基于非光滑约束的Langevin算法

Langevin Algorithms Based on Non-Smooth Constraints

摘要: 贝叶斯推断求解反问题越来越流行，得益于它对解不确定性的诠释。在解决实际问题中，算法的计算时间受到一定的关注，而基于Langevin方程的算法利用了一阶导数信息，具有采样效率高的优点，在采样算法中被广泛使用。对于含有急剧跳跃先验信息的图像问题，贝叶斯框架下先验的选取尤为重要，而TV正则化方法等非光滑约束可以很好刻画先验信息。本文回顾了非光滑约束与高斯先验相结合的混合先验，同时重述了近端Langevin算法，并在此基础上提出了基于非光滑约束的对偶Langevin算法。最后应用于CT成像问题，数值结果表明，我们提出算法是有效的，能够更好地利用非光滑约束刻画解。

Abstract: Bayesian inference for solving inverse problems is becoming increasingly popular, thanks to its in-terpretation of solution uncertainty. In solving practical problems, the computational time of algo-rithms has received certain attention, and algorithms based on Langevin equation utilize first-order derivative information, which has the advantage of high sampling efficiency and is widely used in sampling algorithms. For image problems with sharp jump prior information, the selection of prior information in the Bayesian framework is particularly important, and non-smooth constraints such as TV regularization methods can effectively characterize prior information. This article reviews the hybrid prior combining non-smooth constraints with Gaussian prior, while reiterating the proximal Langevin algorithm, and proposes a dual Langevin algorithm based on non-smooth constraints. Fi-nally, it is applied to the CT imaging problem, and numerical results show that our proposed algo-rithm is effective and can better utilize non-smooth constraints to characterize the solution.