# Split-and-augmented Gibbs sampler - Application to large-scale inference problems

arXiv: https://arxiv.org/abs/1804.05809 · HTML: https://mlsanity.com/paper/1804.05809/

## Key facts

- Cited by 47 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Apr 2018.

## Built on by (most cited first, 30 of 47)

- [Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priors](https://mlsanity.com/paper/2405.18782/) (2024)
- [Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction](https://mlsanity.com/paper/2403.17042/) (2024)
- [InverseBench: Benchmarking Plug-and-Play Diffusion Priors for Inverse Problems in Physical Sciences](https://mlsanity.com/paper/2503.11043/) (2025)
- [Plug-and-Play split Gibbs sampler: embedding deep generative priors in Bayesian inference](https://mlsanity.com/paper/2304.11134/) (2023)
- [Split Gibbs Discrete Diffusion Posterior Sampling](https://mlsanity.com/paper/2503.01161/) (2025)
- [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)
- [DG-LMC: A Turn-key and Scalable Synchronous Distributed MCMC Algorithm via Langevin Monte Carlo within Gibbs](https://mlsanity.com/paper/2106.06300/) (2021)
- [The split Gibbs sampler revisited: improvements to its algorithmic structure and augmented target distribution](https://mlsanity.com/paper/2206.13894/) (2022)
- [Regularization by denoising: Bayesian model and Langevin-within-split Gibbs sampling](https://mlsanity.com/paper/2402.12292/) (2024)
- [Empirical Bayesian image restoration by Langevin sampling with a denoising diffusion implicit prior](https://mlsanity.com/paper/2409.04384/) (2024)
- [Strictly Constrained Generative Modeling via Split Augmented Langevin Sampling](https://mlsanity.com/paper/2505.18017/) (2025)
- [Learning few-step posterior samplers by unfolding and distillation of diffusion models](https://mlsanity.com/paper/2507.02686/) (2025)
- [Reward-Guided Discrete Diffusion via Clean-Sample Markov Chain for Molecule and Biological Sequence Design](https://mlsanity.com/paper/2602.09424/) (2026)
- [Blade: A Derivative-free Bayesian Inversion Method using Diffusion Priors](https://mlsanity.com/paper/2510.10968/) (2025)
- [Machine Learning and the Future of Bayesian Computation](https://mlsanity.com/paper/2304.11251/) (2023)
- [Accelerated Bayesian imaging by relaxed proximal-point Langevin sampling](https://mlsanity.com/paper/2308.09460/) (2023)
- [DIGing--SGLD: Decentralized and Scalable Langevin Sampling over Time--Varying Networks](https://mlsanity.com/paper/2511.12836/) (2025)
- [Approximate blocked Gibbs sampling for Bayesian neural networks](https://mlsanity.com/paper/2208.11389/) (2022)
- [Super-Resolution Optical Coherence Tomography Using Diffusion Model-Based Plug-and-Play Priors](https://mlsanity.com/paper/2505.14916/) (2025)
- [PSI3D: Plug-and-Play 3D Stochastic Inference with Slice-wise Latent Diffusion Prior](https://mlsanity.com/paper/2512.18367/) (2025)
- [Sample-efficient evidence estimation of score based priors for model selection](https://mlsanity.com/paper/2602.20549/) (2026)
- [Bayesian Uncertainty-Aware MRI Reconstruction](https://mlsanity.com/paper/2603.13439/) (2026)
- [Scalable Vertical Federated Learning via Data Augmentation and Amortized Inference](https://mlsanity.com/paper/2405.04043/) (2024)
- [Plug-and-Play Posterior Sampling for Blind Inverse Problems](https://mlsanity.com/paper/2505.22923/) (2025)
- [LVTINO: LAtent Video consisTency INverse sOlver for High Definition Video Restoration](https://mlsanity.com/paper/2510.01339/) (2025)
- [Dual-space posterior sampling for Bayesian inference in constrained inverse problems](https://mlsanity.com/paper/2603.00393/) (2026)
- [Noise Conditional Variational Score Distillation](https://mlsanity.com/paper/2506.09416/) (2025)
- [Diffusion-Prior Split Gibbs Sampling for Synthetic Aperture Radar Imaging under Incomplete Measurements](https://mlsanity.com/paper/2512.02768/) (2025)
- [Bayesian Signal Component Decomposition via Diffusion-within-Gibbs Sampling](https://mlsanity.com/paper/2602.10792/) (2026)

## Abstract (arXiv)

This paper derives two new optimization-driven Monte Carlo algorithms inspired from variable splitting and data augmentation. In particular, the formulation of one of the proposed approaches is closely related to the alternating direction method of multipliers (ADMM) main steps. The proposed framework enables to derive faster and more efficient sampling schemes than the current state-of-the-art methods and can embed the latter. By sampling efficiently the parameter to infer as well as the hyperparameters of the problem, the generated samples can be used to approximate Bayesian estimators of the parameters to infer. Additionally, the proposed approach brings confidence intervals at a low cost contrary to optimization methods. Simulations on two often-studied signal processing problems illustrate the performance of the two proposed samplers. All results are compared to those obtained by recent state-of-the-art optimization and MCMC algorithms used to solve these problems.