Trace Quotient Problems Revisited

  • Shuicheng Yan
  • Xiaoou Tang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3952)


The formulation of trace quotient is shared by many computer vision problems; however, it was conventionally approximated by an essentially different formulation of quotient trace, which can be solved with the generalized eigenvalue decomposition approach. In this paper, we present a direct solution to the former formulation. First, considering that the feasible solutions are constrained on a Grassmann manifold, we present a necessary condition for the optimal solution of the trace quotient problem, which then naturally elicits an iterative procedure for pursuing the optimal solution. The proposed algorithm, referred to as Optimal Projection Pursuing (OPP), has the following characteristics: 1) OPP directly optimizes the trace quotient, and is theoretically optimal; 2) OPP does not suffer from the solution uncertainty issue existing in the quotient trace formulation that the objective function value is invariant under any nonsingular linear transformation, and OPP is invariant only under orthogonal transformations, which does not affect final distance measurement; and 3) OPP reveals the underlying equivalence between the trace quotient problem and the corresponding trace difference problem. Extensive experiments on face recognition validate the superiority of OPP over the solution of the corresponding quotient trace problem in both objective function value and classification capability.


Face Recognition Linear Discriminant Analysis Tangent Space Recognition Rate Grassmann Manifold 
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.


  1. 1.
    Belhumeur, P., Hespanha, J., Kriegman, D.: Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection. IEEE Trans. Pattern Analysis and Machine Intelligence 19(7), 711–720 (1997)CrossRefGoogle Scholar
  2. 2.
    Belkin, M., Niyogi, P.: Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering. In: Advances in Neural Information Processing System 15, Vancouver, British Columbia, Canada (2001)Google Scholar
  3. 3.
    Edelman, A., Arias, T.A., Simth, S.T.: The Geometry of Algorithms with Orthogonality Constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353Google Scholar
  4. 4.
    Fukunaga, K.: Statistical Pattern Recognition. Academic Press, London (1990)zbMATHGoogle Scholar
  5. 5.
    Hand, D.: Kernel Discriminant Analysis. Research Studies Press, Chichester (1982)zbMATHGoogle Scholar
  6. 6.
    He, X., Yan, S., Hu, Y., Niyogi, P., Zhang, H.: Face Recognition using Laplacianfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 27(3) (March 2005)Google Scholar
  7. 7.
    Joliffe, I.: Principal Component Analysis. Springer, New York (1986)CrossRefGoogle Scholar
  8. 8.
    Li, H., Jiang, T., Zhang, K.: Efficient and Robust Feature Extraction by Maximum Margin Criterion. In: Advances in Neural Information Processing Systems 16 (2004)Google Scholar
  9. 9.
    Lu, J., Plataniotis, K.N., Venetsanopoulos, N.: Face Recognition Using Kernel Direct Discriminant Analysis Algorithms. IEEE Trans. On Neural Networks (August 2002)Google Scholar
  10. 10.
    Luettin, J., Maitre, G.: Evaluation Protocol for the Extended M2VTS Database (XM2VTS). DMI for Perceptual Artificial Intelligence (1998)Google Scholar
  11. 11.
    Martinez, A., Benavente, R.: The AR Face Database (2003),
  12. 12.
    Shashua, A., Levin, A.: Linear Image Coding for Regression and Classification using the Tensor-rank Principle. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), Hawaii (December 2001)Google Scholar
  13. 13.
    Sim, T., Baker, S., Bsat, M.: The CMU Pose, Illumination, and Expression (PIE) Database. In: Proceedings of the IEEE International Conference on Automatic Face and Gesture Recognition (May 2002)Google Scholar
  14. 14.
    Turk, M., Pentland, A.: Face Recognition Using Eigenfaces. In: IEEE Conference on Computer Vision and Pattern Recognition, Maui, Hawaii (1991)Google Scholar
  15. 15.
    Vasilescu, M., Terzopoulos, D.: Multilinear Subspace Analysis for Image Ensembles”. In: Proc. Computer Vision and Pattern Recognition Conf (CVPR 2003), June 2003, vol. 2, pp. 93–99 (2003)Google Scholar
  16. 16.
    Wang, X., Tang, X.: A unified framework for subspace face recognition. IEEE Trans. Pattern Analysis and Machine Intelligence 26(9), 1222–1228 (2004)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Wilks, W.: Mathematical Statistics. Wiley, New York (1963)zbMATHGoogle Scholar
  18. 18.
    Yang, M.: Kernel Eigenfaces vs. Kernel Fisherfaces: Face Recognition Using Kernel Methods. In: Proc. of the 5th Int. Conf. on Automatic Face and Gesture Recognition, Washington, DC (May 2002)Google Scholar
  19. 19.
    Yan, S., Xu, D., Zhang, B., Zhang, H.: Graph Embedding: A General Framework for Dimensionality Reduction. In: Proc. Computer Vision and Pattern Recognition Conf. (2005)Google Scholar
  20. 20.
    Ye, J., Janardan, R., Park, C., Park, H.: An optimization criterion for generalized discriminant analysis on undersampled problems. IEEE Transactions on Pattern Analysis and Machine Intelligence 26, 982–994 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Shuicheng Yan
    • 1
  • Xiaoou Tang
    • 1
    • 2
  1. 1.Department of Information EngineeringThe Chinese University of Hong KongHong KongHong Kong
  2. 2.Microsoft Research AsiaBeijingChina

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