SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 771-800, March 2024.
Abstract. Tensors are often studied by introducing preorders such as restriction and degeneration. The former describes transformations of the tensors by local linear maps on its tensor factors; the latter describes transformations where the local linear maps may vary along a curve, and the resulting tensor is expressed as a limit along this curve. In this work, we introduce and study partial degeneration, a special version of degeneration where one of the local linear maps is constant while the others vary along a curve. Motivated by algebraic complexity, quantum entanglement, and tensor networks, we present constructions based on matrix multiplication tensors and find examples by making a connection to the theory of prehomogeneous tensor spaces. We highlight the subtleties of this new notion by showing obstruction and classification results for the unit tensor. To this end, we study the notion of aided rank, a natural generalization of tensor rank. The existence of partial degenerations gives strong upper bounds on the aided rank of a tensor, which allows one to turn degenerations into restrictions. In particular, we present several examples, based on the W-tensor and the Coppersmith–Winograd tensors, where lower bounds on aided rank provide obstructions to the existence of certain partial degenerations.

中文翻译：

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张量的部分退化

《SIAM 矩阵分析与应用杂志》，第 45 卷，第 1 期，第 771-800 页，2024 年 3 月。
摘要。张量通常通过引入限制和退化等预序来研究。前者通过张量因子上的局部线性映射描述张量的变换；后者描述了局部线性映射可能沿曲线变化的变换，并且所得张量表示为沿该曲线的极限。在这项工作中，我们介绍并研究了部分退化，这是退化的一种特殊版本，其中一个局部线性图是恒定的，而其他局部线性图沿曲线变化。在代数复杂性、量子纠缠和张量网络的推动下，我们提出了基于矩阵乘法张量的构造，并通过与预齐次张量空间理论的联系找到了例子。我们通过显示单位张量的阻碍和分类结果来强调这个新概念的微妙之处。为此，我们研究了辅助秩的概念，这是张量秩的自然推广。部分简并的存在为张量的辅助秩提供了强大的上限，这使得人们可以将简并转化为限制。特别是，我们提出了几个基于 W 张量和 Coppersmith-Winograd 张量的例子，其中辅助秩的下限阻碍了某些部分退化的存在。