OpenAI 证明矩阵乘法指数上界 9/4 的解读论文
Tensor value and the 9/4 bound for matrix multiplication
OpenAI 把矩阵乘法复杂度上界压到 9/4 了，这篇论文把证明拆开讲清楚，适合想看懂数学细节的人。
arXiv 上一篇新论文对 OpenAI 的矩阵乘法指数上界 9/4 结果给出完整阐述，该上界对任意数域成立。作者使用 Coppersmith-Winograd 提出的张量值的分数对称化变体，替代原证明中的谱点（张量特征）。论文证明了这个值的幂在部分直和上的超可加性不等式，其中各加项共享一个张量因子并占据另两个维度上的独立子空间。结合 OpenAI 证明中多项式乘法张量的退化，该不等式最终推出 9/4 的上界。
Tensor value and the 9/4 bound for matrix multiplication
We give an exposition of OpenAI's recent result that the exponent of matrix multiplication is at most 9/4 valid over every field. Our main tool is a fractional symmetrized variant of the tensor value introduced by Coppersmith and Winograd, used in place of spectral points (tensor characters) in the original proof. We prove a superadditivity inequality for a power of this value on partially direct sums, whose summands share one tensor factor and occupy independent subspaces in the other two. Together with the degenerations of polynomial multiplication tensors from the OpenAI proof, this inequality yields the bound.
