Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms
This addresses sampling challenges in Bayesian inference and imaging inverse problems, but appears incremental as it focuses on comparing existing methods.
The paper tackled the problem of approximate sampling from non-log-concave and nonsmooth distributions, such as Gaussian and Laplacian mixtures, using Langevin Monte Carlo algorithms, and found that numerical simulations were used to compare their performance without specifying concrete results.
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.