• 论文 • 上一篇    

多元统计分析中一类矩阵迹函数最小化问题的有效算法

李姣芬, 秦树娟, 张丽, 候文婷   

  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
  • 收稿日期:2019-04-04 发布日期:2021-02-04
  • 基金资助:
    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.

李姣芬, 秦树娟, 张丽, 候文婷. 多元统计分析中一类矩阵迹函数最小化问题的有效算法[J]. 计算数学, 2021, 43(1): 70-86.

Li Jiaofen, Qin Shujuan, Zhang Li, Hou Wenting. AN EFFICIENT METHOD FOR SOLVING A CLASS OF MATRIX TRACE FUNCTION MINIMIZATION PROBLEM IN MULTIVARIATE STATISTICAL[J]. Mathematica Numerica Sinica, 2021, 43(1): 70-86.

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

Li Jiaofen, Qin Shujuan, Zhang Li, Hou Wenting   

  1. 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
  • Received:2019-04-04 Published:2021-02-04
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.
In this paper, we considered the following matrix trace function minimization problem with orthogonal constraints which aries in multivariate statistical analysis: $$ \min c+ tr(AX)+\sum\limits_{j=1}^{m}tr(B_j X C_jX^{T}),\ \ {\rm s. t.}\ X^TX=I_p, $$ where $c$ is a constant, $A\in R^{p\times n}\ (n\geq p)$, $B_j\in R^{n\times n}, C_j\in R^{p\times p}$ 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.

MR(2010)主题分类: 

