Latent Factor Analysis of Gaussian Distributions under Graphical Constraints

Latent Factor Analysis of Gaussian Distributions under Graphical Constraints Md Mahmudul Hasan, Shuangqing Wei, Ali Moharrer Abstract --We explore the algebraic structure of the solution space of convex optimization problem Constrained Minimum Trace Factor Analysis (CMTF A), when the population covariance matrix Σ x has an additional latent graphical constraint, namely, a latent star topology. In particular, we have shown that CMTF A can have either a rank 1 or a rank n 1 solution and nothing in between. We found explicit conditions for both rank 1 and rank n 1 solutions for CMTF A 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 CMTF A solution, to satisfy a minimum probability to combine together to build a Gaussian tree. T o support our analytical findings we have presented some numerical demonstrating the usefulness of the contributions of our work. Index T erms --Factor Analysis, MTF A, CMTF A, CMDF A I. INTRODUCTION A. Motivation Factor Analysis (FA) is a commonly used tool in multivariate statistics to represent the correlation structure of a set of observables in terms of significantly smaller number of variables called "latent factors". With the growing use in data mining, high dimensional data analytics, factor analysis has already become a prolific area of research [1] [2]. Classical factor analysis models seek to decompose the correlation matrix of an n -dimensional random vector X R n, Σ x, as the sum of a diagonal matrix D and a Gramian matrix Σ x D .