mlsanityData to 7 Oct 2026

Deep Equilibrium Architectures for Inverse Problems in Imaging

arXiv 2102.07944 ↗First posted 16 Feb 2021eess.IV cs.CVcited by 126

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Abstract (arXiv)

Recent efforts on solving inverse problems in imaging via deep neural networks use architectures inspired by a fixed number of iterations of an optimization method. The number of iterations is typically quite small due to difficulties in training networks corresponding to more iterations; the resulting solvers cannot be run for more iterations at test time without incurring significant errors. This paper describes an alternative approach corresponding to an infinite number of iterations, yielding a consistent improvement in reconstruction accuracy above state-of-the-art alternatives and where the computational budget can be selected at test time to optimize context-dependent trade-offs between accuracy and computation. The proposed approach leverages ideas from Deep Equilibrium Models, where the fixed-point iteration is constructed to incorporate a known forward model and insights from classical optimization-based reconstruction methods.

Key facts. Cited by 126 papers in the map (data through 7 Oct 2026). Builds on 26 of its 40 references that are in the map. First posted on arXiv 16 Feb 2021.

Builds on

26 of 40 references.

References · 26

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  2. 2018Neural Ordinary Differential Equationscited by 3,733
  3. 2014Deep Learning Face Attributes in the Wildcited by 3,679
  4. 2019Deep Equilibrium Modelscited by 603
  5. 2018fastMRI: An Open Dataset and Benchmarks for Accelerated MRIcited by 464
  6. 2017Learning Deep CNN Denoiser Prior for Image Restorationcited by 407
  7. 2017MoDL: Model Based Deep Learning Architecture for Inverse Problemscited by 384
  8. 2017Stable Architectures for Deep Neural Networkscited by 340
  9. 2016The Little Engine that Could: Regularization by Denoising (RED)cited by 303
  10. 2020Deep Learning Techniques for Inverse Problems in Imagingcited by 290
  11. 2017Learned Primal-dual Reconstructioncited by 251
  12. 2016Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applicationscited by 244
14 more
  1. 2019On instabilities of deep learning in image reconstruction - Does AI come at a cost?cited by 211
  2. 2019Plug-and-Play Methods Provably Converge with Properly Trained Denoiserscited by 197
  3. 2019Implicit Deep Learningcited by 124
  4. 2017Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problemscited by 107
  5. 2019RARE: Image Reconstruction using Deep Priors Learned without Ground Truthcited by 64
  6. 2020Solving Inverse Problems With Deep Neural Networks -- Robustness Included?cited by 58
  7. 2018Trellis Networks for Sequence Modelingcited by 50
  8. 2018Neural Proximal Gradient Descent for Compressive Imagingcited by 46
  9. 2018Recurrent Stacking of Layers for Compact Neural Machine Translation Modelscited by 38
  10. 2019Momentum-Net: Fast and convergent iterative neural network for inverse problemscited by 38
  11. 2020Improving Robustness of Deep-Learning-Based Image Reconstructioncited by 23
  12. 2020Results of the 2020 fastMRI Challenge for Machine Learning MR Image Reconstructioncited by 14
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  14. 2019Accelerating ADMM for Efficient Simulation and Optimizationcited by 7

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 11; 2022: 26; 2023: 35; 2024: 24; 2025: 19; 2026: 11.

126 papers.

