High-Dimensional Gaussian Graphical Model Selection: Walk Summability and Local Separation Criterion
Animashree Anandkumar, Vincent Y.F. Tan, Furong Huang, Alan S. Willsky.
Year: 2012, Volume: 13, Issue: 76, Pages: 2293−2337
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.