Gradient-based Sparse Principal Component Analysis with Extensions to Online Learning

TL;DR

This paper introduces a computationally efficient convex sparse PCA algorithm suitable for large, high-dimensional, and streaming data, with theoretical guarantees and applications in gene expression analysis.

Contribution

It proposes a new convex FPS-based sparse PCA algorithm that is scalable and extends to online learning, with theoretical bounds and practical gene data application.

Findings

Algorithm is computationally efficient for large datasets.

Provides explicit bounds for optimization error and statistical accuracy.

Successfully applied to gene expression data for functional gene group detection.

Abstract

Sparse principal component analysis (PCA) is an important technique for dimensionality reduction of high-dimensional data. However, most existing sparse PCA algorithms are based on non-convex optimization, which provide little guarantee on the global convergence. Sparse PCA algorithms based on a convex formulation, for example the Fantope projection and selection (FPS), overcome this difficulty, but are computationally expensive. In this work we study sparse PCA based on the convex FPS formulation, and propose a new algorithm that is computationally efficient and applicable to large and high-dimensional data sets. Nonasymptotic and explicit bounds are derived for both the optimization error and the statistical accuracy, which can be used for testing and inference problems. We also extend our algorithm to online learning problems, where data are obtained in a streaming fashion. The proposed algorithm is applied to high-dimensional gene expression data for the detection of functional gene groups.