  In recent years, deep learning has achieved remarkable empirical success for
image reconstruction. This has catalyzed an ongoing quest for precise
characterization of correctness and reliability of data-driven methods in
critical use-cases, for instance in medical imaging. Notwithstanding the
excellent performance and efficacy of deep learning-based methods, concerns
have been raised regarding their stability, or lack thereof, with serious
practical implications. Significant advances have been made in recent years to
unravel the inner workings of data-driven image recovery methods, challenging
their widely perceived black-box nature. In this article, we will specify
relevant notions of convergence for data-driven image reconstruction, which
will form the basis of a survey of learned methods with mathematically rigorous
reconstruction guarantees. An example that is highlighted is the role of ICNN,
offering the possibility to combine the power of deep learning with classical
convex regularization theory for devising methods that are provably convergent.
  This survey article is aimed at both methodological researchers seeking to
advance the frontiers of our understanding of data-driven image reconstruction
methods as well as practitioners, by providing an accessible description of
useful convergence concepts and by placing some of the existing empirical
practices on a solid mathematical foundation.
