Generalized power method for sparse principal component analysis

TL;DR

This paper introduces a new optimization-based approach for sparse PCA, transforming nonconvex problems into convex maximization tasks, and demonstrates superior performance over existing methods in accuracy and speed.

Contribution

It develops a novel convex reformulation of sparse PCA problems and proposes an efficient gradient method with strong convergence properties.

Findings

Outperforms existing algorithms in solution quality

Achieves faster computational speed

Effective on random and gene expression datasets

Abstract

In this paper we develop a new approach to sparse principal component analysis (sparse PCA). We propose two single-unit and two block optimization formulations of the sparse PCA problem, aimed at extracting a single sparse dominant principal component of a data matrix, or more components at once, respectively. While the initial formulations involve nonconvex functions, and are therefore computationally intractable, we rewrite them into the form of an optimization program involving maximization of a convex function on a compact set. The dimension of the search space is decreased enormously if the data matrix has many more columns (variables) than rows. We then propose and analyze a simple gradient method suited for the task. It appears that our algorithm has best convergence properties in the case when either the objective function or the feasible set are strongly convex, which is the case with our single-unit formulations and can be enforced in the block case. Finally, we demonstrate numerically on a set of random and gene expression test problems that our approach outperforms existing algorithms both in quality of the obtained solution and in computational speed.