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This paper introduces Mixed Poisson Diffusion Models (MPDM), a novel mathematical framework designed specifically for discrete data governed by Poisson noise rather than the standard Gaussian processes used in current generative models. By adapting diffusion dynamics to multivariate counting processes, the authors provide new algorithms for posterior sampling in Poisson inverse problems, addressing a significant gap where existing methods fail on discrete observations. This work expands the theoretical boundaries of diffusion models beyond continuous domains, offering a specialized tool for applications involving count data and specific physical inverse problems.
Mixed Poisson diffusion models and application to posterior sampling
arXiv:2610.20218v1 Announce Type: cross Abstract: Diffusion models are a powerful framework for learning complex probability distributions. While widely used diffusion models for continuous data rely on Gaussian noising processes, we instead develop diffusion models based on Poisson noise. More precisely, we introduce and study a new mathematical framework, called Mixed Poisson Diffusion Models (MPDM), with noising dynamics driven by multivariate counting processes. Our methodology is motivated by Poisson inverse problems. The success of Gaussian diffusion models has led to their use as prior models in a growing number of posterior sampling methods for inverse problems. However, most of these approaches focus on Gaussian inverse problems, in which observations are generated through a deterministic forward operator corrupted by additive Gaussian noise. By contrast, Poisson inverse problems, where observations are discrete and Poisson distributed, remain comparatively underexplored in this literature. Building on our Poisson generative dynamics, we develop a natural framework for posterior sampling in Poisson inverse problems and derive several posterior sampling algorithms based on the proposed model. We evaluate our approach on realistic Poisson inverse problems and show that the resulting framework is well suited to this class of problems.
Abstract:Diffusion models are a powerful framework for learning complex probability distributions. While widely used diffusion models for continuous data rely on Gaussian noising processes, we instead develop diffusion models based on Poisson noise. More precisely, we introduce and study a new mathematical framework, called Mixed Poisson Diffusion Models (MPDM), with noising dynamics driven by multivariate counting processes. Our methodology is motivated by Poisson inverse problems. The success of Gaussian diffusion models has led to their use as prior models in a growing number of posterior sampling methods for inverse problems. However, most of these approaches focus on Gaussian inverse problems, in which observations are generated through a deterministic forward operator corrupted by additive Gaussian noise. By contrast, Poisson inverse problems, where observations are discrete and Poisson distributed, remain comparatively underexplored in this literature. Building on our Poisson generative dynamics, we develop a natural framework for posterior sampling in Poisson inverse problems and derive several posterior sampling algorithms based on the proposed model. We evaluate our approach on realistic Poisson inverse problems and show that the resulting framework is well suited to this class of problems.
Machine Learning (stat.ML); Machine Learning (cs.LG); Computation (stat.CO)
arXiv:2610.20218 [stat.ML]
(or arXiv:2610.20218v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2610.20218 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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