The tensor rank of 5x5 matrices multiplication is bounded by 98 and its border rank by 89

TL;DR

This paper introduces new bounds on the tensor rank and border rank for 5x5 matrix multiplication, presenting a non-commutative algorithm that reduces the number of multiplications needed.

Contribution

It provides the first non-commutative algorithm for 3x5 by 5x5 matrix multiplication with 58 multiplications and extends this to larger matrices, also proposing an approximate algorithm with 89 multiplications.

Findings

Tensor rank of 5x5 matrix multiplication is at most 98.

Border rank of 5x5 matrix multiplication is at most 89.

New non-commutative algorithms reduce the number of multiplications.

Abstract

We present a non-commutative algorithm for the product of 3x5 by 5x5 matrices using 58 multiplications. This algorithm allows to construct a non-commutative algorithm for multiplying 5x5 (resp. 10x10, 15x15) matrices using 98 (resp. 686, 2088) multiplications. Furthermore, we describe an approximate algorithm that requires 89 multiplications and computes this product with an arbitrary small error.