Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Institutions: Université de Tours, Université d'Orléans, Centre National de la Recherche Scientifique, Institut Denis Poisson, Université Le Havre Normandie, Laboratoire d'Informatique, du Traitement de l'Information et des Systèmes, Université de Rouen Normandie, Institut National des Sciences Appliquées Rouen Normandie, Université Toulouse III - Paul Sabatier, Institut National Polytechnique de Toulouse, Université Toulouse-I-Capitole, Institut de Recherche en Informatique de Toulouse, Université Toulouse - Jean Jaurès, Institut Polytechnique de Bordeaux, École Nationale Supérieure d'Électrotechnique, d'Électronique, d'Informatique, d'Hydraulique et des Télécommunications, Commissariat à l'Énergie Atomique et aux Énergies Alternatives, Université Paris-Saclay, CEA Paris-Saclay - Etablissement de Saclay
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Agence Nationale de la Recherche
Awards: AI.iO Artificial Intelligence in Orléans: Learning from heterogeneous data and expert knowledge. Applications in geological and environmental sciences, BAyesian nonparametrics, Complex models and Kernels, Uncertainty quantification and deeP methods, G5296751580, Artificial and Natural Intelligence Toulouse Institute, G7080048889
OpenAlex ID: W7106045563
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