Stochastic Normalizing Flows for Inverse Problems: a Markov Chains Viewpoint

TL;DR

This paper presents a Markov chain perspective on stochastic normalizing flows, enhancing their expressiveness and applicability to inverse problems by incorporating general Markov kernels and posterior sampling.

Contribution

It introduces a Markov chain framework for stochastic normalizing flows, allowing for distributions without densities and improving inverse problem sampling.

Findings

Successful numerical demonstrations of conditional stochastic normalizing flows.

Generalization to distributions without densities.

Theoretical foundation for posterior sampling in inverse problems.

Abstract

To overcome topological constraints and improve the expressiveness of normalizing flow architectures, Wu, K\"ohler and No\'e introduced stochastic normalizing flows which combine deterministic, learnable flow transformations with stochastic sampling methods. In this paper, we consider stochastic normalizing flows from a Markov chain point of view. In particular, we replace transition densities by general Markov kernels and establish proofs via Radon-Nikodym derivatives which allows to incorporate distributions without densities in a sound way. Further, we generalize the results for sampling from posterior distributions as required in inverse problems. The performance of the proposed conditional stochastic normalizing flow is demonstrated by numerical examples.