This paper investigates the convergence properties and applications of the three-operator splitting method, also known as Davis-Yin splitting (DYS) method, integrated with extrapolation and Plug-and-Play (PnP) denoiser within a nonconvex framework. We first propose an extrapolated DYS method to effectively solve a class of structural nonconvex optimization problems that involve minimizing the sum of three possible nonconvex functions. Our approach provides an algorithmic framework that encompasses both extrapolated forward-backward splitting and extrapolated Douglas-Rachford splitting methods.To establish the convergence of the proposed method, we rigorously analyze its behavior based on the Kurdyka-{\L}ojasiewicz property, subject to some tight parameter conditions. Moreover, we introduce two extrapolated PnP-DYS methods with convergence guarantee, where the traditional regularization prior is replaced by a gradient step-based denoiser. This denoiser is designed using a differentiable neural network and can be reformulated as the proximal operator of a specific nonconvex functional. We conduct extensive experiments on image deblurring and image super-resolution problems, where our results showcase the advantage of the extrapolation strategy and the superior performance of the learning-based model that incorporates the PnP denoiser in terms of achieving high-quality recovery images.

翻译：本文研究了集成外推策略与即插即用（PnP）去噪器的三算子分裂方法（即Davis-Yin分裂方法，DYS）在非凸框架下的收敛性质与应用。我们首先提出一种外推DYS方法，用于有效求解一类涉及三个可能非凸函数之和最小化的结构化非凸优化问题。该方法构建了一个算法框架，同时涵盖外推前向-后向分裂和外推Douglas-Rachford分裂方法。为证明所提方法的收敛性，我们基于Kurdyka-Łojasiewicz性质，在严格参数条件下对其行为进行了严谨分析。此外，我们引入两种具有收敛保证的外推PnP-DYS方法，其中传统正则化先验被基于梯度步的去噪器替代。该去噪器采用可微分神经网络设计，并可重写为特定非凸泛函的近端算子。我们在图像去模糊和图像超分辨率问题上开展了大量实验，结果表明外推策略的优势以及融合PnP去噪器的学习模型在实现高质量恢复图像方面展现出卓越性能。