Paper ID sheet
- TITLE: Trust-region methods on Riemannian manifolds
- AUTHORS: P.-A. Absil, C. G. Baker, K. A. Gallivan.
- ABSTRACT:
A general scheme for trust-region methods on Riemannian manifolds is
proposed. A truncated conjugate-gradient algorithm is utilized to solve
the trust-region subproblems. The method is illustrated on several
problems from numerical linear algebra. In particular, for computing an
extreme eigenspace of a symmetric/positive-definite matrix pencil, the
method yields an efficient inverse-free superlinear algorithm, with good
global convergence properties and minimal storage space
requirements. This algorithm is related to a Krylov subspace method
proposed by Golub and Ye [SIAM J. Sci.\ Comput.\ 24 (2002), 312--334]
and to the trace minimization method of Sameh and Wisniewski [SIAM
J. Numer.\ Anal.\ 19 (1982), 1243--1259], among others. Numerical
experiments show that our trust-region method outperforms the Krylov
subspace method on certain types of problems.
- STATUS: Foundations of Computational Mathematics, Volume 7,
Number 3, pages 303-330, July 2007.
Download
- Journal article: http://dx.doi.org/10.1007/s10208-005-0179-9
- Preprint
[pdf].
- Companion tech report: see report FSU-SCS-2006-175 at http://scseprints.scs.fsu.edu/
(or see local
copy).
- Preliminary conference paper: "Trust-region methods on Riemannian manifolds with applications
in numerical linear algebra",
Proceedings of the 16th International Symposium on
Mathematical Theory of Networks and Systems (MTNS2004),
Katholieke Universiteit Leuven, Belgium, July 5-9, 2004. PDF file.
- Slides of a talk given at the Dynamical
Systems and Computation Day held at CESAME, Université
catholique de Louvain, Belgium, July 15, 2004: PDF file.
- Other set of slides: pdf, ps 9up.
Errata
- In the last displayed equation of section 3, the first minus sign under the square root should be a plus.
- Two lines above the last displayed equation of section 3, \delta_k should be \delta_j.
Related Software
- Manopt is a free Matlab toolbox for
optimization on manifolds that features a Riemannian Trust-Region solver.
- GenRTR: the
Generic Riemannian Trust-Region package, included in a slightly modified form in Manopt.
- Other related software is listed in Manopt's About page.
BibTeX entry
@ARTICLE{AbsBakGal2007-FoCM,
author = "Absil, P.-A. and Baker, C. G. and Gallivan, K. A.",
title= "Trust-region methods on {Riemannian} manifolds",
month = "July",
year = 2007,
journal = "Found. Comput. Math.",
fjournal = "Foundations of Computational Mathematics",
volume = 7,
number = 3,
pages = "303--330",
doi = "10.1007/s10208-005-0179-9",
}
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