Theoretical Guarantees for Sparse Principal Component Analysis based on the Elastic Net

Sparse principal component analysis (SPCA) has been widely used for dimensionality reduction and feature extraction in high-dimensional data analysis. Despite there are many methodological and theoretical developments in the past two decades, the theoretical guarantees of the popular SPCA algorithm proposed by Zou, Hastie & Tibshirani (2006) based on the elastic net are still unknown. We aim to close this important theoretical gap in this paper. We first revisit the SPCA algorithm of Zou et al. (2006) and present our implementation. Also, we study a computationally more efficient variant of the SPCA algorithm in Zou et al. (2006) that can be considered as the limiting case of SPCA. We provide the guarantees of convergence to a stationary point for both algorithms. We prove that, under a sparse spiked covariance model, both algorithms can recover the principal subspace consistently under mild regularity conditions. We show that their estimation error bounds match the best available bounds of existing works or the minimax rates up to some logarithmic factors. Moreover, we demonstrate the numerical performance of both algorithms in simulation studies.