On generic identifiability of symmetric tensors of subgeneric rank

TL;DR

This paper proves that most symmetric tensors of subgeneric rank have unique decompositions into sums of powers, with a focus on cubics, extending the understanding of tensor identifiability.

Contribution

The paper establishes generic identifiability for symmetric tensors of subgeneric rank, especially providing new results for cubic tensors, and clarifies known exceptional cases.

Findings

Most symmetric tensors of subgeneric rank are identifiable.

Identifiability holds for cubic tensors with new proof techniques.

Only three known exceptional cases exist, all previously documented.

Abstract

We prove that the general symmetric tensor in $S^d {\mathbb C}^{n+1}$ of rank r is identifiable, provided that r is smaller than the generic rank. That is, its Waring decomposition as a sum of r powers of linear forms is unique. Only three exceptional cases arise, all of which were known in the literature. Our original contribution regards the case of cubics ($d=3$), while for $d\ge 4$ we rely on known results on weak defectivity by Ballico, Ciliberto, Chiantini, and Mella.