A criterion for detecting the identifiability of symmetric tensors of size three

TL;DR

This paper introduces a new criterion for determining when symmetric tensors of size 3x...x3 are identifiable, based on their rank and the Hilbert function of associated point sets.

Contribution

It provides a novel criterion for tensor identifiability that applies to symmetric tensors of specific size and rank bounds, advancing tensor decomposition theory.

Findings

Established a criterion for symmetric tensor identifiability.

Linked tensor rank to the Hilbert function of point sets.

Applicable to tensors with rank up to (d^2+2d)/8.

Abstract

We prove a criterion for the identifiability of symmetric tensors $P$ of type $3\times ...\times 3$, $d$ times, whose rank $k$ is bounded by $(d^2+2d)/8$. The criterion is based on the study of the Hilbert function of a set of points $P_1,..., P_k$ which computes the rank of the tensor $P$.