We study the problem of approximate sampling from non-log-concave distributions, e.g., Gaussian mixtures, which is often challenging even in low dimensions due to their multimodality. We focus on performing this task via Markov chain Monte Carlo (MCMC) methods derived from discretizations of the overdamped Langevin diffusions, which are commonly known as Langevin Monte Carlo algorithms. Furthermore, we are also interested in two nonsmooth cases for which a large class of proximal MCMC methods have been developed: (i) a nonsmooth prior is considered with a Gaussian mixture likelihood; (ii) a Laplacian mixture distribution. Such nonsmooth and non-log-concave sampling tasks arise from a wide range of applications to Bayesian inference and imaging inverse problems such as image deconvolution. We perform numerical simulations to compare the performance of most commonly used Langevin Monte Carlo algorithms.

翻译：我们研究了非对数凹分布（例如高斯混合分布）的近似采样问题，这类分布由于多模态特性即使在低维情况下也常具挑战性。我们聚焦于通过过阻尼Langevin扩散离散化方法（即通常所称的Langevin蒙特卡罗算法）构建的马尔可夫链蒙特卡罗（MCMC）方法完成该任务。此外，我们还关注两类已发展出大量邻近MCMC方法的非光滑情形：（i）具有高斯混合似然的非光滑先验；（ii）拉普拉斯混合分布。此类非光滑且非对数凹的采样问题广泛存在于贝叶斯推断及图像反卷积等成像逆问题应用中。我们通过数值模拟比较了最常用的Langevin蒙特卡罗算法的性能。