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A relaxed proximal gradient descent algorithm for convergent plug-and-play with proximal denoiser

2023-01-31 16:11:47
Samuel Hurault, Antonin Chambolle, Arthur Leclaire, Nicolas Papadakis

Abstract

This paper presents a new convergent Plug-and-Play (PnP) algorithm. PnP methods are efficient iterative algorithms for solving image inverse problems formulated as the minimization of the sum of a data-fidelity term and a regularization term. PnP methods perform regularization by plugging a pre-trained denoiser in a proximal algorithm, such as Proximal Gradient Descent (PGD). To ensure convergence of PnP schemes, many works study specific parametrizations of deep denoisers. However, existing results require either unverifiable or suboptimal hypotheses on the denoiser, or assume restrictive conditions on the parameters of the inverse problem. Observing that these limitations can be due to the proximal algorithm in use, we study a relaxed version of the PGD algorithm for minimizing the sum of a convex function and a weakly convex one. When plugged with a relaxed proximal denoiser, we show that the proposed PnP-$\alpha$PGD algorithm converges for a wider range of regularization parameters, thus allowing more accurate image restoration.

Abstract (translated)

本 paper 提出了一种新的可重复插值算法(PnP)。PnP 方法是一种高效的迭代算法,用于解决图像逆问题,其表述为数据模态与正则项之和的最小化。PnP 方法通过将训练好的噪声去除了的近邻算法(如 Proximal Gradient Descent(PGD))的插值端点与噪声去除算法相连接来实现正则项的添加。为了确保 PnP 方案的收敛,许多工作研究了深度噪声去除算法的具体参数设定。然而,现有的结果需要不可验证或最优假设,或者需要对逆问题参数的限定条件。观察这些限制可能是由于正在使用的近邻算法导致的,因此我们研究了一种放松了的 PGD 算法,以最小化一个凸函数和一个弱凸函数之和。当使用放松了的近邻噪声去除算法时,我们展示了,所提出的 PnP-$alpha$ PGD 算法收敛于更广泛范围内的正则项参数,从而允许更精确的图像恢复。

URL

https://arxiv.org/abs/2301.13731

PDF

https://arxiv.org/pdf/2301.13731.pdf


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