Latent Factor Analysis of Gaussian Distributions under Graphical Constraints
This work addresses a theoretical problem in factor analysis and graphical models for researchers in statistics and machine learning, but it appears incremental as it builds on existing CMTFA methods with specific constraints.
The paper tackles the problem of analyzing the solution space of Constrained Minimum Trace Factor Analysis (CMTFA) under latent graphical constraints, specifically a star topology, showing that solutions can only have rank 1 or n-1 and providing explicit conditions for these cases, with numerical demonstrations supporting the findings.
We explore the algebraic structure of the solution space of convex optimization problem Constrained Minimum Trace Factor Analysis (CMTFA), when the population covariance matrix $Σ_x$ has 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 $Σ_x$. 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. To support our analytical findings we have presented some numerical demonstrating the usefulness of the contributions of our work.