A convex programming approach to the trace quotient problem

The trace quotient problem arises in many applications in pattern classification and computer vision, e.g., manifold learning, low-dimension embedding, etc. The task is to solve a optimization problem involving maximizing the ratio of two traces, i.e., maxiy Tr(f(W))/Tr(h(W)). This optimization problem itself is non-convex in general, hence it is hard to solve it directly. Conventionally, the trace quotient objective function is replaced by a much simpler quotient trace formula, i.e., maxw Tr (h(W)^-1 f (W)), which accommodates a much simpler solution. However, the result is no longer optimal for the original problem setting, and some desirable properties of the original problem are lost. In this paper we proposed a new formulation for solving the trace quotient problem directly. We reformulate the original non-convex problem such that it can be solved by efficiently solving a sequence of semidefinite feasibility problems. The solution is therefore globally optimal. Besides global optimality, our algorithm naturally generates orthonormal projection matrix. Moreover it relaxes the restriction of linear discriminant analysis that the projection matrix's rank can only be at most c - 1, where c is the number of classes. Our approach is more flexible. Experiments show the advantages of the proposed algorithm.