|Latent Factor Analysis of Gaussian Distributions under Graphical Constraints
|Md Mahmudul Hasan, Shuangqing Wei, Ali Moharrer, Louisiana State University, United States
|L.4: Distribution Learning
|Statistics and Learning Theory
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|We explore the algebraic structure of the solution space of convex optimization problem Constrained Minimum Trace Factor Analysis (CMTFA), when the population covariance matrix $\Sigma_x$ 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 $\Sigma_x$. 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.