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Paper Title Algebraic Properties of Wyner Common Information Solution under Graphical Constraints
Paper IdentifierTU1.R2.4
Authors Md Mahmudul Hasan, Shuangqing Wei, Ali Moharrer, Louisiana State University, United States
Session Information Theory and Statistics I
Location Saint Germain, Level 3
Session Time Tuesday, 09 July, 09:50 - 11:10
Presentation Time Tuesday, 09 July, 10:50 - 11:10
Manuscript  Click here to download the manuscript
Abstract The Constrained Minimum Determinant Factor Analysis (CMDFA) setting was motivated by Wyner's common information problem where we seek a latent representation of a given Gaussian vector distribution with the minimum mutual information under certain generative constraints. In this paper, we explore the algebraic structures of the solution space of the CMDFA, when the underlying covariance matrix $\Sigma_x$ has an additional latent graphical constraint, namely, a latent star topology. In particular, sufficient and necessary conditions in terms of the relationships between edge weights of the star graph have been found. Under such conditions and constraints, we have shown that the CMDFA problem has either a rank one solution or a rank $n-1$ solution where $n$ is the dimension of the observable vector. Numerical results are provided to demonstrate the difference between the optimal mutual information and that derived under a naive star constraint.