A Hamiltonian Monte Carlo Method for Non-Smooth Energy Sampling

Lotfi Chaari (1, 2) , Jean-Yves Tourneret (3, 2) , Caroline Chaux (4) , Hadj Batatia (1)

1 IRIT-TCI - Traitement et Compréhension d’Images
2 Toulouse INP - Institut National Polytechnique (Toulouse)
3 IRIT-MINDS - CoMputational imagINg anD viSion
4 I2M - Institut de Mathématiques de Marseille

Efficient sampling from high-dimensional distributions is a challenging issue which is encountered in many large data recovery problems. 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 distributions from the exponential family with smooth energy functions. In this paper, we address the problem of using Hamiltonian dynamics to sample from probability distributions having non-differentiable energy functions such as those based on the 1 norm. Such distributions are being used intensively in sparse signal and image recovery applications. The technique studied in this paper uses a modified leapfrog transform involving a proximal step. The resulting non-smooth Hamiltonian Monte Carlo 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.

Mots clés: Bayesian methods; Sparse sampling; MCMC; Hamiltonian; leapfrog; proximity operator

Domaines: Traitement du signal et de l'image [eess.SP]; Imagerie

Lotfi Chaari, Jean-Yves Tourneret, Caroline Chaux, Hadj Batatia. A Hamiltonian Monte Carlo Method for Non-Smooth Energy Sampling. 2016. ⟨hal-01291840⟩