解耦多元多项式：张量化之间的关联
Decoupling multivariate polynomials: interconnections between tensorizations
AI总结：
本文研究了多元多项式解耦问题，通过展示张量之间的线性变换关系，揭示了CP分解和唯一性性质的关联，并指出考虑张量分解中的内在结构可以提升分解的唯一性，从而扩大方法的应用范围。
摘要：
解耦多元多项式对于理解非线性映射的工作原理、进行参数缩减或近似非线性函数是有用的。已经提出了几种不同的基于张量的方法来完成此任务，涉及不同的函数张量表示，最终导致了典型的聚类分解。我们首先展示所涉及的张量之间存在线性变换关系，并且它们的CP分解和唯一性属性密切相关。这种联系提供了一种更好地评估在某些问题设置中应优先采用哪种方法的方法，可能成为统一这两种方法的起点。其次，我们展示考虑张量分解中之前忽略的内在结构可以提高分解的唯一性属性，从而扩大方法的应用范围。
英文摘要：
Decoupling multivariate polynomials is useful for obtaining an insight into the workings of a nonlinear mapping, performing parameter reduction, or approximating nonlinear functions. Several different tensor-based approaches have been proposed independently for this task, involving different tensor representations of the functions, and ultimately leading to a canonical polyadic decomposition. We first show that the involved tensors are related by a linear transformation, and that their CP decompositions and uniqueness properties are closely related. This connection provides a way to better assess which of the methods should be favored in certain problem settings, and may be a starting point to unify the two approaches. Second, we show that taking into account the previously ignored intrinsic structure in the tensor decompositions improves the uniqueness properties of the decompositions and thus enlarges the applicability range of the methods.
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