Latent Factor Analysis of Gaussian Distributions under Graphical Constraints
Mahmudul Hasan, Shuangqing Wei, Ali Moharrer · 2020
We explore the algebraic structure of the solution space of convex optimization problem Constrained Minimum Trace Factor Analysis (CMTFA), when the population covariance matrix Σxhas an additional latent graphical constraint, namely, a latent star topology. In particular, we have shown that CMTFA can have either a rank 1 or a rank n - 1 solution and nothing in between. The special case of a rank 1 solution, corresponds to the case where just one latent variable captures all the dependencies among the observables, giving rise to a star topology. We found explicit conditions for both rank 1 and rank n - 1 solutions for CMTFA solution of Σx. As a basic attempt towards building a more general Gaussian tree, we have found a necessary and a sufficient condition for multiple clusters each having rank 1 CMTFA solution to satisfy a minimum probability, to combine together to build a Gaussian tree.