Locally associated graphical models
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The notion of multivariate total positivity has proved to be useful in finance and psychology but may be too restrictive in other applications. In this paper we propose a concept of local association, where highly connected components in a graphical model are positively associated and study its properties. Our main motivation comes from gene expression data, where graphical models have become a popular exploratory tool. We focus the exposition on Gaussian distributions but our methods readily extend to non-paranormal distributions. Motivated by a convex optimization problem that arises in this context, we develop a GOLAZO approach that generalizes a number of optimization procedures that arise in the context of graphical models (e.g. the GLASSO) and propose a simple block-coordinate descent optimization procedure for solving the dual problem. Our results on existence of the optimum for such problems are of separate interest.
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