Optimal Latent Transport

It is common to assume that the latent space of a generative model is a lower-dimensional Euclidean space. We instead endow the latent space with a Riemannian structure. Previous work endows this Riemannian structure by pulling back the Euclidean metric of the observation space or the Fisher-Rao metric on the decoder distributions to the latent space. We instead investigate pulling back the Wasserstein metric tensor on the decoder distributions to the latent space. We develop an efficient realization of this metric, and, through proof of concept experiments, demonstrate that the approach is viable.