  Principal component analysis (PCA) is a popular dimension reduction technique
often used to visualize high-dimensional data structures. In genomics, this can
involve millions of variables, but only tens to hundreds of observations.
Theoretically, such extreme high-dimensionality will cause biased or
inconsistent eigenvector estimates, but in practice the principal component
scores are used for visualization with great success. In this paper, we explore
when and why the classical principal component scores can be used to visualize
structures in high-dimensional data, even when there are few observations
compared to the number of variables. Our argument is two-fold: First, we argue
that eigenvectors related to pervasive signals will have eigenvalues scaling
linearly with the number of variables. Second, we prove that for linearly
increasing eigenvalues, the sample component scores will be scaled and rotated
versions of the population scores, asymptotically. Thus the visual information
of the sample scores will be unchanged, even though the sample eigenvectors are
biased. In the case of pervasive signals, the principal component scores can be
used to visualize the population structures, even in extreme high-dimensional
situations.
