Marnissi Yosra, Chouzenoux Emilie, Benazza-Benyahia Amel, Pesquet Jean-Christophe
SAFRAN TECH, Groupe Safran, 78772 Magny-les-Hameaux, France.
Laboratoire Informatique Gaspard Monge (LIGM)-UMR 8049 CNRS, University Paris-East, 93162 Noisy-le-Grand, France.
Entropy (Basel). 2018 Feb 7;20(2):110. doi: 10.3390/e20020110.
In this paper, we are interested in Bayesian inverse problems where either the data fidelity term or the prior distribution is Gaussian or driven from a hierarchical Gaussian model. Generally, Markov chain Monte Carlo (MCMC) algorithms allow us to generate sets of samples that are employed to infer some relevant parameters of the underlying distributions. However, when the parameter space is high-dimensional, the performance of stochastic sampling algorithms is very sensitive to existing dependencies between parameters. In particular, this problem arises when one aims to sample from a high-dimensional Gaussian distribution whose covariance matrix does not present a simple structure. Another challenge is the design of Metropolis-Hastings proposals that make use of information about the local geometry of the target density in order to speed up the convergence and improve mixing properties in the parameter space, while not being too computationally expensive. These two contexts are mainly related to the presence of two heterogeneous sources of dependencies stemming either from the prior or the likelihood in the sense that the related covariance matrices cannot be diagonalized in the same basis. In this work, we address these two issues. Our contribution consists of adding auxiliary variables to the model in order to dissociate the two sources of dependencies. In the new augmented space, only one source of correlation remains directly related to the target parameters, the other sources of correlations being captured by the auxiliary variables. Experiments are conducted on two practical image restoration problems-namely the recovery of multichannel blurred images embedded in Gaussian noise and the recovery of signal corrupted by a mixed Gaussian noise. Experimental results indicate that adding the proposed auxiliary variables makes the sampling problem simpler since the new conditional distribution no longer contains highly heterogeneous correlations. Thus, the computational cost of each iteration of the Gibbs sampler is significantly reduced while ensuring good mixing properties.
在本文中，我们关注贝叶斯逆问题，其中数据保真项或先验分布是高斯分布，或者由分层高斯模型驱动。一般来说，马尔可夫链蒙特卡罗（MCMC）算法使我们能够生成样本集，用于推断基础分布的一些相关参数。然而，当参数空间是高维时，随机抽样算法的性能对参数之间存在的依赖性非常敏感。特别是，当目标是从协方差矩阵不具有简单结构的高维高斯分布中抽样时，就会出现这个问题。另一个挑战是设计Metropolis-Hastings提议，该提议利用目标密度的局部几何信息来加速收敛并改善参数空间中的混合特性，同时计算成本不会过高。这两种情况主要与两种不同的依赖源的存在有关，这两种依赖源要么源于先验，要么源于似然，即相关的协方差矩阵在同一基下不能对角化。在这项工作中，我们解决这两个问题。我们的贡献包括向模型中添加辅助变量，以分离这两种依赖源。在新的扩充空间中，只有一种相关源直接与目标参数相关，其他相关源由辅助变量捕获。我们针对两个实际的图像恢复问题进行了实验，即嵌入高斯噪声中的多通道模糊图像的恢复以及被混合高斯噪声损坏的信号的恢复。实验结果表明，添加所提出的辅助变量使抽样问题变得更简单，因为新的条件分布不再包含高度不同的相关性。因此，在确保良好混合特性的同时，吉布斯采样器每次迭代的计算成本显著降低。
