AI 中文总结
本文通过推广张量秩到划分秩，建立与乘法复杂度的联系，从而在任意常数的多重线性度下给出算术电路下界，并应用于超团猜想等细粒度算法问题。
AI 中文摘要
Strassen的双线性复杂度理论通过张量秩为矩阵乘法等原语的算术复杂度提供了数学刻画。然而，已知这种与张量秩的联系在更高次的多重线性度下会失效。本文揭示了Naslund（JCTA 2020）的划分秩框架中定义的广义张量秩与乘法复杂度之间一个未被探索的联系。这些划分秩使我们能够从下方控制任意常数多重线性度下的乘法复杂度（从而控制算术复杂度），同时在双线性情形下恢复了Strassen的开创性刻画。这为基于秩的方法在细粒度算法和复杂度问题中的新应用提供了可能，例如Lincoln-Williams-Vassilevska Williams（SODA 2018）的超团猜想。此外，我们展示了与已有秩概念（如Tao和Sawin意义下的张量切片秩）及其对称变体的联系。对于计算后一种对称变体，我们给出了一个简单的NP困难性证明，这与Bläser等人（SODA 2021）关于普通非对称张量切片秩的相当复杂的NP困难性证明形成对比。
英文摘要
Strassen's theory of bilinear complexity provides a mathematical characterization of the arithmetic complexity of primitives such as matrix multiplication via the rank of tensors. However, the connection to tensor rank is known to break down in higher degrees of multilinearity. In this work, we highlight an unexplored connection between a generalized notion of tensor rank, which can be defined in Naslund's framework of partition ranks (JCTA 2020), and multiplicative complexity. These partition ranks allow us to control the multiplicative complexity, and thus arithmetic complexity, in any constant degree of multilinearity from below, while recovering Strassen's seminal characterization in the bilinear case. This enables novel potential applications of the rank-based approaches to problems in fine-grained algorithms and complexity, such as the hyperclique conjecture of Lincoln-Williams-Vassilevska Williams (SODA 2018). Moreover, we exhibit connections to established notions of rank, such as tensor slice rank (in the sense of Tao and Sawin), as well as its symmetric variant. For computing the latter symmetric variant, we point out a simple NP-hardness proof, contrasting the rather involved NP-hardness proof for ordinary, non-symmetric tensor slice rank by Bläser et al. (SODA 2021).
Comments12 pages. Accepted to ESA 2026
