Effective identifiability criteria for tensors and polynomials

TL;DR

This paper develops and improves effective criteria for determining the unique decomposition of tensors and polynomials, including symmetric cases like ternary quintic polynomials, with practical algorithms implemented in Macaulay2.

Contribution

The paper introduces new effective $h$-identifiability criteria for a broad class of tensors and enhances these criteria for symmetric tensors, notably for ternary quintic polynomials.

Findings

Provided effective identifiability criteria for large classes of tensors.

Improved criteria specifically for symmetric tensors.

Implemented algorithms in Macaulay2 for practical identification.

Abstract

A tensor $T$, in a given tensor space, is said to be $h$-identifiable if it admits a unique decomposition as a sum of $h$ rank one tensors. A criterion for $h$-identifiability is called effective if it is satisfied in a dense, open subset of the set of rank $h$ tensors. In this paper we give effective $h$-identifiability criteria for a large class of tensors. We then improve these criteria for some symmetric tensors. For instance, this allows us to give a complete set of effective identifiability criteria for ternary quintic polynomial. Finally, we implement our identifiability algorithms in Macaulay2.