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 lack of meaningful 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 has a long range dependence among its entries, 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：本文研究低秩高维张量信号的两类假设检验问题：张量信号对齐问题与张量信号匹配问题。由于张量的高维特性及缺乏有效的检验统计量，这些问题具有显著挑战性。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。该矩阵的条目间存在长程依赖性，使得矩阵分析具有挑战性且复杂，并显著区别于经典随机矩阵理论。本方法为解决高维张量信号背景下的假设检验问题提供了新颖的理论框架。