Proper Quaternion Gaussian Graphical Models

In this paper, we extend Gaussian graphical models to proper quaternion Gaussian distributions. The properness assumption reduces the number of unknowns by a factor of four while graphical models reduce the number of degrees of freedom via sparsity. Each of the methods allows accurate estimation using a small number of samples. To enjoy both gains, we show that the proper quaternion Gaussian inverse covariance estimation problem is convex and has a closed form solution. We proceed to demonstrate that the additional sparsity constraints on the inverse covariance matrix also lead to a convex problem, and the optimizations can be efficiently solved by standard numerical methods. In the special but practical case of a chordal graph, we provide a closed form solution. We demonstrate the improved performance of our suggested estimators on both synthetic and real data.