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Least-squares approximation by elements from matrix orbits achieved by gradient flows on compact lie groups

Authors: Chi-Kwong Li, Yiu-Tung Poon and Thomas Schulte-Herbrüggen
Journal: Math. Comp. 80 (2011), 1601-1621
MSC (2010): Primary 15A18, 15A60, 15A90; Secondary 37N30
Published electronically: December 13, 2010
MathSciNet review: 2785470
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Abstract: Let $ S(A)$ denote the orbit of a complex or real matrix $ A$ under a certain equivalence relation such as unitary similarity, unitary equivalence, unitary congruences etc. Efficient gradient-flow algorithms are constructed to determine the best approximation of a given matrix $ A_0$ by the sum of matrices in $ S(A_1), \dots, S(A_N)$ in the sense of finding the Euclidean least-squares distance

$\displaystyle \min\Big\{\big\Vert X_1+ \cdots + X_N - A_0\big\Vert: X_j \in S(A_j), j = 1, \dots, N\Big\}.$

Connections of the results to different pure and applied areas are discussed.

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Additional Information

Chi-Kwong Li
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187

Yiu-Tung Poon
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50051

Thomas Schulte-Herbrüggen
Affiliation: Department of Chemistry, Technical University of Munich, D-85747, Garching, Germany.

Keywords: Complex Hermitian matrices, real symmetric matrices, eigenvalues, singular values, gradient flows.
Received by editor(s): September 15, 2008
Received by editor(s) in revised form: May 20, 2010
Published electronically: December 13, 2010
Additional Notes: The author is an honorary professor of the University of Hong Kong and an honorary professor of the Taiyuan University of Technology. His research was partially supported by USA NSF and the William and Mary Plumeri Award.
The second author’s research was partially supported by USA NSF
The third author is supported in part by the EU-programmes QAP, Q-ESSENCE and the exchange with COQUIT as well as by the excellence network of Bavaria through QCCC
Article copyright: © Copyright 2010 American Mathematical Society