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Algorithms for the Weighted Orthogonal Procrustes Problem and other Least Squares Problems
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Datavetenskap.
2006 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
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.

sted, utgiver, år, opplag, sider
Umeå: Datavetenskap , 2006.
Serie
Report / UMINF, ISSN 0348-0542 ; 06.10
Emneord [en]
Procrustes, weighted, orthogonal, algorithms, global optimization.
HSV kategori
Identifikatorer
URN: urn:nbn:se:umu:diva-730OAI: oai:DiVA.org:umu-730DiVA, id: diva2:144340
Disputas
2006-04-07, MA121, MIT, Umeå Universitet, Umeå, 10:15
Opponent
Veileder
Tilgjengelig fra: 2006-03-15 Laget: 2006-03-15bibliografisk kontrollert
Delarbeid
1. Algorithms for 3-dimensional Weighted Orthogonal Procrustes Problems
Åpne denne publikasjonen i ny fane eller vindu >>Algorithms for 3-dimensional Weighted Orthogonal Procrustes Problems
Manuskript (Annet vitenskapelig)
Identifikatorer
urn:nbn:se:umu:diva-4996 (URN)
Tilgjengelig fra: 2006-03-15 Laget: 2006-03-15 Sist oppdatert: 2010-01-13bibliografisk kontrollert
2. Algorithms for Linear Least Squares Problems on the Stiefel manifold
Åpne denne publikasjonen i ny fane eller vindu >>Algorithms for Linear Least Squares Problems on the Stiefel manifold
Manuskript (Annet vitenskapelig)
Identifikatorer
urn:nbn:se:umu:diva-4997 (URN)
Tilgjengelig fra: 2006-03-15 Laget: 2006-03-15 Sist oppdatert: 2010-01-13bibliografisk kontrollert
3. On the number of minima to weighted orthogonal procrustes problems
Åpne denne publikasjonen i ny fane eller vindu >>On the number of minima to weighted orthogonal procrustes problems
(engelsk)Manuskript (preprint) (Annet vitenskapelig)
HSV kategori
Identifikatorer
urn:nbn:se:umu:diva-4998 (URN)
Tilgjengelig fra: 2006-03-15 Laget: 2006-03-15 Sist oppdatert: 2018-03-15bibliografisk kontrollert
4. On global minimization of weighted orthogonal procrustes problems
Åpne denne publikasjonen i ny fane eller vindu >>On global minimization of weighted orthogonal procrustes problems
(engelsk)Manuskript (preprint) (Annet vitenskapelig)
HSV kategori
Identifikatorer
urn:nbn:se:umu:diva-4999 (URN)
Tilgjengelig fra: 2006-03-15 Laget: 2006-03-15 Sist oppdatert: 2018-03-15bibliografisk kontrollert
5. A Cubic Convergent Iteration Method
Åpne denne publikasjonen i ny fane eller vindu >>A Cubic Convergent Iteration Method
Manuskript (Annet vitenskapelig)
Identifikatorer
urn:nbn:se:umu:diva-5000 (URN)
Tilgjengelig fra: 2006-03-15 Laget: 2006-03-15 Sist oppdatert: 2010-01-13bibliografisk kontrollert
6. Optimization tools for solving nonlinear ill-posed problems
Åpne denne publikasjonen i ny fane eller vindu >>Optimization tools for solving nonlinear ill-posed problems
2001 Inngår i: Internat. Ser. Numer. Math., Vol. 138, s. 255-264Artikkel i tidsskrift (Fagfellevurdert) Published
Identifikatorer
urn:nbn:se:umu:diva-5001 (URN)
Tilgjengelig fra: 2006-03-15 Laget: 2006-03-15bibliografisk kontrollert

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