# HyDRA: Hybrid Denoising Regularization for Measurement-Only DEQ Training

arXiv: https://arxiv.org/abs/2601.01228 · HTML: https://mlsanity.com/paper/2601.01228/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 44 references that are in the map.
- First posted on arXiv 3 Jan 2026.

## Builds on (most cited first, 26 of 26)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [Noise2Self: Blind Denoising by Self-Supervision](https://mlsanity.com/paper/1901.11365/) (2019)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [A Deep Cascade of Convolutional Neural Networks for Dynamic MR Image Reconstruction](https://mlsanity.com/paper/1704.02422/) (2017)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Noisier2Noise: Learning to Denoise from Unpaired Noisy Data](https://mlsanity.com/paper/1910.11908/) (2019)
- [JFB: Jacobian-Free Backpropagation for Implicit Networks](https://mlsanity.com/paper/2103.12803/) (2021)
- [Equivariant Imaging: Learning Beyond the Range Space](https://mlsanity.com/paper/2103.14756/) (2021)
- [One Network to Solve Them All --- Solving Linear Inverse Problems using Deep Projection Models](https://mlsanity.com/paper/1703.09912/) (2017)
- [Robust Equivariant Imaging: a fully unsupervised framework for learning to image from noisy and partial measurements](https://mlsanity.com/paper/2111.12855/) (2021)
- [A Bayesian Perspective on the Deep Image Prior](https://mlsanity.com/paper/1904.07457/) (2019)
- [A theoretical framework for self-supervised MR image reconstruction using sub-sampling via variable density Noisier2Noise](https://mlsanity.com/paper/2205.10278/) (2022)
- [Deep Decomposition Learning for Inverse Imaging Problems](https://mlsanity.com/paper/1911.11028/) (2019)
- [Regularized Nonlinear Acceleration](https://mlsanity.com/paper/1606.04133/) (2016)
- [UNSURE: self-supervised learning with Unknown Noise level and Stein's Unbiased Risk Estimate](https://mlsanity.com/paper/2409.01985/) (2024)
- [Neural Network-based Reconstruction in Compressed Sensing MRI Without Fully-sampled Training Data](https://mlsanity.com/paper/2007.14979/) (2020)
- [Unsupervised Deep Basis Pursuit: Learning inverse problems without ground-truth data](https://mlsanity.com/paper/1910.13110/) (2019)
- [Sparse2Inverse: Self-supervised inversion of sparse-view CT data](https://mlsanity.com/paper/2402.16921/) (2024)
- [Noisier2Inverse: Self-Supervised Learning for Image Reconstruction with Correlated Noise](https://mlsanity.com/paper/2503.19468/) (2025)
- [Equivariance2Inverse: A Practical Self-Supervised CT Reconstruction Method Benchmarked on Real, Limited-Angle, and Blurred Data](https://mlsanity.com/paper/2510.23317/) (2025)
- [TomoSelfDEQ: Self-Supervised Deep Equilibrium Learning for Sparse-Angle CT Reconstruction](https://mlsanity.com/paper/2502.21320/) (2025)
- [Unsupervised Deep Learning for Inverse Problems in Computed Tomography](https://mlsanity.com/paper/2508.05321/) (2025)

## Abstract (arXiv)

Solving image reconstruction problems of the form (A x = y) remains challenging due to ill-posedness and the lack of large-scale supervised datasets. Deep Equilibrium (DEQ) models have been used successfully but typically require supervised pairs ((x,y)). In many practical settings, only measurements (y) are available. We introduce HyDRA (Hybrid Denoising Regularization Adaptation), a measurement-only framework for DEQ training that combines measurement consistency with an adaptive denoising regularization term, together with a data-driven early stopping criterion. Experiments on sparse-view CT demonstrate competitive reconstruction quality and fast inference.