  Tensor time series, which is a time series consisting of tensorial
observations, has become ubiquitous. It typically exhibits high dimensionality.
One approach for dimension reduction is to use a factor model structure, in a
form similar to Tucker tensor decomposition, except that the time dimension is
treated as a dynamic process with a time dependent structure. In this paper we
introduce two approaches to estimate such a tensor factor model by using
iterative orthogonal projections of the original tensor time series. These
approaches extend the existing estimation procedures and improve the estimation
accuracy and convergence rate significantly as proven in our theoretical
investigation. Our algorithms are similar to the higher order orthogonal
projection method for tensor decomposition, but with significant differences
due to the need to unfold tensors in the iterations and the use of
autocorrelation. Consequently, our analysis is significantly different from the
existing ones. Computational and statistical lower bounds are derived to prove
the optimality of the sample size requirement and convergence rate for the
proposed methods. Simulation study is conducted to further illustrate the
statistical properties of these estimators.
