On Sparse High-Dimensional Graphical Model Learning For Dependent Time Series
This addresses the challenge of graphical model learning for dependent time series, which is important for fields like neuroscience and finance, though it appears incremental as it builds on existing sparse-group lasso and ADMM techniques.
The paper tackles the problem of inferring conditional independence graphs for high-dimensional Gaussian time series by proposing a sparse-group lasso method in the frequency domain with ADMM optimization. It provides theoretical convergence guarantees and demonstrates empirical performance on synthetic and real data.
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.