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

TL;DR

This paper provides finite sample bounds and a matrix perturbation perspective on PCA eigenvalues and eigenvectors, analyzing their relation to population PCA and phase transition phenomena in high-dimensional settings.

Contribution

It introduces a nonasymptotic, high-probability theorem for sample PCA eigenvalues and eigenvectors under a spiked covariance model, and offers a matrix perturbation view of phase transitions.

Findings

Finite sample bounds for PCA eigenvalues and eigenvectors.

Analysis of phase transition and eigenvector loss in high-dimensional PCA.

Eigenvector stability depends on noise level and sample size.

Abstract

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 $n\to\infty$. 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 $p,n\to\infty$, 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.