  The failure of the Euclidean norm to reliably distinguish between nearby and
distant points in high dimensional space is well-known. This phenomenon of
distance concentration manifests in a variety of data distributions, with iid
or correlated features, including centrally-distributed and clustered data.
Unsupervised learning based on Euclidean nearest-neighbors and more general
proximity-oriented data mining tasks like clustering, might therefore be
adversely affected by distance concentration for high-dimensional applications.
While considerable work has been done developing clustering algorithms with
reliable high-dimensional performance, the problem of cluster validation--of
determining the natural number of clusters in a dataset--has not been carefully
examined in high-dimensional problems. In this work we investigate how the
sensitivities of common Euclidean norm-based cluster validity indices scale
with dimension for a variety of synthetic data schemes, including
well-separated and noisy clusters, and find that the overwhelming majority of
indices have improved or stable sensitivity in high dimensions. The curse of
dimensionality is therefore dispelled for this class of fairly generic data
schemes.
