Distributed covariance estimation in Gaussian graphical models

We consider distributed covariance estimation in Gaussian graphical models. A typical motivation is learning the potential functions for inference via belief propagation in large scale networks. The classical approach based on a centralized maximum likelihood principle is infeasible, and suboptimal distributed alternatives which tradeoff performance with communication costs are required. We begin with a natural solution where each node performs independent estimation of its local covariance with its neighbors. We show that these local solutions are consistent, and can be interpreted as a pseudo-likelihood method. Based on this interpretation, we propose to enhance the performance by introducing additional symmetry constraints. We enforce these using the methodology of the Alternating Direction Method of Multipliers. This results in a flexible message passing protocol between neighboring nodes which can be implemented in large scale networks.