  We consider the problem of inferring the conditional independence graph (CIG)
of a sparse, high-dimensional stationary multivariate Gaussian time series. A
sparse-group lasso-based frequency-domain formulation of the problem based on
frequency-domain sufficient statistic for the observed time series is
presented. We investigate an alternating direction method of multipliers (ADMM)
approach for optimization of the sparse-group lasso penalized log-likelihood.
We provide sufficient conditions for convergence in the Frobenius norm of the
inverse PSD estimators to the true value, jointly across all frequencies, where
the number of frequencies are allowed to increase with sample size. This
results also yields a rate of convergence. We also empirically investigate
selection of the tuning parameters based on Bayesian information criterion, and
illustrate our approach using numerical examples utilizing both synthetic and
real data.
