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More Asymmetry Yields Faster Matrix Multiplication

Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, Renfei Zhou

Abstract

We present a new improvement on the laser method for designing fast matrix multiplication algorithms. The new method further develops the recent advances by [Duan, Wu, Zhou FOCS 2023] and [Vassilevska Williams, Xu, Xu, Zhou SODA 2024]. Surprisingly the new improvement is achieved by incorporating more asymmetry in the analysis, circumventing a fundamental tool of prior work that requires two of the three dimensions to be treated identically. The method yields a new bound on the square matrix multiplication exponent $$ω<2.371339,$$ improved from the previous bound of $ω<2.371552$. We also improve the bounds of the exponents for multiplying rectangular matrices of various shapes.

More Asymmetry Yields Faster Matrix Multiplication

Abstract

We present a new improvement on the laser method for designing fast matrix multiplication algorithms. The new method further develops the recent advances by [Duan, Wu, Zhou FOCS 2023] and [Vassilevska Williams, Xu, Xu, Zhou SODA 2024]. Surprisingly the new improvement is achieved by incorporating more asymmetry in the analysis, circumventing a fundamental tool of prior work that requires two of the three dimensions to be treated identically. The method yields a new bound on the square matrix multiplication exponent improved from the previous bound of . We also improve the bounds of the exponents for multiplying rectangular matrices of various shapes.
Paper Structure (59 sections, 11 theorems, 115 equations, 2 figures, 3 tables)

This paper contains 59 sections, 11 theorems, 115 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Technical Overview
  3. The Laser Method and Asymptotic Sum Inequality
  4. Recursive Applications of the Laser Method to Tensor Powers of CWq
  5. Asymmetry and Combination Loss
  6. Our Contributions: More Asymmetry
  7. Limitations to Our Techniques
  8. Preliminaries
  9. Tensors and Tensor Operations
  10. Tensors.
  11. Tensor operations.
  12. Tensor Rank
  13. Degenerations, Restrictions, Zero-outs
  14. Degeneration.
  15. Restriction.
  16. ...and 44 more sections

Key Result

Theorem 3.1

For positive integers $r > m$ and $a_i, b_i,c_i$ for $i\in [m]$, if then $\omega\le 3\tau$ where $\tau\in [2/3,1]$ is the solution to the equation

Figures (2)

  • Figure 1: Main differences between the global stage algorithm in VXXZ24 and this work as outlined in \ref{['subsec:outline']}. Other technical differences are omitted.
  • Figure : Procedure of degeneration (similar to VXXZ24)

Theorems & Definitions (53)

  • Theorem 3.1: Asymptotic Sum Inequality Schonhage81
  • Theorem 3.2: Asymptotic Sum Inequality for $\omega(a,b,c)$ Schonhage81blaser
  • Lemma 3.3
  • Definition 3.4: Complete Split Distribution
  • Definition 3.5
  • Definition 3.6
  • Theorem 3.7: behrend1946setselsholtz2024improving
  • Definition 4.1: Interface Tensor
  • Proposition 5.1
  • Remark 5.2: VXXZ24
  • ...and 43 more