On Sparse High-Dimensional Graphical Model Learning For Dependent Time Series

TL;DR

This paper introduces a frequency-domain sparse-group lasso approach for inferring the conditional independence graph of high-dimensional Gaussian time series, providing convergence guarantees and empirical validation.

Contribution

It presents a novel frequency-domain formulation and ADMM-based optimization method for sparse graph learning in high-dimensional time series, with theoretical convergence analysis.

Findings

Convergence of inverse PSD estimators to true values under certain conditions

Effective tuning parameter selection using Bayesian information criterion

Successful application to synthetic and real data demonstrating method efficacy

Abstract

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.