张量运算和张量秩
Tensor surgery and tensor rank
摘要：
我们介绍了一种将低阶张量转化为高阶张量的方法，并将其应用于由图和超图定义的张量。变换过程类似于外科手术，它分割顶点，创建和吸收虚拟边，并插入新的顶点和边。我们表明，张量手术是能够保持初始张量分解的低秩结构，从而允许证明非平凡的上界张量秩，边界秩和渐近秩的最终张量。我们用一些例子说明我们的方法。三角图上的张量手术，它对应于矩阵乘法张量，导致所有奇循环图的非平凡的秩上界，它对应于迭代矩阵乘法的张量。在渐近设置中，我们得到的矩阵乘法指数$ω$和矩形矩阵乘法参数$α$的上界。如果$ω$等于2，则这些界限是最优的。我们还给出了例子，说明张量手术一般图可能涉及虚拟超边的吸收，我们提供了一个超图张量手术的例子。除了在代数复杂性理论中的相关性外，我们的工作在量子信息论和通信复杂性中也有应用。
英文摘要：
We introduce a method for transforming low-order tensors into higher-order tensors and apply it to tensors defined by graphs and hypergraphs. The transformation proceeds according to a surgery-like procedure that splits vertices, creates and absorbs virtual edges and inserts new vertices and edges. We show that tensor surgery is capable of preserving the low rank structure of an initial tensor decomposition and thus allows to prove nontrivial upper bounds on tensor rank, border rank and asymptotic rank of the final tensors. We illustrate our method with a number of examples. Tensor surgery on the triangle graph, which corresponds to the matrix multiplication tensor, leads to nontrivial rank upper bounds for all odd cycle graphs, which correspond to the tensors of iterated matrix multiplication. In the asymptotic setting we obtain upper bounds in terms of the matrix multiplication exponent $ω$ and the rectangular matrix multiplication parameter $α$. These bounds are optimal if $ω$ equals two. We also give examples that illustrate that tensor surgery on general graphs might involve the absorption of virtual hyperedges and we provide an example of tensor surgery on a hypergraph. Besides its relevance in algebraic complexity theory, our work has applications in quantum information theory and communication complexity.
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