High-dimensional Gaussian graphical model selection: walk summability and local separation criterion

Authors:
Animashree Anandkumar;Vincent Y. F. Tan;Furong Huang;Alan S. Willsky
Affiliations:
Electrical Engineering and Computer Science, University of California, Irvine, Irvine, CA;Data Mining Department, Institute for Infocomm Research, Singapore, Electrical and Computer Engineering, National University of Singapore;Electrical Engineering and Computer Science, University of California, Irvine, Irvine, CA;Stochastic Systems Group, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA
Venue:
The Journal of Machine Learning Research
Year:
2012

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Abstract

We consider the problem of high-dimensional Gaussian graphical model selection. We identify a set of graphs for which an efficient estimation algorithm exists, and this algorithm is based on thresholding of empirical conditional covariances. Under a set of transparent conditions, we establish structural consistency (or sparsistency) for the proposed algorithm, when the number of samples n = Ω(Jmin-2 log p), where p is the number of variables and Jmin is the minimum (absolute) edge potential of the graphical model. The sufficient conditions for sparsistency are based on the notion of walk-summability of the model and the presence of sparse local vertex separators in the underlying graph. We also derive novel non-asymptotic necessary conditions on the number of samples required for sparsistency.