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Article: A constructive algorithm for decomposing a tensor into a finite sum of orthonormal rank1 terms
Title  A constructive algorithm for decomposing a tensor into a finite sum of orthonormal rank1 terms 

Authors  
Issue Date  2015 
Publisher  Society 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. 13151337 How to Cite? 
Abstract  We propose a constructive algorithm that decomposes an arbitrary real tensor into a finite sum of orthonormal rank1 outer products. The algorithm, called TTr1SVD, works by converting the tensor into a tensortrain rank1 (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 rank1 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 Identifier  http://hdl.handle.net/10722/216994 
ISSN  2015 Impact Factor: 1.883 2015 SCImago Journal Rankings: 2.052 
DC Field  Value  Language 

dc.contributor.author  Batselier, K   
dc.contributor.author  Liu, H   
dc.contributor.author  Wong, N   
dc.date.accessioned  20150918T05:45:33Z   
dc.date.available  20150918T05:45:33Z   
dc.date.issued  2015   
dc.identifier.citation  SIAM Journal on Matrix Analysis and Applications, 2015, v. 36 n. 3, p. 13151337   
dc.identifier.issn  08954798   
dc.identifier.uri  http://hdl.handle.net/10722/216994   
dc.description.abstract  We propose a constructive algorithm that decomposes an arbitrary real tensor into a finite sum of orthonormal rank1 outer products. The algorithm, called TTr1SVD, works by converting the tensor into a tensortrain rank1 (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 rank1 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.language  eng   
dc.publisher  Society for Industrial and Applied Mathematics. The Journal's web site is located at http://www.siam.org/journals/simax.php   
dc.relation.ispartof  SIAM Journal on Matrix Analysis and Applications   
dc.rights  SIAM Journal on Matrix Analysis and Applications. Copyright © Society for Industrial and Applied Mathematics.   
dc.rights  Creative Commons: Attribution 3.0 Hong Kong License   
dc.title  A constructive algorithm for decomposing a tensor into a finite sum of orthonormal rank1 terms   
dc.type  Article   
dc.identifier.email  Batselier, K: kbatseli@hku.hk   
dc.identifier.email  Liu, H: htliu@eee.hku.hk   
dc.identifier.email  Wong, N: nwong@eee.hku.hk   
dc.identifier.authority  Wong, N=rp00190   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.1137/141000658   
dc.identifier.hkuros  253238   
dc.identifier.volume  36   
dc.identifier.issue  3   
dc.identifier.spage  1315   
dc.identifier.epage  1337   
dc.publisher.place  United States   