mlsanityData to 14 Oct 2026

A Neural-Network-Based Convex Regularizer for Inverse Problems

arXiv 2211.12461 ↗First posted 22 Nov 2022eess.IV cs.LGcited by 41

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Abstract (arXiv)

The emergence of deep-learning-based methods to solve image-reconstruction problems has enabled a significant increase in reconstruction quality. Unfortunately, these new methods often lack reliability and explainability, and there is a growing interest to address these shortcomings while retaining the boost in performance. In this work, we tackle this issue by revisiting regularizers that are the sum of convex-ridge functions. The gradient of such regularizers is parameterized by a neural network that has a single hidden layer with increasing and learnable activation functions. This neural network is trained within a few minutes as a multistep Gaussian denoiser. The numerical experiments for denoising, CT, and MRI reconstruction show improvements over methods that offer similar reliability guarantees.

Key facts. Cited by 41 papers in the map (data through 14 Oct 2026). Builds on 43 of its 80 references that are in the map. First posted on arXiv 22 Nov 2022.

Builds on

43 of 80 references.

References · 43

  1. 2018Spectral Normalization for Generative Adversarial Networkscited by 1,636
  2. 2016Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoisingcited by 1,358
  3. 2019Deep Equilibrium Modelscited by 615
  4. 2016Deep Convolutional Neural Network for Inverse Problems in Imagingcited by 433
  5. 2016Input Convex Neural Networkscited by 414
  6. 2017MoDL: Model Based Deep Learning Architecture for Inverse Problemscited by 387
  7. 2016The Little Engine that Could: Regularization by Denoising (RED)cited by 307
  8. 2020Deep Learning Techniques for Inverse Problems in Imagingcited by 291
  9. 2016Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applicationscited by 246
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31 more
  1. 2019Plug-and-Play Methods Provably Converge with Properly Trained Denoiserscited by 198
  2. 2021Deep Equilibrium Architectures for Inverse Problems in Imagingcited by 127
  3. 2021Gradient Step Denoiser for convergent Plug-and-Playcited by 108
  4. 2018Adversarial Regularizers in Inverse Problemscited by 103
  5. 2017Deep learning for undersampled MRI reconstructioncited by 91
  6. 2018NETT: Solving Inverse Problems with Deep Neural Networkscited by 89
  7. 2021JFB: Jacobian-Free Backpropagation for Implicit Networkscited by 86
  8. 2022Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularizationcited by 80
  9. 2019Adaptive Gradient Descent without Descentcited by 79
  10. 2017CNN-Based Projected Gradient Descent for Consistent Image Reconstructioncited by 77
  11. 2020Learning Maximally Monotone Operators for Image Recoverycited by 62
  12. 2020Learned convex regularizers for inverse problemscited by 55
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  16. 2020Convolutional Proximal Neural Networks and Plug-and-Play Algorithmscited by 36
  17. 2019Deep Generalization of Structured Low-Rank Algorithms (Deep-SLR)cited by 30
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  19. 2022Improving Lipschitz-Constrained Neural Networks by Learning Activation Functionscited by 24
  20. 2021Recovery Analysis for Plug-and-Play Priors using the Restricted Eigenvalue Conditioncited by 22
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  24. 2020Boosting the Performance of Plug-and-Play Priors via Denoiser Scalingcited by 12
  25. 2021Learning convex regularizers satisfying the variational source condition for inverse problemscited by 8
  26. 2022PnP-ReG: Learned Regularizing Gradient for Plug-and-Play Gradient Descentcited by 7
  27. 2022Memory-efficient model-based deep learning with convergence and robustness guaranteescited by 7
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 5; 2024: 11; 2025: 15; 2026: 10.

41 papers.

