High-dimensional covariance estimation based on Gaussian graphical models

TL;DR

This paper introduces a novel method combining regression, thresholding, and refitting to estimate high-dimensional covariance matrices and graphical models, achieving consistency and fast convergence under sparsity assumptions.

Contribution

It proposes a new approach that integrates multiple regression, thresholding, and refitting for high-dimensional graphical model estimation, with proven consistency and convergence rates.

Findings

Consistent estimation of graphical structure under sparsity.

Fast convergence rates for covariance and inverse covariance matrices.

Explicit bounds for Kullback-Leibler divergence.

Abstract

Undirected graphs are often used to describe high dimensional distributions. Under sparsity conditions, the graph can be estimated using $\ell_1$-penalization methods. We propose and study the following method. We combine a multiple regression approach with ideas of thresholding and refitting: first we infer a sparse undirected graphical model structure via thresholding of each among many $\ell_1$-norm penalized regression functions; we then estimate the covariance matrix and its inverse using the maximum likelihood estimator. We show that under suitable conditions, this approach yields consistent estimation in terms of graphical structure and fast convergence rates with respect to the operator and Frobenius norm for the covariance matrix and its inverse. We also derive an explicit bound for the Kullback Leibler divergence.