Provable diffusion-based posterior sampling for linear inverse problems via DDIM

2026-07-21

AI Summary

This work addresses the lack of theoretical guarantees and high computational cost in posterior sampling with diffusion models for linear inverse problems by proposing the pddim algorithm. Building upon DDIM, pddim introduces lightweight coordinate-wise corrections informed by the observation model and adaptively selects between prior-driven and measurement-driven prediction strategies based on the signal-to-noise ratio along each singular direction. This approach provides the first rigorous theoretical guarantee of posterior consistency for DDIM-based methods and reduces posterior sampling to an efficient coordinate update rule. Experimental results demonstrate that pddim outperforms existing diffusion-based posterior sampling techniques across multiple image restoration tasks, achieving state-of-the-art performance on most evaluation metrics.

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.

Problem

• diffusion-based posterior sampling
• linear inverse problems
• theoretical guarantees
• computational overhead
• DDIM

Innovation

• posterior sampling
• diffusion models
• linear inverse problems
• DDIM
• provable guarantees