Majorize-Minimize adapted Metropolis-Hastings algorithm. Application to multichannel image recovery

International audienceOne challenging task in MCMC methods is the choice of the proposal density. It should ideally provide an accurate approximation of the target density with a low computational cost. In this paper, we are interested in Langevin diffusion where the proposal accounts for a directional component. We propose a novel method for tuning the related drift term. This term is preconditioned by an adaptive matrix based on a Majorize-Minimize strategy. This new procedure is shown to exhibit a good performance in a multispectral image restoration example

Proximal Markov chain Monte Carlo algorithms

This paper presents a new Metropolis-adjusted Langevin algorithm (MALA) that uses convex analysis to simulate efficiently from high-dimensional densities that are log-concave, a class of probability distributions that is widely used in modern high-dimensional statistics and data analysis. The method is based on a new first-order approximation for Langevin diffusions that exploits log-concavity to construct Markov chains with favourable convergence properties. This approximation is closely related to Moreau--Yoshida regularisations for convex functions and uses proximity mappings instead of gradient mappings to approximate the continuous-time process. The proposed method complements existing MALA methods in two ways. First, the method is shown to have very robust stability properties and to converge geometrically for many target densities for which other MALA are not geometric, or only if the step size is sufficiently small. Second, the method can be applied to high-dimensional target densities that are not continuously differentiable, a class of distributions that is increasingly used in image processing and machine learning and that is beyond the scope of existing MALA and HMC algorithms. To use this method it is necessary to compute or to approximate efficiently the proximity mappings of the logarithm of the target density. For several popular models, including many Bayesian models used in modern signal and image processing and machine learning, this can be achieved with convex optimisation algorithms and with approximations based on proximal splitting techniques, which can be implemented in parallel. The proposed method is demonstrated on two challenging high-dimensional and non-differentiable models related to image resolution enhancement and low-rank matrix estimation that are not well addressed by existing MCMC methodology

A Survey of Stochastic Simulation and Optimization Methods in Signal Processing

International audienceModern signal processing (SP) methods rely very heavily on probability and statistics to solve challenging SP problems. SP methods are now expected to deal with ever more complex models, requiring ever more sophisticated computational inference techniques. This has driven the development of statistical SP methods based on stochastic simulation and optimization. Stochastic simulation and optimization algorithms are computationally intensive tools for performing statistical inference in models that are anal ytically intractable and beyond the scope of deterministic inference methods. They have been recently successfully applied to many difficult problems involving complex statistical models and sophisticated (often Bayesian) statistical inference techniques. This survey paper offers an introduction to stochastic simulation and optimization methods in signal and image processing. The paper addresses a variety of high-dimensional Markov chain Monte Carlo (MCMC) methods as well as deterministic surrogate methods, such as variational Bayes, the Bethe approach, belief and expectation propagation and approximate message passing algorithms. It also discusses a range of optimization methods that have been adopted to solve stochastic problems, as well as stochastic methods for deterministic optimization. Subsequently, area as of overlap between simulation and optimization, in particular optimization-within-MCMC and MCMC-driven optimization are discussed

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

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.Comment: 28 pages, 13 figures. Accepted for publication in SIAM Journal on Imaging Sciences (SIIMS

Accelerating Bayesian computation in imaging

The dimensionality and ill-posedness often encountered in imaging inverse problems are a challenge for Bayesian computational methods, particularly for state-of-the-art sampling alternatives based on the Euler-Maruyama discretisation of the Langevin diffusion process. In this thesis, we address this difficulty and propose alternatives to accelerate Bayesian computation in imaging inverse problems, focusing on its computational aspects. We introduce, as our first contribution, a highly efficient proximal Markov chain Monte Carlo (MCMC) methodology, based on a state-of-the-art approximation known as the proximal stochastic orthogonal Runge-Kutta-Chebyshev (SK-ROCK) method. It has the advantage of cleverly combining multiple gradient evaluations to significantly speed up convergence, similar to accelerated gradient optimisation techniques. We rigorously demonstrate the acceleration of the Markov chains in the 2-Wasserstein distance for Gaussian models as a function of the condition number κ. In our second contribution, we propose a more sophisticated MCMC sampler, based on the careful integration of two advanced proximal Langevin MCMC methods, SK-ROCK and split Gibbs sampling (SGS), each of which uses a unique approach to accelerate convergence. More precisely, we show how to integrate the proximal SK-ROCK sampler with the model augmentation and relaxation method used by SGS at the level of the Langevin diffusion process, to speed up Bayesian computation at the expense of asymptotic bias. This leads to a new, faster proximal SK-ROCK sampler that combines the accelerated quality of the original sampler with the computational advantages of augmentation and relaxation. Additionally, we propose the augmented and relaxed model to be considered a generalisation of the target model rather than an approximation that situates relaxation in a bias-variance trade-off. As a result, we can carefully calibrate the amount of relaxation to boost both model accuracy (as determined by model evidence) and sampler convergence speed. To achieve this, we derive an empirical Bayesian method that automatically estimates the appropriate level of relaxation via maximum marginal likelihood estimation. The proposed methodologies are demonstrated in several numerical experiments related to image deblurring, hyperspectral unmixing, tomographic reconstruction and inpainting. Comparisons with Euler-type proximal Monte Carlo approaches confirm that the Markov chains generated with our methods exhibit significantly faster convergence speeds, achieve larger effective sample sizes, and produce lower mean square estimation errors with the same computational budget