使用 Plug & Play 先验的贝叶斯成像：当 Langevin 遇到 Tweedie
Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie
AI总结：
本文针对贝叶斯成像中 Plug & Play (PnP) 先验方法缺乏收敛证明的问题，提出了两种可证明收敛的算法 PnP-ULA 和 PnP-SGD，并建立了详细的收敛保证，适用于图像去模糊、修复和去噪等任务。
摘要：
自 Venkatakrishnan 等人 2013 年的开创性工作以来，Plug & Play (PnP) 方法在贝叶斯成像中已变得无处不在。这些方法通过将显式似然函数与由图像去噪算法隐式定义的先验相结合，推导出成像中逆问题的最小均方误差 (MMSE) 或最大后验 (MAP) 估计器。文献中提出的 PnP 算法主要区别在于它们用于优化或采样的迭代方案。在优化方案的情况下，一些近期工作保证了收敛到不动点，尽管不一定是 MAP 估计。在采样方案的情况下，据我们所知，尚无已知的收敛证明。关于底层的贝叶斯模型和估计器是否定义明确、适定，以及是否具备支持这些数值方案所需的基本正则性属性，仍然存在重要的开放性问题。为了解决这些局限性，本文发展了理论、方法和可证明收敛的算法，用于执行带有 PnP 先验的贝叶斯推断。我们引入了两种算法：1) 用于 Monte Carlo 采样和 MMSE 推断的 PnP-ULA (Unadjusted Langevin Algorithm)；2) 用于 MAP 推断的 PnP-SGD (Stochastic Gradient Descent)。利用关于 Markov 链定量收敛的最新结果，我们在对所使用去噪算子的现实假设下，为这两种算法建立了详细的收敛保证，特别关注基于深度神经网络去噪器。我们还表明，这些算法近似地以一个在决策论上最优且适定的贝叶斯模型为目标。所提出的算法在图像去模糊、修复和去噪等几个典型问题上进行了演示，用于点估计以及不确定性的可视化和量化。
英文摘要：
Since the seminal work of Venkatakrishnan et al. in 2013, Plug & Play (PnP) methods have become ubiquitous in Bayesian imaging. These methods derive Minimum Mean Square Error (MMSE) or Maximum A Posteriori (MAP) estimators for inverse problems in imaging by combining an explicit likelihood function with a prior that is implicitly defined by an image denoising algorithm. The PnP algorithms proposed in the literature mainly differ in the iterative schemes they use for optimisation or for sampling. In the case of optimisation schemes, some recent works guarantee the convergence to a fixed point, albeit not necessarily a MAP estimate. In the case of sampling schemes, to the best of our knowledge, there is no known proof of convergence. There also remain important open questions regarding whether the underlying Bayesian models and estimators are well defined, well-posed, and have the basic regularity properties required to support these numerical schemes. To address these limitations, this paper develops theory, methods, and provably convergent algorithms for performing Bayesian inference with PnP priors. We introduce two algorithms: 1) PnP-ULA (Unadjusted Langevin Algorithm) for Monte Carlo sampling and MMSE inference; and 2) PnP-SGD (Stochastic Gradient Descent) for MAP inference. Using recent results on the quantitative convergence of Markov chains, we establish detailed convergence guarantees for these two algorithms under realistic assumptions on the denoising operators used, with special attention to denoisers based on deep neural networks. We also show that these algorithms approximately target a decision-theoretically optimal Bayesian model that is well-posed. The proposed algorithms are demonstrated on several canonical problems such as image deblurring, inpainting, and denoising, where they are used for point estimation as well as for uncertainty visualisation and quantification.
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