Thresholded Graphical Lasso Adjusts for Latent Variables

Abstract

Structural learning of Gaussian graphical models in the presence of latent variables has long been a challenging problem. Chandrasekaran et al. (2012) proposed a convex program to estimate a sparse graph plus low-rank term that adjusts for latent variables; but, this approach poses challenges from both a computational and statistical perspective. We propose an alternative and incredibly simple solution: apply a hard thresholding operator to existing graph selection methods. Conceptually simple and computationally attractive, we show that thresholding the graphical lasso is graph selection consistent in the presence of latent variables under a simpler minimum edge strength condition and at an improved statistical rate. We also extend results to thresholded neighbourhood selection and CLIME estimators as well. We show that our simple thresholded graph estimators enjoy stronger empirical results than existing approaches for the latent variable graphical model problem and conclude with a neuroscience case study to estimate functional neural connections.