Stat Colloquium [Virtual]: Dr. Yao Zheng

Title: High-dimensional low-rank tensor autoregressive time series modeling

Abstract: 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. A new modelling framework is provided to accomplish this task via autoregression (AR). By considering a low-rank Tucker decomposition for the transition tensor, the proposed tensor AR can flexibly capture the underlying low-dimensional tensor dynamics, providing both substantial dimension reduction and meaningful multi-dimensional dynamic factor interpretations. Several nuclear-norm-regularized estimation methods are first studied and their non-asymptotic properties are derived under the approximate low-rank setting. In particular, by leveraging the special balanced structure of the transition tensor, a novel convex regularization approach based on the sum of nuclear norms of square matriculation is proposed to efficiently encourage low rankness of the coefficient tensor. To further improve the estimation efficiency under exact low-rankness, a non-convex estimator is proposed with a gradient descent algorithm, and its computational and statistical convergence guarantees are established. Simulation studies and an empirical analysis of tensor-valued time series data from multi-category import-export networks demonstrate the advantages of the proposed approach.