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Article: A constructive algorithm for decomposing a tensor into a finite sum of orthonormal rank-1 terms
Title | A constructive algorithm for decomposing a tensor into a finite sum of orthonormal rank-1 terms |
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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. 1315-1337 How to Cite? |
Abstract | We 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 Identifier | http://hdl.handle.net/10722/216994 |
ISSN | 2017 Impact Factor: 1.682 2015 SCImago Journal Rankings: 2.052 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | Batselier, K | - |
dc.contributor.author | Liu, H | - |
dc.contributor.author | Wong, N | - |
dc.date.accessioned | 2015-09-18T05:45:33Z | - |
dc.date.available | 2015-09-18T05:45:33Z | - |
dc.date.issued | 2015 | - |
dc.identifier.citation | SIAM Journal on Matrix Analysis and Applications, 2015, v. 36 n. 3, p. 1315-1337 | - |
dc.identifier.issn | 0895-4798 | - |
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 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.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 | © 2015 Society for Industrial and Applied Mathematics. First Published in SIAM Journal on Matrix Analysis and Applications in volume 36, issue 3, published by the Society for Industrial and Applied Mathematics (SIAM). | - |
dc.title | A constructive algorithm for decomposing a tensor into a finite sum of orthonormal rank-1 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.scopus | eid_2-s2.0-84944607394 | - |
dc.identifier.hkuros | 253238 | - |
dc.identifier.volume | 36 | - |
dc.identifier.issue | 3 | - |
dc.identifier.spage | 1315 | - |
dc.identifier.epage | 1337 | - |
dc.identifier.isi | WOS:000362418800019 | - |
dc.publisher.place | United States | - |