Maximization of the sum of the trace ratio on the Stiefel manifold, I: Theory  

作　　者：ZHANG LeiHong LI RenCang 

机构地区：[1]Department of Applied Mathematics, Shanghai University of Finance and Economics [2]Department of Mathematics, University of Texas at Arlington

出　　处：《Science China Mathematics》2014年第12期2495-2508,共14页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11101257 and 11371102);the Basic Academic Discipline Program;the 11th Five Year Plan of 211 Project for Shanghai University of Finance and Economics;a visiting scholar at the Department of Mathematics,University of Texas at Arlington from February 2013 toJanuary 2014;supported by National Science Foundation of USA(Grant Nos.1115834and 1317330);a Research Gift Grant from Intel Corporation

摘　　要：We are concerned with the maximization of tr（V T AV）/tr（V T BV）＋tr（V T CV） over the Stiefel manifold {V ∈ R m×l ｜ V T V = Il} （l 〈 m）, where B is a given symmetric and positive definite matrix, A and C are symmetric matrices, and tr（. ） is the trace of a square matrix. This is a subspace version of the maximization problem studied in Zhang （2013）, which arises from real-world applications in, for example, the downlink of a multi-user MIMO system and the sparse Fisher discriminant analysis in pattern recognition. We establish necessary conditions for both the local and global maximizers and connect the problem with a nonlinear extreme eigenvalue problem. The necessary condition for the global maximizers offers deep insights into the problem, on the one hand, and, on the other hand, naturally leads to a self-consistent-field （SCF） iteration to be presented and analyzed in detail in Part II of this paper.We are concerned with the maximization of tr(VTAV)/tr(VT BV)+ tr(VT CV)over the Stiefel manifold {V ∈ Rm×| V T V = It}(t < m), where B is a given symmetric and positive definite matrix, A and C are symmetric matrices, and tr() is the trace of a square matrix. This is a subspace version of the maximization problem studied in Zhang(2013), which arises from real-world applications in, for example,the downlink of a multi-user MIMO system and the sparse Fisher discriminant analysis in pattern recognition.We establish necessary conditions for both the local and global maximizers and connect the problem with a nonlinear extreme eigenvalue problem. The necessary condition for the global maximizers offers deep insights into the problem, on the one hand, and, on the other hand, naturally leads to a self-consistent-field(SCF)iteration to be presented and analyzed in detail in Part II of this paper.

关 键 词：trace ratio Rayleigh quotient Stiefel manifold nonlinear eigenvalue problem optimality condition EIGENSPACE 

分 类 号：O151.21[理学—数学]