Identifiability beyond Kruskal's bound for symmetric tensors of degree 4

TL;DR

This paper develops algebraic geometry-based criteria for symmetric tensor identifiability beyond Kruskal's bound, introducing an effective algorithm for quartic tensors with specific decompositions.

Contribution

It extends identifiability criteria beyond Kruskal's limit for symmetric tensors of degree 4 using advanced geometric tools and provides a practical linear algebra-based algorithm.

Findings

Criteria surpass Kruskal's bound for certain symmetric tensors.

Introduces an efficient algorithm for minimality and uniqueness verification.

Extends classical geometric results like Castelnuovo's lemma.

Abstract

We show how methods of algebraic geometry can produce criteria for the identifiability of specific tensors that reach beyond the range of applicability of the celebrated Kruskal criterion. More specifically, we deal with the symmetric identifiability of symmetric tensors in Sym$^4(\mathbb{C}^{n+1})$, i.e., quartic hypersurfaces in a projective space $\mathbb{P}^n$, that have a decomposition in 2n+1 summands of rank 1. This is the first case where the reshaped Kruskal criterion no longer applies. We present an effective algorithm, based on efficient linear algebra computations, that checks if the given decomposition is minimal and unique. The criterion is based on the application of advanced geometric tools, like Castelnuovo's lemma for the existence of rational normal curves passing through a finite set of points, and the Cayley-Bacharach condition on the postulation of finite sets. In order to apply these tools to our situation, we prove a reformulation of these results, hereby extending classical results such as Castelnuovo's lemma and the analysis of Geramita, Kreuzer, and Robbiano, "Cayley-Bacharach schemes and their canonical modules", Trans. Amer. Math. Soc. 339:443-452, 1993.