PCA consistency in high dimension, low sample size context

TL;DR

This paper studies the behavior of PCA in high-dimensional, low-sample-size settings, establishing conditions for consistency and inconsistency of principal component directions as dimension grows.

Contribution

It provides a comprehensive analysis of PCA consistency in HDLSS contexts, including conditions for convergence and inconsistency of PC directions.

Findings

Large leading eigenvalues ensure PC direction consistency.

Most other PC directions are strongly inconsistent.

Geometric representation of HDLSS data holds under broad conditions.

Abstract

Principal Component Analysis (PCA) is an important tool of dimension reduction especially when the dimension (or the number of variables) is very high. Asymptotic studies where the sample size is fixed, and the dimension grows [i.e., High Dimension, Low Sample Size (HDLSS)] are becoming increasingly relevant. We investigate the asymptotic behavior of the Principal Component (PC) directions. HDLSS asymptotics are used to study consistency, strong inconsistency and subspace consistency. We show that if the first few eigenvalues of a population covariance matrix are large enough compared to the others, then the corresponding estimated PC directions are consistent or converge to the appropriate subspace (subspace consistency) and most other PC directions are strongly inconsistent. Broad sets of sufficient conditions for each of these cases are specified and the main theorem gives a catalogue of possible combinations. In preparation for these results, we show that the geometric representation of HDLSS data holds under general conditions, which includes a $\rho$-mixing condition and a broad range of sphericity measures of the covariance matrix.