The Annals of Statistics

Finite sample approximation results for principal component analysis: A matrix perturbation approach

Boaz Nadler

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Principal component analysis (PCA) is a standard tool for dimensional reduction of a set of n observations (samples), each with p variables. In this paper, using a matrix perturbation approach, we study the nonasymptotic relation between the eigenvalues and eigenvectors of PCA computed on a finite sample of size n, and those of the limiting population PCA as n→∞. As in machine learning, we present a finite sample theorem which holds with high probability for the closeness between the leading eigenvalue and eigenvector of sample PCA and population PCA under a spiked covariance model. In addition, we also consider the relation between finite sample PCA and the asymptotic results in the joint limit p, n→∞, with p/n=c. We present a matrix perturbation view of the “phase transition phenomenon,” and a simple linear-algebra based derivation of the eigenvalue and eigenvector overlap in this asymptotic limit. Moreover, our analysis also applies for finite p, n where we show that although there is no sharp phase transition as in the infinite case, either as a function of noise level or as a function of sample size n, the eigenvector of sample PCA may exhibit a sharp “loss of tracking,” suddenly losing its relation to the (true) eigenvector of the population PCA matrix. This occurs due to a crossover between the eigenvalue due to the signal and the largest eigenvalue due to noise, whose eigenvector points in a random direction.

Article information

Ann. Statist. Volume 36, Number 6 (2008), 2791-2817.

First available in Project Euclid: 5 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H25: Factor analysis and principal components; correspondence analysis 62E17: Approximations to distributions (nonasymptotic)
Secondary: 15A42: Inequalities involving eigenvalues and eigenvectors

Principal component analysis spiked covariance model random matrix theory matrix perturbation phase transition


Nadler, Boaz. Finite sample approximation results for principal component analysis: A matrix perturbation approach. Ann. Statist. 36 (2008), no. 6, 2791--2817. doi:10.1214/08-AOS618.

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