A trust-region framework for iteration solution of the direct INDSCAL problem in metric multidimensional scaling

IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED Numerical Algorithms Pub Date : 2024-09-11 DOI:10.1007/s11075-024-01921-w
Xue-lin Zhou, Chao-qian Li
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Abstract

The well-known INdividual Differences SCALing (INDSCAL) model is intended for the simultaneous metric multidimensional scaling (MDS) of several doubly centered matrices of squared dissimilarities. An alternative approach, called for short DINDSCAL (direct INDSCAL), is proposed for analyzing directly the input matrices of squared dissimilarities. In the present work, the problem of fitting the DINDSCAL model to the data is formulated as a Riemannian optimization problem on a product matrix manifold comprised of the Stiefel sub-manifold of zero-sum matrices and non-negative diagonal matrices. A practical algorithm, based on the generic Riemannian trust-region method by Absil et al., is presented to address the underlying problem, which is characterized by global convergence and local superlinear convergence rate. Numerical experiments are conducted to illustrate the efficiency of the proposed method. Furthermore, comparisons with the existing projected gradient approach and some classical methods in the MATLAB toolbox Manopt are also provided to demonstrate the merits of the proposed approach.

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度量多维标度中直接 INDSCAL 问题迭代求解的信任区域框架
众所周知的 INdividual Differences SCALing(INDSCAL)模型用于同时对多个双中心异同平方矩阵进行度量多维标度(MDS)。我们提出了另一种方法,简称为 DINDSCAL(直接 INDSCAL),用于直接分析输入的差异平方矩阵。在本研究中,DINDSCAL 模型与数据的拟合问题被表述为一个乘积矩阵流形上的黎曼优化问题,乘积矩阵流形由零和矩阵和非负对角矩阵的 Stiefel 子流形组成。在 Absil 等人提出的通用黎曼信任区域法基础上,提出了一种实用算法来解决基本问题,该算法具有全局收敛性和局部超线性收敛率的特点。通过数值实验说明了所提方法的效率。此外,还与现有的投影梯度法和 MATLAB 工具箱 Manopt 中的一些经典方法进行了比较,以证明所提方法的优点。
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来源期刊
Numerical Algorithms
Numerical Algorithms 数学-应用数学
CiteScore
4.00
自引率
9.50%
发文量
201
审稿时长
9 months
期刊介绍: The journal Numerical Algorithms is devoted to numerical algorithms. It publishes original and review papers on all the aspects of numerical algorithms: new algorithms, theoretical results, implementation, numerical stability, complexity, parallel computing, subroutines, and applications. Papers on computer algebra related to obtaining numerical results will also be considered. It is intended to publish only high quality papers containing material not published elsewhere.
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