Latent Factor Analysis of Gaussian Distributions under Graphical Constraints

In this paper, we explore the algebraic structures of solution spaces for Gaussian latent factor analysis when the population covariance matrix $\Sigma_x$ has an additional latent graphical constraint, namely, a latent star topology. In particular, we give sufficient and necessary conditions under which the solutions to constrained minimum trace factor analysis (CMTFA) is still star. We further show 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, and characterize the solution for both the cases.