Hilbert functions and tensor analysis

TL;DR

This paper applies algebraic geometry tools to analyze the uniqueness of symmetric tensor decompositions, providing new methods to determine identifiability beyond traditional criteria like Kruskal's.

Contribution

It introduces novel algebraic geometry techniques to assess tensor decomposition uniqueness, extending the range of cases where identifiability can be effectively determined.

Findings

Effective determination of tensor decomposition uniqueness in new cases

Extension of identifiability analysis beyond Kruskal's criterion

Application of algebraic geometry tools to symmetric tensors

Abstract

We show how well known tools of algebraic geometry for the study of finite sets can be fruitfully applied to the study of Waring decompositions of symmetric tensors (forms). We mainly focus on the uniqueness of a given decomposition (the identifiability problem), and show how, in some cases, one can effectively determine the uniqueness even in some range in which the Kruskal's criterion does not apply.