Developing efficient Bayesian computation algorithms for imaging inverse problems is challenging due to the dimensionality involved and because Bayesian imaging models are often not smooth. Current state-of-the-art methods often address these difficulties by replacing the posterior density with a smooth approximation that is amenable to efficient exploration by using Langevin Markov chain Monte Carlo (MCMC) methods. An alternative approach is based on data augmentation and relaxation, where auxiliary variables are introduced in order to construct an approximate augmented posterior distribution that is amenable to efficient exploration by Gibbs sampling. This paper proposes a new accelerated proximal MCMC method called latent space SK-ROCK (ls SK-ROCK), which tightly combines the benefits of the two aforementioned strategies. Additionally, instead of viewing the augmented posterior distribution as an approximation of the original model, we propose to consider it as a generalisation of this model. Following on from this, we empirically show that there is a range of values for the relaxation parameter for which the accuracy of the model improves, and propose a stochastic optimisation algorithm to automatically identify the optimal amount of relaxation for a given problem. In this regime, ls SK-ROCK converges faster than competing approaches from the state of the art, and also achieves better accuracy since the underlying augmented Bayesian model has a higher Bayesian evidence. The proposed methodology is demonstrated with a range of numerical experiments related to image deblurring and inpainting, as well as with comparisons with alternative approaches from the state of the art. An open-source implementation of the proposed MCMC methods is available from https://github.com/luisvargasmieles/ls-MCMC.

翻译：针对成像逆问题开发高效的贝叶斯计算算法极具挑战性，这源于问题的高维特性以及贝叶斯成像模型通常非光滑。当前最先进的方法通常通过用光滑近似替代后验密度来应对这些困难，从而利用Langevin马尔可夫链蒙特卡洛（MCMC）方法实现高效探索。另一种替代方案基于数据增广和松弛技术，引入辅助变量以构建近似增广后验分布，使其适用于吉布斯采样的高效探索。本文提出了一种名为潜在空间SK-ROCK（ls SK-ROCK）的新型加速近端MCMC方法，该方法紧密融合了上述两种策略的优势。此外，我们不再将增广后验分布视为原始模型的近似，而是将其看作该模型的泛化形式。基于此，我们通过实验证明存在一个松弛参数区间，在该区间内模型精度得到提升，并提出了随机优化算法以自动识别给定问题的最优松弛量。在该框架下，ls SK-ROCK不仅比现有竞争方法收敛更快，而且由于底层增广贝叶斯模型具有更高的贝叶斯证据，其精度也更高。通过图像去模糊和修复等一系列数值实验，以及与现有先进方法的比较，验证了所提方法的有效性。所提出的MCMC方法开源实现可访问https://github.com/luisvargasmieles/ls-MCMC获取。