  Inverse problems are fundamental in fields like medical imaging, geophysics,
and computerized tomography, aiming to recover unknown quantities from observed
data. However, these problems often lack stability due to noise and
ill-conditioning, leading to inaccurate reconstructions. To mitigate these
issues, regularization methods are employed, introducing constraints to
stabilize the inversion process and achieve a meaningful solution. Recent
research has shown that the application of regularizing filters to diagonal
frame decompositions (DFD) yields regularization methods. These filters dampen
some frame coefficients to prevent noise amplification. This paper introduces a
non-linear filtered DFD method combined with a learning strategy for
determining optimal non-linear filters from training data pairs. In our
experiments, we applied this approach to the inversion of the Radon transform
using 500 image-sinogram pairs from real CT scans. Although the learned filters
were found to be strictly increasing, they did not satisfy the
non-expansiveness condition required to link them with convex regularizers and
prove stability and convergence in the sense of regularization methods in
previous works. Inspired by this, the paper relaxes the non-expansiveness
condition, resulting in weakly convex regularization. Despite this relaxation,
we managed to derive stability, convergence, and convergence rates with respect
to the absolute symmetric Bregman distance for the learned non-linear
regularizing filters. Extensive numerical results demonstrate the effectiveness
of the proposed method in achieving stable and accurate reconstructions.
