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, presented in arXiv paper 2507.17079. While diffusion-based methods have shown strong empirical success on inverse problems, existing posterior samplers often lack rigorous theoretical guarantees or impose heavy computational costs. pDDIM 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: 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. The authors prove that the proposed sampler converges to the Bayesian posterior conditioned on the measurements, providing theoretical guarantees largely missing in prior work. Empirically, 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

Field: Machine Learning
Authors: Yuchen Jiao, Na Li, Changxiao Cai
arXiv: 2507.17079

Motivation

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 paper addresses this gap with a method that is both provably correct and computationally lightweight.

The Proposed Method: pDDIM

The authors propose a simple and efficient algorithm, pDDIM, for solving linear inverse problems with diffusion priors via a DDIM-type sampler. Key characteristics:

Requires only lightweight, coordinate-wise modifications to the standard DDIM update, while explicitly incorporating the measurement model.
The core idea is to perform posterior sampling 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.

Theoretical and Empirical Results

Theory: The proposed sampler is proven to converge to the Bayesian posterior conditioned on the measurements, providing posterior consistency guarantees.
Experiments: On a range of image restoration tasks, pDDIM outperforms existing diffusion-based posterior samplers, achieving the best performance on most evaluation metrics.

Takeaway

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

---

*Auto-collected on 2026-07-23.*