Radon-Sobolev Variational Auto-Encoders

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.