  Using a modest amount of data from a large population, subgroup discovery
(SGD) identifies outstanding subsets of data with respect to a certain property
of interest of that population. The SGs are described by "rules". These are
constraints on key descriptive parameters that characterize the material or the
environment. These parameters and constraints are obtained by maximizing a
quality function that establishes a tradeoff between SG size and utility, i.e.,
between generality and exceptionality. The utility function measures how
outstanding a SG is. However, this approach does not give a unique solution,
but typically many SGs have similar quality-function values. Here, we identify
coherent collections of SGs of a "Pareto region" presenting various
size-utility tradeoffs and define a SG similarity measure based on the Jaccard
index, which allows us to hierarchically cluster these optimal SGs. These
concepts are demonstrated by learning rules that describe perovskites with high
bulk modulus. We show that SGs focusing on exceptional materials exhibit a high
quality-function value but do not necessarily maximize it. We compare the mean
shift with the cumulative Jensen-Shannon divergence ($D_{sJS}$) as utility
functions and show that the SG rules obtained with $D_{cJS}$ are more focused
than those obtained with the mean shift.
