# Algebraic Properties of Wyner Common Information Solution under Graphical Constraints

## TL;DR

This paper investigates the algebraic structure of the CMDFA solution space for Gaussian vectors with a star-structured covariance matrix, revealing conditions for rank-one or rank-(n-1) solutions under graphical constraints.

## Contribution

It characterizes the solution space of CMDFA with star topology constraints, providing necessary and sufficient conditions for different rank solutions.

## Key findings

- Solution is either rank one or rank n-1 under certain conditions.
- Conditions are expressed in terms of edge weights of the star graph.
- Results specify solutions for the case with n-1 latent factors.

## 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. Further results are given in regards to the solution to the CMDFA with $n-1$ latent factors.