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Article: A constructive algorithm for decomposing a tensor into a finite sum of orthonormal rank-1 terms

TitleA constructive algorithm for decomposing a tensor into a finite sum of orthonormal rank-1 terms
Authors
Issue Date2015
PublisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://www.siam.org/journals/simax.php
Citation
SIAM Journal on Matrix Analysis and Applications, 2015, v. 36 n. 3, p. 1315-1337 How to Cite?
AbstractWe propose a constructive algorithm that decomposes an arbitrary real tensor into a finite sum of orthonormal rank-1 outer products. The algorithm, called TTr1SVD, works by converting the tensor into a tensor-train rank-1 (TTr1) series via the singular value decomposition (SVD). TTr1SVD naturally generalizes the SVD to the tensor regime with properties such as uniqueness for a fixed order of indices, orthogonal rank-1 outer product terms, and easy truncation error quantification. Using an outer product column table it also allows, for the first time, a complete characterization of all tensors orthogonal with the original tensor. Incidentally, this leads to a strikingly simple constructive proof showing that the maximum rank of a real $2 imes 2 imes 2$ tensor over the real field is 3. We also derive a conversion of the TTr1 decomposition into a Tucker decomposition with a sparse core tensor. Numerical examples illustrate each of the favorable properties of the TTr1 decomposition.
Persistent Identifierhttp://hdl.handle.net/10722/216994
ISSN
2015 Impact Factor: 1.883
2015 SCImago Journal Rankings: 2.052

 

DC FieldValueLanguage
dc.contributor.authorBatselier, K-
dc.contributor.authorLiu, H-
dc.contributor.authorWong, N-
dc.date.accessioned2015-09-18T05:45:33Z-
dc.date.available2015-09-18T05:45:33Z-
dc.date.issued2015-
dc.identifier.citationSIAM Journal on Matrix Analysis and Applications, 2015, v. 36 n. 3, p. 1315-1337-
dc.identifier.issn0895-4798-
dc.identifier.urihttp://hdl.handle.net/10722/216994-
dc.description.abstractWe propose a constructive algorithm that decomposes an arbitrary real tensor into a finite sum of orthonormal rank-1 outer products. The algorithm, called TTr1SVD, works by converting the tensor into a tensor-train rank-1 (TTr1) series via the singular value decomposition (SVD). TTr1SVD naturally generalizes the SVD to the tensor regime with properties such as uniqueness for a fixed order of indices, orthogonal rank-1 outer product terms, and easy truncation error quantification. Using an outer product column table it also allows, for the first time, a complete characterization of all tensors orthogonal with the original tensor. Incidentally, this leads to a strikingly simple constructive proof showing that the maximum rank of a real $2 imes 2 imes 2$ tensor over the real field is 3. We also derive a conversion of the TTr1 decomposition into a Tucker decomposition with a sparse core tensor. Numerical examples illustrate each of the favorable properties of the TTr1 decomposition.-
dc.languageeng-
dc.publisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://www.siam.org/journals/simax.php-
dc.relation.ispartofSIAM Journal on Matrix Analysis and Applications-
dc.rightsSIAM Journal on Matrix Analysis and Applications. Copyright © Society for Industrial and Applied Mathematics.-
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.titleA constructive algorithm for decomposing a tensor into a finite sum of orthonormal rank-1 terms-
dc.typeArticle-
dc.identifier.emailBatselier, K: kbatseli@hku.hk-
dc.identifier.emailLiu, H: htliu@eee.hku.hk-
dc.identifier.emailWong, N: nwong@eee.hku.hk-
dc.identifier.authorityWong, N=rp00190-
dc.description.naturepublished_or_final_version-
dc.identifier.doi10.1137/141000658-
dc.identifier.hkuros253238-
dc.identifier.volume36-
dc.identifier.issue3-
dc.identifier.spage1315-
dc.identifier.epage1337-
dc.publisher.placeUnited States-

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