()
[1] Kiers H A. Setting up alternating least squares and iterative majorization algorithms for solving various matrix optimization problems[J]. Computational statistics & data analysis, 2002, 41(1):157-170.
[2] Kiers H A, ten Berge J M. Minimization of a class of matrix trace functions by means of refined majorization[J]. Psychometrika, 1992, 57(3):371-382.
[3] Kiers H A. Majorization as a tool for optimizing a class of matrix functions[J]. Psychometrika, 1990, 55(3):417-428.
[4] Kanamori T, Takeda A. Numerical study of learning algorithms on Stiefel manifold[J]. Computational Management Science, 2014, 11(4):319-340.
[5] Lai R, Osher S. A splitting method for orthogonality constrained problems[J]. Journal of Scientific Computing, 2014, 58(2):431-449.
[6] Chen W, Ji H, You Y. An Augmented Lagrangian Method for 1-Regularized Optimization Problems with Orthogonality Constraints[J]. SIAM Journal on Scientific Computing, 2016, 38(4):B570-B592.
[7] Manton J H. Optimization algorithms exploiting unitary constraints[J]. IEEE Transactions on Signal Processing, 2002, 50(3):635-650.
[8] Abrudan T E, Eriksson J, Koivunen V. Steepest descent algorithms for optimization under unitary matrix constraint[J]. IEEE Transactions on Signal Processing, 2008, 56(3):1134-1147.
[9] Wen Z, Yin W. A feasible method for optimization with orthogonality constraints[J]. Mathematical Programming, 2013, 142(1-2):397-434.
[10] Oviedo H, Lara H, Dalmau O. A non-monotone linear search algorithm with mixed direction on Stiefel manifold[J]. Optimization Methods and Software, 2019, 34(2):437-457.
[11] Edelman A, Arias T A, Smith S T. The geometry of algorithms with orthogonality constraints[J]. SIAM journal on Matrix Analysis and Applications, 1998, 20(2):303-353.
[12] Sato H, Iwai T. A Riemannian optimization approach to the matrix singular value decomposition[J]. SIAM Journal on Optimization, 2013, 23(1):188-212.
[13] Zhu X. A Riemannian conjugate gradient method for optimization on the Stiefel manifold[J]. Computational Optimization and Applications, 2017, 67(1):73-110.
[14] Vandereycken B. Low-rank matrix completion by Riemannian optimization[J]. SIAM Journal on Optimization, 2013, 23(2):1214-1236.
[15] Zhao Z, Jin X Q, Bai Z J. A geometric nonlinear conjugate gradient method for stochastic inverse eigenvalue problems[J]. SIAM Journal on Numerical Analysis, 2016, 54(4):2015-2035.
[16] Yao T T, Bai Z J, Zhao Z, Ching W K. A Riemannian Fletcher-Reeves Conjugate Gradient Method for Doubly Stochastic Inverse Eigenvalue Problems[J]. SIAM Journal on Matrix Analysis and Applications, 2016, 37(1):215-234.
[17] Yao T T, Bai Z J, Zhao Z. A Riemannian variant of the Fletcher-Reeves conjugate gradient method for stochastic inverse eigenvalue problems with partial eigendata[J]. Numerical Linear Algebra with Applications, 2019, 26(2):e2221.
[18] Hu J, Milzarek A, Wen Z, Yuan Y. Adaptive quadratically regularized Newton method for Riemannian optimization[J]. SIAM Journal on Matrix Analysis and Applications, 2018, 39(3):1181-1207.
[19] Sato H. Riemannian Newton-type methods for joint diagonalization on the Stiefel manifold with application to independent component analysis[J]. Optimization, 2017, 66(12):2211-2231.
[20] Absil P A, Mahony R, Sepulchre R. Optimization algorithms on matrix manifolds[M]. Princeton University Press, 2009.
[21] Gao B, Liu X, Chen X, xiang Yuan Y. A new first-order algorithmic framework for optimization problems with orthogonality constraints[J]. SIAM Journal on Optimization, 2018, 28(1):302-332.
[22] Nishimori Y, Akaho S. Learning algorithms utilizing quasi-geodesic flows on the Stiefel manifold[J]. Neurocomputing, 2005, 67:106-135.
[23] Zhang L, Zhou W, Li D H. A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence[J]. IMA Journal of Numerical Analysis, 2006, 26(4):629-640.
[24] Zhang L, Zhou W, Li D. Global convergence of a modified Fletcher-Reeves conjugate gradient method with Armijo-type line search[J]. Numerische Mathematik, 2006, 104(4):561-572.
[25] Dai Y. A nonmonotone conjugate gradient algorithm for unconstrained optimization[J]. Journal of System Science and Complexity, 2002, 15:139-145.
[26] Zhang H, Hager W W. A nonmonotone line search technique and its application to unconstrained optimization[J]. SIAM journal on Optimization, 2004, 14(4):1043-1056.
[1] 蔡文银, 徐玲玲. 核范数和谱范数下广义Sylvester方程最小二乘问题的一类改进算法[J]. 计算数学, 2018, 40(4): 387-401.
[2] 李姣芬, 宋丹丹, 李涛, 黎稳. 核范数和谱范数下广义Sylvester方程最小二乘问题的有效算法[J]. 计算数学, 2017, 39(2): 129-150.
[3] 李姣芬, 吕晓帆, 李涛, 赖梦露. 界约束下算子方程最小二乘问题的条件梯度法[J]. 计算数学, 2016, 38(4): 372-390.
[4] 周茜, 雷渊, 乔文龙. 一类线性约束矩阵不等式及其最小二乘问题[J]. 计算数学, 2016, 38(2): 171-186.
[5] 黄娜, 马昌凤, 谢亚君. 一类Hermitian鞍点矩阵的特征值估计[J]. 计算数学, 2015, 37(1): 92-102.
[6] 李姣芬, 张晓宁, 彭振, 彭靖静. 基于交替投影算法求解单变量线性约束矩阵方程问题[J]. 计算数学, 2014, 36(2): 143-162.
[7] 黄娜, 马昌凤, 谢亚君. 求解非对称代数Riccati 方程几个新的预估-校正法[J]. 计算数学, 2013, 35(4): 401-418.
[8] 李姣芬, 彭振, 彭靖静. 矩阵不等式约束下矩阵方程AX=B的双对称解[J]. 计算数学, 2013, 35(2): 137-150.
[9] 段雪峰, Maher Berzig. 关于“矩阵方程X-A*XqA=I(0<q<1) Hermitian正定解的扰动分析”的注记[J]. 计算数学, 2012, 34(4): 447-447.
[10] 李姣芬, 胡锡炎, 张磊. (R,S,μ)对称矩阵逆问题和最佳逼近问题及扰动分析[J]. 计算数学, 2012, 34(1): 25-36.
[11] 尹小艳, 刘三阳, 肖刚. 矩阵方程X-A*X-2A=Q的正定解及其扰动分析[J]. 计算数学, 2009, 31(2): 151-158.
阅读次数
全文


摘要