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

TL;DR

This paper introduces an accelerated proximal Markov chain Monte Carlo method using a stabilised Runge-Kutta-Chebyshev approximation, significantly improving convergence speed and efficiency in Bayesian imaging computations.

Contribution

It presents a novel MCMC approach that replaces the Euler-Maruyama approximation with a Runge-Kutta-Chebyshev method, enhancing convergence in Bayesian imaging tasks.

Findings

Faster convergence compared to Euler-type methods

Larger effective sample sizes achieved

Lower mean square estimation errors at same computational cost

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.