# Monte Carlo guided Diffusion for Bayesian linear inverse problems

arXiv: https://arxiv.org/abs/2308.07983 · HTML: https://mlsanity.com/paper/2308.07983/

## Key facts

- Cited by 66 papers in the map (data through 14 Oct 2026).
- Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models.
- Builds on 30 of its 56 references that are in the map.
- First posted on arXiv 15 Aug 2023.

## Placed in

- Diffusion models / [Inverse problems and image restoration](https://mlsanity.com/diffusion/inverse-problems-and-image-restoration/)

## Builds on (most cited first, 30 of 30)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Generative Image Inpainting with Contextual Attention](https://mlsanity.com/paper/1801.07892/) (2018)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problem](https://mlsanity.com/paper/2206.04119/) (2022)
- [Practical and Asymptotically Exact Conditional Sampling in Diffusion Models](https://mlsanity.com/paper/2306.17775/) (2023)
- [A Review on Generative Adversarial Networks: Algorithms, Theory, and Applications](https://mlsanity.com/paper/2001.06937/) (2020)
- [Pluralistic Image Completion](https://mlsanity.com/paper/1903.04227/) (2019)
- [High-Fidelity Pluralistic Image Completion with Transformers](https://mlsanity.com/paper/2103.14031/) (2021)
- [Generating Diverse Structure for Image Inpainting With Hierarchical VQ-VAE](https://mlsanity.com/paper/2103.10022/) (2021)
- [Aggregated Contextual Transformations for High-Resolution Image Inpainting](https://mlsanity.com/paper/2104.01431/) (2021)
- [Variational Autoencoder with Arbitrary Conditioning](https://mlsanity.com/paper/1806.02382/) (2018)
- [Towards Coherent Image Inpainting Using Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2304.03322/) (2023)
- [From Denoising Diffusions to Denoising Markov Models](https://mlsanity.com/paper/2211.03595/) (2022)
- [Deep Mean-Shift Priors for Image Restoration](https://mlsanity.com/paper/1709.03749/) (2017)
- [An invitation to sequential Monte Carlo samplers](https://mlsanity.com/paper/2007.11936/) (2020)
- [Semi-supervised Invertible Neural Operators for Bayesian Inverse Problems](https://mlsanity.com/paper/2209.02772/) (2022)
- [Deep Unfolding with Normalizing Flow Priors for Inverse Problems](https://mlsanity.com/paper/2107.02848/) (2021)
- [A Variational Bayesian Approach for Image Restoration. Application to Image Deblurring with Poisson-Gaussian Noise](https://mlsanity.com/paper/1610.07519/) (2016)
- [A domain-decomposed VAE method for Bayesian inverse problems](https://mlsanity.com/paper/2301.05708/) (2023)
- [Utilizing variational autoencoders in the Bayesian inverse problem of photoacoustic tomography](https://mlsanity.com/paper/2204.06270/) (2022)

