  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.
