多元统计分析中一类矩阵迹函数最小化问题的有效算法

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$\min c+ tr(AX)+\sum\limits_{j=1}^{m}tr(B_j X C_jX^{T}),\ \ {\rm s. t.} \ X^TX=I_p,$其中$c$为常数, $A\in R^{p\times n}\ (n\geq p)$, $B_j\in R^{n\times n}, C_j\in R^{p\times p}$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.

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

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^TX=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.