神经学习快速矩阵乘法算法：StrassenNet方法
Neural Learning of Fast Matrix Multiplication Algorithms: A StrassenNet Approach
AI总结：
StrassenNet通过神经学习重现Strassen算法，并发现3×3矩阵乘法张量的有效最小秩为23。
摘要：
快速矩阵乘法可以描述为寻找矩阵乘法张量的低秩分解。我们设计了一种神经架构，StrassenNet，其能够重现Strassen算法用于2×2乘法。在多次独立运行中，网络总是收敛到秩为7的张量，从而数值上恢复了Strassen的最优算法。然后我们用相同的架构在3×3乘法中训练，使用秩r∈{19,…,23}。我们的实验揭示了一个明显的数值阈值：r=23的模型比r≤22的模型具有显著更低的验证误差，这表明r=23可能是3×3矩阵乘法张量的有效最小秩。我们还勾勒了通过ε参数化扩展该方法到边秩分解的途径，并报告了初步结果，这些结果与已知的3×3矩阵乘法张量的边秩界限相一致。
英文摘要：
Fast matrix multiplication can be described as searching for low-rank decompositions of the matrix--multiplication tensor. We design a neural architecture, \textsc{StrassenNet}, which reproduces the Strassen algorithm for $2\times 2$ multiplication. Across many independent runs the network always converges to a rank-$7$ tensor, thus numerically recovering Strassen's optimal algorithm. We then train the same architecture on $3\times 3$ multiplication with rank $r\in\{19,\dots,23\}$. Our experiments reveal a clear numerical threshold: models with $r=23$ attain significantly lower validation error than those with $r\le 22$, suggesting that $r=23$ could actually be the smallest effective rank of the matrix multiplication tensor $3\times 3$. We also sketch an extension of the method to border-rank decompositions via an $\varepsilon$--parametrisation and report preliminary results consistent with the known bounds for the border rank of the $3\times 3$ matrix--multiplication tensor.
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