Latent Factor Analysis of Gaussian Distributions under Graphical Constraints

In this paper, we explored the algebraic structures of solution spaces for Gaussian latent factor analysis when the population covariance matrix Σx is generated due to a latent Gaussian star graph. In particular, we found sufficient and necessary conditions under which the solutions to constrained minimum trace factor analysis (CMTFA) and constrained minimum determinant factor analysis (CMDFA) is still star. The later one (CMDFA) is also the problem of Wyner’s common information, which has been under extensive study in recent years. In addition, we further showed that the solution to CMTFA under the star constraint can only have two cases, i.e. the number of latent variable can be only one (star) or n− 1, where n is the dimension of the observable vector.