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  1. 2021JFB: Jacobian-Free Backpropagation for Implicit Networkscited by 84
  2. 2024DMPlug: A Plug-in Method for Solving Inverse Problems with Diffusion Modelscited by 60
  3. 2022Physics-Driven Deep Learning for Computational Magnetic Resonance Imagingcited by 49
  4. 2022Learned reconstruction methods with convergence guaranteescited by 46
  5. 2022A Neural-Network-Based Convex Regularizer for Inverse Problemscited by 40
  6. 2022HUMUS-Net: Hybrid unrolled multi-scale network architecture for accelerated MRI reconstructioncited by 30
  7. 2021SHINE: SHaring the INverse Estimate from the forward pass for bi-level optimization and implicit modelscited by 26
  8. 2021Learning the optimal Tikhonov regularizer for inverse problemscited by 26
  9. 2023What's in a Prior? Learned Proximal Networks for Inverse Problemscited by 24
  10. 2021Recovery Analysis for Plug-and-Play Priors using the Restricted Eigenvalue Conditioncited by 22
  11. 2022Online Deep Equilibrium Learning for Regularization by Denoisingcited by 22
  12. 2023TorchDEQ: A Library for Deep Equilibrium Modelscited by 21
38 more
  1. 2021Robust Inverse Framework using Knowledge-guided Self-Supervised Learning: An application to Hydrologycited by 15
  2. 2021Joint inference and input optimization in equilibrium networkscited by 15
  3. 2022Explainable AI via Learning to Optimizecited by 14
  4. 2024Fixed Point Diffusion Modelscited by 14
  5. 2022Near-Exact Recovery for Tomographic Inverse Problems via Deep Learningcited by 13
  6. 2023Differentiating Through Integer Linear Programs with Quadratic Regularization and Davis-Yin Splittingcited by 11
  7. 2022Global Convergence of Over-parameterized Deep Equilibrium Modelscited by 10
  8. 2023Lyapunov-Stable Deep Equilibrium Modelscited by 10
  9. 2023Prior Mismatch and Adaptation in PnP-ADMM with a Nonconvex Convergence Analysiscited by 10
  10. 2023Adapt and Diffuse: Sample-adaptive Reconstruction via Latent Diffusion Modelscited by 9
  11. 2023PtychoDV: Vision Transformer-Based Deep Unrolling Network for Ptychographic Image Reconstructioncited by 9
  12. 2022Deep equilibrium networks are sensitive to initialization statisticscited by 8
  13. 2022Deep Model-Based Architectures for Inverse Problems under Mismatched Priorscited by 8
  14. 2023Block Coordinate Plug-and-Play Methods for Blind Inverse Problemscited by 8
  15. 2023Self-Supervised Scalable Deep Compressed Sensingcited by 8
  16. 2022Memory-efficient model-based deep learning with convergence and robustness guaranteescited by 7
  17. 2023PINQI: An End-to-End Physics-Informed Approach to Learned Quantitative MRI Reconstructioncited by 7
  18. 2023Unfolded proximal neural networks for robust image Gaussian denoisingcited by 7
  19. 2024Positive concave deep equilibrium modelscited by 7
  20. 2024On Logical Extrapolation for Mazes with Recurrent and Implicit Networkscited by 7
  21. 2025Deep Equilibrium models for Poisson Imaging Inverse problems via Mirror Descentcited by 7
  22. 2025MRI Super-Resolution with Deep Learning: A Comprehensive Surveycited by 7
  23. 2023Model-corrected learned primal-dual models for fast limited-view photoacoustic tomographycited by 6
  24. 2023Spatially Varying Exposure with 2-by-2 Multiplexing: Optimality and Universalitycited by 6
  25. 2024Deep Equilibrium Models are Almost Equivalent to Not-so-deep Explicit Models for High-dimensional Gaussian Mixturescited by 6
  26. 2021Fixed points of nonnegative neural networkscited by 5
  27. 2021Equilibrated Zeroth-Order Unrolled Deep Networks for Accelerated MRIcited by 5
  28. 2022Deep Equilibrium Models for Video Snapshot Compressive Imagingcited by 5
  29. 2022NESTANets: Stable, accurate and efficient neural networks for analysis-sparse inverse problemscited by 5
  30. 2024Deep Implicit Optimization enables Robust Learnable Features for Deformable Image Registrationcited by 5
  31. 2025Reversible Deep Equilibrium Modelscited by 5
  32. 2021Improved Model based Deep Learning using Monotone Operator Learning (MOL)cited by 4
  33. 2022Learning Proximal Operators to Discover Multiple Optimacited by 4
  34. 2022Connections between Deep Equilibrium and Sparse Representation Models with Application to Hyperspectral Image Denoisingcited by 4
  35. 2022Learning Sparsity-Promoting Regularizers using Bilevel Optimizationcited by 4
  36. 2022Loop Unrolled Shallow Equilibrium Regularizer (LUSER) -- A Memory-Efficient Inverse Problem Solvercited by 4
  37. 2022Robustness of Deep Equilibrium Architectures to Changes in the Measurement Modelcited by 4
  38. 2022Deep equilibrium models as estimators for continuous latent variablescited by 4

50 most cited of 126 All 126 (JSON)