Deep learning had remarkable empirical success in recent years in solving a variety of inverse problems in

imaging, including image denoising, deblurring, tomographic image reconstruction, and image inpainting, just

to name a few. This has catalyzed an ongoing quest for precise characterization of correctness and reliability of

such data-driven methods in critical use-cases, e.g., those arising in medical imaging.

Notwithstanding the excellent empirical performance of data-driven methods for image reconstruction, concerns

have been raised regarding their stability, or lack thereof, with serious practical implications. In applications

where imaging is used for discovering new scientific phenomena, it is important to have mathematical

guarantees to ensure the correctness and reliability of the reconstructed images.

In this talk, we will introduce different notions of convergence pertaining to image reconstruction problems

and provide a broad overview of recent data-driven techniques that satisfy some of those convergence properties.

In particular, the talk will focus on data-driven regularization methods through explicit functionals parametrized

by neural networks and via plug-and-play deeply-learned denoisers]. We will highlight the requisite

properties that such a regularization functional or a denoiser must satisfy to establish convergence guarantees

of different types.