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On Subspace Distance

  • Xichen Sun
  • Qiansheng Cheng
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4142)

Abstract

As pattern recognition methods, subspace methods have attracted much attention in the fields of face, object and video-based recognition in recent years. In subspace methods, each instance is characterized by a subspace that is spanned by a set of vectors. Thus, the distance between instances reduces to the distance between subspaces. Herein, the subspace distance designing problem is considered mathematically. Any distance designed according the method presented here can be embedded into associated recognition algorithms. The main contributions in this paper include:

– Solving the open problem proposed by Wang, Wang and Feng (2005), that is, we proved that their dissimilarity is a distance;

– Presenting a general framework of subspace construction, concretely speaking, we pointed out a view that subspace distance also could be regarded as the classical distance in vector space;

– Proposing two types of kernel subspace distances;

– Comparing some known subspace (dis)similarities mathematically.

Keywords

Face Recognition Classical Distance Subspace Method Subspace Analysis Principal Angle 
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.

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References

  1. 1.
    Basri, R., Jacobs, D.: Lambertian reflectance and linear subspaces. IEEE Trans. Pattern Analysis and Machine Intelligence 25(2), 218–233 (2003)CrossRefGoogle Scholar
  2. 2.
    Birkhoff, G., Mac Lane, S.: A Survey of Modern Algebra (Akp Classics). AK Peters Ltd., Wellesley (1997)Google Scholar
  3. 3.
    Etemad, K., Chellappa, R.: Discriminant analysis for recognition of human face images. Journal of Optical Society of America A 14, 1724–1733 (1997)Google Scholar
  4. 4.
    Golub, G.H., Ye, Q.: An inverse free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems. SIAM Journal on Scientific Computing 24, 312–334 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 2nd edn. The Johns Hopkins University Press, Baltimore, London (1989)MATHGoogle Scholar
  6. 6.
    Hotelling, H.: Relations between two sets of variates. Biometrika 28(3/4), 321–377 (1936)MATHCrossRefGoogle Scholar
  7. 7.
    Martinez, A.M., Kak, A.C.: PCA versus LDA. IEEE Trans. Pattern Recognition and Machine Intelligence 23(2), 228–233 (2001)CrossRefGoogle Scholar
  8. 8.
    Moghaddam, B., Pentland, A.: Probabilistic visual learning for object representation. IEEE Trans. Pattern Recognition and Machine Intelligence 19, 696–710 (1997)CrossRefGoogle Scholar
  9. 9.
    Moghaddam, B.: Bayesian face recognition. Pattern Recognition 13(11), 1771–1782 (2000)CrossRefGoogle Scholar
  10. 10.
    Müller, K.-R., Mika, S., Rätsch, G., Tsuda, K., Schölkopf, B.: An introduction to kernel-based learning algorithms. IEEE Trans Neural Networks 12(2), 181–202 (2001)CrossRefGoogle Scholar
  11. 11.
    Oja, E.: Subspace Methods of Pattern Recognition. Research Studies Press, England (1983)Google Scholar
  12. 12.
    Seung, H.S., Lee, D.D.: The manifold ways of perception. Science 290, 2268–2269 (2000)CrossRefGoogle Scholar
  13. 13.
    Simard, P., Le Cun, Y., Dender, J.: Efficient pattern recognition using a new transformation distance. In: Hanson, S., Cowan, J., Giles, C. (eds.) Advances in Neural Information Processing Systems, pp. 50–58. Morgan Kaufman, San Mateo (1993)Google Scholar
  14. 14.
    Turk, M., Pentland, A.: Eigenfaces for recognition. Journal of Cognitive Neuroscience 3, 72–86 (1991)CrossRefGoogle Scholar
  15. 15.
    Wang, L., Wang, X., Feng, J.: Intrapersonal subspace analysis with application to adaptive bayesian face recognition. Pattern Recognition 38, 617–621 (2005)CrossRefGoogle Scholar
  16. 16.
    Wang, L., Wang, X., Feng, J.: Subspace distance analysis with application to adaptive Bayesian algorithm for face recognition. Pattern Recognition 39(3), 456–464 (2006)MATHCrossRefGoogle Scholar
  17. 17.
    Wolf, L., Shashua, A.: Kernel principal angles for classification machines with applications to image sequence interpretation. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2003)Google Scholar
  18. 18.
    Wolf, L., Shashua, A.: Learning over sets using kernel principal angles. Journal of Machine Learning Research 4(6), 913–931 (2004)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Yamaguchi, O., Fukui, K., Maeda, K.: Face recognition using temporal image sequence. In: IEEE International Conference on Automatic Face & Gesture Recognition (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Xichen Sun
    • 2
  • Qiansheng Cheng
    • 1
  1. 1.LMAM, School of Mathematical SciencesPeking UniversityBeijingChina
  2. 2.National Lab on Machine PerceptionPeking UniversityBeijingChina

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