Obtaining superlinear lower bounds on tensor rank is a major open problem in complexity theory. In this paper we propose a generalization of the approach used by Strassen in the proof of his 3n/2 border rank lower bound. Our approach revolves around a problem on commuting matrices: Given matrices Z_1,...,Z_p of size n and an integer r>n, are there commuting matrices Z'_1,...,Z'_p of size r such that every Z_k is embedded as a submatrix in the top-left corner of Z'_k? As one of our main results, we show that this question always has a positive answer for r larger than rank(T)+n, where T denotes the tensor with slices Z_1,..,Z_p. Taking the contrapositive, if one can show for some specific matrices Z_1,...,Z_p and a specific integer r that this question has a negative answer, this yields the lower bound rank(T) > r-n. There is a little bit of slack in the above rank(T)+n bound, but we also provide a number of exact characterizations of tensor rank and symmetric rank, for ordinary and symmetric tensors, over the fields of real and complex numbers. Each of these characterizations points to a corresponding variation on the above approach. In order to explain how Strassen's theorem fits within this framework we also provide a self-contained proof of his lower bound.

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关于张量秩和交换矩阵

获得张量秩的超线性下界是复杂性理论中的一个主要开放问题。在本文中，我们提出了 Strassen 在他的 3n/2 边界秩下界证明中使用的方法的概括。我们的方法围绕交换矩阵的问题：给定大小为 n 的矩阵 Z_1,...,Z_p 和一个整数 r>n，是否存在大小为 r 的交换矩阵 Z'_1,...,Z'_p 使得每个Z_k 作为子矩阵嵌入 Z'_k 的左上角？作为我们的主要结果之一，我们表明对于大于 rank(T)+n 的 r，这个问题总是有肯定的答案，其中 T 表示切片 Z_1,..,Z_p 的张量。取反证法，如果可以针对某些特定矩阵 Z_1,...,Z_p 和特定整数 r 表明该问题的答案是否定的，则这会产生下限 rank(T) > rn。在上述 rank(T)+n 界限中有一点松弛，但我们还提供了许多张量秩和对称秩的精确表征，对于普通张量和对称张量，在实数和复数领域。这些特征中的每一个都指向上述方法的相应变体。为了解释施特拉森定理如何适合这个框架，我们还提供了他的下界的独立证明。