  In this work, we propose to study the global geometrical properties of
generative models. We introduce a new Riemannian metric to assess the
similarity between any two data points. Importantly, our metric is agnostic to
the parametrization of the generative model and requires only the evaluation of
its data likelihood. Moreover, the metric leads to the conceptual definition of
generative distances and generative geodesics, whose computation can be done
efficiently in the data space. Their approximations are proven to converge to
their true values under mild conditions. We showcase three proof-of-concept
applications of this global metric, including clustering, data visualization,
and data interpolation, thus providing new tools to support the geometrical
understanding of generative models.
