Manifold lifting: scaling MCMC to the vanishing noise regime

Understand

Standard Markov chain Monte Carlo methods struggle to explore distributions that are concentrated in the neighbourhood of low-dimensional structures.

• These pathologies naturally occur in a number of situations.
• For example, they are common to Bayesian inverse problem modelling and Bayesian neural networks, when observational data are highly informative, or when a subset of the statistical parameters of interest are non-identifiable.
• In this paper, we propose a strategy that transforms the original sampling problem into the task of exploring a distribution supported on a manifold embedded in a higher dimensional space; in contrast to the original posterior this lifted distribution remains diffuse in the vanishing noise limit.