  Linear projections are widely used in the analysis of high-dimensional data.
In unsupervised settings where the data harbour latent classes/clusters, the
question of whether class discriminatory signals are retained under projection
is crucial. In the case of mean differences between classes, this question has
been well studied. However, in many contemporary applications, notably in
biomedicine, group differences at the level of covariance or graphical model
structure are important. Motivated by such applications, in this paper we ask
whether linear projections can preserve differences in second order structure
between latent groups. We focus on unsupervised projections, which can be
computed without knowledge of class labels. We discuss a simple theoretical
framework to study the behaviour of such projections which we use to inform an
analysis via quasi-exhaustive enumeration. This allows us to consider the
performance, over more than a hundred thousand sets of data-generating
population parameters, of two popular projections, namely random projections
(RP) and Principal Component Analysis (PCA). Across this broad range of
regimes, PCA turns out to be more effective at retaining second order signals
than RP and is often even competitive with supervised projection. We complement
these results with fully empirical experiments showing 0-1 loss using simulated
and real data. We study also the effect of projection dimension, drawing
attention to a bias-variance trade-off in this respect. Our results show that
PCA can indeed be a suitable first-step for unsupervised analysis, including in
cases where differential covariance or graphical model structure are of
interest.
