A Hamiltonian Monte Carlo Method for Non-Smooth Energy Sampling

Efficient sampling from high-dimensional distributions is a challenging issue which is encountered in many large data recovery problems involving Markov chain Monte Carlo schemes. In this context, sampling using Hamiltonian dynamics is one of the recent techniques that have been proposed to exploit the target distribution geometry. Such schemes have clearly been shown to be efficient for multi-dimensional sampling, but are rather adapted to the exponential families of distributions with smooth energy function. In this paper, we address the problem of using Hamiltonian dynamics to sample from probability distributions having non-differentiable energy functions such as $\ell_1$. Such distributions are being more and more used in sparse signal and image recovery applications. The proposed technique uses a modified leapfrog transform involving a proximal step. The resulting non-smooth Hamiltonian Monte Carlo (ns-HMC) method is tested and validated on a number of experiments. Results show its ability to accurately sample according to various multivariate target distributions. The proposed technique is illustrated on synthetic examples and is applied to an image denoising problem.