# Posterior Sampling by Combining Diffusion Models with Annealed Langevin Dynamics

arXiv: https://arxiv.org/abs/2510.26324 · HTML: https://mlsanity.com/paper/2510.26324/

## Key facts

- Cited by 5 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 5 of its 47 references that are in the map.
- First posted on arXiv 30 Oct 2025.

## Placed in

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

## Builds on (most cited first, 5 of 5)

- [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)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Solving Linear Inverse Problems Provably via Posterior Sampling with Latent Diffusion Models](https://mlsanity.com/paper/2307.00619/) (2023)
- [Decoupled Data Consistency with Diffusion Purification for Image Restoration](https://mlsanity.com/paper/2403.06054/) (2024)

## Built on by (most cited first, 5 of 5)

- [On the Robustness of Langevin Dynamics to Score Function Error](https://mlsanity.com/paper/2603.11319/) (2026)
- [Query Lower Bounds for Diffusion Sampling](https://mlsanity.com/paper/2604.10857/) (2026)
- [The tractability landscape of diffusion alignment: regularization, rewards, and computational primitives](https://mlsanity.com/paper/2605.11361/) (2026)
- [Composing diffusion priors with explicit physical context via generative Gibbs sampling](https://mlsanity.com/paper/2605.10642/) (2026)
- [Omega: Operator-based Mixture Ensemble for Generative Assimilation](https://mlsanity.com/paper/2606.20920/) (2026)

## Abstract (arXiv)

Given a noisy linear measurement y = Ax + xi of a distribution p(x), and a good approximation to the prior p(x), when can we sample from the posterior p(x mid y)? Posterior sampling provides an accurate and fair framework for tasks such as inpainting, deblurring, and MRI reconstruction, and several heuristics attempt to approximate it. Unfortunately, approximate posterior sampling is computationally intractable in general. To sidestep this hardness, we focus on (local or global) log-concave distributions p(x). In this regime, Langevin dynamics yields posterior samples when the exact scores of p(x) are available, but it is brittle to score--estimation error, requiring an MGF bound (sub-exponential error). By contrast, in the unconditional setting, diffusion models succeed with only an L2 bound on the score error. We prove that combining diffusion models with an annealed variant of Langevin dynamics achieves conditional sampling in polynomial time using merely an L4 bound on the score error.