Learning Sparse High-Dimensional Matrix-Valued Graphical Models From Dependent Data
This work addresses a gap in graphical modeling for dependent matrix data, which is incremental as it extends existing i.i.d. methods to time series contexts.
The paper tackles the problem of inferring conditional independence graphs for high-dimensional matrix-variate Gaussian time series with dependent data, developing a sparse-group lasso method that achieves local convergence with a specified rate in the Frobenius norm.
We consider the problem of inferring the conditional independence graph (CIG) of a sparse, high-dimensional, stationary matrix-variate Gaussian time series. All past work on high-dimensional matrix graphical models assumes that independent and identically distributed (i.i.d.) observations of the matrix-variate are available. Here we allow dependent observations. We consider a sparse-group lasso-based frequency-domain formulation of the problem with a Kronecker-decomposable power spectral density (PSD), and solve it via an alternating direction method of multipliers (ADMM) approach. The problem is bi-convex which is solved via flip-flop optimization. We provide sufficient conditions for local convergence in the Frobenius norm of the inverse PSD estimators to the true value. This result also yields a rate of convergence. We illustrate our approach using numerical examples utilizing both synthetic and real data.