Sparse Probabilistic Principal Component Analysis

Abstract

Principal component analysis (PCA) is a popular dimensionality reduction algorithm. However, it is not easy to interpret which of the original features are important based on the principal components. Recent methods improve interpretability by sparsifying PCA through adding an $L_1$ regularizer. In this paper, we introduce a probabilistic formulation for sparse PCA. By presenting sparse PCA as a probabilistic Bayesian formulation, we gain the benefit of automatic model selection. We examine three different priors for achieving sparsification: (1) a two-level hierarchical prior equivalent to a Laplacian distribution and consequently to an $L_1$ regularization, (2) an inverse-Gaussian prior, and (3) a Jeffrey’s prior. We learn these models by applying variational inference. Our experiments verify that indeed our sparse probabilistic model results in a sparse PCA solution.