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Symmetric tensors and symmetric tensor rank

Pierre Comon 1 Gene Golub 2 Lek-Heng Lim 2 Bernard Mourrain 3 
3 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis (1965 - 2019), CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : 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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Submitted on : Wednesday, October 8, 2008 - 9:41:56 PM
Last modification on : Saturday, June 25, 2022 - 10:59:57 PM
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Pierre Comon, Gene Golub, Lek-Heng Lim, Bernard Mourrain. Symmetric tensors and symmetric tensor rank. SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 2008, 30 (3), pp.1254-1279. ⟨hal-00327599⟩



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