A Stabilized Path-Space Approach to Diffusion-Based Posterior Sampling

• URL: http://arxiv.org/abs/2606.12710v1
• Date: Wed, 10 Jun 2026 21:55:55 GMT
• Title: A Stabilized Path-Space Approach to Diffusion-Based Posterior Sampling
• Abstract summary: Diffusion models provide expressive data-driven priors for inverse problems.<n>Many diffusion posterior samplers rely on guidance approximations that can fail for nonlinear operators and multimodal posteriors.<n>We develop a stabilized path-space framework for diffusion-based posterior sampling.
• Score: 12.537644240443086
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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