Unsupervised deep learning approaches have recently become one of the crucial research areas in imaging owing to their ability to learn expressive and powerful reconstruction operators even when paired high-quality training data is scarcely available. In this chapter, we review theoretically principled unsupervised learning schemes for solving imaging inverse problems, with a particular focus on methods rooted in optimal transport and convex analysis. We begin by reviewing the optimal transport-based unsupervised approaches such as the cycle-consistency-based models and learned adversarial regularization methods, which have clear probabilistic interpretations. Subsequently, we give an overview of a recent line of works on provably convergent learned optimization algorithms applied to accelerate the solution of imaging inverse problems, alongside their dedicated unsupervised training schemes. We also survey a number of provably convergent plug-and-play algorithms (based on gradient-step deep denoisers), which are among the most important and widely applied unsupervised approaches for imaging problems. At the end of this survey, we provide an overview of a few related unsupervised learning frameworks that complement our focused schemes. Together with a detailed survey, we provide an overview of the key mathematical results that underlie the methods reviewed in the chapter to keep our discussion self-contained.

翻译：无监督深度学习方法近年来已成为图像处理领域的关键研究方向之一，因其能够在高质量配对训练数据稀缺时，仍能学习具有表达力和强大性能的重建算子。本章从理论角度综述用于求解图像逆问题的无监督学习方案，特别关注基于最优输运和凸分析的方法。我们首先回顾基于最优输运的无监督方法，例如循环一致性模型和学习型对抗正则化方法，这些方法具有清晰的概率解释。随后，我们概述近期关于可证明收敛的学习优化算法的一系列工作，这些算法用于加速图像逆问题的求解，并结合其专门的无监督训练方案。我们还综述了若干可证明收敛的即插即用算法（基于梯度步深度去噪器），这些算法是图像问题中最重要且应用最广泛的无监督方法之一。在本综述末尾，我们简要介绍若干补充本章重点框架的相关无监督学习框架。结合详细综述，我们提供本章所评述方法的关键数学结果概述，以保持讨论的自洽性。