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.

翻译：张量因子模型是高阶大维张量时间序列中极具吸引力的降维工具，在经济学、金融学和医学影像领域具有广泛应用。本文提出了一种基于Tucker分解的张量因子模型的投影估计量，并给出了其最小二乘解释，该解释与主成分分析在向量因子模型中的最小二乘解释相呼应。投影技术同时降低了信号分量维度和特异分量张量的量级，从而提升了信噪比。我们推导出载荷和公共因子张量的投影估计量的收敛速度，该速度优于基于朴素主成分分析的估计量。本文的结果在允许特异分量存在弱交叉相关和自相关的温和条件下成立。此外，我们提出了一种基于特征值比准则的迭代方法来确定因子数量。通过大量数值实验，本文系统评估了所提投影估计量相较于现有最优估计量的实证表现。