Principal component analysis for big data

Abstract

Big data is transforming our world, revolutionizing operations and analytics everywhere, from financial engineering to biomedical sciences. The complexity of big data often makes dimension reduction techniques necessary before conducting statistical inference. Principal component analysis, commonly referred to as PCA, has become an essential tool for multivariate data analysis and unsupervised dimension reduction, the goal of which is to find a lower dimensional subspace that captures most of the variation in the dataset. This article provides an overview of methodological and theoretical developments of PCA over the last decade, with focus on its applications to big data analytics. We first review the mathematical formulation of PCA and its theoretical development from the view point of perturbation analysis. We then briefly discuss the relationship between PCA and factor analysis as well as its applications to large covariance estimation and multiple testing. PCA also finds important applications in many modern machine learning problems, and we focus on community detection, ranking, mixture model and manifold learning in this paper.

review article, in press with Wiley StatsRef

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Cited by in corpus (9)

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• Tensor Methods for Additive Index Models under Discordance and Heterogeneity
• Hypothesis testing for eigenspaces of covariance matrix
• Bayesian inference for spectral projectors of the covariance matrix
• Feature Learning and Classification in Neuroimaging: Predicting Cognitive Impairment from Magnetic Resonance Imaging
• Central Limit Theorems for Classical Multidimensional Scaling
• Finite sample Bernstein-von Mises theorems for functionals and spectral projectors of the covariance matrix
• Low-Rank Principal Eigenmatrix Analysis