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A Procrustes Problem on the Stiefel Manifold
, 1997
"... An orthogonal Procrustes problem on the Stiefel manifold is studied, where a matrix Q with orthonormal columns is to be found that minimizes kAQ \Gamma BkF for an l \Theta m matrix A and an l \Theta n matrix B with l m and m ? n. Based on the normal and secular equations and the properties of the ..."
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An orthogonal Procrustes problem on the Stiefel manifold is studied, where a matrix Q with orthonormal columns is to be found that minimizes kAQ \Gamma BkF for an l \Theta m matrix A and an l \Theta n matrix B with l m and m ? n. Based on the normal and secular equations and the properties of the Stiefel manifold, necessary conditions for a global minimum, as well as necessary and sufficient conditions for a local minimum, are derived. Key words. constraint, Lagrange multiplier, least squares problems, normal equations, orthogonal Procrustes problem, secular equations, singular value decomposition, Stiefel manifold Department of Mathematics, Linkoping University, S581 83 Linkoping, Sweden. Email: elden@math.liu.se. y Computer Science Department, University of Minnesota, Minneapolis, MN 55455, U.S.A. Email: hpark@cs.umn.edu. The work of this author was supported in part by the National Science Foundation grant CCR9507307. 1 Introduction Let two matrices A 0 2 R l\Thetam w...
The Procrustes Problem for Orthogonal Stiefel Matrices
 SIAM J. Scientific Computing
, 1998
"... In this paper we consider the Procrustes problem on the manifold of orthogonal Stiefel matrices. Given matrices A 2 R m\Thetak , B 2 R m\Thetap , m p k, we seek the minimum of kA \Gamma BQk 2 for all matrices Q 2 R p\Thetak , Q T Q = I k\Thetak . We introduce a class of relaxation methods for genera ..."
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In this paper we consider the Procrustes problem on the manifold of orthogonal Stiefel matrices. Given matrices A 2 R m\Thetak , B 2 R m\Thetap , m p k, we seek the minimum of kA \Gamma BQk 2 for all matrices Q 2 R p\Thetak , Q T Q = I k\Thetak . We introduce a class of relaxation methods for generating minimizing sequences and offer a geometric interpretation of these methods. Results of numerical experiments illustrating the convergence of the methods are given.
The Procrustes Problem for Orthogonal Kronecker Products
, 2001
"... Abstract Procrustes problem for orthogonal Kronecker products is considered. Given matrices A 2 Rn 2 \Theta k2, T 2 Rn2 \Theta n2; n * k we seek the minimum of the Frobenius norm kT (Q \Omega Q) \Gamma Ak for all orthogonal Stiefel matrices Q 2 Rn\Theta k, QT Q = Ik. We introduce and implement left ..."
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Abstract Procrustes problem for orthogonal Kronecker products is considered. Given matrices A 2 Rn 2 \Theta k2, T 2 Rn2 \Theta n2; n * k we seek the minimum of the Frobenius norm kT (Q \Omega Q) \Gamma Ak for all orthogonal Stiefel matrices Q 2 Rn\Theta k, QT Q = Ik. We introduce and implement left and right relaxation methods for minimization. Numerical results illustrating performance of both methods are given. 1 Introduction Let A 2 Rm\Theta k and B 2 Rm\Theta p where m * p * k be given matrices. Let kAk = (trace (AT A)) 1 2 denote the standard Frobenius matrix norm. The