Latent Factor Analysis of Gaussian Distributions under Graphical Constraints

• URL: http://arxiv.org/abs/2001.02712v2
• Date: Sat, 11 Jan 2020 05:13:24 GMT
• Title: Latent Factor Analysis of Gaussian Distributions under Graphical Constraints
• Authors: Md Mahmudul Hasan, Shuangqing Wei, Ali Moharrer
• Abstract summary: We show that CMTFA can have either a rank $ 1 $ or a rank $ n-1 $ solution and nothing in between. In particular, we have shown that CMTFA can have either a rank $ 1 $ or a rank $ n-1 $ solution and nothing in between.
• Score: 5.575141499952048
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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. To support our analytical findings we have presented some numerical demonstrating the usefulness of the contributions of our work.

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