  Recently, the use of deep equilibrium methods has emerged as a new approach
for solving imaging and other ill-posed inverse problems. While learned
components may be a key factor in the good performance of these methods in
practice, a theoretical justification from a regularization point of view is
still lacking. In this paper, we address this issue by providing stability and
convergence results for the class of equilibrium methods. In addition, we
derive convergence rates and stability estimates in the symmetric Bregman
distance. We strengthen our results for regularization operators with
contractive residuals. Furthermore, we use the presented analysis to gain
insight into the practical behavior of these methods, including a lower bound
on the performance of the regularized solutions. In addition, we show that the
convergence analysis leads to the design of a new type of loss function which
has several advantages over previous ones. Numerical simulations are used to
support our findings.
