# The split Gibbs sampler revisited: improvements to its algorithmic structure and augmented target distribution

arXiv: https://arxiv.org/abs/2206.13894 · HTML: https://mlsanity.com/paper/2206.13894/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 58 references that are in the map.
- First posted on arXiv 28 Jun 2022.

## Builds on (most cited first, 19 of 19)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Expectation Propagation for approximate Bayesian inference](https://mlsanity.com/paper/1301.2294/) (2013)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [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)
- [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)
- [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)
- [Maximum-a-posteriori estimation with Bayesian confidence regions](https://mlsanity.com/paper/1602.08590/) (2016)
- [Uncertainty quantification for radio interferometric imaging: II. MAP estimation](https://mlsanity.com/paper/1711.04819/) (2017)
- [Patch-Based Image Restoration using Expectation Propagation](https://mlsanity.com/paper/2106.15327/) (2021)
- [Efficient Gaussian Sampling for Solving Large-Scale Inverse Problems using MCMC Methods](https://mlsanity.com/paper/1409.0606/) (2014)
- [Template-Based Image Reconstruction from Sparse Tomographic Data](https://mlsanity.com/paper/1810.08596/) (2018)
- [Stochastic Proximal Gradient Algorithms for Penalized Mixed Models](https://mlsanity.com/paper/1704.08891/) (2017)

## Built on by (most cited first, 11 of 11)

- [Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priors](https://mlsanity.com/paper/2405.18782/) (2024)
- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [LATINO-PRO: LAtent consisTency INverse sOlver with PRompt Optimization](https://mlsanity.com/paper/2503.12615/) (2025)
- [Empirical Bayesian image restoration by Langevin sampling with a denoising diffusion implicit prior](https://mlsanity.com/paper/2409.04384/) (2024)
- [Learning few-step posterior samplers by unfolding and distillation of diffusion models](https://mlsanity.com/paper/2507.02686/) (2025)
- [Accelerated Bayesian imaging by relaxed proximal-point Langevin sampling](https://mlsanity.com/paper/2308.09460/) (2023)
- [Bayesian Uncertainty-Aware MRI Reconstruction](https://mlsanity.com/paper/2603.13439/) (2026)
- [Sampling Strategies in Bayesian Inversion: A Study of RTO and Langevin Methods](https://mlsanity.com/paper/2406.16658/) (2024)
- [Bregman geometry-aware split Gibbs sampling for Bayesian Poisson inverse problems](https://mlsanity.com/paper/2511.12257/) (2025)
- [Deep unfolding of MCMC kernels: scalable, modular & explainable GANs for high-dimensional posterior sampling](https://mlsanity.com/paper/2602.20758/) (2026)
- [FlowSGS: Improving Flow Matching Priors for Inverse Imaging with Stochastic Interpolants](https://mlsanity.com/paper/2609.20769/) (2026)

## Abstract (arXiv)

Developing efficient Bayesian computation algorithms for imaging inverse problems is challenging due to the dimensionality involved and because Bayesian imaging models are often not smooth. Current state-of-the-art methods often address these difficulties by replacing the posterior density with a smooth approximation that is amenable to efficient exploration by using Langevin Markov chain Monte Carlo (MCMC) methods. An alternative approach is based on data augmentation and relaxation, where auxiliary variables are introduced in order to construct an approximate augmented posterior distribution that is amenable to efficient exploration by Gibbs sampling. This paper proposes a new accelerated proximal MCMC method called latent space SK-ROCK (ls SK-ROCK), which tightly combines the benefits of the two aforementioned strategies. Additionally, instead of viewing the augmented posterior distribution as an approximation of the original model, we propose to consider it as a generalisation of this model. Following on from this, we empirically show that there is a range of values for the relaxation parameter for which the accuracy of the model improves, and propose a stochastic optimisation algorithm to automatically identify the optimal amount of relaxation for a given problem. In this regime, ls SK-ROCK converges faster than competing approaches from the state of the art, and also achieves better accuracy since the underlying augmented Bayesian model has a higher Bayesian evidence. The proposed methodology is demonstrated with a range of numerical experiments related to image deblurring and inpainting, as well as with comparisons with alternative approaches from the state of the art. An open-source implementation of the proposed MCMC methods is available from https://github.com/luisvargasmieles/ls-MCMC.