Learning Graph Structures with Autoregressive Graph Signal Models

This paper presents a novel approach to graph learning, GL-AR, which leverages estimated autoregressive coefficients to recover undirected graph structures from time-series graph signals with propagation delay. GL-AR can discern graph structures where propagation between vertices is delayed, mirroring the dynamics of many real-world systems. This is achieved by utilizing the autoregressive coefficients of time-series graph signals in GL-AR's learning algorithm. Existing graph learning techniques typically minimize the smoothness of a graph signal on a recovered graph structure to learn instantaneous relationships. GL-AR extends this approach by showing that minimizing smoothness with autoregressive coefficients can additionally recover relationships with propagation delay. The efficacy of GL-AR is demonstrated through applications to both synthetic and real-world datasets. Specifically, this work introduces the Graph-Tensor Method, a novel technique for generating synthetic time-series graph signals that represent edges as transfer functions. This method, along with real-world data from the National Climatic Data Center, is used to evaluate GL-AR's performance in recovering undirected graph structures. Results indicate that GL-AR's use of autoregressive coefficients enables it to outperform state-of-the-art graph learning techniques in scenarios with nonzero propagation delays. Furthermore, GL-AR's performance is optimized by a new automated parameter selection algorithm, which eliminates the need for computationally intensive trial-and-error methods. This paper has supplementary MATLAB code available at https://github.com/kyle-donoghue/GL-AR .