We consider two hypothesis testing problems for low-rank and high-dimensional tensor signals, namely the tensor signal alignment and tensor signal matching problems. These problems are challenging due to the high dimension of tensors and the lack of suitable test statistics. By exploiting a recent tensor contraction method, we propose and validate relevant test statistics using eigenvalues of a data matrix resulting from the tensor contraction. The matrix entries exhibit long-range dependence, which makes the analysis of the matrix challenging, involved, and distinct from standard random matrix theory. Our approach provides a novel framework for addressing hypothesis testing problems in the context of high-dimensional tensor signals.

翻译：本文研究了低秩高维张量信号的两种假设检验问题：张量信号对齐与张量信号匹配问题。由于张量的高维特性以及缺乏合适的检验统计量，这些问题具有挑战性。通过运用最新的张量收缩方法，我们提出并验证了基于张量收缩生成数据矩阵特征值的相关检验统计量。该矩阵元素呈现长程依赖性，这使得矩阵分析具有挑战性、复杂性，并区别于标准随机矩阵理论。我们的方法为解决高维张量信号背景下的假设检验问题提供了一个新颖的框架。