Toward a generalization of Kruskal's theorem on tensor decomposition

TL;DR

This paper proposes a conjecture that generalizes Kruskal's theorem by weakening the k-rank condition to standard rank, leading to broader criteria for tensor decomposition uniqueness.

Contribution

It introduces a conjecture extending Kruskal's theorem, proves it for specific multipartite vector spaces, and improves bounds on tensor circuit ranks.

Findings

Proves the conjecture over arbitrary fields for certain tensor spaces.

Generalizes Kruskal's theorem to weaker rank conditions.

Provides a sharp quadratic bound on tensor circuit ranks.

Abstract

Kruskal's theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. In this work, we propose a conjecture in which the k-rank condition of Kruskal's theorem is weakened to the standard notion of rank, and the conclusion is relaxed to a statement on the linear dependence of the product tensors. Our conjecture would imply a generalization of Kruskal's theorem. Several adaptations and generalizations of Kruskal's theorem have already been obtained, but these results still cannot certify uniqueness when the k-ranks are below a certain threshold. Our generalization would contain several of these results, and could certify uniqueness below this threshold. We prove our conjecture over an arbitrary field $\mathbb{F}$ when the underlying multipartite vector space takes any one of three forms: ${\mathbb{F}^{d_1}\otimes \mathbb{F}^{d_2}}, \;{\mathbb{F}^{d_1}\otimes\mathbb{F}^{d_2}\otimes \mathbb{F}^2,}$ or $\mathbb{F}^{d_1}\otimes \mathbb{F}^2 \otimes\cdots \otimes \mathbb{F}^2$. As a corollary to the third case, we prove that if $n$ product tensors form a circuit, then they have rank greater than one in at most $n-2$ subsystems. This is a quadratic improvement over a recent bound obtained by Ballico, and is sharp.