# Neumann Networks for Inverse Problems in Imaging

arXiv: https://arxiv.org/abs/1901.03707 · HTML: https://mlsanity.com/paper/1901.03707/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jan 2019.

## Built on by (most cited first, 9 of 9)

- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [Model-Based Deep Learning: On the Intersection of Deep Learning and Optimization](https://mlsanity.com/paper/2205.02640/) (2022)
- [Solving Inverse Problems by Joint Posterior Maximization with Autoencoding Prior](https://mlsanity.com/paper/2103.01648/) (2021)
- [On Maximum-a-Posteriori estimation with Plug & Play priors and stochastic gradient descent](https://mlsanity.com/paper/2201.06133/) (2022)
- [Deep Unrolled Network for Video Super-Resolution](https://mlsanity.com/paper/2102.11720/) (2021)
- [Video Restoration with a Deep Plug-and-Play Prior](https://mlsanity.com/paper/2209.02854/) (2022)
- [Solving the Wide-band Inverse Scattering Problem via Equivariant Neural Networks](https://mlsanity.com/paper/2212.06068/) (2022)
- [Comprehensive Examination of Unrolled Networks for Solving Linear Inverse Problems](https://mlsanity.com/paper/2501.04608/) (2025)
- [Integrating Reweighted Least Squares with Plug-and-Play Diffusion Priors for Noisy Image Restoration](https://mlsanity.com/paper/2511.06823/) (2025)

## Abstract (arXiv)

Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, deconvolution, inpainting, compressed sensing, and superresolution all lie in this framework. Traditional inverse problem solvers minimize a cost function consisting of a data-fit term, which measures how well an image matches the observations, and a regularizer, which reflects prior knowledge and promotes images with desirable properties like smoothness. Recent advances in machine learning and image processing have illustrated that it is often possible to learn a regularizer from training data that can outperform more traditional regularizers. We present an end-to-end, data-driven method of solving inverse problems inspired by the Neumann series, which we call a Neumann network. Rather than unroll an iterative optimization algorithm, we truncate a Neumann series which directly solves the linear inverse problem with a data-driven nonlinear regularizer. The Neumann network architecture outperforms traditional inverse problem solution methods, model-free deep learning approaches, and state-of-the-art unrolled iterative methods on standard datasets. Finally, when the images belong to a union of subspaces and under appropriate assumptions on the forward model, we prove there exists a Neumann network configuration that well-approximates the optimal oracle estimator for the inverse problem and demonstrate empirically that the trained Neumann network has the form predicted by theory.