X-Ray Sobolev Variational Auto-Encoders

The quality of the generative models (Generative adversarial networks, Variational Auto-Encoders, ...) depends heavily on the choice of a good probability distance. However some popular metrics 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, reproducing kernel Hilbert spaces, energy distances).The distances are shown to posses fast implementations andare included in an adapted Variational Auto-Encoder termed X-ray Sobolev Variational Auto-Encoder (XS-VAE) which produces good quality resultson standard generative datasets.