  Clustering aims to divide a set of points into groups. The current paradigm
assumes that the grouping is well-defined (unique) given the probability model
from which the data is drawn. Yet, recent experiments have uncovered several
high-dimensional datasets that form different binary groupings after projecting
the data to randomly chosen one-dimensional subspaces. This paper describes a
probability model for the data that could explain this phenomenon. It is a
simple model to serve as a proof of concept for understanding the geometry of
high-dimensional data. We start by building a rescaled multivariate Bernouilli
model (stretched hypercube) so to create several overlapping grouping
structures in the data. The size of each scaling parameter is related to the
likelihood of uncovering the corresponding grouping by random 1D projection.
Clusters in the original space are then created by adding noise to this
cluster-free model. In high dimension, these clusters would hardly be
observable given a sample set from the distribution because of the curse of
dimensionality, but the binary groupings are clear. Our construction makes it
clear that one needs to make a distinction between "groupings" and "clusters"
in the original space. It also highlights the need to interpret any clustering
found in projected data as merely one among potentially many other groupings in
a dataset.
