High-Dimensional Low-Rank Tensor Autoregressive Time Series Modeling

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.