张量及其Tucker核心：不变性关系
Tensor and Its Tucker Core: the Invariance Relationships
AI总结：
本文研究了张量的Tucker核心保留原张量性质如CP秩、边秩、张量Schatten准范数和Z特征值，通过数值实验验证了其计算优势。
摘要：
在[13]中，Hillar和Lim证明了数值线性代数中许多高效可计算问题的多线性（张量）类比是NP难的。尽管近年来有诸多进展，计算此类`张量类比'的最先进方法仍严重受维度诅咒影响。本文表明，张量的Tucker核心保留了原张量的许多性质，包括CP秩、边秩、张量Schatten准范数和Z特征值。当核心张量比原张量小时，这一性质带来了显著的计算优势，这通过我们的数值实验得到了验证。在分析中，我们实际上使用了一种广义的Tucker-like分解，可以容纳任何列满秩因子矩阵。
英文摘要：
In [13], Hillar and Lim famously demonstrated that "multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard". Despite many recent advancements, the state-of-the-art methods for computing such `tensor analogues' still suffer severely from the curse of dimensionality. In this paper we show that the Tucker core of a tensor however, retains many properties of the original tensor, including the CP rank, the border rank, the tensor Schatten quasi norms, and the Z-eigenvalues. When the core tensor is smaller than the original tensor, this property leads to considerable computational advantages as confirmed by our numerical experiments. In our analysis, we in fact work with a generalized Tucker-like decomposition that can accommodate any full column-rank factor matrices.
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