This paper proposes a criterion for detecting change structures in tensor data. To accommodate tensor structure with structural mode that is not suitable to be equally treated and summarized in a distance to measure the difference between any two adjacent tensors, we define a mode-based signal-screening Frobenius distance for the moving sums of slices of tensor data to handle both dense and sparse model structures of the tensors. As a general distance, it can also deal with the case without structural mode. Based on the distance, we then construct signal statistics using the ratios with adaptive-to-change ridge functions. The number of changes and their locations can then be consistently estimated in certain senses, and the confidence intervals of the locations of change points are constructed. The results hold when the size of the tensor and the number of change points diverge at certain rates, respectively. Numerical studies are conducted to examine the finite sample performances of the proposed method. We also analyze two real data examples for illustration.

翻译：本文提出了一种检测张量数据变化结构的准则。为适应具有结构模态的张量结构（该模态不宜被平等对待或通过距离度量来概括任意两个相邻张量间的差异），我们定义了一种基于模态的信号筛选Frobenius距离，用于张量数据切片的移动求和，以处理张量的密集与稀疏模型结构。作为一种通用距离，它也能处理无结构模态的情形。基于该距离，我们利用自适应变化岭函数的比值构造信号统计量。由此，变化点数量及其位置可在特定意义下得到一致估计，并构建了变化点位置的置信区间。当张量规模与变化点数量分别以特定速率发散时，上述结论成立。通过数值研究检验了所提方法的有限样本性能，并分析两个真实数据示例加以说明。