Robust sparse covariance estimation by thresholding Tyler’s M-estimator
针对重尾或含异常值的数据，提出基于阈值化泰勒M估计量的稀疏形状矩阵估计方法，理论证明在维度和样本量同阶增长时达到极小极大最优速率，模拟实验支持理论结果。
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.