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 distribu- tions is a challenging issue that is encountered in many large data recovery problems. In this context, sampling using Hamil- tonian 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 multidimensional sam- pling but, rather, are 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 probabil- ity distributions having non-differentiable energy functions such as those based on the l1 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 nonsmooth Hamiltonian Monte Carlo method is tested and validated on a number of exper- iments. 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: Sparse sampling; Proximity operator; Hamiltonian; Leapfrog; Bayesian methods; MCMC

Domaines: Géométrie algorithmique [cs.CG]; Traitement du signal et de l'image [eess.SP]

Lotfi Chaari, Jean-Yves Tourneret, Caroline Chaux, Hadj Batatia. A hamiltonian Monte Carlo method for non-smooth energy sampling. IEEE Transactions on Signal Processing, 2016, vol. 64 (n° 21), pp. 5585-5594. ⟨10.1109/TSP.2016.2585120⟩. ⟨hal-01376544⟩