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An Alternating Least Squares Method for the Weighted Approximation of a Symmetric Matrix

Published online by Cambridge University Press:  01 January 2025

Jos M. F. Berge  and
Henk A. L. Kiers
Jos M. F. Berge*
Affiliation:
University of Groningen
Henk A. L. Kiers
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Jos M. F. ten Berge, Department of Psychology, University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, THE NETHERLANDS.
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Abstract

Bailey and Gower examined the least squares approximation C to a symmetric matrix B, when the squared discrepancies for diagonal elements receive specific nonunit weights. They focussed on mathematical properties of the optimal C, in constrained and unconstrained cases, rather than on how to obtain C for any given B. In the present paper a computational solution is given for the case where C is constrained to be positive semidefinite and of a fixed rank r or less. The solution is based on weakly constrained linear regression analysis.

Keywords

matrix approximationweakly constrained regression
Type
Original Paper
Information
Psychometrika , Volume 58 , Issue 1 , March 1993 , pp. 115 - 118
Copyright
Copyright © 1993 The Psychometric Society

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Footnotes

The authors are obliged to John C. Gower for stimulating this research.

References

Bailey, R. A., Gower, J. C. (1990). Approximating a symmetric matrix. Psychometrika, 55, 665–675.CrossRefGoogle Scholar
Browne, M. W. (1987). The Young-Householder algorithm and the least-squares multidimensional scaling of squared distances. Journal of Classification, 4, 175–190.CrossRefGoogle Scholar
Eckart, C., Young, G. (1936). The approximation of one matrix by another of lower rank. Psychometrika, 1, 211–218.CrossRefGoogle Scholar
Gabriel, K. R., Zamir, S. (1979). Lower rank approximation of matrices by least squares with any choice of weights. Technometrics, 21, 489–498.CrossRefGoogle Scholar
Golub, G. H., Von Matt, U. (1991). Quadratically constrained least squares and quadratic problems. Numerische Mathematik, 59, 561–580.CrossRefGoogle Scholar
Harman, H. H., Jones, W. H. (1966). Factor analysis by minimizing residuals (minres). Psychometrika, 31, 351–368.CrossRefGoogle ScholarPubMed
Knol, D. L., Berge, J. M. F. (1989). Least-squares approximation of an improper correlation matrix by a proper one. Psychometrika, 54, 53–61.CrossRefGoogle Scholar
Nosal, M. (1977). A note on the MINRES method. Psychometrika, 42, 149–151.CrossRefGoogle Scholar
Berge, J. M. F. (1991). A general solution for a class of weakly constrained linear regression problems. Psychometrika, 56, 601–609.CrossRefGoogle Scholar
Berge, J. M. F., Zegers, F. E. (1990). Convergence properties of certain MINRES algorithms. Multivariate Behavioral Research, 25, 421–425.CrossRefGoogle ScholarPubMed