## Built on by (most cited first, 30 of 66)

- [A General Framework for Inference-time Scaling and Steering of Diffusion Models](https://mlsanity.com/paper/2501.06848/) (2025)
- [Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing](https://mlsanity.com/paper/2407.01521/) (2024)
- [Derivative-Free Guidance in Continuous and Discrete Diffusion Models with Soft Value-Based Decoding](https://mlsanity.com/paper/2408.08252/) (2024)
- [Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priors](https://mlsanity.com/paper/2405.18782/) (2024)
- [Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts](https://mlsanity.com/paper/2503.02819/) (2025)
- [Inference-Time Alignment in Diffusion Models with Reward-Guided Generation: Tutorial and Review](https://mlsanity.com/paper/2501.09685/) (2025)
- [Fine-Tuning Discrete Diffusion Models via Reward Optimization with Applications to DNA and Protein Design](https://mlsanity.com/paper/2410.13643/) (2024)
- [Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction](https://mlsanity.com/paper/2403.17042/) (2024)
- [Particle Denoising Diffusion Sampler](https://mlsanity.com/paper/2402.06320/) (2024)
- [Provable Probabilistic Imaging using Score-Based Generative Priors](https://mlsanity.com/paper/2310.10835/) (2023)
- [Inference-Time Scaling for Flow Models via Stochastic Generation and Rollover Budget Forcing](https://mlsanity.com/paper/2503.19385/) (2025)
- [Dynamic Search for Inference-Time Alignment in Diffusion Models](https://mlsanity.com/paper/2503.02039/) (2025)
- [Guided Diffusion Sampling on Function Spaces with Applications to PDEs](https://mlsanity.com/paper/2505.17004/) (2025)
- [Variational Diffusion Posterior Sampling with Midpoint Guidance](https://mlsanity.com/paper/2410.09945/) (2024)
- [Reward-Guided Iterative Refinement in Diffusion Models at Test-Time with Applications to Protein and DNA Design](https://mlsanity.com/paper/2502.14944/) (2025)
- [Training-Free Guidance Beyond Differentiability: Scalable Path Steering with Tree Search in Diffusion and Flow Models](https://mlsanity.com/paper/2502.11420/) (2025)
- [Learning Image Priors through Patch-based Diffusion Models for Solving Inverse Problems](https://mlsanity.com/paper/2406.02462/) (2024)
- [CoDe: Blockwise Control for Denoising Diffusion Models](https://mlsanity.com/paper/2502.00968/) (2025)
- [Inference-Time Scaling of Discrete Diffusion Models via Importance Weighting and Optimal Proposal Design](https://mlsanity.com/paper/2505.22524/) (2025)
- [DiffusionBlend: Learning 3D Image Prior through Position-aware Diffusion Score Blending for 3D Computed Tomography Reconstruction](https://mlsanity.com/paper/2406.10211/) (2024)
- [Training-Free Adaptation of Diffusion Models via Doob's h-Transform](https://mlsanity.com/paper/2602.16198/) (2026)
- [Decoupled Diffusion Sampling for Inverse Problems on Function Spaces](https://mlsanity.com/paper/2601.23280/) (2026)
- [Local Manifold Approximation and Projection for Manifold-Aware Diffusion Planning](https://mlsanity.com/paper/2506.00867/) (2025)
- [MIRA: Towards Mitigating Reward Hacking in Inference-Time Alignment of T2I Diffusion Models](https://mlsanity.com/paper/2510.01549/) (2025)
- [ESS-Flow: Training-free guidance of flow-based models as inference in source space](https://mlsanity.com/paper/2510.05849/) (2025)
- [Inverse Problem Sampling in Latent Space Using Sequential Monte Carlo](https://mlsanity.com/paper/2502.05908/) (2025)
- [Wasserstein Convergence of Critically Damped Langevin Diffusions](https://mlsanity.com/paper/2511.02419/) (2025)
- [Enhancing Diffusion Models for Inverse Problems with Covariance-Aware Posterior Sampling](https://mlsanity.com/paper/2412.20045/) (2024)
- [Efficient Zero-Shot Inpainting with Decoupled Diffusion Guidance](https://mlsanity.com/paper/2512.18365/) (2025)
- [Contrastive Distribution Matching for Amortized Sequential Monte Carlo in Discrete Diffusion](https://mlsanity.com/paper/2605.23346/) (2026)

## Abstract (arXiv)

Ill-posed linear inverse problems arise frequently in various applications, from computational photography to medical imaging. A recent line of research exploits Bayesian inference with informative priors to handle the ill-posedness of such problems. Amongst such priors, score-based generative models (SGM) have recently been successfully applied to several different inverse problems. In this study, we exploit the particular structure of the prior defined by the SGM to define a sequence of intermediate linear inverse problems. As the noise level decreases, the posteriors of these inverse problems get closer to the target posterior of the original inverse problem. To sample from this sequence of posteriors, we propose the use of Sequential Monte Carlo (SMC) methods. The proposed algorithm, MCGDiff, is shown to be theoretically grounded and we provide numerical simulations showing that it outperforms competing baselines when dealing with ill-posed inverse problems in a Bayesian setting.