Mixed Poisson diffusion models and application to posterior sampling
Standard diffusion models rely on additive Gaussian noise, yet many imaging modalities---including PET and astronomical photon counting---produce discrete Poisson observations. This paper introduces Mixed Poisson Diffusion Models (MPDMs), a generative framework in which the forward corruption is driven by multivariate counting processes rather than Brownian motion. The core payoff is a family of posterior samplers structurally matched to Poisson inverse problems, featuring exact sampling for Poisson denoising and principled adaptations of DPS, PnP-DM, and DAPS.
The paper develops a mathematically rigorous Poissonian counterpart to Gaussian diffusion, complete with early-stopping and discretization error bounds. The identification of the mixed Poisson compensator with a learnable denoiser is elegant, and the exact posterior sampler for Poisson denoising is a genuine structural advance. That said, the empirical gains over Gaussian baselines on realistic imaging tasks are incremental rather than decisive, and the evaluation leans heavily on reconstruction metrics that do not assess true posterior fidelity.
The theoretical construction is the paper's strongest asset. Theorem 2 cleanly identifies the Poisson denoiser $K^\pi_t$ with the stochastic intensity of the forward noising process, providing a simulation-free learning objective via Bregman divergence minimization. Theorem 3 then yields an exact posterior sampler for Poisson denoising---a direct analogue of the closed-form Gaussian case that requires no retraining or approximation. The counting-process formulation also admits an exact thinning sampler, avoiding the discretization bias inherent in SDE-based diffusion models.
First, the theoretical exactness of posterior sampling collapses outside the simple denoising case. For general forward operators, the MPDM adaptations of DPS, PnP-DM and DAPS rely on heuristic approximations of discrete derivatives and ad-hoc projections onto the positive orthant, reintroducing the very modeling arbitrariness the Poisson framework is meant to eliminate. The authors concede that 'non-negativity of the denoiser is not necessarily guaranteed with this approximation: in practice, we project it to the positive orthant' (Section 3).\n\nSecond, the empirical evaluation conflates reconstruction quality with posterior sampling quality. Aside from a low-dimensional toy Gamma mixture, the main experiments report only PSNR, SSIM and LPIPS averaged over validation images---metrics that measure point-estimate fidelity rather than distributional matching. The authors note that 'We provide a complementary evaluation of posterior sampling quality in Appendix E.4' (Section 4), yet this assessment remains peripheral to the paper's central quantitative claims.
The evidence supports the claim that MPDMs are viable generative models and that PDPS is exact for Poisson denoising, but the comparison to Gaussian diffusion on realistic tasks is less conclusive. On BrainWeb and EAGLE, MPDM-based methods are competitive and occasionally lead on LPIPS, yet Gaussian-based DPS and DAPS---using either exact or approximate Poisson likelihoods---frequently achieve comparable or better PSNR and SSIM. The authors train both model classes 'from scratch for each dataset, with identical architectures and training budgets' (Section 4), which is fair, but the chosen Gaussian diffusion formulation is non-standard. The reader is left uncertain whether the gains stem from the Poisson noising itself or from implementation details such as the Riemannian Langevin steps used only in the MPDM variants.
The paper includes detailed pseudocode for all algorithms and reports training hyperparameters in the appendices, which aids reproducibility. However, no public code repository is mentioned, and critical hyperparameters---notably the likelihood weight $\eta$ and Langevin step sizes---are tuned empirically per experiment without systematic sensitivity analysis. Furthermore, the exact thinning sampler has uncertain wall-clock cost, and the discretized Euler scheme used in the imaging experiments introduces an unquantified bias that is not reflected in the reported metrics.
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.
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