Scale-Consistent Posterior Dynamics for Diffusion Inverse Problems

2026-08-15

AI Summary

This study addresses the challenges of intractable likelihoods and inconsistent transport scales in posterior sampling for diffusion inverse problems by proposing a scale-consistent posterior SDE framework. The method integrates a Langevin corrector with a variance-matched IMEX splitting-step integrator, providing theoretical guarantees of marginal invariance and convergence. Experiments on FFHQ and ImageNet benchmarks demonstrate that high-fidelity reconstructions for super-resolution and deblurring are achievable with only 100 function evaluations. These results indicate significant improvements in both sampling efficiency and generation quality, establishing a novel paradigm for efficiently solving diffusion inverse problems.

Abstract

Posterior sampling with a pretrained diffusion prior is governed by a conditional score whose intermediate likelihood component is generally intractable. We begin from an ideal one-parameter posterior SDE family in which a stochasticity parameter controls probability-flow transport and stochastic exploration without changing the posterior marginals. To obtain a tractable model, we express the likelihood in a rescaled clean-image coordinate and use log-SNR to organize the resulting posterior proxies. Projecting the diffusion uncertainty through the forward operator then yields a noise-conditioned covariance path whose targets approach the clean posterior. Because endpoint consistency of these targets does not ensure that a surrogate transport follows them, we interleave the transport with a frozen-target Langevin corrector, producing a continuous surrogate SDE. We discretize this model with an outer Lie--Trotter splitting and a variance-matched split-step IMEX predictor that treats the learned prior explicitly, the linear likelihood implicitly, and the stochastic innovation after the implicit solve. We prove marginal invariance of the ideal family, posterior convergence of the continuous surrogate under mixing and transport-defect conditions, and a first-order weak error bound for the discrete algorithm. Experiments on FFHQ and ImageNet with 100 score evaluations demonstrate competitive reconstruction fidelity for super-resolution and deblurring. A controlled 100-image ablation separates scale consistency from the finite-step effects of stochastic-increment placement, continuation, and corrector allocation. A separate noiseless box-inpainting study shows that large exploration reaches a performance plateau only when the matched innovation is injected after the stiff likelihood solve.

Problem

• Diffusion Inverse Problems
• Posterior Sampling
• Intractable Likelihood
• Conditional Score

Innovation

• Scale-Consistent Posterior Dynamics
• Stochasticity Parameter
• Frozen-Target Langevin Corrector
• Variance-Matched IMEX Predictor
• Log-SNR Organization