Robust high-dimensional precision matrix estimation

TL;DR

This paper introduces a robust method for estimating high-dimensional precision matrices that remains effective under data contamination, overcoming limitations of traditional approaches like the graphical lasso.

Contribution

It proposes replacing the sample covariance with an elementwise robust covariance estimator within the graphical lasso, ensuring robustness and computational efficiency in high dimensions.

Findings

The estimator is positive definite and sparse.

It has a high breakdown point under contamination.

The method is computationally fast.

Abstract

The dependency structure of multivariate data can be analyzed using the covariance matrix $\Sigma$. In many fields the precision matrix $\Sigma^{-1}$ is even more informative. As the sample covariance estimator is singular in high-dimensions, it cannot be used to obtain a precision matrix estimator. A popular high-dimensional estimator is the graphical lasso, but it lacks robustness. We consider the high-dimensional independent contamination model. Here, even a small percentage of contaminated cells in the data matrix may lead to a high percentage of contaminated rows. Downweighting entire observations, which is done by traditional robust procedures, would then results in a loss of information. In this paper, we formally prove that replacing the sample covariance matrix in the graphical lasso with an elementwise robust covariance matrix leads to an elementwise robust, sparse precision matrix estimator computable in high-dimensions. Examples of such elementwise robust covariance estimators are given. The final precision matrix estimator is positive definite, has a high breakdown point under elementwise contamination and can be computed fast.