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High-Dimensional Gaussian Graphical Model Selection: Walk Summability and Local Separation Criterion

  • Author(s): Anandkumar, Animashree;
  • Tan, Vincent Y.F.;
  • Huang, Furong;
  • Willsky, Alan S.
  • et al.

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(J−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.

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