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：我们研究了从非对数凹分布（例如高斯混合分布）中近似采样的问题，这类分布由于多峰特性，即使在低维情况下也往往具有挑战性。我们重点探讨通过马尔可夫链蒙特卡洛方法完成此任务，这些方法源自过阻尼朗之万扩散过程的离散化，通常被称为朗之万蒙特卡洛算法。此外，我们还关注两类非光滑情形，针对它们已发展出大量近端MCMC方法：（i）考虑具有高斯混合似然的非光滑先验分布；（ii）拉普拉斯混合分布。此类非光滑且非对数凹的采样任务广泛出现在贝叶斯推断与成像反问题（如图像去卷积）等应用中。我们通过数值模拟比较了最常用的朗之万蒙特卡洛算法的性能。