Motivated by Tucker tensor decomposition, this paper imposes low-rank structures to the column and row spaces of coefficient matrices in a multivariate infinite-order vector autoregression (VAR), which leads to a supervised factor model with two factor modelings being conducted to responses and predictors simultaneously. Interestingly, the stationarity condition implies an intrinsic weak group sparsity mechanism of infinite-order VAR, and hence a rank-constrained group Lasso estimation is considered for high-dimensional linear time series. Its non-asymptotic properties are discussed thoughtfully by balancing the estimation, approximation and truncation errors. Moreover, an alternating gradient descent algorithm with thresholding is designed to search for high-dimensional estimates, and its theoretical justifications, including statistical and convergence analysis, are also provided. Theoretical and computational properties of the proposed methodology are verified by simulation experiments, and the advantages over existing methods are demonstrated by two real examples.

翻译：受Tucker张量分解启发，本文在多变量无穷阶向量自回归（VAR）的系数矩阵列空间和行空间中施加低秩结构，从而构建了一个监督因子模型，该模型同时对响应变量和预测变量进行两种因子建模。有趣的是，平稳性条件隐含了无穷阶VAR的内在弱群稀疏性机制，因此针对高维线性时间序列，本文提出了一种秩约束的群Lasso估计方法。通过平衡估计误差、逼近误差和截断误差，本文深入讨论了该方法的非渐近性质。此外，我们设计了一种带阈值的交替梯度下降算法来搜索高维估计量，并提供了其理论论证，包括统计分析和收敛性分析。通过仿真实验验证了所提方法在理论和计算上的性质，并通过两个真实案例展示了其相较于现有方法的优势。