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Title: Optimization algorithms on the Grassmann manifold with application to matrix eigenvalue problems
Authors: Sato, Hiroyuki
Iwai, Toshihiro
Author's alias: 佐藤, 寛之
Keywords: Grassmann manifold
Riemannian optimization
Steepest descent method
Newton’s method
Rayleigh quotient
Lyapunov equation
Issue Date: Jun-2014
Publisher: Springer Japan
Journal title: Japan Journal of Industrial and Applied Mathematics
Volume: 31
Issue: 2
Start page: 355
End page: 400
Abstract: This article deals with the Grassmann manifold as a submanifold of the matrix Euclidean space, that is, as the set of all orthogonal projection matrices of constant rank, and sets up several optimization algorithms in terms of such matrices. Interest will center on the steepest descent and Newton’s methods together with applications to matrix eigenvalue problems. It is shown that Newton’s equation in the proposed Newton’s method applied to the Rayleigh quotient minimization problem takes the form of a Lyapunov equation, for which an existing efficient algorithm can be applied, and thereby the present Newton’s method works efficiently. It is also shown that in case of degenerate eigenvalues the optimal solutions form a submanifold diffeomorphic to a Grassmann manifold of lower dimension. Furthermore, to generate globally converging sequences, this article provides a hybrid method composed of the steepest descent and Newton’s methods on the Grassmann manifold together with convergence analysis.
Rights: The final publication is available at Springer via http://dx.doi.org/10.1007/s13160-014-0141-9.
This is not the published version. Please cite only the published version. この論文は出版社版でありません。引用の際には出版社版をご確認ご利用ください。
URI: http://hdl.handle.net/2433/199668
DOI(Published Version): 10.1007/s13160-014-0141-9
Appears in Collections:Journal Articles

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