多维标度问题的一类直接拟合法  

A CLASS OF DIRECT FITTING METHOD FOR MULTIDIMENSIONAL SCALING PROBLEM

作　　者：宋佳铄 周学林[1,2] 李姣芬 Song Jiashuo;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》2024年第3期291-311,共21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(2023GXNSFAA026067,2024GXNSFAA010521);2023年广西研究生教育创新计划项目(YCSW2023316);2022年桂林电子科技大学位与研究生教育改革项目(2022YXW01);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)是一种用于分析和可视化数据之间相似性或距离关系的统计方法,它通过将数据点映射到低维空间中的坐标来表示它们之间的相对距离或相似性.多维标度问题的古典解通过对(非欧氏型)距离矩阵平方进行双中心化处理,进而通过截断特征值分解寻求低维的拟合构造点.本文对距离矩阵平方进行直接拟合,重构问题为零列和Stiefel子流形和线性流形约束下的矩阵优化模型,并结合乘积流形几何性质,设计一类自适应问题模型的基于Zhang-Hager技术拓展的黎曼梯度下降求解算法.数值实验说明通过直接拟合能得到误差更小的拟合欧氏距离矩阵,且所提算法与已有投影梯度流算法及黎曼优化工具箱中的黎曼一阶和二阶算法在迭代效率上有一定的优势.Multidimensional scaling(MDS)is a statistical method used to analyze and visualize the similarity or distance relationships between data points.It represents the relative distances or similarities between data points by mapping them to coordinates in a low-dimensional space.The classical solution to the multidimensional scaling problem involves a double centering process on the squared(non-Euclidean)distance matrix,followed by truncating the eigenvalue decomposition to seek a low-dimensional approximate configuration of points.In this paper,we directly fit the squared distance matrix and reformulate the reconstruction problem as a constrained matrix optimization model in the product manifold composed of zero column and column orthogonal matrices and diagonal matrices.By leveraging the geometric properties of the product manifold and incorporating an extended Riemannian gradient descent algorithm based on Zhang-Hager technique,we design a class of adaptive problem models.Numerical experiments demonstrate that direct fitting yields a smaller error in fitting the Euclidean distance matrix.Moreover,the proposed algorithm exhibits certain advantages in terms of iterative efficiency compared to existing projection gradient flow algorithms and first-order and second-order Riemannian algorithms in the Riemannian optimization toolbox.

关 键 词：多维标度分析 直接拟合 黎曼梯度法 乘积流形 

分 类 号：O24[理学—计算数学]