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成像领域的重要研究方向之一，其优势在于即使缺乏配对的高质量训练数据，也能学习到具有表达力和强大性能的重构算子。本章从理论角度系统梳理了用于求解成像逆问题的无监督学习方案，特别聚焦于基于最优传输和凸分析的方法。首先回顾基于最优传输的无监督方法，例如基于循环一致性模型和学习的对抗正则化方法，这些方法具有清晰的概率解释。随后概述近期关于可证明收敛的学习优化算法系列工作及其在加速成像逆问题求解中的应用，同时介绍相应的专用无监督训练方案。此外，本文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），这些算法是成像领域最重要且应用最广泛的无监督方法之一。在本综述的结尾，我们概述了若干补充性无监督学习框架。通过详尽的综述，我们系统介绍了本章方法所依托的关键数学结论，以确保讨论的自洽性。