Ihara Coefficients: A Flexible Tool for Higher Order Learning

  • Peng Ren
  • Tatjana Aleksić
  • Richard C. Wilson
  • Edwin R. Hancock
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6218)

Abstract

The aim of this paper is to seek a compact characterization of irregular unweighted hypergraphs for the purposes of clustering. To this end, we propose a novel hypergraph characterization method by using the Ihara coefficients, i.e. the characteristic polynomial coefficients extracted from the Ihara zeta function. We investigate the flexibility of the Ihara coefficients for learning relational structures with different relational orders. Furthermore, we introduce an efficient method for computing the coefficients. Our representation for hypergraphs takes into account not only the vertex connections but also the hyperedge cardinalities, and thus can distinguish different relational orders, which is prone to ambiguity in the hypergraph Laplacian. In experiments we demonstrate the effectiveness of the proposed characterization for clustering irregular unweighted hypergraphs and its advantages over the spectral characterization of the hypergraph Laplacian.

Keywords

Bipartite Graph Adjacency Matrix Laplacian Matrix Rand Index Flexible Tool 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Agarwal, S., Branson, K., Belongie, S.: Higher-order learning with graphs. In: ICML (2006)Google Scholar
  2. 2.
    Bass, H.: The ihara-selberg zeta function of a tree lattice. International Journal of Mathematics 6, 717–797 (1992)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bretto, A., Cherifi, H., Aboutajdine, D.: Hypergraph imaging: an overview. Pattern Recognition 35(3), 651–658 (2002)MATHCrossRefGoogle Scholar
  4. 4.
    Ren, P., Aleksić, T., Wilson, R.C., Hancock, E.R.: Hypergraphs, characteristic polynomials and the ihara zeta function. In: CAIP (2009)Google Scholar
  5. 5.
    Ren, P., Wilson, R.C., Hancock, E.R.: Spectral embedding of feature hypergraphs. In: da Vitoria Lobo, N., Kasparis, T., Roli, F., Kwok, J.T., Georgiopoulos, M., Anagnostopoulos, G.C., Loog, M. (eds.) S+SSPR 2008. LNCS, vol. 5342, pp. 308–317. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Rota-Bullo, S., Albarelli, A., Pelillo, M., Torsello, A.: A hypergraph-based approach to affine parameters estimation. In: ICPR (2008)Google Scholar
  7. 7.
    Shashua, A., Zass, R., Hazan, T.: Multi-way clustering using super-symetric non-negtive tensor factorization. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3954, pp. 595–608. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Storm, C.K.: The zeta function of a hypergraph. Electronic Journal of Combinatorics 13 (2006)Google Scholar
  9. 9.
    Zass, R., Shashua, A.: Probabilistic graph and hypergraph matching. In: CVPR (2008)Google Scholar
  10. 10.
    Zhou, D., Huang, J., Scholkopf, B.: Learning with hypergraphs: Clustering, classification, and embedding. In: NIPS (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Peng Ren
    • 1
  • Tatjana Aleksić
    • 2
  • Richard C. Wilson
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceThe University of YorkYorkUK
  2. 2.Faculty of ScienceUniversity of KragujevacKragujevacSerbia

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