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  1. 2023Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithmscited by 31
  2. 2023Manifold Learning by Mixture Models of VAEs for Inverse Problemscited by 17
  3. 2025Learning Regularization Functionals for Inverse Problems: A Comparative Studycited by 15
  4. 2024Do Bayesian imaging methods report trustworthy probabilities?cited by 6
  5. 2025DEALing with Image Reconstruction: Deep Attentive Least Squarescited by 6
  6. 2024Stability of Data-Dependent Ridge-Regularization for Inverse Problemscited by 5
  7. 2025Good Things Come in Pairs: Paired Autoencoders for Inverse Problemscited by 5
  8. 2023Accelerated Bayesian imaging by relaxed proximal-point Langevin samplingcited by 4
  9. 2023An adaptively inexact first-order method for bilevel optimization with application to hyperparameter learningcited by 4
  10. 2024Iteratively Refined Image Reconstruction with Learned Attentive Regularizerscited by 3
  11. 2024The Star Geometry of Critic-Based Regularizer Learningcited by 3
  12. 2024A primal-dual algorithm for image reconstruction with input-convex neural network regularizerscited by 3
29 more
  1. 2024Learning of Patch-Based Smooth-Plus-Sparse Models for Image Reconstructioncited by 3
  2. 2025A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithmcited by 3
  3. 2024Joint Edge Optimization Deep Unfolding Network for Accelerated MRI Reconstructioncited by 2
  4. 2024Bilevel Learning with Inexact Stochastic Gradientscited by 2
  5. 2025Efficient Bayesian Computation Using Plug-and-Play Priors for Poisson Inverse Problemscited by 2
  6. 2025Multivariate Fields of Experts for Convergent Image Reconstructioncited by 2
  7. 2025Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoiserscited by 2
  8. 2023Local monotone operator learning using non-monotone operators: MnM-MOLcited by 1
  9. 2024Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overviewcited by 1
  10. 2024Controlled Learning of Pointwise Nonlinearities in Neural-Network-Like Architecturescited by 1
  11. 2025Universal Architectures for the Learning of Polyhedral Norms and Convex Regularizerscited by 1
  12. 2025Why the noise model matters: A performance gap in learned regularizationcited by 1
  13. 2025Equivariant Deep Equilibrium Models for Imaging Inverse Problemscited by 1
  14. 2026Weakly Convex Ridge Regularization for 3D Non-Cartesian MRI Reconstructioncited by 1
  15. 2024Gradient Networkscited by 0
  16. 2025MAP Image Recovery with Guarantees using Locally Convex Multi-Scale Energy (LC-MUSE) Modelcited by 0
  17. 2025Deep regularization networks for inverse problems with noisy operatorscited by 0
  18. 2025Stochastic Adaptive Gradient Descent Without Descentcited by 0
  19. 2025Breaking the Weak Recovery Limit in Random Phase Retrieval with Learned Regularizerscited by 0
  20. 2025Bilevel Learning via Inexact Stochastic Gradient Descentcited by 0
  21. 2026Deep unfolding of MCMC kernels: scalable, modular & explainable GANs for high-dimensional posterior samplingcited by 0
  22. 2026UOTIP: Unbalanced Optimal Transport Map for Unpaired Inverse Problemscited by 0
  23. 2026Bridging data-driven priors via the score function for posterior sampling -- Comparative review and experimental studycited by 0
  24. 2026A Distributionally Robust Framework for Learned Reconstructions in Inverse Problemscited by 0
  25. 2026Stabilizing Deep Reconstruction Operators with Contractive Anchoringcited by 0
  26. 2026Trainable Nonexpansive Denoisers for Contractive Image Reconstructioncited by 0
  27. 2026CLEAR: Complex Learned Explicit Analytical Regularization for Ultra-Accelerated 4D Flow CMR Reconstructioncited by 0
  28. 2026Plug-and-Play Methods Provably Converge Even with Improperly Trained Denoisers: Convergence by Architectural Designcited by 0
  29. 2026Taming Nonconvexity with Epigraph Liftingcited by 0