Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie-猫眼课题宝

DOI：

10.1137/21m1406349

发表时间：

2021-03

期刊：

ArXiv

影响因子：

通讯作者：

中科院分区：

其他

文献类型：

作者：

R. Laumont;Valentin De Bortoli;Andrés Almansa;J. Delon;Alain Durmus;M. Pereyra

自Venkatakrishnan等人的开创性工作以来。[89]在2013年，即插即用（PAPK）方法在贝叶斯成像中无处不在。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来获得用于成像中的逆问题的最小均方误差（MMSE）或最大后验（MAP）估计。在文献中提出的Pestrian算法主要不同的迭代方案，他们使用的优化或采样。在优化方案的情况下，最近的一些作品保证收敛到一个固定点，虽然不一定是MAP估计。在抽样方案的情况下，据我们所知，没有已知的收敛性证明。还有一些重要的悬而未决的问题，关于基础的贝叶斯模型和估计量是否定义良好，适定性，并具有支持这些数值方案所需的基本规律性。为了解决这些局限性，本文开发的理论，方法和可证明收敛的算法进行贝叶斯推理与Pennsylvania先验。我们介绍两种算法：1）PnPULA（Plug & Play Unadjusted Langevin Algorithm，即插即用未调整朗之万算法），用于蒙特卡罗采样和MMSE推断;以及2）PnP-SGD（Plug & Play Stochastic Gradient Descent，即插即用随机梯度下降），用于MAP推断。利用最近关于马尔可夫链定量收敛的结果，我们在对所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，特别注意基于深度神经网络的去噪器。我们还表明，这些算法近似目标的决策理论上最优的贝叶斯模型，是适定的。所提出的算法演示了几个典型的问题，如图像去模糊，修复和去噪，在那里他们被用于点估计以及不确定性可视化和量化。该数据库得到了EPSRC赠款EP/R 034710/1的部分支持。RL得到了伊勒-德-法国大区的赠款。AD感谢拉格朗日数学和计算研究中心的支持。《蒙特利尔议定书》得到了EPSRC赠款EP/T007346/1的部分支持。JD和AA感谢法国研究机构通过Postalleap项目（ANR-19-CE 23 -0027-01）提供的支持。这项工作的计算机实验运行在NVIDIA捐赠的Titan Xp GPU上，以及GENCI-IDRIS的HPC资源上（Grant 2020-AD 011011641）。†These authors contributed equally <$Université de Paris，MAP 5 UMR 8145，F-75006 Paris，France §Department of Statistics University of Oxford 24-29 St Giles OX 1 3LB，Oxford United Kingdom <$Institut Universitaire de France（IUF）Centre Borelli，UMR 9010，École Normale Supérière Paris-Saclay巴黎高等师范学校数学与计算机科学学院赫瑞瓦特大学和麦克斯韦数学科学研究所，爱丁堡，英国1 ar X iv：2 10 3。04 71 5v 4 [ st at .M E ] 1 9 M ar 2 02 1

Since the seminal work of Venkatakrishnan et al. [89] 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) PnPULA (Plug & Play Unadjusted Langevin Algorithm) for Monte Carlo sampling and MMSE inference; and 2) PnP-SGD (Plug & Play 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. ∗VDB was partially supported by EPSRC grant EP/R034710/1. RL was partially supported by grants from Région Ile-De-France. AD acknowledges support of the Lagrange Mathematical and Computing Research Center. MP was partially supported by EPSRC grant EP/T007346/1. JD and AA acknowledge support from the French Research Agency through the PostProdLEAP project (ANR-19-CE23-0027-01). Computer experiments for this work ran on a Titan Xp GPU donated by NVIDIA, as well as on HPC resources from GENCI-IDRIS (Grant 2020-AD011011641). †These authors contributed equally ‡Université de Paris, MAP5 UMR 8145, F-75006 Paris, France §Department of Statistics University of Oxford 24-29 St Giles OX1 3LB, Oxford United Kingdom ¶Institut Universitaire de France (IUF) ‖Centre Borelli, UMR 9010, École Normale Supérieure Paris-Saclay ∗∗School of Mathematical and Computer Sciences, Heriot-Watt University & Maxwell Institute for Mathematical Sciences, Edinburgh, United Kingdom 1 ar X iv :2 10 3. 04 71 5v 4 [ st at .M E ] 1 9 M ar 2 02 1

Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie

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