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

TL;DR

This paper introduces an efficient method for high-dimensional Gaussian graphical model selection based on thresholding empirical covariances, with theoretical guarantees under walk-summability and local separation conditions.

Contribution

It proposes a new algorithm for model selection that is both computationally efficient and theoretically sound in high-dimensional settings, leveraging walk-summability and local separators.

Findings

Algorithm achieves structural consistency under specified conditions.

Sample complexity is characterized as n=omega(J_{min}^{-2} log p).

Derived non-asymptotic necessary sample size conditions.

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=omega(J_{min}^{-2} log p), where p is the number of variables and J_{min} 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.