Exploiting the Structure in Tensor Decompositions for Matrix Multiplication

TL;DR

This paper introduces a novel tensor decomposition-based algorithm for matrix multiplication that achieves a lower computational exponent than previous methods, specifically improving the exponent for 6x6 matrices.

Contribution

It presents a new algorithm leveraging special features of tensor decompositions to reduce the matrix multiplication exponent below the tensor rank suggestion.

Findings

Reduced the exponent for 6x6 matrix multiplication from 2.8075 to 2.8019.

Maintains a reasonable leading coefficient in the new algorithm.

Demonstrates the effectiveness of tensor structure exploitation in matrix multiplication.

Abstract

We present a new algorithm for fast matrix multiplication using tensor decompositions which have special features. Thanks to these features we obtain exponents lower than what the rank of the tensor decomposition suggests. In particular for $6\times 6$ matrix multiplication we reduce the exponent of the recent algorithm by Moosbauer and Poole from $2.8075$ to $2.8019$, while retaining a reasonable leading coefficient.