Spatial-Sign based Robust Sparse Precision Matrix Estimation and its Applications

We address the problem of robust sparse estimation of the inverse shape 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 l1 Inverse Matrix Estimation (SCLIME) and the Spatial-Sign Graphical Lasso Estimation (SGLASSO), to estimate the inverse shape 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 are competitive with existing methods under normal data and outperform them under heavy-tailed distributions.