Successive projection method for solving the unbalanced Procrustes problem
type: Item
instance of: scholarly article
title: Successive projection method for solving the unbalanced Procrustes problem
zbMATH Open document ID: 1112.65039
author: Zhenyue Zhang
author: Keqin Du
published in: Science in China. Series A
publication date: 2007-02-16T00:00:00Z
review text: The Procrustes problem (PP) considered here is to minimize the Frobenius norm \(f(Q)=\| AQ-B\| _F\) over all matrices \(Q\) satisfying \(Q^TQ=I\) when \(A\) and \(B\) are given matrices. When \(Q\) is square, the PP is called balanced, when \(Q\) is \(n\times k\) with \(k<n\), the PP is unbalanced. An unbalanced PP with optimal or approximate solution \(Q\), can be expanded to a balanced one by taking the orthogonal complement \(H\) of \(Q\) and minimizing the norm of \(A G - [B,AH]\) with \(G\) orthogonal. Besides a survey of alternatives methods, a successive projection method is proposed which updates the columns of \(Q\) by solving PPs with \(k=1\) for each column separately. Such a PP with \(k=1\) is equivalent to a linear least squares problem with a quadratic equality constraint. After each cycle of the \(k\) columns, a global correction is obtained e.g., by solving an expanded balanced PP. It is shown that the method converges to a matrix \(Q\) which satisfies the necessary (but not the sufficient) conditions for an optimal solution of the PP.
Mathematics Subject Classification ID: 65F20
zbMATH DE Number: 5127929
zbMATH Keywords: least squares problems
zbMATH Keywords: orthogonal constraint
reviewed by: Adhemar Bultheel
MaRDI profile type: Publication
full work available at URL: s11425-006-0971-2
OpenAlex ID: W1894340352
cites work: A parallel algorithm for the unbalanced orthogonal Procrustes problem
cites work: The Procrustes Problem for Orthogonal Stiefel Matrices
cites work: A generalized solution of the orthogonal Procrustes problem
cites work: Q5689624
cites work: A Projection Method for Least Squares Problems with a Quadratic Equality Constraint
DOI: 10.1007/S11425-006-0971-2
description: scientific article; zbMATH DE number 5127929