Abstract

The formulation of trace quotient is shared by many computer vision  problems; however, it was conventionally approximated by an essentially dif- ferent 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 character- istics: 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 non- singular 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 recog- nition validate the superiority of OPP over the solution of the corresponding  quotient trace problem in both objective function value and classification   capability.