Flip Graphs for Matrix Multiplication

TL;DR

This paper presents a novel graph-based approach called flip graphs for discovering efficient matrix multiplication schemes, successfully reducing the number of multiplications for specific matrix formats over various fields.

Contribution

The paper introduces flip graphs as a new method for finding matrix multiplication schemes, achieving reductions in multiplications for (4,4,5) and (5,5,5) formats.

Findings

Reduced multiplications for (4,4,5) and (5,5,5) matrix formats.

Applicable over characteristic two and arbitrary fields.

Introduced a graph-based search method for matrix multiplication schemes.

Abstract

We introduce a new method for discovering matrix multiplication schemes based on random walks in a certain graph, which we call the flip graph. Using this method, we were able to reduce the number of multiplications for the matrix formats (4, 4, 5) and (5, 5, 5), both in characteristic two and for arbitrary ground fields.