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Perturbation Theory and Optimality Conditions for the Best Multilinear Rank Approximation of a Tensor
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.ORCID iD: 0000-0003-2281-856X
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-1542-2690
2011 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 32, no 4, 1422-1450 p.Article in journal (Refereed) Published
Abstract [en]

The problem of computing the best rank-(p,q,r) approximation of a third order tensor is considered. First the problem is reformulated as a maximization problem on a product of three Grassmann manifolds. Then expressions for the gradient and the Hessian are derived in a local coordinate system at a stationary point, and conditions for a local maximum are given. A first order perturbation analysis is performed using the Grassmann manifold framework. The analysis is illustrated in a few examples, and it is shown that the perturbation theory for the singular value decomposition is a special case of the tensor theory.

Place, publisher, year, edition, pages
SIAM , 2011. Vol. 32, no 4, 1422-1450 p.
Keyword [en]
tensor, multilinear rank, best rank-(p, q, r) approximation, perturbation theory, first order optimality conditions, second order optimality conditions, Grassmann manifold, stationary point
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-72910DOI: 10.1137/110823298ISI: 000298373400017OAI: oai:DiVA.org:liu-72910DiVA: diva2:463582
Note
funding agencies|Swedish Research Council||Institute for Computational Engineering and Sciences at The University of Texas at Austin||| Dnr 2008-7145 |Available from: 2011-12-16 Created: 2011-12-09 Last updated: 2017-12-08Bibliographically approved

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