Stiefel流形约束下矩阵迹函数最小化问题的黎曼共轭梯度算法  

An efficient algorithm for minimizing matrix trace function under Stiefel manifold constraint

作　　者：秦树娟 周学林 李姣芬[1] QIN Shujuan;ZHOU Xuelin;LI Jiaofen(School of Mathematics and Computational Science,Guilin University of Electronic Technology,Guilin 541004,China;Academic Affairs Office,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,广西桂林541004 [2]桂林电子科技大学教务处,广西桂林541004

出　　处：《桂林电子科技大学学报》2020年第6期539-544,共6页Journal of Guilin University of Electronic Technology

基　　金：国家自然科学基金(11761024);广西自然科学基金(2016GXNSFAA380074)。

摘　　要：为求解机器学习特征提取中的一类Stiefel流形约束下矩阵迹函数最小化问题,提出了一种黎曼非线性共轭梯度算法。将该问题转化为乘积流形约束下的最小化问题,围绕乘积流形的切空间、正交投影及目标函数等进行展开,采用收缩算子和向量转移算子的方式来更新迭代,将Dai的非单调共轭梯度法推广至黎曼流形上,并采用Armijo型非单调线性搜索条件来保证算法的全局收敛性。收敛性分析表明,该算法是可行的。In order to solve the problem of matrix trace function minimization under a class of Stiefel manifold constraints in machine learning feature extraction,a Riemannian nonlinear conjugate gradient algorithm is proposed.The problem is transformed into a minimization problem under the constraint of the product manifold,and expanded around the tangent space,orthogonal projection and objective function of the product manifold.The retraction operator and the vectortransport operator are used to update the iteration,and the Dai’s non-monotonic conjugate gradient method is extended to Riemannian manifolds,and Armijo-type non-monotonic linear search conditions are used to ensure the global convergence of the algorithm.Convergence analysis shows that the algorithm is feasible.

关 键 词：黎曼共轭梯度法 Stiefel流形 矩阵迹函数 乘积流形 

分 类 号：O241.7[理学—计算数学]