We address the problem of robust sparse estimation of the precision matrix for heavy-tailed distributions in high-dimensional settings. In such high-dimensional contexts, we observe that the covariance matrix can be approximated by a spatial-sign covariance matrix, scaled by a constant. Based on this insight, we introduce two new procedures, the Spatial-Sign Constrained $l_1$ Inverse Matrix Estimation (SCLIME) and the Spatial-sign Graphic LASSO Estimation (SGLASSO), to estimate the precision matrix. Under mild regularity conditions, we establish that the consistency rate of these estimators matches that of existing estimators from the literature. To demonstrate its practical utility, we apply the proposed estimator to two classical problems: the elliptical graphical model and linear discriminant analysis. Through extensive simulation studies and real data applications, we show that our estimators outperforms existing methods, particularly in the presence of heavy-tailed distributions.

翻译：本文针对高维重尾分布下的鲁棒稀疏精度矩阵估计问题展开研究。在高维背景下，我们观察到协方差矩阵可由空间符号协方差矩阵经常数缩放近似。基于这一发现，我们提出了两种新方法——空间符号约束$l_1$逆矩阵估计（SCLIME）与空间符号图LASSO估计（SGLASSO）——用于估计精度矩阵。在温和的正则性条件下，我们证明了这些估计量的一致性速率与现有文献中的估计量相当。为展示其实际应用价值，我们将所提出的估计量应用于两个经典问题：椭圆图模型与线性判别分析。通过大量仿真研究与实际数据应用，我们证明所提出的估计方法优于现有方法，尤其在处理重尾分布时表现更为突出。