Decomposition of Low Rank Multi-symmetric Tensor

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10693)


We study the decomposition of a multi-symmetric tensor T as a sum of powers of product of linear forms in correlation with the decomposition of its dual \(T^{*}\) as a weighted sum of evaluations. We use the properties of the associated Artinian Gorenstein Algebra \(A_{\tau }\) to compute the decomposition of its dual \(T^{*}\) which is defined via a formal power series \(\tau \). We use the low rank decomposition of the Hankel operator \(H_{\tau }\) associated to the symbol \(\tau \) into a sum of indecomposable operators of low rank. A basis of \(A_{\tau }\) is chosen such that the multiplication by some variables is possible. We compute the sub-coordinates of the evaluation points and their weights using the eigen-structure of multiplication matrices. The new algorithm that we propose works for small rank. We give a theoretical generalized approach of the method in n dimensional space. We show a numerical example of the decomposition of a multi-linear tensor of rank 3 in 3 dimensional space.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Laboratory of Mathematics and Its Applications LaMA-LebanonLebanese UniversityBeirutLebanon
  2. 2.UCA, Inria, AROMATHSophia AntipolisFrance

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