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 decompositions or factorizations, 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 the estimation of various low-rank structures, which are customary to play the role of signals of tensor factor models. In this paper, we attempt to ``decode" the estimation of a higher-order tensor factor model by leveraging tensor matricization. Specifically, we recast it into mode-wise traditional high-dimensional vector/fiber factor models, enabling the deployment of conventional principal components analysis (PCA) for estimation. Demonstrated by the Tucker tensor factor model (TuTFaM), which is induced from the noisy version of the widely-used 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 extent. We establish the inferential theory of the proposed estimators, conduct rich simulation experiments, and illustrate how the proposed estimations can work in tensor reconstruction, and clustering for independent video and dependent economic datasets, respectively.

翻译：高维、高阶张量数据在计算机视觉和网络分析等多个领域日益受到重视。张量因子模型源于张量分解或因子分解的含噪版本，是研究可能相关或独立的一组张量变量对象的自然且有效工具。然而，针对各种低秩结构（通常作为张量因子模型信号的角色）估计的统计推断理论仍处于早期发展阶段。本文通过利用张量矩阵化，试图"解码"高阶张量因子模型的估计问题。具体而言，我们将该估计重新表述为模态维度的传统高维向量/纤维因子模型，从而能够采用传统主成分分析（PCA）进行估计。以广泛使用的Tucker分解含噪版本导出的Tucker张量因子模型（TuTFaM）为例，我们总结得出信号分量的估计本质上属于模态PCA技术，而投影和迭代的参与将在不同程度上增强信噪比。我们建立了所提出估计量的推断理论，进行了丰富的仿真实验，并分别展示了所提估计方法在张量重建以及独立视频数据和依赖型经济数据集聚类中的实际应用。