小型格式下的快速矩阵乘法：基于开源翻转图框架的新方案探索

摘要

本文提出了一种基于翻转图法的开源C++框架，用于探索快速矩阵乘法方案。该框架支持多种系数环——二元环(Z_2)、模三环(Z_3)和整数三元环(Z_T = {-1,0,1})，并实现了固定维度和元维度两种搜索算子。通过采用高效的系数向量比特级编码和OpenMP并行技术，该工具可在商用硬件上实现大规模探索。研究覆盖了从(2×2×2)到(16×16×16)的680种方案，其中276个方案采用Z_T系数，117个采用整数系数。借助该框架，79个矩阵乘法方案的乘法复杂度（秩）得到改进。特别值得一提的是，新发现的4×4×10方案仅需115次乘法运算，实现ω≈2.80478，在该特定维度上超越了Strassen指数。此外，在 ternary 系数下重新发现了93个先前仅存在于有理数或整数域的方案，并在整数系数下重现了68个原本需要分数运算的方案。所有工具及发现的方案均已公开，以确保研究的可复现性。

English

An open-source C++ framework for discovering fast matrix multiplication schemes using the flip graph approach is presented. The framework supports multiple coefficient rings -- binary (Z_2), modular ternary (Z_3) and integer ternary (Z_T = {-1,0,1}) -- and implements both fixed-dimension and meta-dimensional search operators. Using efficient bit-level encoding of coefficient vectors and OpenMP parallelism, the tools enable large-scale exploration on commodity hardware. The study covers 680 schemes ranging from (2 times 2 times 2) to (16 times 16 times 16), with 276 schemes now in Z_T coefficients and 117 in integer coefficients. With this framework, the multiplicative complexity (rank) is improved for 79 matrix multiplication schemes. Notably, a new 4 times 4 times 10 scheme requiring only 115 multiplications is discovered, achieving ωapprox 2.80478 and beating Strassen's exponent for this specific size. Additionally, 93 schemes are rediscovered in ternary coefficients that were previously known only over rationals or integers, and 68 schemes in integer coefficients that previously required fractions. All tools and discovered schemes are made publicly available to enable reproducible research.

小型格式下的快速矩阵乘法：基于开源翻转图框架的新方案探索

Fast Matrix Multiplication in Small Formats: Discovering New Schemes with an Open-Source Flip Graph Framework

摘要

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