[Paper Review] Sure Screening for Gaussian Graphical Models

Shikai Luo, Rui Song+1 more|arXiv (Cornell University)|Jul 29, 2014

Statistical Methods and InferenceMathematics26 references17 citations

TL;DR

This paper introduces GRASS (Graphical Sure Screening), a computationally efficient method for estimating Gaussian graphical models in high-dimensional settings by thresholding sample correlations in the covariance matrix. It establishes the sure screening property—guaranteeing with high probability that all true edges are included—and enables control of the expected false positive rate through a data-driven threshold.

ABSTRACT

We propose {graphical sure screening}, or GRASS, a very simple and computationally-efficient screening procedure for recovering the structure of a Gaussian graphical model in the high-dimensional setting. The GRASS estimate of the conditional dependence graph is obtained by thresholding the elements of the sample covariance matrix. The proposed approach possesses the sure screening property: with very high probability, the GRASS estimated edge set contains the true edge set. Furthermore, with high probability, the size of the estimated edge set is controlled. We provide a choice of threshold for GRASS that can control the expected false positive rate. We illustrate the performance of GRASS in a simulation study and on a gene expression data set, and show that in practice it performs quite competitively with more complex and computationally-demanding techniques for graph estimation.

Motivation & Objective

• Address the computational infeasibility of existing sparse precision matrix estimation methods when p is in the tens of thousands.
• Develop a screening procedure that maintains statistical reliability while drastically reducing computation time in high-dimensional graphical modeling.
• Provide theoretical guarantees for edge recovery under high-dimensional asymptotics where p grows exponentially with sample size n.
• Enable control of the expected false positive rate in edge selection through a principled threshold choice.
• Bridge the gap between simple marginal screening and complex precision matrix estimation by applying sure screening to unsupervised graphical models.

Proposed method

• GRASS estimates the conditional dependence graph by thresholding the off-diagonal entries of the sample covariance matrix.
• The method selects edges based on the magnitude of sample correlations: |X_a^T X_b / n| > γ_n, where γ_n is a data-driven threshold.
• The threshold γ_n is set as Φ⁻¹(1 - f/(p(p-1)))/√n to control the expected false positive rate at f/[p(p-1)].
• Theoretical analysis shows that under mild assumptions, the estimated edge set contains the true edge set with high probability.
• The approach leverages the fact that the precision matrix column corresponds to regression coefficients, enabling a screening strategy based on marginal correlations.
• The method achieves O(p²) computational complexity, in contrast to O(p³) for traditional methods like graphical lasso.

Experimental results

Research questions

• RQ1 Can a simple, computationally efficient screening procedure recover the true conditional dependence structure of a Gaussian graphical model in high-dimensional settings?
• RQ2 Does thresholding sample correlations in the covariance matrix yield a method with the sure screening property—i.e., with high probability, including all true edges?
• RQ3 Can the expected false positive rate in edge selection be controlled through a data-driven thresholding rule?
• RQ4 How does the performance of GRASS compare to more complex, computationally intensive methods like graphical lasso in high-dimensional scenarios?
• RQ5 What is the theoretical relationship between GRASS and existing sparse precision matrix estimation techniques such as ℓ₁-penalized likelihood?

Key findings

• GRASS achieves the sure screening property: with high probability, the estimated edge set contains the true edge set, even when p grows exponentially with n.
• The method controls the expected false positive rate at f/[p(p-1)] by choosing a threshold γ_n = Φ⁻¹(1 - f/(p(p-1)))/√n.
• The computational cost of GRASS is O(p²), making it scalable to problems with p = 25,000 or more variables.
• In simulations and a gene expression dataset, GRASS performs competitively with more complex methods like graphical lasso.
• Theoretical analysis reveals a surprising connection between GRASS and ℓ₁-penalized likelihood estimation for precision matrices.
• The method is the first to apply sure screening to unsupervised graphical modeling, extending the framework beyond regression settings.