提出新框架让得分模型更稳定融入优化算法,解决图像修复中的收敛难题。
Taming Score-Based Denoisers in ADMM: A Convergent Plug-and-Play Framework
- 设计三阶段去噪器,自动校正+方向修正+得分去噪,匹配优化迭代路径
- 证明在固定步长下可高概率收敛到解球内,自适应步长下保证全局收敛
- 适合图像恢复等逆问题,尤其对传统方法难处理的噪声或缺失数据有效
尽管基于得分的生成模型已成为求解逆问题的强大先验,但将其直接整合进如ADMM之类的优化算法仍具挑战性。核心问题在于:一、训练时使用的噪声数据流形与ADMM迭代路径几何不匹配,尤其受对偶变量影响;二、使用得分去噪器时缺乏收敛性理论支撑。为此,我们提出一种新型ADMM-PnP框架,引入三阶段去噪器AC-DC:(1)通过加性高斯噪声实现自校正(AC),(2)利用条件Langevin动力学进行方向修正(DC),(3)完成得分去噪。在收敛性方面,我们建立两个结果:首先,在适当参数下,每轮ADMM迭代为弱非扩张算子,可使用常数步长实现高概率固定点“球收敛”;其次,在较宽松条件下,AC-DC去噪器为有界去噪器,支持自适应步长下的收敛。在多种逆问题上的实验表明,该方法在解质量上持续优于各类基线。
原文摘要 · Abstract (English)
While score-based generative models have emerged as powerful priors for solving inverse problems, directly integrating them into optimization algorithms such as ADMM remains nontrivial. Two central challenges arise: i) the mismatch between the noisy data manifolds used to train the score functions and the geometry of ADMM iterates, especially due to the influence of dual variables, and ii) the lack of convergence understanding when ADMM is equipped with score-based denoisers. To address the manifold mismatch issue, we propose ADMM plug-and-play (ADMM-PnP) with the AC-DC denoiser, a new framework that embeds a three-stage denoiser into ADMM: (1) auto-correction (AC) via additive Gaussian noise, (2) directional correction (DC) using conditional Langevin dynamics, and (3) score-based denoising. In terms of convergence, we establish two results: first, under proper denoiser parameters, each ADMM iteration is a weakly nonexpansive operator, ensuring high-probability fixed-point $\textit{ball convergence}$ using a constant step size; second, under more relaxed conditions, the AC-DC denoiser is a bounded denoiser, which leads to convergence under an adaptive step size schedule. Experiments on a range of inverse problems demonstrate that our method consistently improves solution quality over a variety of baselines.
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