The Procrustes Problem for Orthogonal Stiefel Matrices

In this paper we consider the Procrustes problem on the manifold of orthogonal Stiefel matrices. Given matrices ${\cal A}\in {\Bbb R}^{m\times k},$ ${\cal B}\in {\Bbb R}^{m\times p},$ $m\ge p \ge k,$ we seek the minimum of $\|{\cal A}-{\cal B}Q\|^2$ for all matrices $Q\in {\Bbb R}^{p\times k},$ $Q^TQ=I_{k\times k}$. We introduce a class of relaxation methods for generating sequences of approximations to a minimizer and offer a geometric interpretation of these methods. Results of numerical experiments illustrating the convergence of the methods are given.

[1]  K. Fan.  On a Theorem of Weyl Concerning Eigenvalues of Linear Transformations: II. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[2]  G. Forsythe,et al.  On the Stationary Values of a Second-Degree Polynomial on the Unit Sphere , 1965 .

[3]  L. Eldén.  Algorithms for the regularization of ill-conditioned least squares problems , 1977 .

[4]  J. Berge,et al.  A general solution to Mosier's oblique procrustes problem , 1977 .

[5]  Gene H. Golub,et al.  Matrix computations , 1983 .

[6]  Adam Lutoborski,et al.  On the convergence of the euler-jacobi method , 1992 .

[7]  G. Golub,et al.  Quadratically constrained least squares and quadratic problems , 1991 .

[8]  Dirk L. Knol,et al.  Orthogonal rotations to maximal agreement for two or more matrices of different column orders , 1984 .

[9]  G. Forsythe,et al.  The cyclic Jacobi method for computing the principal values of a complex matrix , 1960 .

[10]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[11]  Alan Edelman,et al.  The Geometry of Algorithms with Orthogonality Constraints , 1998, SIAM J. Matrix Anal. Appl..

[12]  W. Gander.  Least squares with a quadratic constraint , 1980 .

[13]  K. Fan.  On a Theorem of Weyl Concerning Eigenvalues of Linear Transformations I. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[14]  Haesun Park,et al.  A parallel algorithm for the unbalanced orthogonal Procrustes problem , 1991, Parallel Comput..