# Conditional Diffusion Guidance under Hard Constraint: A Stochastic Analysis Approach

arXiv: https://arxiv.org/abs/2602.05533 · HTML: https://mlsanity.com/paper/2602.05533/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 60 of its 102 references that are in the map.
- First posted on arXiv 5 Feb 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 60)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Make-A-Video: Text-to-Video Generation without Text-Video Data](https://mlsanity.com/paper/2209.14792/) (2022)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025)
- [Diffusion Model Alignment Using Direct Preference Optimization](https://mlsanity.com/paper/2311.12908/) (2023)
- [DPOK: Reinforcement Learning for Fine-tuning Text-to-Image Diffusion Models](https://mlsanity.com/paper/2305.16381/) (2023)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Universal Guidance for Diffusion Models](https://mlsanity.com/paper/2302.07121/) (2023)
- [Aligning Text-to-Image Models using Human Feedback](https://mlsanity.com/paper/2302.12192/) (2023)
- [Using Human Feedback to Fine-tune Diffusion Models without Any Reward Model](https://mlsanity.com/paper/2311.13231/) (2023)
- [Mercury: Ultra-Fast Language Models Based on Diffusion](https://mlsanity.com/paper/2506.17298/) (2025)
- [Pros and Cons of GAN Evaluation Measures](https://mlsanity.com/paper/1802.03446/) (2018)
- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [Goal-Conditioned Reinforcement Learning: Problems and Solutions](https://mlsanity.com/paper/2201.08299/) (2022)
- [Fine-Tuning of Continuous-Time Diffusion Models as Entropy-Regularized Control](https://mlsanity.com/paper/2402.15194/) (2024)
- [Optimal Goal-Reaching Reinforcement Learning via Quasimetric Learning](https://mlsanity.com/paper/2304.01203/) (2023)
- [Understanding Reinforcement Learning-Based Fine-Tuning of Diffusion Models: A Tutorial and Review](https://mlsanity.com/paper/2407.13734/) (2024)
- [Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models](https://mlsanity.com/paper/2306.09251/) (2023)
- [Optimizing DDPM Sampling with Shortcut Fine-Tuning](https://mlsanity.com/paper/2301.13362/) (2023)
- [Large-scale Reinforcement Learning for Diffusion Models](https://mlsanity.com/paper/2401.12244/) (2024)
- [An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimization](https://mlsanity.com/paper/2404.07771/) (2024)

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

- [Training-Free Adaptation of Diffusion Models via Doob's h-Transform](https://mlsanity.com/paper/2602.16198/) (2026)
- [Diffusion-Based Posterior Sampling: A Feynman-Kac Analysis of Bias and Stability](https://mlsanity.com/paper/2605.06538/) (2026)
- [Constrained Diffusion Models with Primal-Dual Inference](https://mlsanity.com/paper/2606.17192/) (2026)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [On the Robustness of Distribution Support under Diffusion Guidance](https://mlsanity.com/paper/2605.07220/) (2026)
- [DOHF: Online Diffusion Fine-tuning with Doob's h-transform Guidance](https://mlsanity.com/paper/2609.31882/) (2026)
- [WEIRDO: WEak resIdual Regularized DOob's h-transform diffusion alignment](https://mlsanity.com/paper/2609.39531/) (2026)
- [Conditional Diffusion Under Linear Constraints: Langevin Mixing and Information-Theoretic Guarantees](https://mlsanity.com/paper/2605.05387/) (2026)
- [Tweedie's Formulae and Diffusion Generative Models Beyond Gaussian](https://mlsanity.com/paper/2605.19391/) (2026)
- [Commitment Before Realization: When Classifier-Free Guidance Becomes Unnecessary in Masked Diffusion Language Models](https://mlsanity.com/paper/2608.08082/) (2026)
- [Learning to Solve Stochastic Controls with Unknown Drifts and Running Rewards: Theory, Algorithms and Convergence](https://mlsanity.com/paper/2609.14972/) (2026)

## Abstract (arXiv)

We study conditional generation in diffusion models under hard constraints, where generated samples must satisfy prescribed events with probability one. Such constraints arise naturally in safety-critical applications and in rare-event simulation, where soft or reward-based guidance methods offer no guarantee of constraint satisfaction. Building on a probabilistic interpretation of diffusion models, we develop a principled conditional diffusion guidance framework based on Doob's h-transform, martingale representation and quadratic variation process. Specifically, the resulting guided dynamics augment a pretrained diffusion with an explicit drift correction involving the logarithmic gradient of a conditioning function, without modifying the pretrained score network. Leveraging martingale and quadratic-variation identities, we propose two novel off-policy learning algorithms based on a martingale loss and a martingale-covariation loss to estimate h and its gradient using only trajectories from the pretrained model. We provide non-asymptotic guarantees for the resulting conditional sampler in both total variation and Wasserstein distances, explicitly characterizing the impact of score approximation and guidance estimation errors. Numerical experiments demonstrate the effectiveness of the proposed methods in enforcing hard constraints and generating rare-event samples. The code of the numerical experiments can be found at https://github.com/ZhengyiGuo2002/CDG_Finance.