[Paper Review] New ways to multiply 3 x 3-matrices

Marijn J. H. Heule, Manuel Kauers+1 more|arXiv (Cornell University)|May 24, 2019

graph theory and CDMA systems26 references4 citations

TL;DR

This paper presents over 13,000 new, mutually inequivalent schemes for multiplying 3×3 matrices using exactly 23 scalar multiplications, significantly expanding the known set of such algorithms. Using SAT solving and algebraic parameterization over ℤ₂, the authors construct families of solutions with up to 17 free parameters, all with integer coefficients, thereby proving the solution space of the Brent equations has dimension at least 17.

ABSTRACT

It is known since the 1970s that no more than 23 multiplications are required for computing the product of two 3 x 3-matrices. It is not known whether this can also be done with fewer multiplications. However, there are several mutually inequivalent ways of doing the job with 23 multiplications. In this article, we extend this list considerably by providing more than 13 000 new and mutually inequivalent schemes for multiplying 3 x 3-matrices using 23 multiplications. Moreover, we show that the set of all these schemes is a manifold of dimension at least 17.

Motivation & Objective

• To systematically extend the known set of 3×3 matrix multiplication schemes using exactly 23 scalar multiplications.
• To identify and generate new families of such schemes with integer coefficients, ensuring applicability over arbitrary coefficient rings.
• To determine the dimension of the solution space of the Brent equations for 3×3 matrix multiplication.
• To explore whether schemes with 22 multiplications exist by analyzing structural simplifications in the 23-multiplication schemes.
• To develop and apply a systematic method for parameterizing matrix multiplication schemes using SAT solvers and linear algebra over ℤ₂.

Proposed method

• Employ SAT solvers to search for valid matrix multiplication schemes over the finite field ℤ₂, starting from known solutions such as Laderman’s and Courtois et al.’s.
• Use a systematic algorithm to introduce free parameters into the schemes by replacing entries in the A, B, and C tensors with fresh variables and solving the resulting linear systems.
• Apply algebraic techniques, including Gröbner bases and variable substitutions, to eliminate denominators and verify parameter independence in the resulting families.
• Iteratively apply the parameterization process to previously generated schemes to increase the number of free parameters, achieving up to 17 parameters in some families.
• Lift solutions from ℤ₂ to arbitrary coefficient rings while preserving the bilinear structure and integer coefficients.
• Use tensor product arithmetic to test whether any of the 23-multiplication schemes can be simplified to 22 multiplications by merging summands via linear dependencies.

Experimental results

Research questions

• RQ1 How many distinct, mutually inequivalent 3×3 matrix multiplication schemes exist using exactly 23 scalar multiplications?
• RQ2 What is the maximal number of free parameters in a family of 3×3 matrix multiplication schemes with 23 multiplications and integer coefficients?
• RQ3 Can any of the 23-multiplication schemes be simplified to use only 22 multiplications through linear dependencies in the tensor factors?
• RQ4 What is the dimension of the solution variety defined by the Brent equations for 3×3 matrix multiplication?
• RQ5 Are there additional 3×3 matrix multiplication schemes with coefficients in {-1, 0, 1} beyond those already discovered?

Key findings

• The authors discovered more than 13,000 new, mutually inequivalent 3×3 matrix multiplication schemes using exactly 23 scalar multiplications.
• All new schemes have coefficients in {-1, 0, 1}, ensuring applicability over any coefficient ring, unlike previous schemes with fractional coefficients.
• The solution space of the Brent equations for 3×3 matrix multiplication has dimension at least 17, as demonstrated by the existence of parameterized families with up to 17 independent parameters.
• The authors successfully constructed families with up to 17 parameters using a SAT-based search and algebraic parameterization, with no denominators in the final expressions.
• None of the 23-multiplication schemes could be simplified to 22 multiplications by merging summands via linear dependencies in the A, B, or C tensors.
• The method enabled the discovery of new families based on Laderman’s, Courtois et al.’s, Oh et al.’s, and Smirnov’s schemes, with 0, 6, 10, and 14 parameters respectively, significantly extending prior results.