High-Dimensional Low-Rank Tensor Autoregressive Time Series Modeling
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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.
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