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We consider the problem of covariance matrix estimation in the presence oflatent variables. Under suitable conditions, it is possible to learn themarginal covariance matrix of the observed variables via a tractable convexprogram, where the concentration matrix of the observed variables is decomposedinto a sparse matrix (representing the graphical structure of the observedvariables) and a low rank matrix (representing the marginalization effect oflatent variables). We present an efficient first-order method based on splitBregman to solve the convex problem. The algorithm is guaranteed to convergeunder mild conditions. We show that our algorithm is significantly faster thanthe state-of-the-art algorithm on both artificial and real-world data. Applyingthe algorithm to a gene expression data involving thousands of genes, we showthat most of the correlation between observed variables can be explained byonly a few dozen latent factors.

Efficient Latent Variable Graphical Model Selection via Split Bregman Method