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Flip Graphs with Symmetry and New Matrix Multiplication Schemes

Jakob Moosbauer, Michael Poole

TL;DR

The paper tackles the problem of discovering fast, small-rank matrix multiplication algorithms by combining a flip-graph search with symmetry constraints. It introduces orbit-based symmetry reductions and an orbit-aware lifting framework to guide searches from diagonal starting points, while preserving group-invariant structure. The key results are new symmetric schemes for $5×5$ and $6×6$ with $93$ and $153$ multiplications, respectively, along with successful lifting to integers and improved bounds over prior work. The work demonstrates practical scalability through a Python/C++ implementation and highlights the potential of symmetry to yield asymptotically faster algorithms than Strassen.

Abstract

The flip graph algorithm is a method for discovering new matrix multiplication schemes by following random walks on a graph. We introduce a version of the flip graph algorithm for matrix multiplication schemes that admit certain symmetries. This significantly reduces the size of the search space, allowing for more efficient exploration of the flip graph. The symmetry in the resulting schemes also facilitates the process of lifting solutions from $F_2$ to $\mathbb{Z}$. Our results are new schemes for multiplying $5\times 5$ matrices using $93$ multiplications and $6\times 6$ matrices using $153$ multiplications over arbitrary ground fields.

Flip Graphs with Symmetry and New Matrix Multiplication Schemes

TL;DR

The paper tackles the problem of discovering fast, small-rank matrix multiplication algorithms by combining a flip-graph search with symmetry constraints. It introduces orbit-based symmetry reductions and an orbit-aware lifting framework to guide searches from diagonal starting points, while preserving group-invariant structure. The key results are new symmetric schemes for and with and multiplications, respectively, along with successful lifting to integers and improved bounds over prior work. The work demonstrates practical scalability through a Python/C++ implementation and highlights the potential of symmetry to yield asymptotically faster algorithms than Strassen.

Abstract

The flip graph algorithm is a method for discovering new matrix multiplication schemes by following random walks on a graph. We introduce a version of the flip graph algorithm for matrix multiplication schemes that admit certain symmetries. This significantly reduces the size of the search space, allowing for more efficient exploration of the flip graph. The symmetry in the resulting schemes also facilitates the process of lifting solutions from to . Our results are new schemes for multiplying matrices using multiplications and matrices using multiplications over arbitrary ground fields.

Paper Structure

This paper contains 9 sections, 2 theorems, 32 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Matrix Multiplication and its Symmetry Group
  3. Flip Graph
  4. Search Algorithm
  5. Results
  6. Lifting
  7. Identifying Starting Points
  8. Implementation
  9. Conclusion

Key Result

Theorem 4

Let $n\in \mathbb{N}$ and let $G\leq S_{3}\rtimes \operatorname{GL}(n)^{\times 3}$ be a finite subgroup of the symmetry group of matrix multiplication. Let $S$ be a $G$-invariant $n\times n$ matrix multiplication scheme and let $T$ be a set of representatives for the orbits of $G$ in $S$. Let $A\oti is a $G$-invariant $n\times n$ matrix multiplication scheme. We call $S'$ an orbit flip of $S$.

Figures (1)

  • Figure 1: The component of the $C_{3}\times \mathbb{Z}_{2}$-symmetric $(2,2,2)$ flip graph that contains the starting point with diagonal partition $\{\{1,2\}\}$, marked in green. Vertices that can be reduced to Strassen's algorithm are marked in red.

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 4
  • proof
  • Definition 5: Flip Graph with Symmetry
  • Definition 6
  • Lemma 7
  • proof