Upper bound of typical ranks of m x n x ((m-1)n-1) tensors over the real number field

TL;DR

This paper investigates the typical ranks of certain three-dimensional tensors over the real numbers, establishing upper bounds and conditions under which two typical ranks occur, with implications for tensor decomposition complexity.

Contribution

It provides an upper bound for the typical rank of m x n x ((m-1)n-1) tensors and characterizes when two typical ranks appear based on the Hurwitz-Radon function.

Findings

Typical rank is less than or equal to (m-1)n.

Tensors have two typical ranks when m ≤ ρ(n).

The minimal typical rank is (m-1)n-1.

Abstract

Let $3\leq m\leq n$. We study typical ranks of $m\times n\times ((m-1)n-1)$ tensors over the real number field. The number $(m-1)n-1$ is a minimal typical rank of $m\times n\times ((m-1)n-1)$ tensors over the real number field. We show that a typical rank of $m\times n\times ((m-1)n-1)$ tensors over the real number field is less than or equal to $(m-1)n$ and in particular, $m\times n\times ((m-1)n-1)$ tensors over the real number field has two typical ranks $(m-1)n-1, (m-1)n$ if $m\leq \rho(n)$, where $\rho$ is the Hurwitz-Radon function defined as $\rho(n)=2^b+8c$ for nonnegative integers $a,b,c$ such that $n=(2a+1)2^{b+4c}$ and $0\leq b<4$.