Provable diffusion-based posterior sampling for linear inverse problems via DDIM
2026-07-21 • Machine Learning
Machine LearningArtificial Intelligence
AI summaryⓘ
The authors present a new algorithm called PDDIM to solve linear inverse problems using diffusion models, which are a type of method to generate data. Their approach modifies an existing sampler (DDIM) in a simple way by handling each measurement direction separately and switching between learned priors and measurement-based predictions depending on noise levels. They prove their method converges correctly to the true underlying distribution given the measurements. Experiments show their method works better or as well as current samplers on image restoration tasks and is easy to implement with theoretical guarantees.
diffusion modelsinverse problemsposterior samplingDDIM samplerlinear measurement operatorsingular value decompositionsignal-to-noise ratioBayesian posteriorimage restorationcoordinate-wise updates
Authors
Yuchen Jiao, Na Li, Changxiao Cai, Yuxin Chen, Gen Li
Abstract
Diffusion-based methods have achieved remarkable empirical success in solving inverse problems. However, many existing posterior samplers either lack rigorous theoretical guarantees or incur substantial computational overhead. We propose a simple and efficient algorithm, called \pddim, for solving linear inverse problems with diffusion priors via a DDIM-type sampler. Our method requires only lightweight, coordinate-wise modifications to the standard DDIM update, while explicitly incorporating the measurement model. The key idea is to perform posterior sampling separately along each singular direction of the measurement operator: for each direction, the sampler follows the learned diffusion prior when the observation signal-to-noise ratio (SNR) is below the corresponding diffusion SNR, and switches to a calibrated measurement-based predictor otherwise. We prove that the proposed sampler converges to the Bayesian posterior conditioned on the measurements. Empirical results show that the proposed sampler performs favorably against existing diffusion-based posterior samplers across a range of image restoration tasks, achieving the best performance on the majority of evaluation metrics considered. Overall, our results convert posterior sampling for noisy linear inverse problems to simple coordinate-wise DDIM updates, yielding an efficient, easy-to-implement algorithm with provable posterior consistency.