Title: Efficient Bayesian Computation Using Plug-and-Play Priors for Poisson Inverse Problems Show Chinese title

Title: 基于即插即用先验的泊松逆问题高效贝叶斯计算

Abstract: This paper introduces a novel plug-and-play (PnP) Langevin sampling methodology for Bayesian inference in low-photon Poisson imaging problems, a challenging class of problems with significant applications in astronomy, medicine, and biology. PnP Langevin sampling algorithms offer a powerful framework for Bayesian image restoration, enabling accurate point estimation as well as advanced inference tasks, including uncertainty quantification and visualization analyses, and empirical Bayesian inference for automatic model parameter tuning. However, existing PnP Langevin algorithms are not well-suited for low-photon Poisson imaging due to high solution uncertainty and poor regularity properties, such as exploding gradients and non-negativity constraints. To address these challenges, we propose two strategies for extending Langevin PnP sampling to Poisson imaging models: (i) an accelerated PnP Langevin method that incorporates boundary reflections and a Poisson likelihood approximation and (ii) a mirror sampling algorithm that leverages a Riemannian geometry to handle the constraints and the poor regularity of the likelihood without approximations. The effectiveness of these approaches is demonstrated through extensive numerical experiments and comparisons with state-of-the-art methods. Show Chinese abstract

Abstract: 本文介绍了一种新颖的即插即用（PnP）兰格朗日采样方法，用于低光子泊松成像问题中的贝叶斯推断，这是一类在天文学、医学和生物学中具有重要意义的挑战性问题。 PnP兰格朗日采样算法为贝叶斯图像恢复提供了一个强大的框架，能够实现准确的点估计以及高级推断任务，包括不确定性量化与可视化分析，以及自动模型参数调节的经验贝叶斯推断。 然而，由于高解不确定性及不良正则性属性（如梯度爆炸和非负性约束），现有的PnP兰格朗日算法并不适合低光子泊松成像。 为了解决这些挑战，我们提出了两种扩展兰格朗日PnP采样到泊松成像模型的策略：(i) 一种结合边界反射和泊松似然近似的加速PnP兰格朗日方法；(ii) 一种利用黎曼几何处理约束和似然不良正则性的镜像采样算法。 通过广泛的数值实验和与最先进方法的比较，验证了这些方法的有效性。