# Equivalent Polyadic Decompositions of Matrix Multiplication Tensors

## TL;DR

This paper introduces an efficient algorithm to determine the equivalence of polyadic decompositions of matrix multiplication tensors, analyzes their classes for various sizes, and explores conditions for integer-based decompositions to improve fast matrix multiplication algorithms.

## Contribution

It presents a novel algorithm for deciding equivalence of tensor decompositions and analyzes their classes for different matrix sizes, advancing understanding of fast matrix multiplication methods.

## Key findings

- For larger matrices, decompositions are likely essentially different.
- The algorithm effectively distinguishes equivalent decompositions.
- Necessary conditions for integer-based decompositions are identified.

## Abstract

Invariance transformations of polyadic decompositions of matrix multiplication tensors define an equivalence relation on the set of such decompositions. In this paper, we present an algorithm to efficiently decide whether two polyadic decompositions of a given matrix multiplication tensor are equivalent. With this algorithm, we analyze the equivalence classes of decompositions of several matrix multiplication tensors. This analysis is relevant for the study of fast matrix multiplication as it relates to the question of how many essentially different fast matrix multiplication algorithms there exist. This question has been first studied by de~Groote, who showed that for the multiplication of $2\times2$ matrices with $7$ active multiplications, all algorithms are essentially equivalent to Strassen's algorithm. In contrast, the results of our analysis show that for the multiplication of larger matrices, (e.g., $2\times3$ by $3\times2$ or $3\times3$ by $3\times3$ matrices), two decompositions are very likely to be essentially different. We further provide a necessary criterion for a polyadic decomposition to be equivalent to a polyadic decomposition with integer entries. Decompositions with specific integer entries, e.g., powers of two, provide fast matrix multiplication algorithms with better efficiency and stability properties. This condition can be tested algorithmically and we present the conclusions obtained for the decompositions of small/medium matrix multiplication tensors.