This research investigates the effectiveness of a non-convex
clustering criterion with the ability to discriminate clusters by
means of quadratic boundaries that take into account cluster
variances. Since no algorithms have been shown to work efficiently and
effectively for this kind of criterion, we introduce and evaluate a
generalized version of the incremental one-by-one clustering algorithm
of MacQueen (1967) that is suitable for general variance-based
criteria, whether convex or not. An experimental evaluation shows that
the criterion performs remarkably well with a variety of heterogeneous
data sets, both synthetic and real-world. Two novel applications of
unsupervised learning to problems in the financial domain are then
developed to test the method further. First, given a portfolio of
investments with potentially hundreds of assets, we pose the problem
of reducing the number of such assets while preserving the risk-return
characteristics of the original portfolio. Next, we tackle the problem
of finding risk-reward opportunities to short-sell securities. For
both problems, we offer novel clustering-based solutions and proceed
to show that the clustering criterion addressed in the present
research is an excellent choice. We conclude this work by arguing that
the financial applications just described exhibit the phenomenon of
volatility clustering, which should be more properly targeted by a
variance-aware criterion such as the one addressed in this work.