Decomposition of Low Rank Multi-symmetric Tensor

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 $\tau$. 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.