# Learning Regularization Functionals for Inverse Problems: A Comparative Study

arXiv: https://arxiv.org/abs/2510.01755 · HTML: https://mlsanity.com/paper/2510.01755/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 98 of its 165 references that are in the map.
- First posted on arXiv 2 Oct 2025.

## Builds on (most cited first, 30 of 98)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Making Convolutional Networks Shift-Invariant Again](https://mlsanity.com/paper/1904.11486/) (2019)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients](https://mlsanity.com/paper/2010.07468/) (2020)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications](https://mlsanity.com/paper/1605.01710/) (2016)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Truncated Back-propagation for Bilevel Optimization](https://mlsanity.com/paper/1810.10667/) (2018)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [On the Iteration Complexity of Hypergradient Computation](https://mlsanity.com/paper/2006.16218/) (2020)

## Built on by (most cited first, 15 of 15)

- [Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoisers](https://mlsanity.com/paper/2510.27211/) (2025)
- [A Stability Benchmark of Generative Regularizers for Inverse Problems](https://mlsanity.com/paper/2605.10076/) (2026)
- [Graph Neural Regularizers for PDE Inverse Problems](https://mlsanity.com/paper/2510.21012/) (2025)
- [Weakly Convex Ridge Regularization for 3D Non-Cartesian MRI Reconstruction](https://mlsanity.com/paper/2603.27158/) (2026)
- [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026)
- [Data-Consistent Learning of Inverse Problems](https://mlsanity.com/paper/2601.12831/) (2026)
- [Monotone Optimisation with Learned Projections](https://mlsanity.com/paper/2601.20983/) (2026)
- [Diffusion Graph Posterior Sampling for Nonlinear Inverse Problems with Application to Electrical Impedance Tomography](https://mlsanity.com/paper/2605.19621/) (2026)
- [Self-Tuning Regularization for Image Scanning Microscopy](https://mlsanity.com/paper/2605.31426/) (2026)
- [A Distributionally Robust Framework for Learned Reconstructions in Inverse Problems](https://mlsanity.com/paper/2606.30230/) (2026)
- [Learning spatially varying regularisation parameters of low regularity for image reconstruction](https://mlsanity.com/paper/2608.25127/) (2026)
- [CLEAR: Complex Learned Explicit Analytical Regularization for Ultra-Accelerated 4D Flow CMR Reconstruction](https://mlsanity.com/paper/2609.22950/) (2026)
- [Latent Score-Based Bayesian Cram'er-Rao Bound Estimation for High-Dimensional Imaging Systems](https://mlsanity.com/paper/2610.03956/) (2026)
- [The Geometry of Anisotropic Dilation for Optimal Regularization](https://mlsanity.com/paper/2610.09310/) (2026)
- [Taming Nonconvexity with Epigraph Lifting](https://mlsanity.com/paper/2610.16429/) (2026)

## Abstract (arXiv)

In recent years, a variety of learned regularization frameworks for solving inverse problems in imaging have emerged. These offer flexible modeling together with mathematical insights. The proposed methods differ in their architectural design and training strategies, making direct comparison challenging due to non-modular implementations. We address this gap by collecting and unifying the available code into a common framework. This unified view allows us to systematically compare the approaches and highlight their strengths and limitations, providing valuable insights into their future potential. We also provide concise descriptions of each method, complemented by practical guidelines.