Robust sparse covariance estimation by thresholding Tyler’s M-estimator
John Goes, Gilad Lerman, Boaz Nadler
The Annals of Statistics · 2020 · Volume 48 · Issue 1
Abstract
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 $p/n\to\gamma>0$, our estimators are minimax rate optimal. Results on simulated data support our theoretical analysis.
Keywords
62G20, 62G35, 62H12, covariance matrix estimation, elliptical distribution, Sparsity, spectral norm, thresholding, Tyler’s M-estimator
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