# A Stabilized Path-Space Approach to Diffusion-Based Posterior Sampling arXiv: https://arxiv.org/abs/2606.12710 ยท HTML: https://mlsanity.com/paper/2606.12710/ ## Key facts - Cited by 1 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 38 of its 58 references that are in the map. - First posted on arXiv 10 Jun 2026. ## 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 38) - [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (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) - [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022) - [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019) - [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022) - [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022) - [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023) - [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021) - [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021) - [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022) - [Universal Guidance for Diffusion Models](https://mlsanity.com/paper/2302.07121/) (2023) - [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017) - [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024) - [Central Moment Discrepancy (CMD) for Domain-Invariant Representation Learning](https://mlsanity.com/paper/1702.08811/) (2017) - [Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing](https://mlsanity.com/paper/2407.01521/) (2024) - [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022) - [Fine-Tuning of Continuous-Time Diffusion Models as Entropy-Regularized Control](https://mlsanity.com/paper/2402.15194/) (2024) - [Improved sampling via learned diffusions](https://mlsanity.com/paper/2307.01198/) (2023) - [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014) - [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025) - [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) - [Non-Denoising Forward-Time Diffusions](https://mlsanity.com/paper/2312.14589/) (2023) - [Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processes](https://mlsanity.com/paper/2403.13724/) (2024) - [DEFT: Efficient Fine-Tuning of Diffusion Models by Learning the Generalised h-transform](https://mlsanity.com/paper/2406.01781/) (2024) - [Fine-tuning of diffusion models via stochastic control: entropy regularization and beyond](https://mlsanity.com/paper/2403.06279/) (2024) - [Stochastic Optimal Control for Collective Variable Free Sampling of Molecular Transition Paths](https://mlsanity.com/paper/2207.02149/) (2022) - [A mean-field games laboratory for generative modeling](https://mlsanity.com/paper/2304.13534/) (2023) - [Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inference](https://mlsanity.com/paper/2508.12511/) (2025) ## Built on by (most cited first, 1 of 1) - [Posterior Information Dynamics of Diffusion Models for Linear Inverse Problems](https://mlsanity.com/paper/2608.21709/) (2026) ## Abstract (arXiv) Diffusion models provide expressive data-driven priors for Bayesian inverse problems, but many diffusion posterior samplers rely on heuristic guidance approximations that can fail for nonlinear operators and multimodal posteriors. In this work, we develop a stabilized path-space framework for diffusion-based posterior sampling. Starting from a base diffusion process whose terminal marginal represents the prior, we define a likelihood-weighted target measure on trajectories and cast posterior sampling as learning a controlled stochastic process whose path measure matches this target. This formulation connects diffusion posterior sampling to stochastic optimal control while preserving the Bayesian structure needed for uncertainty quantification. We introduce a time reparameterization that makes the path-space control problem well posed by removing the bias induced by the unknown initial value function, without auxiliary training. We then learn the control via a trust-region path-space optimization method with log-variance objectives. The path-space perspective also unifies our learned control approach with existing guidance-based samplers, quantifies the sampling error induced by approximate controls, and yields importance sampling corrections for asymptotically exact posterior expectations. We evaluate the proposed framework on a suite of benchmark inverse problems with analytically characterized or high-quality reference posteriors, enabling principled assessment of sampling accuracy and uncertainty quantification. These experiments provide insight into the behavior of diffusion-based posterior samplers and demonstrate improved accuracy and robustness over leading approaches.