  Precision and Recall are two prominent metrics of generative performance,
which were proposed to separately measure the fidelity and diversity of
generative models. Given their central role in comparing and improving
generative models, understanding their limitations are crucially important. To
that end, in this work, we identify a critical flaw in the common approximation
of these metrics using k-nearest-neighbors, namely, that the very
interpretations of fidelity and diversity that are assigned to Precision and
Recall can fail in high dimensions, resulting in very misleading conclusions.
Specifically, we empirically and theoretically show that as the number of
dimensions grows, two model distributions with supports at equal point-wise
distance from the support of the real distribution, can have vastly different
Precision and Recall regardless of their respective distributions, hence an
emergent asymmetry in high dimensions. Based on our theoretical insights, we
then provide simple yet effective modifications to these metrics to construct
symmetric metrics regardless of the number of dimensions. Finally, we provide
experiments on real-world datasets to illustrate that the identified flaw is
not merely a pathological case, and that our proposed metrics are effective in
alleviating its impact.
