Linear Convergence of Gradient Methods for Estimating Structured Transition Matrices in High-dimensional Vector Autoregressive Models

In this paper, we present non-asymptotic optimization guarantees of gradient descent methods for estimating structured transition matrices in high-dimensional vector autoregressive (VAR) models. We adopt the projected gradient descent (PGD) for single-structured transition matrices and the alternating projected gradient descent (AltPGD) for superposition-structured ones. Our analysis demonstrates that both gradient algorithms converge linearly to the statistical error even though the strong convexity of the objective function is absent under the high-dimensional settings. Moreover our result is sharp (up to a constant factor) in the sense of matching the phase transition theory of the corresponding model with independent samples. To the best of our knowledge, this analysis constitutes first non-asymptotic optimization guarantees of the linear rate for regularized estimation in high-dimensional VAR models. Numerical results are provided to support our theoretical analysis.