  It's well-known that inverse problems are ill-posed and to solve them
meaningfully one has to employ regularization methods. Traditionally, popular
regularization methods have been the penalized Variational approaches. In
recent years, the classical regularized-reconstruction approaches have been
outclassed by the (deep-learning-based) learned reconstruction algorithms.
However, unlike the traditional regularization methods, the theoretical
underpinnings, such as stability and regularization, have been insufficient for
such learned reconstruction algorithms. Hence, the results obtained from such
algorithms, though empirically outstanding, can't always be completely trusted,
as they may contain certain instabilities or (hallucinated) features arising
from the learned process. In fact, it has been shown that such learning
algorithms are very susceptible to small (adversarial) noises in the data and
can lead to severe instabilities in the recovered solution, which can be quite
different than the inherent instabilities of the ill-posed (inverse) problem.
Whereas, the classical regularization methods can handle such (adversarial)
noises very well and can produce stable recovery. Here, we try to present
certain regularization methods to stabilize such (unstable) learned
reconstruction methods and recover a regularized solution, even in the presence
of adversarial noises. For this, we need to extend the classical notion of
regularization and incorporate it in the learned reconstruction algorithms. We
also present some regularization techniques to regularize two of the most
popular learning reconstruction algorithms, the Learned Post-Processing
Reconstruction and the Learned Unrolling Reconstruction.
