  Variational regularization methods are commonly used to approximate solutions
of inverse problems. In recent years, model-based variational regularization
methods have often been replaced with data-driven ones such as the
fields-of-expert model (Roth and Black, 2009). Training the parameters of such
data-driven methods can be formulated as a bilevel optimization problem. In
this paper, we compare the framework of bilevel learning for the training of
data-driven variational regularization models with the novel framework of deep
equilibrium models (Bai, Kolter, and Koltun, 2019) that has recently been
introduced in the context of inverse problems (Gilton, Ongie, and Willett,
2021). We show that computing the lower-level optimization problem within the
bilevel formulation with a fixed point iteration is a special case of the deep
equilibrium framework. We compare both approaches computationally, with a
variety of numerical examples for the inverse problems of denoising, inpainting
and deconvolution.
