  The purpose of this paper is to study the fractal phenomena in large data
sets and the associated questions of dimension reduction. We examine situations
where the classical Principal Component Analysis is not effective in
identifying the salient underlying fractal features of the data set. Instead,
we employ the discrete energy, a technique borrowed from geometric measure
theory, to limit the number of points of a given data set that lie near a
$k$-dimensional hyperplane, or, more generally, near a set of a given upper
Minkowski dimension. Concrete motivations stemming from naturally arising data
sets are described and future directions outlined.
