On the geometry of border rank algorithms for n x 2 by 2 x 2 matrix multiplication

TL;DR

This paper investigates the geometric properties of border rank algorithms for multiplying n x 2 matrices by 2 x 2 matrices, providing insights into their structure and efficiency.

Contribution

It offers a detailed geometric analysis of border rank algorithms specific to n x 2 by 2 x 2 matrix multiplication, advancing understanding of their complexity.

Findings

Characterization of border rank algorithms' geometric structure

Identification of limitations in existing algorithms

Potential pathways for more efficient algorithms

Abstract

We make an in-depth study of the known border rank (i.e. approximate) algorithms for the matrix multiplication tensor encoding the multiplication of an n x 2 matrix by a 2 x 2 matrix.