High-dimensional, higher-order tensor data are gaining prominence in a variety of fields, including but not limited to computer vision and network analysis. Tensor factor models, induced from noisy versions of tensor decomposition or factorization, are natural potent instruments to study a collection of tensor-variate objects that may be dependent or independent. However, it is still in the early stage of developing statistical inferential theories for estimation of various low-rank structures, which are customary to play the role of signals of tensor factor models. In this paper, starting from tensor matricization, we aim to ``decode" estimation of a higher-order tensor factor model in the sense that, we recast it into mode-wise traditional high-dimensional vector/fiber factor models so as to deploy the conventional estimation of principle components analysis (PCA). Demonstrated by the Tucker tensor factor model (TuTFaM), which is induced from most popular Tucker decomposition, we summarize that estimations on signal components are essentially mode-wise PCA techniques, and the involvement of projection and iteration will enhance the signal-to-noise ratio to various extend. We establish the inferential theory of the proposed estimations and conduct rich simulation experiments under TuTFaM, and illustrate how the proposed estimations can work in tensor reconstruction, clustering for video and economic datasets, respectively.

翻译：高维、高阶张量数据在包括计算机视觉和网络分析在内的多个领域中日益凸显其重要性。基于张量分解或因子分解的含噪版本衍生的张量因子模型，是研究相互依赖或独立的张量变量集合的有效工具。然而，针对各种低秩结构（通常扮演张量因子模型信号角色）估计的统计推断理论仍处于早期发展阶段。本文从张量矩阵化出发，旨在“解码”高阶张量因子模型的估计问题，即通过将其重新表述为模-wise的传统高维向量/纤维因子模型，从而应用传统主成分分析（PCA）估计方法。以最流行的Tucker分解衍生的Tucker张量因子模型（TuTFaM）为例，我们总结出信号成分的估计本质上是模-wise的PCA技术，而投影与迭代的引入会在不同程度上增强信噪比。我们建立了所提出估计的推断理论，并在TuTFaM框架下进行了丰富的模拟实验，同时展示了所提出估计方法在张量重建、视频聚类及经济数据集分析中的实际应用。