Expectation-maximization for sparse and non-negative PCA

Abstract

We study the problem of finding the dominant eigenvector of the sample covariance matrix, under additional constraints on the vector: a cardinality constraint limits the number of non-zero elements, and non-negativity forces the elements to have equal sign. This problem is known as sparse and non-negative principal component analysis (PCA), and has many applications including dimensionality reduction and feature selection. Based on expectation-maximization for probabilistic PCA, we present an algorithm for any combination of these constraints. Its complexity is at most quadratic in the number of dimensions of the data. We demonstrate significant improvements in performance and computational efficiency compared to other constrained PCA algorithms, on large data sets from biology and computer vision. Finally, we show the usefulness of non-negative sparse PCA for unsupervised feature selection in a gene clustering task.