Accelerating proximal Markov chain Monte Carlo by using an explicit stabilised method

Abstract

We present a highly efficient proximal Markov chain Monte Carlo methodology to perform Bayesian computation in imaging problems. Similarly to previous proximal Monte Carlo approaches, the proposed method is derived from an approximation of the Langevin diffusion. However, instead of the conventional Euler-Maruyama approximation that underpins existing proximal Monte Carlo methods, here we use a state-of-the-art orthogonal Runge-Kutta-Chebyshev stochastic approximation that combines several gradient evaluations to significantly accelerate its convergence speed, similarly to accelerated gradient optimisation methods. The proposed methodology is demonstrated via a range of numerical experiments, including non-blind image deconvolution, hyperspectral unmixing, and tomographic reconstruction, with total-variation and ${\ell }_1$-type priors. Comparisons with Euler-type proximal Monte Carlo methods confirm that the Markov chains generated with our method exhibit significantly faster convergence speeds, achieve larger effective sample sizes, and produce lower mean square estimation errors at equal computational budget.

arXiv comment28 pages, 13 figures. Accepted for publication in SIAM Journal on Imaging Sciences (SIIMS)

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Cited by in corpus (3)

• Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetry
• Fast Scalable Image Restoration using Total Variation Priors and Expectation Propagation
• Maximum likelihood estimation of regularisation parameters in high-dimensional inverse problems: an empirical Bayesian approach. Part I: Methodology and Experiments