Typical rank of $m\times n\times (m-1)n$ tensors with $3\leq m\leq n$ over the real number field

TL;DR

This paper characterizes the typical rank of certain three-dimensional tensors over the real numbers, showing a precise threshold based on the Hurwitz-Radon function that determines whether the typical rank is unique or not.

Contribution

It establishes a complete characterization of the typical rank for $m imes n imes (m-1)n$ tensors, revealing a sharp threshold at the Hurwitz-Radon function value.

Findings

If $m \

m \

the set of tensors has only one typical rank $(m-1)n$.

Abstract

Tensor type data are used recently in various application fields, and then a typical rank is important. Let $3\leq m\leq n$. We study typical ranks of $m\times n\times (m-1)n$ tensors over the real number field. Let $\rho$ be 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 $0\leq b<4$. If $m \leq \rho(n)$, then the set of $m\times n\times (m-1)n$ tensors has two typical ranks $(m-1)n,(m-1)n+1$. In this paper, we show that the converse is also true: if $m > \rho(n)$, then the set of $m\times n\times (m-1)n$ tensors has only one typical rank $(m-1)n$.