We study the algorithmic problem of multiplying large matrices that are rectangular. We prove that the method that has been used to construct the fastest algorithms for rectangular matrix multiplication cannot give optimal algorithms. In fact, we prove a precise numerical barrier for this method. Our barrier improves the previously known barriers, both in the numerical sense, as well as in its generality. We prove our result using the asymptotic spectrum of tensors. More precisely, we crucially make use of two families of real tensor parameters with special algebraic properties: the quantum functionals and the support functionals. In particular, we prove that any lower bound on the dual exponent of matrix multiplication $\alpha$ via the big Coppersmith-Winograd tensors cannot exceed 0.625.

中文翻译：

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矩形矩阵乘法的障碍

我们研究将矩形大矩阵相乘的算法问题。我们证明了用于构造矩形矩阵乘法最快算法的方法不能给出最优算法。事实上，我们证明了这种方法的精确数值障碍。我们的障碍在数值意义上和一般性方面都改进了先前已知的障碍。我们使用张量的渐近谱证明了我们的结果。更准确地说，我们至关重要地利用了具有特殊代数性质的两个实张量参数族：量子泛函和支持泛函。特别是，我们证明了通过大 Coppersmith-Winograd 张量对矩阵乘法 $\alpha$ 的对偶指数的任何下限不能超过 0.625。