The Graphical Lasso: New Insights and Alternatives
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The graphical lasso FHT2007a is an algorithm for learning the structure in an undirected Gaussian graphical model, using ℓ_1 regularization to control the number of zeros in the precision matrix Θ=Σ^-1 BGA2008,yuan_lin_07. The R package FHT2007a is popular, fast, and allows one to efficiently build a path of models for different values of the tuning parameter. Convergence of can be tricky; the converged precision matrix might not be the inverse of the estimated covariance, and occasionally it fails to converge with warm starts. In this paper we explain this behavior, and propose new algorithms that appear to outperform . By studying the "normal equations" we see that, is solving the dual of the graphical lasso penalized likelihood, by block coordinate ascent; a result which can also be found in BGA2008. In this dual, the target of estimation is Σ, the covariance matrix, rather than the precision matrix Θ. We propose similar primal algorithms and , that also operate by block-coordinate descent, where Θ is the optimization target. We study all of these algorithms, and in particular different approaches to solving their coordinate sub-problems. We conclude that is superior from several points of view.
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