Spatial Sign based Principal Component Analysis for High Dimensional Data

TL;DR

This paper introduces a robust, computationally efficient sparse PCA method based on spatial signs for high-dimensional data with elliptical distributions, outperforming existing techniques in accuracy and robustness.

Contribution

The paper proposes a novel SSPCA method using spatial signs, achieving optimal convergence rates and robustness in high-dimensional sparse settings.

Findings

SSPCA achieves optimal convergence rates in sparse high-dimensional data.

The method is robust against heavy-tailed distributions.

Simulation and real data validate superior performance.

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.