Spatial Sign based Principal Component Analysis for High Dimensional Data

Abstract: This article focuses on the robust principal component analysis (PCA) of high-dimensional data with elliptical distributions. We investigate the PCA of the sample spatial-sign covariance matrix in both nonsparse and sparse contexts, referring to them as SPCA and SSPCA, respectively. We present both nonasymptotic and asymptotic analyses to quantify the theoretical performance of SPCA and SSPCA. In sparse settings, we demonstrate that SSPCA, implemented through a combinatoric program, achieves the optimal rate of convergence. Our proposed SSPCA method is computationally efficient and exhibits robustness against heavy-tailed distributions compared to existing methods. Simulation studies and real-world data applications further validate the superiority of our approach.