Gaussian graphical models for proper quaternion distributions

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 and allow for improved accuracy. We begin by showing that the unconstrained proper quaternion maximum likelihood problem is convex and has a closed form solution that resembles the classical sample covariance. Then, we proceed and add convex sparsity constraints to the inverse covariance matrix and minimize them using convex optimization toolboxes. Finally, we show that in the special case of chordal graphs, the estimates follow a simple closed form which aggregates the unconstrained solutions in each of the cliques. We demonstrate the performance of our suggested estimators on both synthetic and real data.