mlsanityData to 7 Oct 2026

Lipschitz Certificates for Layered Network Structures Driven by Averaged Activation Operators

arXiv 1903.01014 ↗First posted 3 Mar 2019math.OCcited by 54

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Abstract (arXiv)

Obtaining sharp Lipschitz constants for feed-forward neural networks is essential to assess their robustness in the face of perturbations of their inputs. We derive such constants in the context of a general layered network model involving compositions of nonexpansive averaged operators and affine operators. By exploiting this architecture, our analysis finely captures the interactions between the layers, yielding tighter Lipschitz constants than those resulting from the product of individual bounds for groups of layers. The proposed framework is shown to cover in particular many practical instances encountered in feed-forward neural networks. Our Lipschitz constant estimates are further improved in the case of structures employing scalar nonlinear functions, which include standard convolutional networks as special cases.

Key facts. Cited by 54 papers in the map (data through 7 Oct 2026). First posted on arXiv 3 Mar 2019.

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Citing papers per year: 2021: 13; 2022: 12; 2023: 9; 2024: 11; 2025: 5; 2026: 4.

54 papers.

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  1. 2021A Dynamical System Perspective for Lipschitz Neural Networkscited by 59
  2. 2023When Deep Learning Meets Polyhedral Theory: A Surveycited by 39
  3. 2021CLIP: Cheap Lipschitz Training of Neural Networkscited by 29
  4. 2021Robust Implicit Networks via Non-Euclidean Contractionscited by 19
  5. 2023Certified Robustness via Dynamic Margin Maximization and Improved Lipschitz Regularizationcited by 19
  6. 2021On the expressivity of bi-Lipschitz normalizing flowscited by 15
  7. 2022Chordal Sparsity for Lipschitz Constant Estimation of Deep Neural Networkscited by 15
  8. 2021Convergence of Deep ReLU Networkscited by 14
  9. 2022Explainable AI via Learning to Optimizecited by 14
  10. 2021Convergence of Deep Convolutional Neural Networkscited by 13
  11. 2022Lipschitz constant estimation for 1D convolutional neural networkscited by 13
  12. 2022Adversarial robustness of sparse local Lipschitz predictorscited by 11
38 more
  1. 2021Connections between Numerical Algorithms for PDEs and Neural Networkscited by 10
  2. 2023A path-norm toolkit for modern networks: consequences, promises and challengescited by 10
  3. 2024ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networkscited by 10
  4. 2021Robustness Certificates for Implicit Neural Networks: A Mixed Monotone Contractive Approachcited by 8
  5. 2022Provably Tightest Linear Approximation for Robustness Verification of Sigmoid-like Neural Networkscited by 8
  6. 2024Lipschitz constant estimation for general neural network architectures using control toolscited by 8
  7. 2025An Adaptive Orthogonal Convolution Scheme for Efficient and Flexible CNN Architecturescited by 8
  8. 2021Robustness against Adversarial Attacks in Neural Networks using Incremental Dissipativitycited by 7
  9. 2022On Adversarial Robustness of Deep Image Deblurringcited by 7
  10. 2023Unfolded proximal neural networks for robust image Gaussian denoisingcited by 7
  11. 2024Positive concave deep equilibrium modelscited by 7
  12. 2023CertViT: Certified Robustness of Pre-Trained Vision Transformerscited by 6
  13. 2021What training reveals about neural network complexitycited by 5
  14. 2021Fixed points of nonnegative neural networkscited by 5
  15. 2022NESTANets: Stable, accurate and efficient neural networks for analysis-sparse inverse problemscited by 5
  16. 2023Uniform Convergence of Deep Neural Networks with Lipschitz Continuous Activation Functions and Variable Widthscited by 5
  17. 2023Bridging Dimensions: Confident Reachability for High-Dimensional Controllerscited by 5
  18. 2024Policy Verification in Stochastic Dynamical Systems Using Logarithmic Neural Certificatescited by 3
  19. 2025Sounding that Object: Interactive Object-Aware Image to Audio Generationcited by 3
  20. 2022Comparative Analysis of Interval Reachability for Robust Implicit and Feedforward Neural Networkscited by 2
  21. 2024Learning truly monotone operators with applications to nonlinear inverse problemscited by 2
  22. 2024Computable Lipschitz Bounds for Deep Neural Networkscited by 2
  23. 2024LipKernel: Lipschitz-Bounded Convolutional Neural Networks via Dissipative Layerscited by 2
  24. 2022Federated Graph-based Networks with Shared Embeddingcited by 1
  25. 2024Evaluating Adversarial Robustness of Low dose CT Recoverycited by 1
  26. 2024Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overviewcited by 1
  27. 2024Stability Bounds for the Unfolded Forward-Backward Algorithmcited by 1
  28. 2025Noise-adapted Neural Operator for Robust Non-Line-of-Sight Imagingcited by 1
  29. 2025ECLipsE-Gen-Local: Efficient Compositional Local Lipschitz Estimates for Deep Neural Networkscited by 1
  30. 2026Tight Long-Term Tail Decay of (Clipped) SGD in Non-Convex Optimizationcited by 1
  31. 2021Learning Lipschitz Feedback Policies from Expert Demonstrations: Closed-Loop Guarantees, Generalization and Robustnesscited by 0
  32. 2021On the Robustness and Generalization of Deep Learning Driven Full Waveform Inversioncited by 0
  33. 2022Convergence of Deep Neural Networks with General Activation Functions and Poolingcited by 0
  34. 2022Robust Training and Verification of Implicit Neural Networks: A Non-Euclidean Contractive Approachcited by 0
  35. 2022DualApp: Tight Over-Approximation for Neural Network Robustness Verification via Under-Approximationcited by 0
  36. 2023CQnet: convex-geometric interpretation and constraining neural-network trajectoriescited by 0
  37. 2023Generalized Activation via Multivariate Projectioncited by 0
  38. 2024Transfer Learning for Control Systems via Neural Simulation Relationscited by 0

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