In plug-and-play (PnP) regularization, the proximal operator in algorithms such as ISTA and ADMM is replaced by a powerful denoiser. This formal substitution works surprisingly well in practice. In fact, PnP has been shown to give state-of-the-art results for various imaging applications. The empirical success of PnP has motivated researchers to understand its theoretical underpinnings and, in particular, its convergence. It was shown in prior work that for kernel denoisers such as the nonlocal means, PnP-ISTA provably converges under some strong assumptions on the forward model. The present work is motivated by the following questions: Can we relax the assumptions on the forward model? Can the convergence analysis be extended to PnP-ADMM? Can we estimate the convergence rate? In this letter, we resolve these questions using the contraction mapping theorem: (i) for symmetric denoisers, we show that (under mild conditions) PnP-ISTA and PnP-ADMM exhibit linear convergence; and (ii) for kernel denoisers, we show that PnP-ISTA and PnP-ADMM converge linearly for image inpainting. We validate our theoretical findings using reconstruction experiments.

翻译：在插件即用（PnP）正则化方法中，ISTA和ADMM等算法中的近端算子被替换为强大的去噪器。这种形式上的替换在实际应用中取得了惊人的效果，事实上，PnP已在多种图像处理任务中展现出最优性能。PnP的经验成功促使研究者探索其理论基础，特别是收敛性问题。既有研究表明，对于非局部均值等核去噪器，PnP-ISTA在前向模型强约束条件下具有可证明的收敛性。本文旨在解决以下问题：能否放宽前向模型约束条件？收敛性分析能否推广至PnP-ADMM？能否估计收敛速率？通过压缩映射定理，本文解决了上述问题：（i）对于对称去噪器，我们证明（在温和条件下）PnP-ISTA和PnP-ADMM具有线性收敛性；（ii）对于核去噪器，我们证明PnP-ISTA和PnP-ADMM在图像修复中线性收敛。我们通过重建实验验证了理论结论。