  We advocate the use of a notion of entropy that reflects the relative
abundances of the symbols in an alphabet, as well as the similarities between
them. This concept was originally introduced in theoretical ecology to study
the diversity of ecosystems. Based on this notion of entropy, we introduce
geometry-aware counterparts for several concepts and theorems in information
theory. Notably, our proposed divergence exhibits performance on par with
state-of-the-art methods based on the Wasserstein distance, but enjoys a
closed-form expression that can be computed efficiently. We demonstrate the
versatility of our method via experiments on a broad range of domains: training
generative models, computing image barycenters, approximating empirical
measures and counting modes.
