mlsanityData to 14 Oct 2026

Total Deep Variation for Linear Inverse Problems

arXiv 2001.05005 ↗First posted 14 Jan 2020math.OC cs.CV cs.LGcited by 53

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Abstract (arXiv)

Diverse inverse problems in imaging can be cast as variational problems composed of a task-specific data fidelity term and a regularization term. In this paper, we propose a novel learnable general-purpose regularizer exploiting recent architectural design patterns from deep learning. We cast the learning problem as a discrete sampled optimal control problem, for which we derive the adjoint state equations and an optimality condition. By exploiting the variational structure of our approach, we perform a sensitivity analysis with respect to the learned parameters obtained from different training datasets. Moreover, we carry out a nonlinear eigenfunction analysis, which reveals interesting properties of the learned regularizer. We show state-of-the-art performance for classical image restoration and medical image reconstruction problems.

Key facts. Cited by 53 papers in the map (data through 14 Oct 2026). First posted on arXiv 14 Jan 2020.

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 15; 2022: 13; 2023: 8; 2024: 8; 2025: 5; 2026: 4.

53 papers.

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  1. 2021Bayesian imaging using Plug & Play priors: when Langevin meets Tweediecited by 97
  2. 2022A Neural-Network-Based Convex Regularizer for Inverse Problemscited by 41
  3. 2021End-to-end reconstruction meets data-driven regularization for inverse problemscited by 32
  4. 2023Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithmscited by 31
  5. 2021Bayesian Uncertainty Estimation of Learned Variational MRI Reconstructioncited by 30
  6. 2021Learning the optimal Tikhonov regularizer for inverse problemscited by 26
  7. 2023What's in a Prior? Learned Proximal Networks for Inverse Problemscited by 25
  8. 2021Bayesian Imaging With Data-Driven Priors Encoded by Neural Networks: Theory, Methods, and Algorithmscited by 24
  9. 2021Generalized Normalizing Flows via Markov Chainscited by 23
  10. 2022PatchNR: Learning from Very Few Images by Patch Normalizing Flow Regularizationcited by 22
  11. 2022Stable Deep MRI Reconstruction using Generative Priorscited by 21
  12. 2025Learning Regularization Functionals for Inverse Problems: A Comparative Studycited by 15
38 more
  1. 2024Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisationcited by 14
  2. 2021Coupling Model-Driven and Data-Driven Methods for Remote Sensing Image Restoration and Fusioncited by 11
  3. 2021Connections between Numerical Algorithms for PDEs and Neural Networkscited by 10
  4. 2022WPPNets and WPPFlows: The Power of Wasserstein Patch Priors for Superresolutioncited by 10
  5. 2023Uncertainty Estimation and Propagation in Accelerated MRI Reconstructioncited by 10
  6. 2022Computed Tomography Reconstruction using Generative Energy-Based Priorscited by 9
  7. 2021Learning convex regularizers satisfying the variational source condition for inverse problemscited by 8
  8. 2022PnP-ReG: Learned Regularizing Gradient for Plug-and-Play Gradient Descentcited by 7
  9. 2024QN-Mixer: A Quasi-Newton MLP-Mixer Model for Sparse-View CT Reconstructioncited by 6
  10. 2025Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes [conference paper]cited by 6
  11. 2021Adversarially learned iterative reconstruction for imaging inverse problemscited by 5
  12. 2022Deep unfolding as iterative regularization for imaging inverse problemscited by 5
  13. 2022Convergent Data-driven Regularizations for CT Reconstructioncited by 5
  14. 2023Solving Low-Dose CT Reconstruction via GAN with Local Coherencecited by 5
  15. 2024Stability of Data-Dependent Ridge-Regularization for Inverse Problemscited by 5
  16. 2025Good Things Come in Pairs: Paired Autoencoders for Inverse Problemscited by 5
  17. 2022Fixed-Point Automatic Differentiation of Forward--Backward Splitting Algorithms for Partly Smooth Functionscited by 4
  18. 2023Provably Convergent Data-Driven Convex-Nonconvex Regularizationcited by 4
  19. 2021Designing Rotationally Invariant Neural Networks from PDEs and Variational Methodscited by 3
  20. 2022Learning Variational Models with Unrolling and Bilevel Optimizationcited by 3
  21. 2024Unsupervised Training of Convex Regularizers using Maximum Likelihood Estimationcited by 3
  22. 2024Iteratively Refined Image Reconstruction with Learned Attentive Regularizerscited by 3
  23. 2024The Star Geometry of Critic-Based Regularizer Learningcited by 3
  24. 2024A primal-dual algorithm for image reconstruction with input-convex neural network regularizerscited by 3
  25. 2021CDLNet: Noise-Adaptive Convolutional Dictionary Learning Network for Blind Denoising and Demosaicingcited by 2
  26. 2025Multivariate Fields of Experts for Convergent Image Reconstructioncited by 2
  27. 2021Deep learning based dictionary learning and tomographic image reconstructioncited by 1
  28. 2021Supervised learning of analysis-sparsity priors with automatic differentiationcited by 1
  29. 2022Multimodal learning-based inversion models for the space-time reconstruction of satellite-derived geophysical fieldscited by 1
  30. 2022Gabor is Enough: Interpretable Deep Denoising with a Gabor Synthesis Dictionary Priorcited by 1
  31. 2023Learned Interferometric Imaging for the SPIDER Instrumentcited by 1
  32. 2023Learned reconstruction methods for inverse problems: sample error estimatescited by 1
  33. 2024Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overviewcited by 1
  34. 2025Learning Binary Sampling Patterns for Single-Pixel Imaging using Bilevel Optimisationcited by 1
  35. 2026On the Generalization Behavior of Deep Residual Networks From a Dynamical System Perspectivecited by 1
  36. 2026Weakly Convex Ridge Regularization for 3D Non-Cartesian MRI Reconstructioncited by 1
  37. 2021Accelerated First Order Methods for Variational Imagingcited by 0
  38. 2022Explainable bilevel optimization: an application to the Helsinki deblur challengecited by 0

50 most cited of 53 All 53 (JSON)