Abstract
Deep Equilibrium Models (DEQs) are implicit neural networks with fixed points, which have recently gained attention for learning image regularization functionals, particularly in settings involving Gaussian fidelities, where assumptions on the forward operator ensure contractiveness of standard (proximal) Gradient Descent operators. In this work, we extend the application of DEQs to Poisson inverse problems, where the data fidelity term is more appropriately modeled by the Kullback-Leibler divergence. To this end, we introduce a novel DEQ formulation based on Mirror Descent defined in terms of a tailored non-Euclidean geometry that naturally adapts with the structure of the data term. This enables the learning of neural regularizers within a principled training framework. We derive sufficient conditions to guarantee the convergence of the learned reconstruction scheme and propose computational strategies that enable both efficient training and fully parameter-free inference. Numerical experiments show that our method outperforms traditional model-based approaches and it is comparable to the performance of Bregman Plug-and-Play methods, while mitigating their typical drawbacks - namely, sensitivity to initialization and careful tuning of hyperparameters. The code is publicly available at this https URL.
Abstract (translated)
深度平衡模型（DEQs）是一种隐式神经网络，具有固定点特性，在学习图像正则化函数方面近期受到了广泛关注，特别是在涉及高斯保真度的情况下。在这种情况下，关于前向操作符的假设保证了标准梯度下降算子的收敛性。在这项工作中，我们将DEQ的应用扩展到了泊松逆问题领域，在此背景下数据保真度项更适宜用Kullback-Leibler散度来建模。 为此，我们提出了一种新的基于镜像下降（Mirror Descent）的DEQ模型，该模型根据特定于任务而非欧几里得几何结构进行定义。这种设计能够自然地适应数据项的特性，并允许在网络框架内学习神经正则化器。我们推导了保证学习重构方案收敛性的充分条件，并提出计算策略以实现高效训练和完全无需参数调整的推理过程。 数值实验表明，与传统模型驱动的方法相比，我们的方法性能更优；同时，在表现上也与Bregman Plug-and-Play方法相当，但避免了后者常见的缺点——即对初始化敏感以及需要精细调参。代码在提供的URL中公开发布。
URL
https://arxiv.org/abs/2507.11461
