The Necessary and Sufficient Conditions for the Existence of the Optimal Solution of Trace Ratio Problems

Many dimensionality reduction problems can be formulated as a trace ratio form, i.e. argmax(W)Tr(W-T SpW)/Tr(W-T StW), where S-p and S-t represent the (dis) similarity between data, W is the projection matrix, and Tr(.) is the trace of a matrix. Some representative algorithms of this category include principal component analysis (PCA), linear discriminant analysis (LDA) and marginal Fisher analysis (MFA). Previous research focuses on how to solve the trace ratio problems with either (generalized) eigenvalue decomposition or iterative algorithms. In this paper, we analyze an algorithm that transforms the trace ratio problems into a series of trace difference problems, i.e. argmax(W)Tr[(W-T (S-p -lambda S-t) W], and propose the necessary and sufficient conditions for the existence of the optimal solution of trace ratio problems. The correctness of this theoretical result is proved. To evaluate the applied algorithm, we tested it on three face recognition applications. Experimental results demonstrate its convergence and effectiveness.