We present a novel technique for sparse principal component analysis. This
method, named Eigenvectors from Eigenvalues Sparse Principal Component Analysis
(EESPCA), is based on the recently detailed formula for computing normed,
squared eigenvector loadings of a Hermitian matrix from the eigenvalues of the
full matrix and associated sub-matrices. Relative to the state-of-the-art
LASSO-based sparse PCA method of Witten, Tibshirani and Hastie, the EESPCA
technique offers a two-orders-of-magnitude improvement in computational speed,
does not require estimation of tuning parameters, and can more accurately
identify true zero principal component loadings across a range of data matrix
sizes and covariance structures. Importantly, EESPCA achieves these performance
benefits while maintaining a reconstruction error close to that generated by
the Witten et al. approach. EESPCA is a practical and effective technique for
sparse PCA with particular relevance to computationally demanding problems such
as the analysis of large data matrices or statistical techniques like
resampling that involve the repeated application of sparse PCA