The Sparse Principal Component Analysis Problem: Optimality Conditions and Algorithms

TL;DR

This paper introduces a new optimality condition for sparse PCA, develops algorithms based on it, and demonstrates that these algorithms outperform existing methods that rely on weaker optimality conditions.

Contribution

It proposes a coordinate-wise optimality condition for sparse PCA and develops algorithms that guarantee convergence to this stronger condition, improving solution quality.

Findings

Algorithms based on the new optimality condition outperform traditional methods.

Numerical experiments support the superiority of convergence to stronger optimality conditions.

The new condition provides better interpretability and solution quality in sparse PCA.

Abstract

Sparse principal component analysis addresses the problem of finding a linear combination of the variables in a given data set with a sparse coefficients vector that maximizes the variability of the data. This model enhances the ability to interpret the principal components, and is applicable in a wide variety of fields including genetics and finance, just to name a few. We suggest a necessary coordinate-wise-based optimality condition, and show its superiority over the stationarity-based condition that is commonly used in the literature, and which is the basis for many of the algorithms designed to solve the problem. We devise algorithms that are based on the new optimality condition, and provide numerical experiments that support our assertion that algorithms, which are guaranteed to converge to stronger optimality conditions, perform better than algorithms that converge to points satisfying weaker optimality conditions.