The Tensor Rank Problem over the Quaternions

TL;DR

This paper investigates the tensor rank over quaternions for small tensor sizes, providing bounds, explicit decompositions, and demonstrating optimality in certain cases, advancing understanding of quaternion tensor structure.

Contribution

It establishes new bounds on quaternion tensor ranks, offers explicit decompositions for specific cases, and proves optimality of these bounds in some scenarios.

Findings

Bound on tensor rank over quaternions for small tensors

Explicit decomposition of a 2x2x2 quaternion tensor

Optimality of the upper bounds in certain cases

Abstract

We provide a nontrivial bound on the rank of any tensor $T$ over the quaternions $\mathbb{H}$ in the $n_1\times n_2\times n_3$ cases where $2\leq n_i\leq 3$. We describe a decomposition of $T$ into $3$ simple tensors in the $2\times 2\times 2$ case. We also show that the upper bound is the best possible for some of the cases, and we provide various partial results involving tensor decompositions over $\mathbb{C}$ and $\mathbb{H}$.