多维标度中个体差异标度模型的一类黎曼共轭梯度法  

A RIEMANNIAN CONJUGATE GRADIENT APPROACH FOR THE INDSCAL MODEL PROBLEM IN MULTIDIMENSIONAL SCALING

作　　者：周菁 陈新 周学林[1,2] 李姣芬 Zhou Jing;Chen Xin;Zhou Xuelin;Li Jiaofen(School of Mathematics and Computing Science,Guilin University of Electronic Technology,Guilin 541004,China;School of Mathematics and Statistics,Yunan University,Kunming 650000,China;School of Mathematics and Computing Science,Center for Applied Mathematics of Guangxi(GUET),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,桂林541004 [2]云南大学数学与统计学院,昆明650000 [3]桂林电子科技大学数学与计算科学学院,广西应用数学中心(桂林电子科技大学),广西高校数据分析与计算重点实验室,桂林541004

出　　处：《计算数学》2025年第1期98-121,共24页Mathematica Numerica Sinica

基　　金：国家自然科学基金(12261026,12361079,12201149);广西自然科学基金项目(2023GXNSFAA026067,2024GXNSFAA010521);2023年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,YQ24105);广西科技项目(Guike AD23023002)资助。

摘　　要：多维标度分析(MDS)是在低维空间展示和分析多维数据结构的一种数据分析技术,其中的个体差异标度模型(INDSCAL)是一类针对多个数据矩阵同时度量多维标度的特定模型,它不仅对所要分析对象的结构进行分析,还能兼顾到判断主体之间的尺度差异.本文将正交INDSCAL模型拟合问题重构为Stiefel流形和对角矩阵线性流形约束下的矩阵优化模型,结合乘积流形几何性质,设计一类自适应问题模型的强Wolfe型混合黎曼共轭梯度求解算法,并给出算法的全局收敛性.数值实验说明所提算法对问题模型是可行有效的,且较黎曼优化工具箱中已有算法及其他黎曼一阶算法在迭代效率上有一定的优势.Multidimensional scaling(MDS)is a data analysis technology that displays and ana-lyzes the corresponding multidimensional data structure in the low-dimensional space.The Individual Difference Scaling(INDSCAL)is a specific model for simultaneous metric multi-dimensional scaling(MDS)of several data matrices,which not only analyzes the structure of the analysis object,but also takes into account the difference in scales between subjects.In the present work the orthogonal INDSCAL(O-INDSCAL)problem is considered and the problem of fitting the O-INDSCAL model is constructed as a matrix optimization model constrained by Stiefel manifold and linear manifolds.By leveraging the geometric proper-ties of the product manifold,basing on the strong Wolfe line search,we design an adaptive extended hybrid Riemannian conjugate gradient algorithm for the underlying problem and its global convergence is further discussed.Numerical experiments demonstrate that the hybrid method is feasible and effective for the model.Moreover,the proposed algorithm exhibits certain advantages in terms of iterative efficiency compared to the algorithms in the Riemannian optimization toolbox and other Riemannian first-order algorithms.

关 键 词：多维标度分析 个体差异标度模型 乘积流形 黎曼共轭梯度法 强Wolfe条件 

分 类 号：O224[理学—运筹学与控制论]