Gradient Flows: Applications to Classification, Image Denoising, and Riemannian MCMC

A pressing question in Bayesian statistics and machine learning is to introduce a unified theoretical framework that brings together some of the many statistical models and algorithmic methodologies employed by practitioners. In this paper we suggest that the variational formulation of the Bayesian update, coupled with the theory of gradient flows, provides both an overarching structure and a powerful tool for the analysis of many such models and algorithms. As particular instances of our general framework, we provide three variational formulations of the Bayesian update with three associated gradient flows that converge to the posterior. We highlight the key unifying role of the concept of geodesic convexity, as it determines ---in all three instances--- the rate of convergence of the flows to the posterior. These gradient flows naturally suggest stochastic processes to be used to build proposals for Markov chain Monte Carlo (MCMC) algorithms. Moreover, by construction the processes are guaranteed to satisfy certain optimality criteria. A core part of the paper is to explore three areas of application: the variational formulation of high-dimensional classification and Bayesian image denoising, and the optimal choice of metric in Riemannian MCMC methods.