  Tensor Factor Models (TFM) are appealing dimension reduction tools for
high-order large-dimensional tensor time series, and have wide applications in
economics, finance and medical imaging. In this paper, we propose a projection
estimator for the Tucker-decomposition based TFM, and provide its least-square
interpretation which parallels to the least-square interpretation of the
Principal Component Analysis (PCA) for the vector factor model. The projection
technique simultaneously reduces the dimensionality of the signal component and
the magnitudes of the idiosyncratic component tensor, thus leading to an
increase of the signal-to-noise ratio. We derive a convergence rate of the
projection estimator of the loadings and the common factor tensor which are
faster than that of the naive PCA-based estimator. Our results are obtained
under mild conditions which allow the idiosyncratic components to be weakly
cross- and auto- correlated. We also provide a novel iterative procedure based
on the eigenvalue-ratio principle to determine the factor numbers. Extensive
numerical studies are conducted to investigate the empirical performance of the
proposed projection estimators relative to the state-of-the-art ones.
