Algebraic properties of solutions to common information of Gaussian vectors under sparse graphical constraints
We formulate Wyner's common information for random vectors x Rn with joint Gaussian density. We show that finding the common information of Gaussian vectors is equivalent to maximizing a log-determinant of the additive Gaussian noise covariance matrix. We coin such optimization problem as a constrained minimum determinant factor analysis (CMDFA) problem. The convexity of such problem with necessary and sufficient conditions on CMDFA solution is shown. We study the algebraic properties of CMDFA solution space, through which we study two sparse Gaussian graphical models, namely, latent Gaussian stars, and explicit Gaussian chains. Interestingly, we show that depending on pairwise covariance values in a Gaussian graphical structure, one may not always end up with the same parameters and structure for CMDFA solution as those found via graphical learning algorithms. In addition, our results suggest that Gaussian chains have little room left for dimension reduction in terms of the number of latent variables required in their CMDFA solutions.
Publication Source (Journal or Book title)
55th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2017
Moharrer, A., & Wei, S. (2017). Algebraic properties of solutions to common information of Gaussian vectors under sparse graphical constraints. 55th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2017, 2018-January, 1040-1047. https://doi.org/10.1109/ALLERTON.2017.8262852