[Paper Review] On the generic rank of 3-tensors

Shmuel Friedland|arXiv (Cornell University)|May 24, 2008

Tensor decomposition and applications15 references14 citations

TL;DR

This paper investigates the generic and typical ranks of 3-tensors using algebraic geometry and matrix theory, proposing a conjecture for the generic rank over complex numbers that is numerically verified up to dimensions 14×14×14. It further identifies an infinite family of 3-tensors over the reals with at least two typical ranks, challenging the uniqueness of typical rank in real tensor decompositions.

ABSTRACT

We study the generic and typical ranks of 3-tensors of dimension l x m x n using results from matrices and algebraic geometry. We state a conjecture about the exact values of the generic rank of 3-tensors over the complex numbers, which is verified numerically for l,m,n not greater than 14. We also discuss the typical ranks over the real numbers, and give an example of an infinite family of 3-tensors of the form l=m, n=(m-1)^2+1, m=3,4,..., which have at least two typical ranks.

Motivation & Objective

• To determine the generic rank of 3-tensors over the complex numbers using algebraic geometry and matrix-theoretic tools.
• To investigate the behavior of typical ranks over the real numbers, particularly when they may not be unique.
• To construct and analyze an infinite family of 3-tensors with dimensions l=m, n=(m−1)²+1 for m≥3 that exhibit at least two typical ranks over the reals.

Proposed method

• Leveraging known results from matrix theory and algebraic geometry to analyze the rank of 3-tensors.
• Formulating a conjecture for the generic rank of 3-tensors over the complex numbers based on structural and symmetry properties.
• Using numerical verification to support the conjecture for tensor dimensions l, m, n ≤ 14.
• Applying algebraic geometry techniques to study the closure of tensor rank varieties over the reals.
• Analyzing the structure of the set of tensors of a given rank to identify cases where multiple typical ranks coexist.
• Constructing an explicit infinite family of 3-tensors with l=m and n=(m−1)²+1 to demonstrate non-uniqueness of typical ranks.

Experimental results

Research questions

• RQ1 What is the generic rank of a 3-tensor of size l×m×n over the complex numbers, and can it be characterized algebraically?
• RQ2 How does the typical rank of a 3-tensor behave over the real numbers, and under what conditions can it fail to be unique?
• RQ3 Can an infinite family of 3-tensors be constructed such that they possess at least two distinct typical ranks over the reals?
• RQ4 To what extent do numerical experiments support the proposed conjecture for generic rank in small dimensions?

Key findings

• The paper proposes a conjecture for the generic rank of 3-tensors over the complex numbers, which is numerically verified for all l, m, n ≤ 14.
• An infinite family of 3-tensors with dimensions l=m, n=(m−1)²+1 for m=3,4,... is shown to have at least two typical ranks over the reals.
• The existence of such a family demonstrates that typical rank is not always unique over the reals, contradicting the expectation of uniqueness in some settings.
• The results suggest that the real tensor rank variety can have multiple irreducible components corresponding to different typical ranks.
• The analysis confirms that the generic rank over the complex numbers aligns with the conjectured formula for all tested small dimensions.
• The study reveals structural differences between real and complex tensor rank behavior, particularly in the multiplicity of typical ranks.