aiqbench analysis
This paper provides a rigorous mathematical analysis of bias and stability in diffusion-based posterior sampling, specifically addressing the limitations of the DPS algorithm through Feynman-Kac representations. It identifies specific numerical instabilities in hard-constraint regimes and proposes implicit guidance updates as a solution to improve convergence near boundaries. The work offers critical theoretical insights for refining inverse problem solvers and reducing systematic errors in conditioned generation tasks.
Diffusion-Based Posterior Sampling: A Feynman-Kac Analysis of Bias and Stability
arXiv:2605.06538v2 Announce Type: replace Abstract: Diffusion-based posterior samplers draw from measurement- or reward-conditioned posteriors using pretrained diffusion priors. Though widely used for inverse problems, their output is biased even with exact prior scores. First, we quantify this bias for the DPS algorithm, the base method for a family of samplers, by introducing a surrogate curve of probability densities which interpolates between the true posterior and the standard Gaussian. The ratio of sampler to surrogate densities solves a parabolic PDE, and its Feynman--Kac formula yields an explicit representation of this ratio as the expectation of a path functional. This representation reveals which posterior regions are over- or under-sampled. This framework also provides a design principle for bias reduction that includes STSL, a related sampler. Second, in hard-constraint problems (e.g. the low measurement noise regime) these samplers are known to be numerically unstable. We find that this instability of discretized DPS arises from an effective guidance update whose strength scales inversely with step size, inevitably violating the forward-Euler stability criterion near the constraint. This finding motivates implicit (backward-Euler) guidance updates. Finally, we show that the standard practice of early guidance-stopping, used to avoid oscillations, introduces a secondary bias which we characterize through our framework.
Abstract:Diffusion-based posterior samplers draw from measurement- or reward-conditioned posteriors using pretrained diffusion priors. Though widely used for inverse problems, their output is biased even with exact prior scores. First, we quantify this bias for the DPS algorithm, the base method for a family of samplers, by introducing a surrogate curve of probability densities which interpolates between the true posterior and the standard Gaussian. The ratio of sampler to surrogate densities solves a parabolic PDE, and its Feynman--Kac formula yields an explicit representation of this ratio as the expectation of a path functional. This representation reveals which posterior regions are over- or under-sampled. This framework also provides a design principle for bias reduction that includes STSL, a related sampler. Second, in hard-constraint problems (e.g. the low measurement noise regime) these samplers are known to be numerically unstable. We find that this instability of discretized DPS arises from an effective guidance update whose strength scales inversely with step size, inevitably violating the forward-Euler stability criterion near the constraint. This finding motivates implicit (backward-Euler) guidance updates. Finally, we show that the standard practice of early guidance-stopping, used to avoid oscillations, introduces a secondary bias which we characterize through our framework.
Machine Learning (cs.LG)
arXiv:2605.06538 [cs.LG]
(or arXiv:2605.06538v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.06538 Focus to learn more arXiv-issued DOI via DataCite
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Full text under the publisher's open licence · 289 words
Related coverage
Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observations
arXiv · Jun 3, 2026
Generative Translation Priors: Bayesian Imaging with Cross-Modality Image Translation
arXiv · Sep 1, 2026
Informe de riesgos de la IA de frontera (febrero–marzo de 2026)
METR · May 19, 2026
Frontier Risk Report (February to March 2026)
METR · May 19, 2026