  Modern technological advances have enabled an unprecedented amount of
structured data with complex temporal dependence, urging the need for new
methods to efficiently model and forecast high-dimensional tensor-valued time
series. This paper provides the first practical tool to accomplish this task
via autoregression (AR). By considering a low-rank Tucker decomposition for the
transition tensor, the proposed tensor autoregression can flexibly capture the
underlying low-dimensional tensor dynamics, providing both substantial
dimension reduction and meaningful dynamic factor interpretation. For this
model, we introduce both low-dimensional rank-constrained estimator and
high-dimensional regularized estimators, and derive their asymptotic and
non-asymptotic properties. In particular, by leveraging the special balanced
structure of the AR transition tensor, a novel convex regularization approach,
based on the sum of nuclear norms of square matricizations, is proposed to
efficiently encourage low-rankness of the coefficient tensor. A truncation
method is further introduced to consistently select the Tucker ranks.
Simulation experiments and real data analysis demonstrate the advantages of the
proposed approach over various competing ones.
