mlsanityData to 7 Oct 2026

Deep Neural Network Structures Solving Variational Inequalities

arXiv 1808.07526 ↗First posted 22 Aug 2018math.OCcited by 25

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Abstract (arXiv)

Motivated by structures that appear in deep neural networks, we investigate nonlinear composite models alternating proximity and affine operators defined on different spaces. We first show that a wide range of activation operators used in neural networks are actually proximity operators. We then establish conditions for the averagedness of the proposed composite constructs and investigate their asymptotic properties. It is shown that the limit of the resulting process solves a variational inequality which, in general, does not derive from a minimization problem.

Key facts. Cited by 25 papers in the map (data through 7 Oct 2026). First posted on arXiv 22 Aug 2018.

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Citing papers per year: 2021: 9; 2022: 4; 2023: 4; 2024: 3; 2025: 2; 2026: 3.

25 papers.

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  1. 2021Nonsmooth Implicit Differentiation for Machine Learning and Optimizationcited by 43
  2. 2024Transductive Zero-Shot and Few-Shot CLIPcited by 31
  3. 2021Generalized Normalizing Flows via Markov Chainscited by 23
  4. 2021Connections between Numerical Algorithms for PDEs and Neural Networkscited by 10
  5. 2022Lifted Bregman Training of Neural Networkscited by 10
  6. 2023Designing Stable Neural Networks using Convex Analysis and ODEscited by 10
  7. 2022Towards Practical Few-Shot Query Sets: Transductive Minimum Description Length Inferencecited by 9
  8. 2023Lipschitz-bounded 1D convolutional neural networks using the Cayley transform and the controllability Gramiancited by 9
  9. 2023CertViT: Certified Robustness of Pre-Trained Vision Transformerscited by 6
  10. 2021Fixed points of nonnegative neural networkscited by 5
  11. 2022Proximal Residual Flows for Bayesian Inverse Problemscited by 5
  12. 2021Unrolled Variational Bayesian Algorithm for Image Blind Deconvolutioncited by 4
13 more
  1. 2021Designing Rotationally Invariant Neural Networks from PDEs and Variational Methodscited by 3
  2. 2026Quantization Robustness of Monotone Operator Equilibrium Networkscited by 3
  3. 2021Deep Transform and Metric Learning Networkscited by 2
  4. 2022Towards Understanding Graph Neural Networks: An Algorithm Unrolling Perspectivecited by 2
  5. 2025An Energy-Based Mechanism for Compositional Behaviorcited by 2
  6. 2021Reframing Neural Networks: Deep Structure in Overcomplete Representationscited by 1
  7. 2024Stability Bounds for the Unfolded Forward-Backward Algorithmcited by 1
  8. 2025Circuit realization and hardware linearization of monotone operator equilibrium networkscited by 1
  9. 2021Variational models for signal processing with Graph Neural Networkscited by 0
  10. 2023CQnet: convex-geometric interpretation and constraining neural-network trajectoriescited by 0
  11. 2024A lifted Bregman strategy for training unfolded proximal neural network Gaussian denoiserscited by 0
  12. 2026A Single-Loop Bilevel Deep Learning Method for Optimal Control of Obstacle Problemscited by 0
  13. 2026Predictive Coding with Bayesian Priors via Proximal Gradientscited by 0