach.


Graph Kernels Manifold Learning Continuous-Time Quantum Walk Quantum Jensen-Shannon Divergence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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)CrossRefGoogle Scholar
  2. 2.
    Siddiqi, K., Shokoufandeh, A., Dickinson, S., Zucker, S.: Shock graphs and shape matching. International Journal of Computer Vision 35, 13–32 (1999)CrossRefGoogle Scholar
  3. 3.
    Jeong, H., Tombor, B., Albert, R., Oltvai, Z., Barabási, A.: The large-scale organization of metabolic networks. Nature 407, 651–654 (2000)CrossRefGoogle Scholar
  4. 4.
    Schölkopf, B., Smola, A.J.: Learning with kernels: Support vector machines, regularization, optimization, and beyond. MIT press (2001)Google Scholar
  5. 5.
    Gaertner, T., Flach, P., 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)CrossRefGoogle Scholar
  6. 6.
    Borgwardt, K., Kriegel, H.: Shortest-path kernels on graphs. In: Fifth IEEE International Conference on Data Mining, p. 8. IEEE (2005)Google Scholar
  7. 7.
    Shervashidze, N., Vishwanathan, S., Petri, T., Mehlhorn, K., Borgwardt, K.: Efficient graphlet kernels for large graph comparison. In: Proceedings of the International Workshop on Artificial Intelligence and Statistics (2009)Google Scholar
  8. 8.
    Haussler, D.: Convolution kernels on discrete structures. Technical report, UC Santa Cruz (1999)Google Scholar
  9. 9.
    Farhi, E., Gutmann, S.: Quantum computation and decision trees. Physical Review A 58, 915 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Emms, D., Wilson, R., Hancock, E.: Graph embedding using a quasi-quantum analogue of the hitting times of continuous time quantum walks. Quantum Information & Computation 9, 231–254 (2009)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Rossi, L., Torsello, A., Hancock, E.R.: Approximate axial symmetries from continuous time quantum walks. In: Gimel’farb, G., Hancock, E., Imiya, A., Kuijper, A., Kudo, M., Omachi, S., Windeatt, T., Yamada, K. (eds.) SSPR&SPR 2012. LNCS, vol. 7626, pp. 144–152. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  12. 12.
    Lamberti, P., Majtey, A., Borras, A., Casas, M., Plastino, A.: Metric character of the quantum Jensen-Shannon divergence. Physical Review A 77, 052311 (2008)CrossRefGoogle Scholar
  13. 13.
    Nielsen, M., Chuang, I.: Quantum computation and quantum information. Cambridge university press (2010)Google Scholar
  14. 14.
    Tenenbaum, J.B., De Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323 (2000)CrossRefGoogle Scholar
  15. 15.
    Czaja, W., Ehler, M.: Schroedinger eigenmaps for the analysis of biomedical data. IEEE Transactions on Pattern Analysis and Machine Intelligence 35, 1274–1280 (2013)CrossRefGoogle Scholar
  16. 16.
    Kendall, D.G.: Abundance matrices and seriation in archaeology. Probability Theory and Related Fields 17, 104–112 (1971)MathSciNetGoogle Scholar
  17. 17.
    Briët, J., Harremoës, P.: Properties of classical and quantum jensen-shannon divergence. Physical review A 79, 052311 (2009)CrossRefGoogle Scholar
  18. 18.
    Nayar, S., Nene, S., Murase, H.: Columbia object image library (coil 100). Technical report, Tech. Report No. CUCS-006-96. Department of Comp. Science, Columbia University (1996)Google Scholar
  19. 19.
    Torsello, A., Rossi, L.: Supervised learning of graph structure. In: Pelillo, M., Hancock, E.R. (eds.) SIMBAD 2011. LNCS, vol. 7005, pp. 117–132. Springer, Heidelberg (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Luca Rossi
    • 1
  • Andrea Torsello
    • 1
  • Edwin R. Hancock
    • 2
  1. 1.Department of Environmental Science, Informatics and StatisticsCa’ Foscari University of VeniceItaly
  2. 2.Department of Computer ScienceUniversity of YorkUK

Personalised recommendations