Bregman geometry-aware split Gibbs sampling for Bayesian Poisson inverse problems

TL;DR

This paper introduces a geometry-aware Bayesian sampling method for Poisson inverse problems, effectively handling non-Euclidean geometry and positivity constraints to improve reconstruction quality.

Contribution

It develops a novel Gibbs sampling framework incorporating Bregman divergence-based data augmentation and Riemannian Langevin Monte Carlo for efficient, geometry-preserving inference.

Findings

Achieves competitive reconstruction quality in denoising, deblurring, and PET tasks.

Handles positivity constraints effectively through Riemannian Langevin dynamics.

Provides explicit Gibbs sampling steps for most conditionals, enhancing computational efficiency.

Abstract

This paper proposes a novel Bayesian framework for solving Poisson inverse problems by devising a Monte Carlo sampling algorithm which accounts for the underlying non-Euclidean geometry. To address the challenges posed by the Poisson likelihood -- such as non-Lipschitz gradients and positivity constraints -- we derive a Bayesian model which leverages exact and asymptotically exact data augmentations. In particular, the augmented model incorporates two sets of splitting variables both derived through a Bregman divergence based on the Burg entropy. Interestingly the resulting augmented posterior distribution is characterized by conditional distributions which benefit from natural conjugacy properties and preserve the intrinsic geometry of the latent and splitting variables. This allows for efficient sampling via Gibbs steps, which can be performed explicitly for all conditionals, except the one incorporating the regularization potential. For this latter, we resort to a Hessian Riemannian Langevin Monte Carlo (HRLMC) algorithm which is well suited to handle priors with explicit or easily computable score functions. By operating on a mirror manifold, this Langevin step ensures that the sampling satisfies the positivity constraints and more accurately reflects the underlying problem structure. Performance results obtained on denoising, deblurring, and positron emission tomography (PET) experiments demonstrate that the method achieves competitive performance in terms of reconstruction quality compared to optimization- and sampling-based approaches.