Sparse Principal Component Analysis

TLDR

PCA is widely used for dimensionality reduction, yet its components combine all variables, making interpretation difficult. The authors propose sparse principal component analysis (SPCA) that employs lasso/elastic‑net regularization to generate components with sparse loadings. By reformulating PCA as a regression‑type optimization and adding a lasso constraint, they develop efficient algorithms for multivariate and gene‑expression data and derive a new formula for total variance. SPCA yields encouraging results on both simulated and real datasets.

Abstract

Principal component analysis (PCA) is widely used in data processing and dimensionality reduction. However, PCA suffers from the fact that each principal component is a linear combination of all the original variables, thus it is often difficult to interpret the results. We introduce a new method called sparse principal component analysis (SPCA) using the lasso (elastic net) to produce modified principal components with sparse loadings. We first show that PCA can be formulated as a regression-type optimization problem; sparse loadings are then obtained by imposing the lasso (elastic net) constraint on the regression coefficients. Efficient algorithms are proposed to fit our SPCA models for both regular multivariate data and gene expression arrays. We also give a new formula to compute the total variance of modified principal components. As illustrations, SPCA is applied to real and simulated data with encouraging results.