This paper proposes a data-driven approach for constructing firmly nonexpansive operators. We demonstrate its applicability in Plug-and-Play (PnP) methods, where classical algorithms such as Forward-Backward splitting, Chambolle-Pock primal-dual iteration, Douglas-Rachford iteration or alternating directions method of multipliers (ADMM), are modified by replacing one proximal map by a learned firmly nonexpansive operator. We provide sound mathematical background to the problem of learning such an operator via expected and empirical risk minimization. We prove that, as the number of training points increases, the empirical risk minimization problem converges (in the sense of Gamma-convergence) to the expected risk minimization problem. Further, we derive a solution strategy that ensures firmly nonexpansive and piecewise affine operators within the convex envelope of the training set. We show that this operator converges to the best empirical solution as the number of points in the envelope increases in an appropriate way. Finally, the experimental section details practical implementations of the method and presents an application in image denoising, where we consider a novel, interpretable PnP Chambolle-Pock primal-dual iteration.

翻译：本文提出了一种数据驱动的方法来构建严格非扩张算子。我们展示了其在即插即用方法中的适用性，其中通过将经典算法（如前向-后向分裂、Chambolle-Pock原始-对偶迭代、Douglas-Rachford迭代或交替方向乘子法）中的一个邻近映射替换为学习得到的严格非扩张算子进行改进。我们为通过期望和经验风险最小化学习此类算子的问题提供了坚实的数学基础。我们证明，随着训练点数量的增加，经验风险最小化问题（在Gamma收敛的意义上）收敛于期望风险最小化问题。进一步，我们推导出一种确保在训练集凸包内得到严格非扩张且分段仿射算子的求解策略。我们证明，当凸包中点数以适当方式增加时，该算子收敛于最佳经验解。最后，实验部分详细阐述了该方法的实际实现，并展示了在图像去噪中的应用，其中我们考虑了一种新颖且可解释的即插即用Chambolle-Pock原始-对偶迭代。