The Gaussian Latent Machine: Efficient Prior and Posterior Sampling for Inverse Problems

TL;DR

This paper introduces the Gaussian latent machine, a novel latent variable model that unifies and improves sampling methods for Bayesian imaging priors and posteriors, demonstrating high efficiency and effectiveness.

Contribution

It proposes a new Gaussian latent machine model that generalizes existing sampling algorithms, enabling efficient two-block Gibbs sampling and direct sampling in Bayesian imaging.

Findings

The approach is highly efficient across various Bayesian imaging problems.

It unifies multiple existing sampling algorithms into a single framework.

Numerical experiments confirm the method's effectiveness and efficiency.

Abstract

We consider the problem of sampling from a product-of-experts-type model that encompasses many standard prior and posterior distributions commonly found in Bayesian imaging. We show that this model can be easily lifted into a novel latent variable model, which we refer to as a Gaussian latent machine. This leads to a general sampling approach that unifies and generalizes many existing sampling algorithms in the literature. Most notably, it yields a highly efficient and effective two-block Gibbs sampling approach in the general case, while also specializing to direct sampling algorithms in particular cases. Finally, we present detailed numerical experiments that demonstrate the efficiency and effectiveness of our proposed sampling approach across a wide range of prior and posterior sampling problems from Bayesian imaging.