# Stein Diffusion Guidance: Training-Free Posterior Correction for Sampling Beyond High-Density Regions

arXiv: https://arxiv.org/abs/2507.05482 · HTML: https://mlsanity.com/paper/2507.05482/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 58 references that are in the map.
- First posted on arXiv 7 Jul 2025.

## Builds on (most cited first, 30 of 41)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [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)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Scaling Autoregressive Models for Content-Rich Text-to-Image Generation](https://mlsanity.com/paper/2206.10789/) (2022)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Universal Guidance for Diffusion Models](https://mlsanity.com/paper/2302.07121/) (2023)
- [Score-based Generative Modeling of Graphs via the System of Stochastic Differential Equations](https://mlsanity.com/paper/2202.02514/) (2022)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [Hierarchical Generation of Molecular Graphs using Structural Motifs](https://mlsanity.com/paper/2002.03230/) (2020)
- [Large-scale Classification of Fine-Art Paintings: Learning The Right Metric on The Right Feature](https://mlsanity.com/paper/1505.00855/) (2015)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Manifold Preserving Guided Diffusion](https://mlsanity.com/paper/2311.16424/) (2023)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [Fine-Tuning of Continuous-Time Diffusion Models as Entropy-Regularized Control](https://mlsanity.com/paper/2402.15194/) (2024)
- [TFG: Unified Training-Free Guidance for Diffusion Models](https://mlsanity.com/paper/2409.15761/) (2024)
- [Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts](https://mlsanity.com/paper/2503.02819/) (2025)
- [Exploring Chemical Space with Score-based Out-of-distribution Generation](https://mlsanity.com/paper/2206.07632/) (2022)
- [Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space](https://mlsanity.com/paper/2005.05409/) (2020)
- [Fine-Tuning Discrete Diffusion Models via Reward Optimization with Applications to DNA and Protein Design](https://mlsanity.com/paper/2410.13643/) (2024)
- [Generating High Fidelity Data from Low-density Regions using Diffusion Models](https://mlsanity.com/paper/2203.17260/) (2022)
- [Particle Guidance: non-I.I.D. Diverse Sampling with Diffusion Models](https://mlsanity.com/paper/2310.13102/) (2023)
- [Hit and Lead Discovery with Explorative RL and Fragment-based Molecule Generation](https://mlsanity.com/paper/2110.01219/) (2021)
- [SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence](https://mlsanity.com/paper/2006.02509/) (2020)

## Built on by (most cited first, 1 of 1)

- [Interacting particle guidance for sampling reward-tilted generative priors](https://mlsanity.com/paper/2609.37227/) (2026)

## Abstract (arXiv)

Training-free diffusion guidance offers a flexible framework for leveraging off-the-shelf classifiers without additional training. Yet, current approaches hinge on posterior approximations via Tweedie's formula, which often yield unreliable guidance, particularly in low-density regions. Stochastic optimal control (SOC), in contrast, enables principled posterior sampling but remains computationally prohibitive for efficient inference. In this work, we reconcile the strengths of these paradigms by introducing Stein Diffusion Guidance (SDG), a novel training-free framework grounded in a surrogate SOC objective. We establish a new theoretical bound on the SOC value function, revealing the necessity of correcting approximate posteriors to reflect true diffusion dynamics. Building on Stein variational inference, SDG computes the steepest descent direction that minimizes the Kullback-Leibler divergence between approximate and true posteriors. By integrating a principled Stein correction mechanism along with a novel running cost functional, SDG enables effective guidance in low-density regions. Our experiments on diverse image-guidance tasks and on challenging small-ligand sampling for protein docking suggest that SDG consistently outperforms standard training-free guidance methods and highlights its potential for broader posterior sampling problems beyond high-density regimes.