---
title: "Provable Diffusion-based Posterior Sampling for Linear Inverse Problems via DDIM"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2607.19333v1"
permalink: "https://feed.astralor.com/items/itm_7749e751c32ff2bc851a72a2"
published: "2026-07-21T17:53:36.000Z"
score: 79
channel: "ai"
tags: ["DDIM","PDDIM","Posterior Sampling","Image Restoration","Diffusion Models","Linear Inverse Problems"]
---

# Provable Diffusion-based Posterior Sampling for Linear Inverse Problems via DDIM

## AI commentary

> This paper reduces diffusion-based posterior sampling to coordinate-wise DDIM updates with a clean theoretical proof, marking a significant step for diffusion-based inverse problem solving. It is particularly valuable for researchers in image restoration and compressed sensing, offering an efficient and provable practical tool.

## At a glance

This paper proposes a simple and efficient algorithm called PDDIM 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. The key idea is to perform posterior sampling separately along each singular direction: when the observation SNR is below the diffusion SNR, it follows the learned prior; otherwise, it switches to a calibrated measurement-based predictor. The sampler is proven to converge to the Bayesian posterior conditioned on measurements. Empirical results show favorable performance on image restoration tasks.

## Key points

- Proposes PDDIM algorithm, requiring only coordinate-wise modifications to DDIM for posterior sampling.
- Provides theoretical proof that the sampler converges to the conditional posterior distribution.
- Achieves state-of-the-art performance on most metrics in image restoration tasks.

## Full text

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.

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