Sparse Generalized Principal Component Analysis for Large-scale Applications beyond Gaussianity

TL;DR

This paper introduces a scalable, sparse generalized PCA method that extends traditional PCA to exponential family distributions, effectively handling missing data and large-scale high-dimensional datasets.

Contribution

It generalizes sparse PCA to exponential family distributions with built-in missing data handling and proposes efficient, scalable algorithms with improved regularization and convergence properties.

Findings

Demonstrates efficiency in high-dimensional simulations

Shows effectiveness on real large-scale datasets

Achieves faster convergence with accelerated gradient

Abstract

Principal Component Analysis (PCA) is a dimension reduction technique. It produces inconsistent estimators when the dimensionality is moderate to high, which is often the problem in modern large-scale applications where algorithm scalability and model interpretability are difficult to achieve, not to mention the prevalence of missing values. While existing sparse PCA methods alleviate inconsistency, they are constrained to the Gaussian assumption of classical PCA and fail to address algorithm scalability issues. We generalize sparse PCA to the broad exponential family distributions under high-dimensional setup, with built-in treatment for missing values. Meanwhile we propose a family of iterative sparse generalized PCA (SG-PCA) algorithms such that despite the non-convexity and non-smoothness of the optimization task, the loss function decreases in every iteration. In terms of ease and intuitive parameter tuning, our sparsity-inducing regularization is far superior to the popular Lasso. Furthermore, to promote overall scalability, accelerated gradient is integrated for fast convergence, while a progressive screening technique gradually squeezes out nuisance dimensions of a large-scale problem for feasible optimization. High-dimensional simulation and real data experiments demonstrate the efficiency and efficacy of SG-PCA.