Meyer, Renate Universal optimality of rank constrained matrix approximation. (English) Zbl 0817.62055 Bock, Hans-Hermann (ed.) et al., Information systems and data analysis. Prospects, foundations, applications. Proceedings of the 17th annual conference of the Gesellschaft für Klassifikation e. V., Univ. of Kaiserslautern, Germany, March 3-5, 1993. Berlin: Springer-Verlag. 332-339 (1994). Summary: The major results of this paper consist of an extension of optimality properties concerning the best rank-\(r\) matrix approximation, and of the solution of classical multidimensional scaling (MDS), in either case from the class of orthogonally invariant norms to a wider class of approximation criteria. Observing that many problems in multivariate analysis exhibit symmetries under a group of transformations, and, furthermore, realizing the relevance of preorderings for matrix approximation problems in multivariate statistics and MDS, leads to a natural approach towards proving universal optimality of their solution via invariant preorderings of matrices.For the entire collection see [Zbl 0812.00033]. MSC: 62H99 Multivariate analysis 62A01 Foundations and philosophical topics in statistics 91C15 One- and multidimensional scaling in the social and behavioral sciences 15A99 Basic linear algebra 62-06 Proceedings, conferences, collections, etc. pertaining to statistics Keywords:best rank \(r\) matrix approximation; multidimensional scaling; orthogonally invariant norms; approximation criteria; symmetries; groups of transformations; invariant preorderings of matrices PDFBibTeX XMLCite \textit{R. Meyer}, in: Information systems and data analysis. Prospects, foundations, applications. Proceedings of the 17th annual conference of the Gesellschaft für Klassifikation e. V., Univ. of Kaiserslautern, Germany, March 3-5, 1993. Berlin: Springer-Verlag. 332--339 (1994; Zbl 0817.62055)