We present an upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on k vertices. For $${k \geq 4}$$k≥4, we show that the exponent per edge is at most 0.77, outperforming the best known upper bound on the exponent per edge for matrix multiplication (k = 3), which is approximately 0.79. We raise the question whether for some k the exponent per edge can be below 2/3, i.e. can outperform matrix multiplication even if the matrix multiplication exponent equals 2. In order to obtain our results, we generalize to higher-order tensors a result by Strassen on the asymptotic subrank of tight tensors and a result by Coppersmith and Winograd on the asymptotic rank of matrix multiplication. Our results have applications in entanglement theory and communication complexity.

中文翻译：

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图张量的渐近张量秩：超越矩阵乘法

我们提出了由 k 个顶点上的完整图定义的张量族的张量秩的渐近行为指数的上限。对于 $${k \geq 4}$$k≥4，我们表明每条边的指数最多为 0.77，优于矩阵乘法的每条边指数的最著名上限 (k = 3)，大约为0.79。我们提出了一个问题，对于某些 k，每条边的指数是否可以低于 2/3，即即使矩阵乘法指数等于 2 也能胜过矩阵乘法。为了获得我们的结果，我们将结果推广到高阶张量Strassen 关于紧张量的渐近子秩以及 Coppersmith 和 Winograd 关于矩阵乘法的渐近秩的结果。我们的结果可应用于纠缠理论和通信复杂性。