  Recent work on interpretability has focused on concept-based explanations,
where deep learning models are explained in terms of high-level units of
information, referred to as concepts. Concept learning models, however, have
been shown to be prone to encoding impurities in their representations, failing
to fully capture meaningful features of their inputs. While concept learning
lacks metrics to measure such phenomena, the field of disentanglement learning
has explored the related notion of underlying factors of variation in the data,
with plenty of metrics to measure the purity of such factors. In this paper, we
show that such metrics are not appropriate for concept learning and propose
novel metrics for evaluating the purity of concept representations in both
approaches. We show the advantage of these metrics over existing ones and
demonstrate their utility in evaluating the robustness of concept
representations and interventions performed on them. In addition, we show their
utility for benchmarking state-of-the-art methods from both families and find
that, contrary to common assumptions, supervision alone may not be sufficient
for pure concept representations.
