Sparse Principal Component Analysis: a Least Squares approximation approach

TL;DR

This paper introduces a novel Sparse PCA method that maintains PCA's optimality by incorporating a sparsity constraint, using branch-and-bound and iterative algorithms for effective sparse component extraction.

Contribution

It presents a new approach to Sparse PCA that preserves PCA's original optimality and introduces algorithms that do not require pre-specifying sparsity levels or component count.

Findings

The proposed method effectively identifies sparse principal components.

Compared to existing methods, it offers competitive or superior performance.

The algorithms are versatile and applicable to real datasets.

Abstract

Sparse Principal Components Analysis aims to find principal components with few non-zero loadings. We derive such sparse solutions by adding a genuine sparsity requirement to the original Principal Components Analysis (PCA) objective function. This approach differs from others because it preserves PCA's original optimality: \uns\ of the components and Least Squares approximation of the data. To identify the best subset of non-zero loadings we propose a Branch-and-Bound search and an iterative elimination algorithm. This last algorithm finds sparse solutions with large loadings and can be run without specifying the cardinality of the loadings and the number of components to compute in advance. We give thorough comparisons with the existing Sparse PCA methods and several examples on real datasets.