  The quality of generative models (such as Generative adversarial networks and
Variational Auto-Encoders) depends heavily on the choice of a good probability
distance. However some popular metrics like the Wasserstein or the Sliced
Wasserstein distances, the Jensen-Shannon divergence, the Kullback-Leibler
divergence, lack convenient properties such as (geodesic) convexity, fast
evaluation and so on. To address these shortcomings, we introduce a class of
distances that have built-in convexity. We investigate the relationship with
some known paradigms (sliced distances - a synonym for Radon distances -,
reproducing kernel Hilbert spaces, energy distances). The distances are shown
to possess fast implementations and are included in an adapted Variational
Auto-Encoder termed Radon Sobolev Variational Auto-Encoder (RS-VAE) which
produces high quality results on standard generative datasets.
  Keywords: Variational Auto-Encoder; Generative model; Sobolev spaces; Radon
Sobolev Variational Auto-Encoder;
