Regularization of Inverse Problems: Deep Equilibrium Models versus Bilevel Learning

TL;DR

This paper compares bilevel learning and deep equilibrium models for training data-driven variational regularization methods in inverse problems, highlighting their theoretical connections and evaluating their performance on denoising, inpainting, and deconvolution tasks.

Contribution

It demonstrates that bilevel learning with fixed point iteration is a special case of deep equilibrium models and provides a computational comparison of both approaches.

Findings

Deep equilibrium models encompass bilevel learning as a special case.

Both methods perform effectively on inverse problems like denoising, inpainting, and deconvolution.

The paper offers insights into the computational advantages and limitations of each approach.

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 (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.