An Axiomatic Approach to Tensor Rank Functions

TL;DR

This paper critiques existing axioms for tensor rank functions, proposes stronger axioms to eliminate undesirable properties, and addresses questions about the submax rank function.

Contribution

It introduces a stronger set of axioms for tensor rank functions and resolves open questions related to the submax rank function.

Findings

Existing axioms allow undesirable properties in tensor rank functions

Stronger axioms successfully eliminate these properties

Questions about the submax rank function are answered

Abstract

Recent work of Qi et al. arXiv:2004.11240v7 proposes a set of axioms for tensor rank functions. The current paper presents examples showing that their axioms allow rank functions to have some undesirable properties, and a stronger set of axioms is suggested that eliminates these properties. Two questions raised by Qi et al. involving the submax rank function are also answered.