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 


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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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