{"title":"A trust-region framework for iteration solution of the direct INDSCAL problem in metric multidimensional scaling","authors":"Xue-lin Zhou, Chao-qian Li","doi":"10.1007/s11075-024-01921-w","DOIUrl":null,"url":null,"abstract":"<p>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.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"2 1","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerical Algorithms","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11075-024-01921-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
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.
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.