[论文解读] Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie

Rémi Laumont, Valentin De Bortoli+4 more|arXiv (Cornell University)|Mar 8, 2021

Sparse and Compressive Sensing Techniques参考文献 71被引用 9

一句话总结

本文提出PnP-ULA，一种贝叶斯成像框架，结合了即插即用先验（Plug & Play priors）与未调整的朗之万算法（Unadjusted Langevin Algorithm），用于马尔可夫链蒙特卡洛采样和最小均方误差估计。在现实假设下建立了严格的收敛性保证——尤其针对深度神经网络去噪器——表明该方法近似于一个适定的、决策理论最优的贝叶斯模型。

ABSTRACT

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.

研究动机与目标

• 建立一个理论基础坚实的贝叶斯推断框架，用于即插即用先验。
• 解决即插即用方法中蒙特卡洛采样缺乏收敛性证明的问题。
• 证明PnP-ULA近似于一个适定的、决策理论最优的贝叶斯模型。
• 为使用深度神经网络去噪器作为先验时的基于朗之万的采样提供收敛性保证。
• 通过后验采样实现成像逆问题中的不确定性量化。

提出的方法

• 提出PnP-ULA，一种基于伊藤-丸山（Euler-Maruyama）离散化朗之万SDE的马尔可夫链，结合即插即用先验。
• 利用特威迪（Tweedie）恒等式将先验势的梯度与MMSE去噪器关联，避免使用近似化的近端算子或基于梯度的近似。
• 使用去噪器作为朗之万更新中近端算子的代理，从而可使用最先进的深度神经网络去噪器。
• 在较弱条件下（包括得分函数的利普希茨连续性）建立总变差范数下向真实后验收敛的理论保证。
• 使用莫拉伊达-约希达（Moreau-Yosida）包络作为理论桥梁，将PnP-ULA与标准贝叶斯模型联系起来。
• 在图像去模糊与图像修复任务中验证该方法，涵盖点估计与不确定性量化。

实验结果

研究问题

• RQ1 能否为即插即用先验的蒙特卡洛采样提供严格的收敛性保证？
• RQ2 PnP-ULA是否近似于一个从频率学派视角有意义的适定贝叶斯模型？
• RQ3 深度神经网络去噪器的性质如何影响PnP-ULA的收敛性与准确性？
• RQ4 在PnP-ULA下，所得贝叶斯模型的决策理论最优性如何？
• RQ5 如何通过PnP-ULA可靠地量化图像重建中的不确定性？

主要发现

• 在去噪器满足温和且可验证的条件下，PnP-ULA在总变差范数下收敛至真实后验。
• 即使先验由深度神经网络隐式定义，该算法仍被证明近似于一个决策理论最优的贝叶斯模型。
• 在图像去模糊任务中，PnP-ULA在Simpson图像上达到PSNR 30.50与SSIM 0.93，优于PnP-SGD（α=1）的结果。
• 在图像修复任务中，PnP-ULA在Alley图像上达到PSNR 28.98与SSIM 0.80，不确定性图谱在边缘与结构化区域附近集中。
• 通过边际标准差进行不确定性量化显示，图像修复任务的不确定性高于去模糊任务，尤其在较高尺度下，这与退化特性有关。
• 该方法实现了可靠的不确定性可视化与量化，通过在下采样后的样本上计算标准差图，以评估不同空间尺度下的不确定性。