The complexity of matrix multiplication is measured in terms of $\omega$, the smallest real number such that two $n\times n$ matrices can be multiplied using $O(n^{\omega+\epsilon})$ field operations for all $\epsilon>0$; the best bound until now is $\omega<2.37287$ [Le Gall'14]. All bounds on $\omega$ since 1986 have been obtained using the so-called laser method, a way to lower-bound the `value' of a tensor in designing matrix multiplication algorithms. The main result of this paper is a refinement of the laser method that improves the resulting value bound for most sufficiently large tensors. Thus, even before computing any specific values, it is clear that we achieve an improved bound on $\omega$, and we indeed obtain the best bound on $\omega$ to date: $$\omega < 2.37286.$$ The improvement is of the same magnitude as the improvement that [Le Gall'14] obtained over the previous bound [Vassilevska W.'12]. Our improvement to the laser method is quite general, and we believe it will have further applications in arithmetic complexity.

翻译：矩阵乘法的复杂度以 $\omega$ 来衡量，$\omega$ 是满足以下条件的最小实数：对于任意 $\epsilon>0$，两个 $n\times n$ 矩阵的乘法可以在 $O(n^{\omega+\epsilon})$ 次域操作内完成；目前的最佳上界是 $\omega<2.37287$ [Le Gall'14]。自1986年以来，所有关于 $\omega$ 的上界都是通过所谓的激光方法获得的，该方法通过下界估计张量的“值”来设计矩阵乘法算法。本文的主要成果是对激光方法的一种改进，该改进提升了对于大多数足够大的张量所能得到的值下界。因此，即使在计算任何具体值之前，我们显然已经得到了一个改进的 $\omega$ 上界，并且我们确实得到了迄今为止最佳的 $\omega$ 上界：$$\omega < 2.37286.$$ 这一改进的幅度与 [Le Gall'14] 相对于先前上界 [Vassilevska W.'12] 所取得的改进幅度相同。我们对激光方法的改进具有相当的通用性，我们相信它将在算术复杂性领域有进一步的应用。