Tensor-valued data arise frequently from a wide variety of scientific applications, and many among them can be translated into an alteration detection problem of tensor dependence structures. In this article, we formulate the problem under the popularly adopted tensor-normal distributions and aim at two-sample correlation/partial correlation comparisons of tensor-valued observations. Through decorrelation and centralization, a separable covariance structure is employed to pool sample information from different tensor modes to enhance the power of the test. Additionally, we propose a novel Sparsity-Exploited Reranking Algorithm (SERA) to further improve the multiple testing efficiency. The algorithm is approached through reranking of the p-values derived from the primary test statistics, by incorporating a carefully constructed auxiliary tensor sequence. Besides the tensor framework, SERA is also generally applicable to a wide range of two-sample large-scale inference problems with sparsity structures, and is of independent interest. The asymptotic properties of the proposed test are derived and the algorithm is shown to control the false discovery at the pre-specified level. We demonstrate the efficacy of the proposed method through intensive simulations and two scientific applications.

翻译：张量值数据在各类科学应用中频繁出现，其中许多问题可转化为张量依赖结构的变化检测问题。本文在广泛采用的张量正态分布框架下构建该问题，旨在对张量值观测进行双样本相关性/偏相关性比较。通过去相关与中心化处理，采用可分离协方差结构从不同张量模式中整合样本信息以增强检验效能。此外，我们提出一种新颖的基于稀疏性重排算法（SERA）以进一步优化多重检验效率。该算法通过整合精心构建的辅助张量序列，对原始检验统计量导出的p值进行重排。除张量框架外，SERA还可普遍适用于具有稀疏结构的各类双样本大规模推断问题，具有独立研究价值。本文推导了所提检验的渐近性质，并证明该算法能在预设水平上控制错误发现率。通过密集模拟实验与两项科学应用，验证了所提方法的有效性。