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

Utilitarian Welfare Optimization in the Generalized Vertex Coloring Games: An Implication to Venue Selection in Events Planning

We consider a general class of multi-agent games in networks, namely the generalized vertex coloring games (G-VCGs), inspired by real-life applications of the venue selection problem in events planning. Certain utility responding to the contemporary coloring assignment will be received by each agent under some particular mechanism, who, striving to maximize his own utility, is restricted to local information thus self-organizing when choosing another color. Our focus is on maximizing some utilitarian-looking welfare objective function concerning the cumulative utilities across the network in a decentralized fashion. Firstly, we investigate on a special class of the G-VCGs, namely Identical Preference VCGs (IP-VCGs) which recovers the rudimentary work by \cite{chaudhuri2008network}. We reveal its convergence even under a completely greedy policy and completely synchronous settings, with a stochastic bound on the converging rate provided. Secondly, regarding the general G-VCGs, a greediness-preserved Metropolis-Hasting based policy is proposed for each agent to initiate with the limited information and its optimality under asynchronous settings is proved using theories from the regular perturbed Markov processes. The policy was also empirically witnessed to be robust under independently synchronous settings. Thirdly, in the spirit of ``robust coloring'', we include an expected loss term in our objective function to balance between the utilities and robustness. An optimal coloring for this robust welfare optimization would be derived through a second-stage MH-policy driven algorithm. Simulation experiments are given to showcase the efficiency of our proposed strategy.Comment: 35 Page

Probabilistic modeling and inference for sequential space-varying blur identification

International audienceThe identification of parameters of spatially variant blurs given a clean image and its blurry noisy version is a challenging inverse problem of interest in many application fields, such as biological microscopy and astronomical imaging. In this paper, we consider a parametric model of the blur and introduce an 1D state-space model to describe the statistical dependence among the neighboring kernels. We apply a Bayesian approach to estimate the posterior distribution of the kernel parameters given the available data. Since this posterior is intractable for most realistic models, we propose to approximate it through a sequential Monte Carlo approach by processing all data in a sequential and efficient manner. Additionally, we propose a new sampling method to alleviate the particle degeneracy problem, which is present in approximate Bayesian filtering, particularly in challenging concentrated posterior distributions. The considered method allows us to process sequentially image patches at a reasonable computational and memory costs. Moreover, the probabilistic approach we adopt in this paper provides uncertainty quantification which is useful for image restoration. The practical experimental results illustrate the improved estimation performance of our novel approach, demonstrating also the benefits of exploiting the spatial structure the parametric blurs in the considered models

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

Stochastic Approximation with Biased MCMC for Expectation Maximization

The expectation maximization (EM) algorithm is a widespread method for empirical Bayesian inference, but its expectation step (E-step) is often intractable. Employing a stochastic approximation scheme with Markov chain Monte Carlo (MCMC) can circumvent this issue, resulting in an algorithm known as MCMC-SAEM. While theoretical guarantees for MCMC-SAEM have previously been established, these results are restricted to the case where asymptotically unbiased MCMC algorithms are used. In practice, MCMC-SAEM is often run with asymptotically biased MCMC, for which the consequences are theoretically less understood. In this work, we fill this gap by analyzing the asymptotics and non-asymptotics of SAEM with biased MCMC steps, particularly the effect of bias. We also provide numerical experiments comparing the Metropolis-adjusted Langevin algorithm (MALA), which is asymptotically unbiased, and the unadjusted Langevin algorithm (ULA), which is asymptotically biased, on synthetic and real datasets. Experimental results show that ULA is more stable with respect to the choice of Langevin stepsize and can sometimes result in faster convergence.Comment: Accepted to AISTATS'2

Rejection Sampling with Vertical Weighted Strips

A number of distributions that arise in statistical applications can be expressed in the form of a weighted density: the product of a base density and a nonnegative weight function. Generating variates from such a distribution may be nontrivial and can involve an intractable normalizing constant. Rejection sampling may be used to generate exact draws, but requires formulation of a suitable proposal distribution. To be practically useful, the proposal must both be convenient to sample from and not reject candidate draws too frequently. A well-known approach to design a proposal involves decomposing the target density into a finite mixture, whose components may correspond to a partition of the support. This work considers such a construction that focuses on majorization of the weight function. This approach may be applicable when assumptions for adaptive rejection sampling and related algorithms are not met. An upper bound for the rejection probability based on this construction can be expressed to evaluate the efficiency of the proposal before sampling. A method to partition the support is considered where regions are bifurcated based on their contribution to the bound. Examples based on the von Mises Fisher distribution and Gaussian Process regression are provided to illustrate the method