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Sparse principal component analysis (SPCA) is widely used for dimensionalityreduction and feature extraction in high-dimensional data analysis. Despitemany methodological and theoretical developments in the past two decades, thetheoretical guarantees of the popular SPCA algorithm proposed by Zou, Hastie &Tibshirani (2006) are still unknown. This paper aims to address this criticalgap. We first revisit the SPCA algorithm of Zou et al. (2006) and present ourimplementation. We also study a computationally more efficient variant of theSPCA algorithm in Zou et al. (2006) that can be considered as the limiting caseof SPCA. We provide the guarantees of convergence to a stationary point forboth algorithms and prove that, under a sparse spiked covariance model, bothalgorithms can recover the principal subspace consistently under mildregularity conditions. We show that their estimation error bounds match thebest available bounds of existing works or the minimax rates up to somelogarithmic factors. Moreover, we demonstrate the competitive numericalperformance of both algorithms in numerical studies.

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