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 a new modeling framework 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. For this model, we first study several nuclear-norm-regularized estimation methods and derive their non-asymptotic properties 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 matricizations 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.

翻译：现代技术进步带来了前所未有的结构化数据，这些数据具有复杂的时间依赖性，迫切需要新方法来高效建模和预测高维张量值时间序列。本文提出了一种通过自回归实现此目标的新建模框架。通过考虑转移张量的低秩Tucker分解，所提出的张量自回归能够灵活捕捉底层低维张量动态特性，既提供了显著的维度约简，又赋予了有意义的多维动态因子解释。针对此模型，我们首先研究了几种基于核范数正则化的估计方法，并推导了它们在近似低秩设置下的非渐近性质。特别地，通过利用转移张量特殊的平衡结构，提出了一种基于平方矩阵化核范数之和的新型凸正则化方法，以高效促进系数张量的低秩性。为进一步在精确低秩条件下提升估计效率，我们提出了一种基于梯度下降算法的非凸估计器，并建立了其计算与统计收敛性保证。通过模拟研究以及对多类别进出口网络张量值时间序列数据的实证分析，验证了所提出方法的优势。