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

Robust Estimators in High-Dimensions Without the Computational Intractability

“…In this work, we study agnostic distribution learning for a number of fundamental classes of distributions: (1) a single Gaussian, (2) a product distribution on the hypercube {0, 1} d , (3) mixtures of two product distributions (under a natural balancedness condition), and (4) mixtures of k Gaussians with spherical covariances. Prior to our work, all known efficient algorithms (e.g., [LT15,BD15]) for these classes required the error guarantee, f (ε, d), to depend polynomially in the dimension d. Hence, previous efficient estimators could only tolerate at most a 1/ poly(d) fraction of errors. In this work, we obtain the first efficient algorithms for the aforementioned problems, where f (ε, d) is completely independent of d and depends polynomially (often, nearly linearly) in the fraction ε of corrupted samples.…”

Robust and sparse Gaussian graphical modelling under cell‐wise contamination

“…For the simpler approach that uses rank correlations, Loh and Tan () were able to give an analysis of the estimation error . Obtaining analogous results for the estimator via γ ‐divergence is an interesting open problem for future work.…”

“…Some other estimators of Σ have been proposed under the cell‐wise contamination. Öllerer and Croux () and Loh and Tan () considered use of rank correlations. Based on the decomposition , Loh and Tan () estimated the scale by MAD and used the Kendall's tau and Spearman's rho to estimate the correlation.…”

“…Öllerer and Croux () and Loh and Tan () considered use of rank correlations. Based on the decomposition , Loh and Tan () estimated the scale by MAD and used the Kendall's tau and Spearman's rho to estimate the correlation. Öllerer and Croux () proposed to use Q n from Rousseeuw and Croux () for the scale and estimate the correlation by the Gaussian rank correlation from Boudt et al ().…”

“…However, while many researchers have considered the traditional “whole‐vector” contamination framework (see, e.g., Maronna et al, , Chapter 6), there are fewer existing methods for cell‐wise contamination. Specifically, we are aware of three approaches, namely, use of alternative t ‐distributions (Finegold & Drton, ; ), use of rank correlations (Öllerer and Croux, ; Loh and Tan, ) and a pairwise covariance estimation method by Tarr et al () who adopt an idea of Gnanadesikan and Kettenring (). In contrast, in this paper, we provide a robust covariance matrix estimator via γ ‐divergence as proposed by Fujisawa and Eguchi ().…”