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On the Number of Minima to Weighted Orthogonal Procrustes Problems
Umeå University, Faculty of Science and Technology, Computing Science.
In: BITArticle in journal (Refereed) Submitted
URN: urn:nbn:se:umu:diva-4998OAI: diva2:144336
Available from: 2006-03-15 Created: 2006-03-15Bibliographically approved
In thesis
1. Algorithms for the Weighted Orthogonal Procrustes Problem and other Least Squares Problems
Open this publication in new window or tab >>Algorithms for the Weighted Orthogonal Procrustes Problem and other Least Squares Problems
2006 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, we present algorithms for local and global minimization of some Procrustes type problems. Typically, these problems are about rotating and scaling a known set of data to fit another set with applications related to determination of rigid body movements, factor analysis and multidimensional scaling. The known sets of data are usually represented as matrices, and the rotation to be determined is commonly a matrix Q with orthonormal columns.

The algorithms presented use Newton and Gauss-Newton search directions with optimal step lengths, which in most cases result in a fast computation of a solution.

Some of these problems are known to have several minima, e.g., the weighted orthogonal Procrustes problem (WOPP). A study on the maximal amount of minima has been done for this problem. Theoretical results and empirical observations gives strong indications that there are not more than 2n minimizers, where n is the number of columns in Q. A global optimization method to compute all 2n minima is presented.

Also considered in this thesis is a cubically convergent iteration method for solving nonlinear equations. The iteration method presented uses second order information (derivatives) when computing a search direction. Normally this is a computational heavy task, but if the second order derivatives are constant, which is the case for quadratic equations, a performance gain can be obtained. This is confirmed by a small numerical study.

Finally, regularization of ill-posed nonlinear least squares problems is considered. The quite well known L-curve for linear least squares problems is put in context for nonlinear problems.

Place, publisher, year, edition, pages
Umeå: Datavetenskap, 2006
Report / UMINF, ISSN 0348-0542 ; 06.10
Procrustes, weighted, orthogonal, algorithms, global optimization.
National Category
Computational Mathematics
urn:nbn:se:umu:diva-730 (URN)
Public defence
2006-04-07, MA121, MIT, Umeå Universitet, Umeå, 10:15
Available from: 2006-03-15 Created: 2006-03-15Bibliographically approved

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