  Deep generative models are universal tools for learning data distributions on
high dimensional data spaces via a mapping to lower dimensional latent spaces.
We provide a study of latent space geometries and extend and build upon
previous results on Riemannian metrics. We show how a class of heuristic
measures gives more flexibility in finding meaningful, problem-specific
distances, and how it can be applied to diverse generator types such as
autoregressive generators commonly used in e.g. language and other sequence
modeling. We further demonstrate how a diffusion-inspired transformation
previously studied in cartography can be used to smooth out latent spaces,
stretching them according to a chosen measure. In addition to providing more
meaningful distances directly in latent space, this also provides a unique tool
for novel kinds of data visualizations. We believe that the proposed methods
can be a valuable tool for studying the structure of latent spaces and learned
data distributions of generative models.
