Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Adaptive Flip Graph Algorithm for Matrix Multiplication

Yamato Arai, Yuma Ichikawa, Koji Hukushima

TL;DR

This study proposes the “adaptive flip graph algorithm”, which combines adaptive searches with the flip graph algorithm for finding fast and efficient methods for matrix multiplication, and a formal proof is provided that the introduction of plus transitions in the proposed algorithm ensures the connectivity of any node in the flip graph, which represents a method of matrix multiplication.

Abstract

This study proposes the "adaptive flip graph algorithm", which combines adaptive searches with the flip graph algorithm for finding fast and efficient methods for matrix multiplication. The adaptive flip graph algorithm addresses the inherent limitations of exploration and inefficient search encountered in the original flip graph algorithm, particularly when dealing with large matrix multiplication. For the limitation of exploration, the proposed algorithm adaptively transitions over the flip graph, introducing a flexibility that does not strictly reduce the number of multiplications. Concerning the issue of inefficient search in large instances, the proposed algorithm adaptively constraints the search range instead of relying on a completely random search, facilitating more effective exploration. Numerical experimental results demonstrate the effectiveness of the adaptive flip graph algorithm, showing a reduction in the number of multiplications for a $4\times 5$ matrix multiplied by a $5\times 5$ matrix from $76$ to $73$, and that from $95$ to $94$ for a $5 \times 5$ matrix multiplied by another $5\times 5$ matrix. These results are obtained in characteristic two.

Adaptive Flip Graph Algorithm for Matrix Multiplication

TL;DR

This study proposes the “adaptive flip graph algorithm”, which combines adaptive searches with the flip graph algorithm for finding fast and efficient methods for matrix multiplication, and a formal proof is provided that the introduction of plus transitions in the proposed algorithm ensures the connectivity of any node in the flip graph, which represents a method of matrix multiplication.

Abstract

This study proposes the "adaptive flip graph algorithm", which combines adaptive searches with the flip graph algorithm for finding fast and efficient methods for matrix multiplication. The adaptive flip graph algorithm addresses the inherent limitations of exploration and inefficient search encountered in the original flip graph algorithm, particularly when dealing with large matrix multiplication. For the limitation of exploration, the proposed algorithm adaptively transitions over the flip graph, introducing a flexibility that does not strictly reduce the number of multiplications. Concerning the issue of inefficient search in large instances, the proposed algorithm adaptively constraints the search range instead of relying on a completely random search, facilitating more effective exploration. Numerical experimental results demonstrate the effectiveness of the adaptive flip graph algorithm, showing a reduction in the number of multiplications for a matrix multiplied by a matrix from to , and that from to for a matrix multiplied by another matrix. These results are obtained in characteristic two.
Paper Structure (17 sections, 3 theorems, 6 equations, 2 figures, 1 table, 4 algorithms)

This paper contains 17 sections, 3 theorems, 6 equations, 2 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation of Matrix Multiplication
  3. Flip Graph Algorithm
  4. Limitation and Practical Issues of Flip Graph Algorithm
  5. Non-Reduction States in Principle
  6. Practical Non-Reducibility for Large Matrix Multiplication
  7. Adaptive Flip Graph Algorithm
  8. Plus Transition
  9. Edge Constraints
  10. Connectivity of flip graphs through plus transitions
  11. Results
  12. Effectiveness of Plus Transition Escaping Non-Reduction States
  13. Acceleration by Edge Constraints in Large Matrix Multiplications
  14. Results of Adaptive Flip Graph Algorithm for Various Matrix Multiplications
  15. Discussion and Summary
  16. ...and 2 more sections

Key Result

lemma 1

In the $(n,m,p)$-matrix multiplication scheme S, it holds that $rank([\alpha^{(1)}, \alpha^{(2)}, ..., \alpha^{(R)}]) = n*m$. Similarly, for $\beta$ and $\gamma$, $rank([\beta^{(1)}, \beta^{(2)}, ..., \beta^{(R)}]) = m*p$ and $rank([\gamma^{(1)}, \gamma^{(2)}, ..., \gamma^{(R)}]) = p*n$.

Figures (2)

  • Figure 1: Dependence of the minimum rank found in the search on the number of iterations in the $(4,4,4)$-matrix multiplication scheme. Results for the flip graph algorithm with the plus transition are marked with circles and those without the plus transition are marked with triangles. Error bars represent the standard error obtained by 10 independent runs.
  • Figure 2: Dependence of the minimum rank found in the search on the number of iterations in the $(5,5,5)$-matrix multiplication scheme. Results for the flip graph algorithm with the edge constraints are marked with circles and those without the edge constraints are marked with triangles. Error bars represent the standard error obtained by 10 independent runs.

Theorems & Definitions (9)

  • definition 1: Matrix Multiplication Tensor
  • definition 2: Matrix Multiplication Scheme
  • definition 3: Flip
  • definition 4: Reduction
  • definition 5: Flip Graph
  • definition 6: Plus
  • lemma 1
  • theorem 1
  • corollary 1