Robust and Sparse PCA for High‐Dimensional Data via Huber Loss and Non‐Convex Regularization

ABSTRACT Principal Component Analysis (PCA) is a popular technique for dimension reduction, but the conventional PCA approach includes many irrelevant variables in the estimates of the principal components (PCs). This makes it difficult to interpret the PCs, especially with a large number of variables. To enhance interpretability, various sparse PCA approaches have been proposed. These approaches mainly impose convex penalties such as the penalty to generate sparse estimates of PCs. However, achieving a sparser solution requires selecting a larger regularization parameter, which increases the bias in the estimation of PC loadings and negatively impacts estimation accuracy. To address this bias issue, this paper extends the sparse PCA method to the case with nonconvex penalties. To simplify the resulting optimization problem, an iterative procedure based on the local linear approximation is developed for estimating the sparse PC loadings. The simulation studies and a real example show that the proposed approach provides more accurate PC estimates than the existing PCA approaches in the presence of rowwise and/or cellwise data contamination.