Characterization of Decomposition of Matrix Multiplication Tensors

TL;DR

This paper studies the algebraic complexity of matrix multiplication tensors by characterizing their decompositions, introducing a new decomposition for 3x3 by 3x6 matrices with rank 40, which could improve practical algorithms.

Contribution

It introduces a novel tensor decomposition for 3x3 by 3x6 matrix multiplication with rank 40, enhancing understanding of tensor decompositions in algebraic complexity.

Findings

Characterized existing tensor decompositions using signature vectors.

Developed a new decomposition for 3x3 by 3x6 matrix multiplication.

Provided insights into tensor rank and decomposition transformations.

Abstract

In this paper, the canonical polyadic (CP) decomposition of tensors that corresponds to matrix multiplications is studied. Finding the rank of these tensors and computing the decompositions is a fundamental problem of algebraic complexity theory. In this paper, we characterize existing decompositions (found by any algorithm) by certain vectors called signature, and transform them in another decomposition which can be more suitable in practical algorithms. In particular, we present a novel decomposition of the tensor multiplication of matrices of the size 3x3 with 3x6 with rank 40.