Recently, the use of deep equilibrium methods has emerged as a new approach for solving imaging and other ill-posed inverse problems. While learned components may be a key factor in the good performance of these methods in practice, a theoretical justification from a regularization point of view is still lacking. In this paper, we address this issue by providing stability and convergence results for the class of equilibrium methods. In addition, we derive convergence rates and stability estimates in the symmetric Bregman distance. We strengthen our results for regularization operators with contractive residuals. Furthermore, we use the presented analysis to gain insight into the practical behavior of these methods, including a lower bound on the performance of the regularized solutions. In addition, we show that the convergence analysis leads to the design of a new type of loss function which has several advantages over previous ones. Numerical simulations are used to support our findings.

翻译：近期，深度均衡方法作为一种解决成像及其他不适定逆问题的新途径崭露头角。尽管学习组件可能是这些方法在实践中表现优异的关键因素，但从正则化角度出发的理论证明仍付阙如。本文针对这一问题，给出了均衡方法类的稳定性与收敛性结果。此外，我们在对称Bregman距离下推导了收敛速率与稳定性估计，并针对具有压缩残差的正则化算子强化了相关结论。进一步地，我们利用所提出的分析揭示了这些方法的实际行为特性，包括正则化解性能的下界。同时，研究表明收敛性分析能够指导设计一种新型损失函数，该函数相比以往方法具有多重优势。数值模拟实验支撑了我们的发现。