An alternating least squares method for the weighted approximation of a symmetric matrix
Tóm tắt
Bailey and Gower examined the least squares approximationC to a symmetric matrixB, when the squared discrepancies for diagonal elements receive specific nonunit weights. They focussed on mathematical properties of the optimalC, in constrained and unconstrained cases, rather than on how to obtainC for any givenB. In the present paper a computational solution is given for the case whereC is constrained to be positive semidefinite and of a fixed rankr or less. The solution is based on weakly constrained linear regression analysis.
Tài liệu tham khảo
Bailey, R. A., & Gower, J. C. (1990). Approximating a symmetric matrix.Psychometrika, 55, 665–675.
Browne, M. W. (1987). The Young-Householder algorithm and the least-squares multidimensional scaling of squared distances.Journal of Classification, 4, 175–190.
Eckart, C., & Young, G. (1936). The approximation of one matrix by another of lower rank.Psychometrika, 1, 211–218.
Gabriel, K. R., & Zamir, S. (1979). Lower rank approximation of matrices by least squares with any choice of weights.Technometrics, 21, 489–498.
Golub, G. H., & Von Matt, U. (1991). Quadratically constrained least squares and quadratic problems.Numerische Mathematik, 59, 561–580.
Harman, H. H., & Jones, W. H. (1966). Factor analysis by minimizing residuals (minres).Psychometrika, 31, 351–368.
Knol, D. L., & ten Berge, J. M. F. (1989). Least-squares approximation of an improper correlation matrix by a proper one.Psychometrika, 54, 53–61.
Nosal, M. (1977). A note on the MINRES method.Psychometrika, 42, 149–151.
ten Berge, J. M. F. (1991). A general solution for a class of weakly constrained linear regression problems.Psychometrika, 56, 601–609.
ten Berge, J. M. F., & Zegers, F. E. (1990). Convergence properties of certain MINRES algorithms.Multivariate Behavioral Research, 25, 421–425.