High-dimensional robust precision matrix estimation: Cellwise corruption under $\epsilon$-contamination

TL;DR

This paper studies the high-dimensional robustness of precision matrix estimators under cellwise contamination, providing error bounds and comparing graphical Lasso and CLIME in terms of statistical consistency and breakdown properties.

Contribution

It offers high-dimensional error bounds for robust precision matrix estimators under cellwise contamination, highlighting differences in breakdown properties between graphical Lasso and CLIME.

Findings

Graphical Lasso has superior breakdown properties compared to CLIME.

Error bounds are derived without assumptions on the contaminating distribution.

Estimators remain consistent under a nonvanishing fraction of cellwise contamination.

Abstract

We analyze the statistical consistency of robust estimators for precision matrices in high dimensions. We focus on a contamination mechanism acting cellwise on the data matrix. The estimators we analyze are formed by plugging appropriately chosen robust covariance matrix estimators into the graphical Lasso and CLIME. Such estimators were recently proposed in the robust statistics literature, but only analyzed mathematically from the point of view of the breakdown point. This paper provides complementary high-dimensional error bounds for the precision matrix estimators that reveal the interplay between the dimensionality of the problem and the degree of contamination permitted in the observed distribution. We also show that although the graphical Lasso and CLIME estimators perform equally well from the point of view of statistical consistency, the breakdown property of the graphical Lasso is superior to that of CLIME. We discuss implications of our work for problems involving graphical model estimation when the uncontaminated data follow a multivariate normal distribution, and the goal is to estimate the support of the population-level precision matrix. Our error bounds do not make any assumptions about the the contaminating distribution and allow for a nonvanishing fraction of cellwise contamination.