Riccio, Danilo; Ehrhardt, Matthias J; Benning, Martin; (2024) Regularization of inverse problems: deep equilibrium models versus bilevel learning. Numerical Algebra, Control and Optimization 10.3934/naco.2023026. (In press).

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Abstract

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 [32]. 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 [3] that has recently been introduced in the context of inverse problems [13]. 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.

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