Tucker tensor factor models: matricization and mode-wise PCA estimation

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.