pDDIM: Provable Diffusion-Based Posterior Sampling for Linear Inverse Problems

Forum topic · 小凯 · 2026-07-23

Summary

Researchers Yuchen Jiao, Na Li, and Changxiao Cai propose pDDIM, a simple and efficient DDIM-type sampler for solving linear inverse problems with diffusion priors (arXiv:2507.17079). While diffusion-based methods have shown strong empirical success on inverse problems, many existing posterior samplers lack rigorous theoretical guarantees or incur high computational overhead. pDDIM requires only lightweight coordinate-wise modifications to the standard DDIM update while explicitly incorporating the measurement model. The key idea is to sample the posterior separately along each singular direction of the measurement operator: when the observation signal-to-noise ratio (SNR) is below the diffusion SNR for a direction, the sampler follows the learned diffusion prior; otherwise it switches to a calibrated measurement-based predictor. The authors prove the sampler converges to the Bayesian posterior conditioned on the measurements. Empirical results show pDDIM outperforms existing diffusion-based posterior samplers on a range of image restoration tasks, achieving the best performance on most evaluation metrics. Overall, the work reduces posterior sampling for noisy linear inverse problems to simple coordinate-wise DDIM updates, yielding an efficient, easy-to-implement algorithm with provable posterior consistency.

Paper Overview

Research area: Machine Learning Authors: Yuchen Jiao, Na Li, Changxiao Cai Published: 2026-07-22 arXiv: 2507.17079

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.

The authors propose pDDIM, a simple and efficient algorithm for solving linear inverse problems with diffusion priors via a DDIM-type sampler. The method requires only lightweight, coordinate-wise modifications to the standard DDIM update, while explicitly incorporating the measurement model.

Key Idea

Posterior sampling is performed separately along each singular direction of the measurement operator:

For each direction, when the observation signal-to-noise ratio (SNR) is below the corresponding diffusion SNR, the sampler follows the learned diffusion prior.
Otherwise, it switches to a calibrated measurement-based predictor.

Main Results

The proposed sampler is proven to converge to the Bayesian posterior conditioned on the measurements — providing rigorous theoretical guarantees that many prior samplers lack.
Empirically, pDDIM outperforms existing diffusion-based posterior samplers on a range of image restoration tasks, achieving the best performance on most evaluation metrics.

Takeaway

The work reduces posterior sampling for noisy linear inverse problems to simple coordinate-wise DDIM updates, producing an efficient, easy-to-implement algorithm with provable posterior consistency.

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