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

Revisiting maximum-a-posteriori estimation in log-concave models

Maximum-a-posteriori (MAP) estimation is the main Bayesian estimation methodology in imaging sciences, where high dimensionality is often addressed by using Bayesian models that are log-concave and whose posterior mode can be computed efficiently by convex optimisation. Despite its success and wide adoption, MAP estimation is not theoretically well understood yet. The prevalent view in the community is that MAP estimation is not proper Bayesian estimation in a decision-theoretic sense because it does not minimise a meaningful expected loss function (unlike the minimum mean squared error (MMSE) estimator that minimises the mean squared loss). This paper addresses this theoretical gap by presenting a decision-theoretic derivation of MAP estimation in Bayesian models that are log-concave. A main novelty is that our analysis is based on differential geometry, and proceeds as follows. First, we use the underlying convex geometry of the Bayesian model to induce a Riemannian geometry on the parameter space. We then use differential geometry to identify the so-called natural or canonical loss function to perform Bayesian point estimation in that Riemannian manifold. For log-concave models, this canonical loss is the Bregman divergence associated with the negative log posterior density. We then show that the MAP estimator is the only Bayesian estimator that minimises the expected canonical loss, and that the posterior mean or MMSE estimator minimises the dual canonical loss. We also study the question of MAP and MSSE estimation performance in large scales and establish a universal bound on the expected canonical error as a function of dimension, offering new insights into the good performance observed in convex problems. These results provide a new understanding of MAP and MMSE estimation in log-concave settings, and of the multiple roles that convex geometry plays in imaging problems.Comment: Accepted for publication in SIAM Imaging Science

Seismic Tomography Using Variational Inference Methods

Seismic tomography is a methodology to image the interior of solid or fluid media, and is often used to map properties in the subsurface of the Earth. In order to better interpret the resulting images it is important to assess imaging uncertainties. Since tomography is significantly nonlinear, Monte Carlo sampling methods are often used for this purpose, but they are generally computationally intractable for large datasets and high-dimensional parameter spaces. To extend uncertainty analysis to larger systems we use variational inference methods to conduct seismic tomography. In contrast to Monte Carlo sampling, variational methods solve the Bayesian inference problem as an optimization problem, yet still provide probabilistic results. In this study, we applied two variational methods, automatic differential variational inference (ADVI) and Stein variational gradient descent (SVGD), to 2D seismic tomography problems using both synthetic and real data and we compare the results to those from two different Monte Carlo sampling methods. The results show that variational inference methods can produce accurate approximations to the results of Monte Carlo sampling methods at significantly lower computational cost, provided that gradients of parameters with respect to data can be calculated efficiently. We expect that the methods can be applied fruitfully to many other types of geophysical inverse problems.Comment: 26 pages, 14 figure

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

Consistent Inversion of Noisy Non-Abelian X-Ray Transforms

For $M$ a simple surface, the non-linear statistical inverse problem of recovering a matrix field $\Phi: M \to \mathfrak{so}(n)$ from discrete, noisy measurements of the $SO(n)$-valued scattering data $C_\Phi$ of a solution of a matrix ODE is considered ($n\geq 2$). Injectivity of the map $\Phi \mapsto C_\Phi$ was established by [Paternain, Salo, Uhlmann; Geom.Funct.Anal. 2012]. A statistical algorithm for the solution of this inverse problem based on Gaussian process priors is proposed, and it is shown how it can be implemented by infinite-dimensional MCMC methods. It is further shown that as the number $N$ of measurements of point-evaluations of $C_\Phi$ increases, the statistical error in the recovery of $\Phi$ converges to zero in $L^2(M)$-distance at a rate that is algebraic in $1/N$, and approaches $1/\sqrt N$ for smooth matrix fields $\Phi$. The proof relies, among other things, on a new stability estimate for the inverse map $C_\Phi \to \Phi$. Key applications of our results are discussed in the case $n=3$ to polarimetric neutron tomography, see [Desai et al., Nature Sc.Rep. 2018] and [Hilger et al., Nature Comm. 2018

Uncertainty quantification in medical image synthesis

Machine learning approaches to medical image synthesis have shown outstanding performance, but often do not convey uncertainty information. In this chapter, we survey uncertainty quantification methods in medical image synthesis and advocate the use of uncertainty for improving clinicians’ trust in machine learning solutions. First, we describe basic concepts in uncertainty quantification and discuss its potential benefits in downstream applications. We then review computational strategies that facilitate inference, and identify the main technical and clinical challenges. We provide a first comprehensive review to inform how to quantify, communicate and use uncertainty in medical synthesis applications