# Accelerated Bayesian imaging by relaxed proximal-point Langevin sampling

arXiv: https://arxiv.org/abs/2308.09460 · HTML: https://mlsanity.com/paper/2308.09460/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 51 references that are in the map.
- First posted on arXiv 18 Aug 2023.

## Builds on (most cited first, 24 of 24)

- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [Deep Bayesian Inversion](https://mlsanity.com/paper/1811.05910/) (2018)
- [Learned convex regularizers for inverse problems](https://mlsanity.com/paper/2008.02839/) (2020)
- [Split-and-augmented Gibbs sampler - Application to large-scale inference problems](https://mlsanity.com/paper/1804.05809/) (2018)
- [Proximal Markov chain Monte Carlo algorithms](https://mlsanity.com/paper/1306.0187/) (2013)
- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm](https://mlsanity.com/paper/1908.10859/) (2019)
- [Scalable Bayesian uncertainty quantification in imaging inverse problems via convex optimization](https://mlsanity.com/paper/1803.00889/) (2018)
- [Efficient MCMC Sampling with Dimension-Free Convergence Rate using ADMM-type Splitting](https://mlsanity.com/paper/1905.11937/) (2019)
- [Uncertainty quantification for radio interferometric imaging: I. proximal MCMC methods](https://mlsanity.com/paper/1711.04818/) (2017)
- [Proximal nested sampling for high-dimensional Bayesian model selection](https://mlsanity.com/paper/2106.03646/) (2021)
- [The split Gibbs sampler revisited: improvements to its algorithmic structure and augmented target distribution](https://mlsanity.com/paper/2206.13894/) (2022)
- [Maximum-a-posteriori estimation with Bayesian confidence regions](https://mlsanity.com/paper/1602.08590/) (2016)
- [Implicit Langevin Algorithms for Sampling From Log-concave Densities](https://mlsanity.com/paper/1903.12322/) (2019)
- [Uncertainty quantification for radio interferometric imaging: II. MAP estimation](https://mlsanity.com/paper/1711.04819/) (2017)
- [Preconditioned P-ULA for Joint Deconvolution-Segmentation of Ultrasound Images -- Extended Version](https://mlsanity.com/paper/1903.08111/) (2019)
- [Patch-Based Image Restoration using Expectation Propagation](https://mlsanity.com/paper/2106.15327/) (2021)

## Built on by (most cited first, 4 of 4)

- [Plug-and-Play Posterior Sampling under Mismatched Measurement and Prior Models](https://mlsanity.com/paper/2310.03546/) (2023)
- [Proximal Interacting Particle Langevin Algorithms](https://mlsanity.com/paper/2406.14292/) (2024)
- [Active-Trace Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling](https://mlsanity.com/paper/2608.13467/) (2026)
- [Bridging data-driven priors via the score function for posterior sampling -- Comparative review and experimental study](https://mlsanity.com/paper/2606.14800/) (2026)

## Abstract (arXiv)

This paper presents a new accelerated proximal Markov chain Monte Carlo methodology to perform Bayesian inference in imaging inverse problems with an underlying convex geometry. The proposed strategy takes the form of a stochastic relaxed proximal-point iteration that admits two complementary interpretations. For models that are smooth or regularised by Moreau-Yosida smoothing, the algorithm is equivalent to an implicit midpoint discretisation of an overdamped Langevin diffusion targeting the posterior distribution of interest. This discretisation is asymptotically unbiased for Gaussian targets and shown to converge in an accelerated manner for any target that is kappa-strongly log-concave (i.e., requiring in the order of √kappa iterations to converge, similarly to accelerated optimisation schemes), comparing favorably to [M. Pereyra, L. Vargas Mieles, K.C. Zygalakis, SIAM J. Imaging Sciences, 13,2 (2020), pp. 905-935] which is only provably accelerated for Gaussian targets and has bias. For models that are not smooth, the algorithm is equivalent to a Leimkuhler-Matthews discretisation of a Langevin diffusion targeting a Moreau-Yosida approximation of the posterior distribution of interest, and hence achieves a significantly lower bias than conventional unadjusted Langevin strategies based on the Euler-Maruyama discretisation. For targets that are kappa-strongly log-concave, the provided non-asymptotic convergence analysis also identifies the optimal time step which maximizes the convergence speed. The proposed methodology is demonstrated through a range of experiments related to image deconvolution with Gaussian and Poisson noise, with assumption-driven and data-driven convex priors. Source codes for the numerical experiments of this paper are available from https://github.com/MI2G/accelerated-langevin-imla.