# Inference-Time Alignment for Diffusion Models via Variationally Stable Doob's Matching

arXiv: https://arxiv.org/abs/2601.06514 · HTML: https://mlsanity.com/paper/2601.06514/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 27 of its 110 references that are in the map.
- First posted on arXiv 10 Jan 2026.

## Placed in

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

## Builds on (most cited first, 27 of 27)

- [Aligning Text-to-Image Models using Human Feedback](https://mlsanity.com/paper/2302.12192/) (2023)
- [Fine-Tuning of Continuous-Time Diffusion Models as Entropy-Regularized Control](https://mlsanity.com/paper/2402.15194/) (2024)
- [Inference-Time Alignment in Diffusion Models with Reward-Guided Generation: Tutorial and Review](https://mlsanity.com/paper/2501.09685/) (2025)
- [Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory](https://mlsanity.com/paper/2403.11968/) (2024)
- [Linear Convergence of Diffusion Models Under the Manifold Hypothesis](https://mlsanity.com/paper/2410.09046/) (2024)
- [Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality](https://mlsanity.com/paper/2410.18784/) (2024)
- [Fine-tuning of diffusion models via stochastic control: entropy regularization and beyond](https://mlsanity.com/paper/2403.06279/) (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)
- [DriftLite: Lightweight Drift Control for Inference-Time Scaling of Diffusion Models](https://mlsanity.com/paper/2509.21655/) (2025)
- [Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distance](https://mlsanity.com/paper/2508.03210/) (2025)
- [Stochastic Control for Fine-tuning Diffusion Models: Optimality, Regularity, and Convergence](https://mlsanity.com/paper/2412.18164/) (2024)
- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [Test-time scaling of diffusions with flow maps](https://mlsanity.com/paper/2511.22688/) (2025)
- [Provable Diffusion Posterior Sampling for Bayesian Inversion](https://mlsanity.com/paper/2512.08022/) (2025)
- [Differentiable Neural Networks with RePU Activation: with Applications to Score Estimation and Isotonic Regression](https://mlsanity.com/paper/2305.00608/) (2023)
- [Deep conditional distribution learning via conditional F"ollmer flow](https://mlsanity.com/paper/2402.01460/) (2024)
- [Nonlinear Assimilation via Score-based Sequential Langevin Sampling](https://mlsanity.com/paper/2411.13443/) (2024)
- [Domain Guidance: A Simple Transfer Approach for a Pre-trained Diffusion Model](https://mlsanity.com/paper/2504.01521/) (2025)
- [Towards a unified framework for guided diffusion models](https://mlsanity.com/paper/2512.04985/) (2025)
- [Non-asymptotic error bounds for probability flow ODEs under weak log-concavity](https://mlsanity.com/paper/2510.17608/) (2025)
- [Iterative Importance Fine-tuning of Diffusion Models](https://mlsanity.com/paper/2502.04468/) (2025)
- [Characteristic Learning for Provable One Step Generation](https://mlsanity.com/paper/2405.05512/) (2024)
- [DogFit: Domain-guided Fine-tuning for Efficient Transfer Learning of Diffusion Models](https://mlsanity.com/paper/2508.05685/) (2025)
- [Iterative Tilting for Diffusion Fine-Tuning](https://mlsanity.com/paper/2512.03234/) (2025)
- [Implicit score matching meets denoising score matching: improved rates of convergence and log-density Hessian estimation](https://mlsanity.com/paper/2512.24378/) (2025)
- [Is RL fine-tuning harder than regression? A PDE learning approach for diffusion models](https://mlsanity.com/paper/2509.02528/) (2025)
- [Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural Networks](https://mlsanity.com/paper/2512.23643/) (2025)

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

- [Slowly Annealed Langevin Dynamics: Theory and Applications to Training-Free Guided Generation](https://mlsanity.com/paper/2605.07950/) (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)
- [Schrödinger--Föllmer Actor--Critic: Diffusion Policy Improvement with Finite-Sample Analysis](https://mlsanity.com/paper/2609.33239/) (2026)
- [Radial Tail Repair: Post-Training Correction for Diffusion and Flow Models](https://mlsanity.com/paper/2610.16851/) (2026)

## Abstract (arXiv)

Inference-time alignment for diffusion models aims to adapt a pre-trained reference diffusion model toward a target distribution without retraining the reference score network, thereby preserving the generative capacity of the reference model while enforcing desired properties at the inference time. A central mechanism for achieving such alignment is guidance, which modifies the sampling dynamics through an additional drift term. In this work, we introduce variationally stable Doob's matching, a novel framework for provable guidance estimation grounded in Doob's h-transform. Our approach formulates guidance as the gradient of logarithm of an underlying Doob's h-function and employs gradient-regularized regression to simultaneously estimate both the h-function and its gradient, resulting in a consistent estimator of the guidance. Theoretically, we establish non-asymptotic convergence rates for the estimated guidance. Moreover, we analyze the resulting controllable diffusion processes and prove non-asymptotic convergence guarantees for the generated distributions in the 2-Wasserstein distance. Finally, we show that variationally stable guidance estimators are adaptive to unknown low dimensionality, effectively mitigating the curse of dimensionality under low-dimensional subspace assumptions.