Decomposition of Low Rank Multi-Symmetric Tensor

Jouhayna Harmouch; Bernard Mourrain; Houssam Khalil

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

2 LARIFA - Faculty of Sciences Libanese University Lebanon

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.

Document type :

Conference papers

Domain :

Mathematics [math] / Commutative Algebra [math.AC]

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https://hal.inria.fr/hal-01648747
Contributor : Jouhayna Harmouch <>
Submitted on : Thursday, November 30, 2017 - 1:37:43 PM
Last modification on : Wednesday, October 10, 2018 - 10:09:20 AM

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