Spectral Partitioning with Indefinite Kernels Using the Nyström Extension

  • Serge Belongie
  • Charless Fowlkes
  • Fan Chung
  • Jitendra Malik
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2352)

Abstract

Fowlkes et al. [7] recently introduced an approximation to the Normalized Cut (NCut) grouping algorithm [18] based on random subsampling and the Nyström extension. As presented, their method is restricted to the case where W, the weighted adjacency matrix, is positive definite. Although many common measures of image similarity (i.e. kernels) are positive definite, a popular example being Gaussian-weighted distance, there are important cases that are not. In this work, we present a modification to Nyström-NCut that does not require W to be positive definite. The modification only affects the orthogonalization step, and in doing so it necessitates one additional O(m3) operation, where m is the number of random samples used in the approximation. As such it is of interest to know which kernels are positive definite and which are indefinite. In addressing this issue, we further develop connections between NCut and related methods in the kernel machines literature. We provide a proof that the Gaussian-weighted chi-squared kernel is positive definite, which has thus far only been conjectured. We also explore the performance of the approximation algorithm on a variety of grouping cues including contour, color and texture.

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References

  1. 1.
    C. Berg, J. P. R. Christensen, and P. Ressel. Harmonic Analysis on Semigroups. Springer-Verlag, New York, 1984.MATHGoogle Scholar
  2. 2.
    R. Bhatia. Matrix Analysis. Springer Verlag, 1997.Google Scholar
  3. 3.
    O. Chapelle, P. Haffner, and V. Vapnik. SVMs for histogram based image classification. IEEE Trans. Neural Networks, 10(5):1055–1064, September 1999.Google Scholar
  4. 4.
    F. R. K. Chung. Spectral Graph Theory. Number 92 in CBMS Regional Conference Series in Mathematics. AMS, 1997.Google Scholar
  5. 5.
    T. H. Cormen, C. E. Leiserson, and R. L. Rivest. Introduction to Algorithms. MIT Press, 1991.Google Scholar
  6. 6.
    J. de Leeuw. Multidimensional scaling. UCLA Dept. of Statistics, Preprint no. 274, 2000.Google Scholar
  7. 7.
    C. Fowlkes, S. Belongie, and J. Malik. Efficient spatiotemporal grouping using the Nyström method. In Proc. IEEE Conf. Comput. Vision and Pattern Recognition, December 2001.Google Scholar
  8. 8.
    Yoram Gdalyahu, Daphna Weinshall, and Michael Werman. Stochastic image segmentation by typical cuts. In CVPR, 1999.Google Scholar
  9. 9.
    R.A. Horn and C.R. Johnson. Matrix Analysis. Cambridge Univ Press, 1985.Google Scholar
  10. 10.
    T. Leung and J. Malik. Contour continuity in region-based image segmentation. In H. Burkhardt and B. Neumann, editors, Proc. Euro. Conf. Computer Vision, volume 1, pages 544–59, Freiburg, Germany, June 1998. Springer-Verlag.Google Scholar
  11. 11.
    J. Malik, S. Belongie, T. Leung, and J. Shi. Contour and texture analysis for image segmentation. Int’l. Journal of Computer Vision, 43(1):7–27, June 2001.Google Scholar
  12. 12.
    R. Mathias. An arithmetic-geometric-harmonic mean inequality involving Hadamard products. Linear Algebra and its Applications, 184:71–78, 1993.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    P. Perona and W. T. Freeman. A factorization approach to grouping. In Proc. 5th Europ. Conf. Comput. Vision, 1998.Google Scholar
  14. 14.
    J. Puzicha and S. Belongie. Model-based halftoning for color image segmentation. In ICPR, volume 3, pages 629–632, 2000.Google Scholar
  15. 15.
    J. Puzicha, T. Hofmann, and J. Buhmann. Non-parametric similarity measures for unsupervised texture segmentation and image retrieval. In Proc. IEEE Conf. Computer Vision and Pattern Recognition, pages 267–72, San Juan, Puerto Rico, Jun. 1997.Google Scholar
  16. 16.
    B. Schölkopf and A. Smola. Learning with Kernels. Cambridge, MA: MIT Press, 2001. in preparation.Google Scholar
  17. 17.
    B. Schölkopf, A. Smola, and K.-R. Müller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10:1299–1319, 1998.CrossRefGoogle Scholar
  18. 18.
    J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Trans. Pattern Analysis and Machine Intelligence, 22(8):888–905, August 2000.Google Scholar
  19. 19.
    Y. Weiss. Segmentation using eigenvectors: a unifying view. In Proc. 7th Int’l. Conf. Computer Vision, pages 975–982, 1999.Google Scholar
  20. 20.
    C. Williams and M. Seeger. Using the Nyström method to speed up kernel machines. In T.K. Leen, T. G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13: Proceedings of the 2000 Conference, pages 682–688, 2001.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Serge Belongie
    • 1
  • Charless Fowlkes
    • 2
  • Fan Chung
    • 1
  • Jitendra Malik
    • 2
  1. 1.University of CaliforniaSan Diego, La JollaUSA
  2. 2.University of California, BerkeleyBerkeleyUSA

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