多元统计分析中一类矩阵迹函数最小化问题的有效算法  

AN EFFICIENT METHOD FOR SOLVING A CLASS OF MATRIX TRACE FUNCTION MINIMIZATION PROBLEM IN MULTIVARIATE STATISTICAL

作　　者：李姣芬 秦树娟 张丽 候文婷 Li Jiaofen;Qin Shujuan;Zhang Li;Hou Wenting(School of Mathematics and Computational Science,Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation,Guilin University of Electronic Technology,Guilin 541004,China)

机构地区：[1]桂林电子科技大学数学与计算科学学院,广西高校数据分析与计算重点实验室,广西自动检测技术与仪器重点实验室,桂林541004

出　　处：《计算数学》2021年第1期70-86,共17页Mathematica Numerica Sinica

基　　金：国家自然科学基金资助项目(No.11761024,11561015,11961012);广西自然科学基金资助项目(No.2016GXNSFAA380074,2016GXNSFFA380009,2017GXNSFBA198082);2018年广西自治区大学生创新训练项目;桂林电子科技大学研究生优秀学位论文培育项目(No.17YJPYSS24,2019YJSPY03)和研究生创新计划项目(No.2020YJSCX02)资助.

摘　　要：研究来源于多元统计分析中的一类矩阵迹函数最小化问题min c+tr(AX)+m∑(j=1)tr(BjXCjX^(T)),s.t.X^(T)X=Ip其中c为常数,A∈R^(p×n)(n≥p),Bj∈R^(n×n),Cj∈R^(p×p)为给定系数矩阵.数值实验表明已有的Majorization算法虽可行,但收敛速度缓慢且精度不高.本文从黎曼流形的角度重新研究该问题,基于Stiefel流形的几何性质,构造一类黎曼非单调共轭梯度迭代求解算法,并给出算法收敛性分析.数值实验和数值比较验证所提出的算法对于问题模型是高效可行的.In this paper,we considered the following matrix trace function minimization problem with orthogonal constraints which aries in multivariate statistical analysis:min c+tr(AX)+m∑(j=1)tr(BjXCjX^(T)),s.t.X^(T)X=Ip where is a constant,A∈R^(p×n)(n≥p),Bj∈R^(n×n),Cj∈R^(p×p),are given matrices.Numerical experiments show that the existing Majorization algorithm is although feasible,but the convergence speed is slow and the accuracy is not high.In this paper we reconsidered this model from the viewpoint of Stiefel manifold.Based on the geometric properties of the Stifel manifold,a class of Riemannian nonmonotone conjugate gradient method is constructed to solve the presented problem.Some numerical tests are given to show the efficiency of the proposed method.Comparisons with some existing methods are also given.

关 键 词：Stiefel流形 矩阵迹函数 黎曼共轭梯度 正交约束 

分 类 号：O224[理学—运筹学与控制论]