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Global Optimality of the Successive Maxbet Algorithm

Published online by Cambridge University Press:  01 January 2025

Mohamed Hanafi  and
Jos M. F. ten Berge
Mohamed Hanafi*
Affiliation:
Departement SMAD, Enitiaa, Nantes
Jos M. F. ten Berge
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Mohamed Hanafi, ENITIAA, Rue de la Geraudiere, 44322 Nantes, CEDEX 03, FRANCE. E-Mail: hanafi@enitiaa-nantes.fr
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Abstract

The Maxbet method is an alternative to the method of generalized canonical correlation analysis and of Procrustes analysis. Contrary to these methods, it does not maximize the inner products (covariances) between linear composites, but also takes their sums of squares (variances) into account. It is well-known that the Maxbet algorithm, which has been proven to converge monotonically, may converge to local maxima. The present paper discusses an eigenvalue criterion which is sufficient, but not necessary for global optimality. However, in two special cases, the eigenvalue criterion is shown to be necessary and sufficient for global optimality. The first case is when there are only two data sets involved; the second case is when the inner products between all variables involved are positive, regardless of the number of data sets.

Keywords

canonical correlationProcrustes rotationmultivariate eigenvalue problemMaxbet
Type
Articles
Information
Psychometrika , Volume 68 , Issue 1 , March 2003 , pp. 97 - 103
Copyright
Copyright © 2003 The Psychometric Society

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Footnotes

The authors are obliged to Henk Kiers for critical comments on a previous draft.

References

Chu, M.T., Watterson, J.L. (1993). On a multivariate eigenvalue problem, Part I: Algebraic theory and a power method. SIAM Journal on Scientific Computing, 14(5), 1089–1106CrossRefGoogle Scholar
Frobenius, G. (1912). Über Matrizen aus nicht negativen Elementen [On matrices of nonnegative elements]. Sitzungsberichte Preussische Akademie der Wissenschaft [Proceedings of the Prussian Academy of Science] (pp. 456–477). Berlin.Google Scholar
Horst, P. (1961). Relations amongm sets of measures. Psychometrika, 26, 129–149CrossRefGoogle Scholar
Hotelling, H. (1936). Relations between two sets of variates. Biometrika, 28, 321–377CrossRefGoogle Scholar
Kettenring, J.R. (1971). Canonical analysis of several sets of variables. Biometrika, 58, 433–451CrossRefGoogle Scholar
Perron, O. (1907).Grundlagen für eine Theorie des Jacobischen Kettenbruchalogrithmus [Foundations for a theory of the Jacobian continued fraction algorithm.]. Mathematische Annalen, 64, 11–76CrossRefGoogle Scholar
ten Berge, J.M.F. (1988). Generalized approaches to the MAXBET problem and the MAXDIFF problem, with applications to canonical correlations. Psychometrika, 53, 487–494CrossRefGoogle Scholar
Tucker, R.L. (1951). A method for synthesis of factor analysis studies. Washington DC: Department of the ArmyCrossRefGoogle Scholar
van de Geer, J.P. (1984). Linear relations amongk sets of variables. Psychometrika, 49, 79–94CrossRefGoogle Scholar