# Convergence analysis of equilibrium methods for inverse problems

arXiv: https://arxiv.org/abs/2306.01421 · HTML: https://mlsanity.com/paper/2306.01421/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 32 references that are in the map.
- First posted on arXiv 2 Jun 2023.

## Builds on (most cited first, 16 of 16)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems](https://mlsanity.com/paper/1704.03488/) (2017)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [Plug-and-Play Methods for Integrating Physical and Learned Models in Computational Imaging](https://mlsanity.com/paper/2203.17061/) (2022)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [A Review of Convolutional Neural Networks for Inverse Problems in Imaging](https://mlsanity.com/paper/1710.04011/) (2017)
- [An Online Plug-and-Play Algorithm for Regularized Image Reconstruction](https://mlsanity.com/paper/1809.04693/) (2018)
- [Imaging with Equivariant Deep Learning](https://mlsanity.com/paper/2209.01725/) (2022)
- [Provably Convergent Plug-and-Play Quasi-Newton Methods](https://mlsanity.com/paper/2303.07271/) (2023)
- [PnP-ReG: Learned Regularizing Gradient for Plug-and-Play Gradient Descent](https://mlsanity.com/paper/2204.13940/) (2022)
- [Regularization of Inverse Problems: Deep Equilibrium Models versus Bilevel Learning](https://mlsanity.com/paper/2206.13193/) (2022)
- [Convergence rates for critical point regularization](https://mlsanity.com/paper/2302.08830/) (2023)

## Built on by (most cited first, 2 of 2)

- [Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation](https://mlsanity.com/paper/2402.01052/) (2024)
- [Invertible ResNets for Inverse Imaging Problems: Competitive Performance with Provable Regularization Properties](https://mlsanity.com/paper/2409.13482/) (2024)

## Abstract (arXiv)

Solving inverse problems (Ax = y) is central to a variety of practically important fields such as medical imaging, remote sensing, and non-destructive testing. The most successful and theoretically best-understood method is convex variational regularization, where approximate but stable solutions are defined as minimizers of ( |A(·) - yδ|2 / 2 + α R(·)), with (R) a regularization functional. Recent methods such as deep equilibrium models and plug-and-play approaches, however, go beyond variational regularization. Motivated by these innovations, we introduce implicit non-variational (INV) regularization, where approximate solutions are defined as solutions of (A^*(A x - yδ) + α R(x) = 0) for some regularization operator (R). When the regularization operator is the gradient of a functional, INV reduces to classical variational regularization. However, in methods like DEQ and PnP, (R) is not a gradient field, and the existing theoretical foundation remains incomplete. To address this, we establish stability and convergence results in this broader setting, including convergence rates and stability estimates measured via a absolute Bregman distance.