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| HAL: hal-00327599, version 1 |
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| SIAM Journal on Matrix Analysis and Applications 30, 3 (2008) 1254-1279 |
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| Symmetric tensors and symmetric tensor rank |
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| Pierre Comon 1Gene Golub 2 |
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| For the France-Stanford collaboration(s) |
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| (2008-09-25) |
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| A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank-1 order k tensor is the outer product of $k$ non-zero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r=1. |
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| 1: | Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe SIGNAL |
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Université Nice Sophia Antipolis [UNS] – CNRS : UMR7271 |
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| 2: | Institute for Computational and Mathematical Engineering (iCME) |
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Stanford University |
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| 3: | GALAAD (INRIA Sophia Antipolis) |
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INRIA – CNRS : UMR6621 – Université Nice Sophia Antipolis [UNS] |
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| Subject | : | Computer Science/Signal and Image Processing Mathematics/Algebraic Geometry Engineering Sciences/Signal and Image processing |
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| Tensors – multiway arrays – outer product decomposition – symmetric outer product decomposition – Candecomp – Parafac – tensor rank – symmetric rank – symmetric tensor rank – generic symmetric rank – maximal symmetric rank – quantics |
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| hal-00327599, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00327599 | |
| oai:hal.archives-ouvertes.fr:hal-00327599 | |
| From: Pierre Comon | |
| Submitted on: Wednesday, 8 October 2008 21:41:56 | |
| Updated on: Friday, 13 November 2009 11:33:09 | |
