Ill-posed inverse problems are fundamental in many domains, ranging from astrophysics to medical imaging. Emerging diffusion models provide a powerful prior for solving these problems. Existing maximum-a-posteriori (MAP) or posterior sampling approaches, however, rely on different computational approximations, leading to inaccurate or suboptimal samples. To address this issue, we introduce a new approach to solving MAP problems with diffusion model priors using a dual ascent optimization framework. Our framework achieves better image quality as measured by various metrics for image restoration problems, it is more robust to high levels of measurement noise, it is faster, and it estimates solutions that represent the observations more faithfully than the state of the art.
DDiff employs the Alternating Direction Method of Multipliers (ADMM) framework, which introduces a slack variable z with the constraint x = z. This decomposition allows the data fidelity and prior terms to be handled separately through three alternating update steps.
The primary innovation of DDiff lies in its novel z-update formulation. Previous methods like DiffPIR use Half-Quadratic Splitting (HQS) without dual variables, while naive implementations of diffusion models in ADMM frameworks suffer from off-manifold inputs to the score function.
DDiff addresses this through two key contributions:
1. Proper Integration of Dual Variables: Unlike HQS-based methods, DDiff maintains dual variables that accumulate constraint violations, leading to improved measurement consistency.
2. On-Manifold Score Evaluation: The specialized noising step ensures that the diffusion model's score function operates on inputs consistent with its training distribution, enabling more accurate prior application.
Ablation studies confirm that both components are essential and must work together - using dual variables without the noising step actually degrades performance compared to not using dual variables at all.
DDiff was evaluated across a comprehensive suite of linear and nonlinear inverse problems using FFHQ256×256 and ImageNet256×256 datasets. The method was compared against leading approaches: Diffusion Posterior Sampling (DPS), Decoupled Annealing Posterior Sampling (DAPS), and DiffPIR.
DDiff achieves significantly lower LPIPS scores (better perceptual quality) across most tasks and exhibits strong performance in PSNR and SSIM metrics.
Lower residual errors indicate reconstructions that more faithfully match the observed data.
A critical finding is DDiff's exceptional robustness to measurement noise. While competing methods show rapidly degrading performance as noise increases, DDiff maintains high-quality reconstructions even under severe noise conditions. This characteristic is particularly valuable for real-world applications such as low-dose CT imaging or cryo-electron microscopy where noise is inherent to the measurement process.
DDiff demonstrates superior computational efficiency compared to sampling-based methods like DAPS:
The efficiency gains stem from:
@misc{kim2025dualascentdiffusioninverse,
title={Dual Ascent Diffusion for Inverse Problems},
author={Minseo Kim and Axel Levy and Gordon Wetzstein},
year={2025},
eprint={2505.17353},
archivePrefix={arXiv},
primaryClass={cs.CV},
url={https://arxiv.org/abs/2505.17353},
}