Inner Rank and Lower Bounds for Matrix Multiplication

TL;DR

This paper introduces the concept of inner rank to analyze matrix multiplication tensors, providing a simple proof for a known lower bound on their rank, and suggests that this new notion warrants further investigation.

Contribution

The paper proposes the inner rank as a new tool for lower bounds on matrix multiplication tensor rank, offering a concise proof of a known bound and highlighting its potential for future research.

Findings

Inner rank offers a new perspective for tensor rank analysis.

A short proof establishes the lower bound of 2n^2 - n + 1 for matrix multiplication tensors.

Inner rank does not currently improve existing bounds but is promising for future work.

Abstract

We develop a notion of {\em inner rank} as a tool for obtaining lower bounds on the rank of matrix multiplication tensors. We use it to give a short proof that the border rank (and therefore rank) of the tensor associated with $n\times n$ matrix multiplication over an arbitrary field is at least $2n^2-n+1$. While inner rank does not provide improvements to currently known lower bounds, we argue that this notion merits further study.