SIAM Journal on Imaging Sciences, Volume 14, Issue 3, Page 1206-1237, January 2021.
We introduce a new paradigm for solving regularized variational problems. These are typically formulated to address ill-posed inverse problems encountered in signal and image processing. The objective function is traditionally defined by adding a regularization function to a data fit term, which is subsequently minimized by using iterative optimization algorithms. Recently, several works have proposed to replace the operator related to the regularization by a more sophisticated denoiser. These approaches, known as plug-and-play (PnP) methods, have shown excellent performance. Although it has been noticed that, under some Lipschitz properties on the denoisers, the convergence of the resulting algorithm is guaranteed, little is known about characterizing the asymptotically delivered solution. In the current article, we propose to address this limitation. More specifically, instead of employing a functional regularization, we perform an operator regularization, where a maximally monotone operator (MMO) is learned in a supervised manner. This formulation is flexible as it allows the solution to be characterized through a broad range of variational inequalities, and it includes convex regularizations as special cases. From an algorithmic standpoint, the proposed approach consists in replacing the resolvent of the MMO by a neural network (NN). We present a universal approximation theorem proving that nonexpansive NNs are suitable models for the resolvent of a wide class of MMOs. The proposed approach thus provides a sound theoretical framework for analyzing the asymptotic behavior of first-order PnP algorithms. In addition, we propose a numerical strategy to train NNs corresponding to resolvents of MMOs. We apply our approach to image restoration problems and demonstrate its validity in terms of both convergence and quality.

中文翻译：

---

学习用于图像恢复的最大单调运算符

SIAM Journal on Imaging Sciences，第 14 卷，第 3 期，第 1206-1237 页，2021 年 1 月。
我们引入了一种解决正则化变分问题的新范式。这些通常用于解决信号和图像处理中遇到的不适定逆问题。目标函数传统上是通过向数据拟合项添加正则化函数来定义的，随后使用迭代优化算法将其最小化。最近，有几项工作提出用更复杂的降噪器替换与正则化相关的算子。这些被称为即插即用 (PnP) 方法的方法已显示出出色的性能。尽管已经注意到，在降噪器的某些 Lipschitz 特性下，可以保证所得算法的收敛性，但对表征渐近传递的解决方案知之甚少。在当前的文章中，我们建议解决这个限制。更具体地说，我们不使用函数正则化，而是执行算子正则化，其中以有监督的方式学习最大单调算子 (MMO)。这个公式很灵活，因为它允许通过广泛的变分不等式来表征解决方案，并且它包括作为特殊情况的凸正则化。从算法的角度来看，所提出的方法包括用神经网络 (NN) 替换 MMO 的解析器。我们提出了一个通用逼近定理，证明非膨胀 NN 是适用于解决各种 MMO 的模型。因此，所提出的方法为分析一阶 PnP 算法的渐近行为提供了一个合理的理论框架。此外，我们提出了一种数值策略来训练与 MMO 的解决方案相对应的 NN。我们将我们的方法应用于图像恢复问题，并证明其在收敛性和质量方面的有效性。