  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.
