Decomposition of Low Rank Multi-Symmetric Tensor

Jouhayna Harmouch 1, 2 Bernard Mourrain 1 Houssam Khalil 2
1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , National and Kapodistrian University of Athens
Abstract : 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 $τ$. 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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Jouhayna Harmouch, Bernard Mourrain, Houssam Khalil. Decomposition of Low Rank Multi-Symmetric Tensor. MACIS 2017 - 7th International Conference on Mathematical Aspects of Computer and Information Sciences, Nov 2017, Vienna, Austria. pp.51--66, ⟨10.1007/978-3-319-72453-9_4⟩. ⟨hal-01648747v2⟩

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