Wasserstein Information Geometry

Optimal transport (Wasserstein metric) nowadays play important roles in data science. In this talk, we brief review its development and applications in machine learning. In particular, we will focus its induced differential structure. We will introduce the Wasserstein natural gradient in parametric models. The metric tensor in probability density space is pulled back to the one on parameter space. We derive the Wasserstein gradient flows and proximal operator in parameter space. We demonstrate that the Wasserstein natural gradient works efficiently in several statistical machine learning problems, including Boltzmann machine, generative adversary models (GANs) and variational Bayesian statistics.