umu.sePublications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Algorithms for the Weighted Orthogonal Procrustes Problem and other Least Squares Problems
Umeå University, Faculty of Science and Technology, Computing Science.
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.
Series
Report / UMINF, ISSN 0348-0542 ; 06.10
Keyword [en]
Procrustes, weighted, orthogonal, algorithms, global optimization.
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-730OAI: oai:DiVA.org:umu-730DiVA: diva2:144340
Public defence
2006-04-07, MA121, MIT, Umeå Universitet, Umeå, 10:15
Opponent
Supervisors
Available from: 2006-03-15 Created: 2006-03-15Bibliographically approved
List of papers
1. Algorithms for 3-dimensional Weighted Orthogonal Procrustes Problems
Open this publication in new window or tab >>Algorithms for 3-dimensional Weighted Orthogonal Procrustes Problems
Manuscript (Other academic)
Identifiers
urn:nbn:se:umu:diva-4996 (URN)
Available from: 2006-03-15 Created: 2006-03-15 Last updated: 2010-01-13Bibliographically approved
2. Algorithms for Linear Least Squares Problems on the Stiefel manifold
Open this publication in new window or tab >>Algorithms for Linear Least Squares Problems on the Stiefel manifold
Manuscript (Other academic)
Identifiers
urn:nbn:se:umu:diva-4997 (URN)
Available from: 2006-03-15 Created: 2006-03-15 Last updated: 2010-01-13Bibliographically approved
3. On the number of minima to weighted orthogonal procrustes problems
Open this publication in new window or tab >>On the number of minima to weighted orthogonal procrustes problems
(English)Manuscript (preprint) (Other academic)
National Category
Computer Science
Identifiers
urn:nbn:se:umu:diva-4998 (URN)
Available from: 2006-03-15 Created: 2006-03-15 Last updated: 2017-01-26Bibliographically approved
4. On global minimization of weighted orthogonal procrustes problems
Open this publication in new window or tab >>On global minimization of weighted orthogonal procrustes problems
(English)Manuscript (preprint) (Other academic)
National Category
Computer Science
Identifiers
urn:nbn:se:umu:diva-4999 (URN)
Available from: 2006-03-15 Created: 2006-03-15 Last updated: 2017-01-26Bibliographically approved
5. A Cubic Convergent Iteration Method
Open this publication in new window or tab >>A Cubic Convergent Iteration Method
Manuscript (Other academic)
Identifiers
urn:nbn:se:umu:diva-5000 (URN)
Available from: 2006-03-15 Created: 2006-03-15 Last updated: 2010-01-13Bibliographically approved
6. Optimization tools for solving nonlinear ill-posed problems
Open this publication in new window or tab >>Optimization tools for solving nonlinear ill-posed problems
2001 In: Internat. Ser. Numer. Math., Vol. 138, 255-264 p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:umu:diva-5001 (URN)
Available from: 2006-03-15 Created: 2006-03-15Bibliographically approved

Open Access in DiVA

fulltext(248 kB)1458 downloads
File information
File name FULLTEXT01.pdfFile size 248 kBChecksum SHA-1
bf778b76257220e42a064030091ae6b5fdec421b3bd05579cecc3f922b730874890de77f
Type fulltextMimetype application/pdf

By organisation
Computing Science
Computational Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 1458 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

urn-nbn

Altmetric score

urn-nbn
Total: 2609 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf