Many-to-Many Graph Matching: A Continuous Relaxation Approach

  • Mikhail Zaslavskiy
  • Francis Bach
  • Jean-Philippe Vert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6323)

Abstract

Graphs provide an efficient tool for object representation in various machine learning applications. Once graph-based representations are constructed, an important question is how to compare graphs. This problem is often formulated as a graph matching problem where one seeks a mapping between vertices of two graphs which optimally aligns their structure. In the classical formulation of graph matching, only one-to-one correspondences between vertices are considered. However, in many applications, graphs cannot be matched perfectly and it is more interesting to consider many-to-many correspondences where clusters of vertices in one graph are matched to clusters of vertices in the other graph. In this paper, we formulate the many-to-many graph matching problem as a discrete optimization problem and propose two approximate algorithms based on alternative continuous relaxations of the combinatorial problem. We compare new methods with other existing methods on several benchmark datasets.

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References

  1. 1.
    Belongie, S., Malik, J., Puzicha, J.: Shape matching and object recognition using shape contexts. IEEE Trans. Pattern Anal. Mach. Intell. 24(4), 509–522 (2002)CrossRefGoogle Scholar
  2. 2.
    Neuhaus, M., Riesen, K., Bunke, H.: Fast suboptimal algorithms for the computation of graph edit distance. In: Yeung, D.-Y., Kwok, J.T., Fred, A.L.N., Roli, F., de Ridder, D. (eds.) SSPR 2006 and SPR 2006. LNCS, vol. 4109, pp. 163–172. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Almohamad, H.A., Duffuaa, S.O.: A linear programming approach for the weighted graph matching problem. IEEE Trans. Pattern Anal. Mach. Intell. 15(5), 522–525 (1993)CrossRefGoogle Scholar
  4. 4.
    Zaslavskiy, M., Bach, F., Vert, J.-P.: A path following algorithm for the graph matching problem. IEEE Trans. Pattern Anal. Mach. Intell. 31(12), 2227–2242 (2009)CrossRefGoogle Scholar
  5. 5.
    Umeyama, S.: An eigendecomposition approach to weighted graph matching problems. IEEE Trans. Pattern Anal. Mach. Intell. 10(5), 695–703 (1988)MATHCrossRefGoogle Scholar
  6. 6.
    Caelli, T., Kosinov, S.: An eigenspace projection clustering method for inexact graph matching. IEEE Trans. Pattern Anal. Mach. Intell. 26(4), 515–519 (2004)CrossRefGoogle Scholar
  7. 7.
    Carcassoni, M., Hancock, E.: Spectral correspondence for point pattern matching. Pattern Recogn. 36(1), 193–204 (2003)MATHCrossRefGoogle Scholar
  8. 8.
    Cour, T., Srinivasan, P., Shi, J.: Balanced graph matching. In: Advanced in Neural Information Processing Systems (2006)Google Scholar
  9. 9.
    Leordeanu, M., Hebert, M.: A spectral technique for correspondence problems using pairwise constraints. In: International Conference of Computer Vision (ICCV), vol. 2, pp. 1482–1489 (October 2005)Google Scholar
  10. 10.
    Duchenne, O., Bach, F., Kweon, I., Ponce, J.: A tensor-based algorithm for high-order graph matching. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition CVPR 2009, June 20-25, pp. 1980–1987 (2009)Google Scholar
  11. 11.
    Berretti, S., Del Bimbo, A., Pala, P.: A graph edit distance based on node merging. In: Proc. of ACM International Conference on Image and Video Retrieval (CIVR), Dublin, Ireland, July 2004, pp. 464–472 (2004)Google Scholar
  12. 12.
    Ambauen, R., Fischer, S., Bunke, H.: Graph edit distance with node splitting and merging, and its application to diatom idenfication. In: GbRPR, pp. 95–106 (2003)Google Scholar
  13. 13.
    Keselman, Y., Shokoufandeh, A., Demirci, M.F., Dickinson, S.: Many-to-many graph matching via metric embedding. In: CVPR, pp. 850–857 (2003)Google Scholar
  14. 14.
    Bertsekas, D.: Nonlinear programming. Athena Scientific (1999)Google Scholar
  15. 15.
    Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Research 2, 83–97 (1955)CrossRefGoogle Scholar
  16. 16.
    Nesterov, Y., Nemirovsky, A.: Interior point polynomial methods in convex programming: Theory and applications. SIAM, Philadelphia (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Mikhail Zaslavskiy
    • 1
    • 2
    • 3
    • 5
  • Francis Bach
    • 4
  • Jean-Philippe Vert
    • 1
    • 2
    • 3
  1. 1.Center for Computational Biology, Mines ParisTech, FontainebleauFrance
  2. 2.Institut Curie 
  3. 3.INSERM, U900ParisFrance
  4. 4.INRIA-WILLOW project, Ecole Normale SupérieureParisFrance
  5. 5.Center for Mathematical Morphology, Mines ParisTech, FontainebleauFrance

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