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 whereas 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.

翻译：张量通常通过引入限制和退化等预序来研究：前者描述通过其张量因子上的局部线性映射对张量进行的变换；后者描述局部线性映射可能沿曲线变化，且所得张量表示为沿该曲线极限的变换。本文引入并研究部分退化——这是退化的一种特殊形式，其中一个局部线性映射保持恒定，而其他映射沿曲线变化。受代数复杂性、量子纠缠和张量网络的启发，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建立联系找到了示例。我们通过展示单位张量的阻碍和分类结果，突显了这一新概念的微妙之处。为此，我们研究了辅助秩的概念，这是张量秩的自然推广。部分退化的存在为张量的辅助秩提供了强上界，从而使得退化可转化为限制。特别地，我们基于W-张量和Coppersmith-Winograd张量提出了若干示例，其中辅助秩的下界为特定部分退化的存在提供了阻碍。