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Implicit Deep Learning

arXiv 1908.06315 ↗First posted 17 Aug 2019cs.LG math.OC stat.MLcited by 124

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Abstract (arXiv)

Implicit deep learning prediction rules generalize the recursive rules of feedforward neural networks. Such rules are based on the solution of a fixed-point equation involving a single vector of hidden features, which is thus only implicitly defined. The implicit framework greatly simplifies the notation of deep learning, and opens up many new possibilities, in terms of novel architectures and algorithms, robustness analysis and design, interpretability, sparsity, and network architecture optimization.

Key facts. Cited by 124 papers in the map (data through 7 Oct 2026). First posted on arXiv 17 Aug 2019.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 24; 2022: 31; 2023: 19; 2024: 24; 2025: 11; 2026: 15.

124 papers.

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  1. 2021Efficient and Modular Implicit Differentiationcited by 199
  2. 2021Deep Equilibrium Architectures for Inverse Problems in Imagingcited by 126
  3. 2023DYffusion: A Dynamics-informed Diffusion Model for Spatiotemporal Forecastingcited by 105
  4. 2021JFB: Jacobian-Free Backpropagation for Implicit Networkscited by 84
  5. 2021On Training Implicit Modelscited by 73
  6. 2021Stabilizing Equilibrium Models by Jacobian Regularizationcited by 71
  7. 2022A Survey on Graph Representation Learning Methodscited by 66
  8. 2022Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modelingcited by 65
  9. 2022Deep Equilibrium Optical Flow Estimationcited by 63
  10. 2022On the Principles of Parsimony and Self-Consistency for the Emergence of Intelligencecited by 50
  11. 2022Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Networkcited by 45
  12. 2022Equivariant Hypergraph Diffusion Neural Operatorscited by 45
38 more
  1. 2021Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustnesscited by 43
  2. 2021Nonsmooth Implicit Differentiation for Machine Learning and Optimizationcited by 43
  3. 2022Deep Equilibrium Approaches to Diffusion Modelscited by 42
  4. 2022Learning differentiable solvers for systems with hard constraintscited by 34
  5. 2022EIGNN: Efficient Infinite-Depth Graph Neural Networkscited by 32
  6. 2024Linear-Time Graph Neural Networks for Scalable Recommendationscited by 31
  7. 2023One-step differentiation of iterative algorithmscited by 28
  8. 2022Automatic differentiation of nonsmooth iterative algorithmscited by 25
  9. 2023TorchDEQ: A Library for Deep Equilibrium Modelscited by 21
  10. 2021Robust Implicit Networks via Non-Euclidean Contractionscited by 19
  11. 2021Hamiltonian Deep Neural Networks Guaranteeing Non-vanishing Gradients by Designcited by 18
  12. 2021A global convergence theory for deep ReLU implicit networks via over-parameterizationcited by 17
  13. 2021Closed-Loop Data Transcription to an LDR via Minimaxing Rate Reductioncited by 17
  14. 2021Multiset-Equivariant Set Prediction with Approximate Implicit Differentiationcited by 17
  15. 2021Almost Surely Stable Deep Dynamicscited by 16
  16. 2022The least-control principle for local learning at equilibriumcited by 16
  17. 2023LazyGNN: Large-Scale Graph Neural Networks via Lazy Propagationcited by 16
  18. 2021Recurrent Neural Network Controllers Synthesis with Stability Guarantees for Partially Observed Systemscited by 15
  19. 2021Joint inference and input optimization in equilibrium networkscited by 15
  20. 2023Wide Neural Networks as Gaussian Processes: Lessons from Deep Equilibrium Modelscited by 14
  21. 2022Revisiting Sparse Convolutional Model for Visual Recognitioncited by 13
  22. 2021Implicit vs Unfolded Graph Neural Networkscited by 12
  23. 2022Transforming PageRank into an Infinite-Depth Graph Neural Networkcited by 11
  24. 2023Differentiating Through Integer Linear Programs with Quadratic Regularization and Davis-Yin Splittingcited by 11
  25. 2021Youla-REN: Learning Nonlinear Feedback Policies with Robust Stability Guaranteescited by 10
  26. 2022Global Convergence of Over-parameterized Deep Equilibrium Modelscited by 10
  27. 2024Differentiation Through Black-Box Quadratic Programming Solverscited by 10
  28. 2024Monotone, Bi-Lipschitz, and Polyak-Lojasiewicz Networkscited by 9
  29. 2021A Recipe for Global Convergence Guarantee in Deep Neural Networkscited by 8
  30. 2021Robustness Certificates for Implicit Neural Networks: A Mixed Monotone Contractive Approachcited by 8
  31. 2022Synthesis of Stabilizing Recurrent Equilibrium Network Controllerscited by 8
  32. 2022Deep equilibrium networks are sensitive to initialization statisticscited by 8
  33. 2024Positive concave deep equilibrium modelscited by 7
  34. 2024On Logical Extrapolation for Mazes with Recurrent and Implicit Networkscited by 7
  35. 2022Scaling up and Stabilizing Differentiable Planning with Implicit Differentiationcited by 6
  36. 2023Implementation and (Inverse Modified) Error Analysis for implicitly-templated ODE-netscited by 6
  37. 2024Deep Equilibrium Models are Almost Equivalent to Not-so-deep Explicit Models for High-dimensional Gaussian Mixturescited by 6
  38. 2024LPGD: A General Framework for Backpropagation through Embedded Optimization Layerscited by 6

50 most cited of 124 All 124 (JSON)