Robust Sparse Covariance Estimation by Thresholding Tyler's M-Estimator

Abstract

Estimating a high-dimensional sparse covariance matrix from a limited number of samples is a fundamental problem in contemporary data analysis. Most proposals to date, however, are not robust to outliers or heavy tails. Towards 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 derive bounds on the difference in spectral norm between our estimators and the shape matrix in the joint limit as the dimension $p$ and sample size $n$ tend to infinity with $p/n\to \gamma >0$. These bounds are minimax rate-optimal. Results on simulated data support our theoretical analysis.

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• Covariance regularization by thresholding
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• Regularized $M$-estimators of scatter matrix
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Cited by in corpus (3)

• R-NL: Covariance Matrix Estimation for Elliptical Distributions based on Nonlinear Shrinkage
• Robust self-tuning semiparametric PCA for contaminated elliptical distribution
• Tyler's and Maronna's M-estimators: Non-Asymptotic Concentration Results