Robust sparse covariance estimation by thresholding Tyler's m-estimator

Estimating a high-dimensional sparse covariance matrix from a limited number of samples is a fundamental task in contemporary data analysis. Most proposals to date, however, are not robust to outliers or heavy tails. Toward bridging this gap, in this work we consider estimating a sparse shape matrix from n samples following a possibly heavy-tailed elliptical distribution. We propose estimators based on thresholding either Tyler's M-estimator or its regularized variant. We prove that in the joint limit as the dimension p and the sample size n tend to infinity with p/n → γ > 0, our estimators are minimax rate optimal. Results on simulated data support our theoretical analysis.