Efficient Bayesian Computation by Proximal Markov Chain Monte Carlo: When Langevin Meets Moreau

DOI10.1137/16M1108340zbMath1401.65016arXiv1612.07471OpenAlexW2566924527MaRDI QIDQ4686924

Alain Durmus, Marcelo Pereyra, Eric Moulines

Publication date: 10 October 2018

Published in: SIAM Journal on Imaging Sciences (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1612.07471

zbMATH Keywords

convex optimizationmodel selectioninverse problemsBayesian inferenceMarkov chain Monte Carlo methodsuncertainty quantificationmathematical imaging

Mathematics Subject Classification ID

Related Items (32)

Asymptotically Exact Data Augmentation: Models, Properties, and Algorithms ⋮ Unnamed Item ⋮ Patch-Based Image Restoration Using Expectation Propagation ⋮ Bayesian Imaging Using Plug & Play Priors: When Langevin Meets Tweedie ⋮ Bayesian Imaging with Data-Driven Priors Encoded by Neural Networks ⋮ PAC-Bayesian risk bounds for group-analysis sparse regression by exponential weighting ⋮ Gradient-based adaptive importance samplers ⋮ Efficient Bayes inference in neural networks through adaptive importance sampling ⋮ Smoothing unadjusted Langevin algorithms for nonsmooth composite potential functions ⋮ Gradient-Based Markov Chain Monte Carlo for Bayesian Inference With Non-differentiable Priors ⋮ Bayesian Inference and Uncertainty Quantification for Medical Image Reconstruction with Poisson Data ⋮ The Split Gibbs Sampler Revisited: Improvements to Its Algorithmic Structure and Augmented Target Distribution ⋮ Self-Supervised Deep Learning for Image Reconstruction: A Langevin Monte Carlo Approach ⋮ A strongly monotonic polygonal Euler scheme ⋮ On maximum a posteriori estimation with Plug \& Play priors and stochastic gradient descent ⋮ Efficient Bayesian Computation for Low-Photon Imaging Problems ⋮ The forward-backward envelope for sampling with the overdamped Langevin algorithm ⋮ Bayesian Trend Filtering via Proximal Markov Chain Monte Carlo ⋮ Weak approximation of transformed stochastic gradient MCMC ⋮ Accelerating Proximal Markov Chain Monte Carlo by Using an Explicit Stabilized Method ⋮ Maximum Likelihood Estimation of Regularization Parameters in High-Dimensional Inverse Problems: An Empirical Bayesian Approach Part I: Methodology and Experiments ⋮ Maximum Likelihood Estimation of Regularization Parameters in High-Dimensional Inverse Problems: An Empirical Bayesian Approach. Part II: Theoretical Analysis ⋮ User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient ⋮ Sparse reconstructions from few noisy data: analysis of hierarchical Bayesian models with generalized gamma hyperpriors ⋮ Unnamed Item ⋮ Efficient stochastic optimisation by unadjusted Langevin Monte Carlo. Application to maximum marginal likelihood and empirical Bayesian estimation ⋮ Solving inverse problems using data-driven models ⋮ Scalable Bayesian Uncertainty Quantification in Imaging Inverse Problems via Convex Optimization ⋮ Unadjusted Langevin algorithm for sampling a mixture of weakly smooth potentials ⋮ Proximal nested sampling for high-dimensional Bayesian model selection ⋮ Nonparametric Bayesian inference for reversible multidimensional diffusions ⋮ Geometric Integration of Measure-Preserving Flows for Sampling

Uses Software

• UNLocBoX
• PhaseLift

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• A simple proof of the Poincaré inequality for a large class of probability measures
• Regularization methods for large-scale problems
• Exponential convergence of Langevin distributions and their discrete approximations
• An algorithm for total variation minimization and applications
• Statistical and computational inverse problems.
• Nonasymptotic convergence analysis for the unadjusted Langevin algorithm
• Proximal Markov chain Monte Carlo algorithms
• Super-Resolution of Positive Sources: The Discrete Setup
• (Non-) asymptotic properties of Stochastic Gradient Langevin Dynamics
• Fast Gibbs sampling for high-dimensional Bayesian inversion
• A Nonlocal Bayesian Image Denoising Algorithm
• A New Semiblind Deconvolution Approach for Fourier-Based Image Restoration: An Application in Astronomy
• Proximal Splitting Methods in Signal Processing
• Dictionary Learning for Noisy and Incomplete Hyperspectral Images
• Alternating Direction Method for Image Inpainting in Wavelet Domains
• Ergodic Properties of Recurrent Diffusion Processes and Stabilization of the Solution to the Cauchy Problem for Parabolic Equations
• Markov Chains and Stochastic Stability
• The Bayesian Lasso
• Variational Analysis
• Photographing Paintings by Image Fusion
• Fast Markov chain Monte Carlo sampling for sparse Bayesian inference in high-dimensional inverse problems using L1-type priors
• Maximum-a-Posteriori Estimation with Bayesian Confidence Regions
• An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems
• Bayesian Inverse Problems with $l_1$ Priors: A Randomize-Then-Optimize Approach
• Theoretical Guarantees for Approximate Sampling from Smooth and Log-Concave Densities
• Expansion of the global error for numerical schemes solving stochastic differential equations
• Phase Retrieval via Matrix Completion
• Compressed sensing
• Finite dimensional convexity and optimization
• Bayesian computation: a summary of the current state, and samples backwards and forwards

This page was built for publication: Efficient Bayesian Computation by Proximal Markov Chain Monte Carlo: When Langevin Meets Moreau