特征提取中一类矩阵迹函数极值问题的黎曼优化算法

数学物理学报 ›› 2024, Vol. 44 ›› Issue (4): 1012-1036.

特征提取中一类矩阵迹函数极值问题的黎曼优化算法

李姣芬,孔鲁源,宋佳铄,文娅琼^*()

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桂林 541004; 广西应用数学中心 (桂林电子科技大学) 广西桂林 541004

收稿日期:2023-10-10 修回日期:2023-12-22 出版日期:2024-08-26 发布日期:2024-07-26
通讯作者: *文娅琼, E-mail: pmxdgmcdsy@163.com
基金资助:
国家自然科学基金(12261026);国家自然科学基金(12361079);国家自然科学基金(11961012);国家自然科学基金(12201149);广西自然科学基金(2023GXNSFAA026067);桂林电子科技大学研究生教育创新计划项目(2022YXW01);桂林电子科技大学研究生教育创新计划项目(2022YCXS142);广西研究生教育创新计划项目(YCSW2023316);广西自动检测技术与仪器重点实验室基金(YQ23104);广西自动检测技术与仪器重点实验室基金(YQ24105)

A Riemannian Optimization Approach for a Class of Matrix Trace Function Extremum Problem in Feature Extraction

Li Jiaofen,Kong Lvyuan,Song Jiashuo,Wen Yaqiong^*()

School of Mathematics and Computational Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guangxi Guilin 541004; Center for Applied Mathematics of Guangxi (GUET), Guangxi Guilin 541004

Received:2023-10-10 Revised:2023-12-22 Online:2024-08-26 Published:2024-07-26
Supported by:
National Natural Science Foundation of China(12261026);National Natural Science Foundation of China(12361079);National Natural Science Foundation of China(11961012);National Natural Science Foundation of China(12201149);Natural Science Foundation of Guangxi(2023GXNSFAA026067);Innovation Project of GUET Graduate Education(2022YXW01);Innovation Project of GUET Graduate Education(2022YCXS142);Innovation Project of Guangxi Graduate Education(YCSW2023316);Guangxi Key Laboratory of Automatic Detecting Technology and Instruments(YQ23104);Guangxi Key Laboratory of Automatic Detecting Technology and Instruments(YQ24105)

摘要/Abstract

摘要：

研究来源于特征提取中的一类鲁棒判别回归模型, 该模型问题可以重构为 Stiefel 流形和线性流形组成的乘积流形约束下的一类矩阵迹函数极小化问题. 整合紧流形和线性流形, 结合乘积流形几何性质, 本文设计适用于求解重构问题简化版本的一类基于 Zhang-Hager 技术拓展的乘积流形黎曼非线性共轭梯度法, 并给出算法全局收敛性分析. 数据实验表明所提算法对于问题求解是高效可行的, 且与已有算法、其它黎曼梯度类算法及黎曼优化工具箱中已有的黎曼一阶和二阶算法相比在迭代解精度或迭代效率上有一定优势.

关键词: 特征提取, 矩阵迹函数, 乘积流形, 黎曼共轭梯度法

Abstract:

The present study focuses on robust discriminant regression models for feature extraction, which can be rephrased as minimizing matrix trace function subject to product manifold constraints. By building upon the Zhang-Hager technique, the authors develop a Riemannian nonlinear conjugate gradient method for solving a simplified version of the reconstruction problem. The method exploits the geometric properties of the product manifold, and the global convergence analysis of the proposed algorithm is provided. Empirical results demonstrate that the proposed algorithm is effective and feasible for solving the underlying problem. In terms of iteration efficiency, the proposed algorithm outperforms the existing method, other Riemannian gradient-like algorithms and Riemannian first-order and second-order algorithms available in the MATLAB toolbox Manopt.

Key words: Feature extraction, Matrix trace function, Product manifold, Riemannian conjugate gradient method

中图分类号:

O151.1

李姣芬, 孔鲁源, 宋佳铄, 文娅琼. 特征提取中一类矩阵迹函数极值问题的黎曼优化算法[J]. 数学物理学报, 2024, 44(4): 1012-1036.

Li Jiaofen, Kong Lvyuan, Song Jiashuo, Wen Yaqiong. A Riemannian Optimization Approach for a Class of Matrix Trace Function Extremum Problem in Feature Extraction[J]. Acta mathematica scientia,Series A, 2024, 44(4): 1012-1036.

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