mlsanityData to 7 Oct 2026

Approximating Lipschitz continuous functions with GroupSort neural networks

arXiv 2006.05254 ↗First posted 9 Jun 2020stat.ML cs.LGcited by 27

Not on the map. Why?

Abstract (arXiv)

Recent advances in adversarial attacks and Wasserstein GANs have advocated for use of neural networks with restricted Lipschitz constants. Motivated by these observations, we study the recently introduced GroupSort neural networks, with constraints on the weights, and make a theoretical step towards a better understanding of their expressive power. We show in particular how these networks can represent any Lipschitz continuous piecewise linear functions. We also prove that they are well-suited for approximating Lipschitz continuous functions and exhibit upper bounds on both the depth and size. To conclude, the efficiency of GroupSort networks compared with more standard ReLU networks is illustrated in a set of synthetic experiments.

Key facts. Cited by 27 papers in the map (data through 7 Oct 2026). First posted on arXiv 9 Jun 2020.

Builds on

No reference list yet.

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 8; 2022: 5; 2023: 3; 2024: 6; 2025: 1; 2026: 4.

27 papers.

Sort:

  1. 2021Generative Modeling with Optimal Transport Mapscited by 80
  2. 2022Impossibility Theorems for Feature Attributioncited by 78
  3. 2022Rethinking Lipschitz Neural Networks and Certified Robustness: A Boolean Function Perspectivecited by 48
  4. 2021Pay attention to your loss: understanding misconceptions about 1-Lipschitz neural networkscited by 30
  5. 2021Learning by Transference: Training Graph Neural Networks on Growing Graphscited by 22
  6. 2022On the Number of Regions of Piecewise Linear Neural Networkscited by 21
  7. 2021Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 16
  8. 2021Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximationcited by 16
  9. 2022Approximation of Lipschitz Functions using Deep Spline Neural Networkscited by 15
  10. 2023DP-SGD Without Clipping: The Lipschitz Neural Network Waycited by 11
  11. 2023Robust One-Class Classification with Signed Distance Function using 1-Lipschitz Neural Networkscited by 10
  12. 2021Universal Approximation of Residual Flows in Maximum Mean Discrepancycited by 7
15 more
  1. 2024Novel Quadratic Constraints for Extending LipSDP beyond Slope-Restricted Activationscited by 6
  2. 2023Convolutional Neural Networks as 2-D systemscited by 5
  3. 2022GAN Estimation of Lipschitz Optimal Transport Mapscited by 4
  4. 2024Fast and scalable Wasserstein-1 neural optimal transport solver for single-cell perturbation predictioncited by 3
  5. 2021Wasserstein Adversarially Regularized Graph Autoencodercited by 2
  6. 2024A Complete Set of Quadratic Constraints for Repeated ReLU and Generalizationscited by 2
  7. 2024Towards Better Multi-head Attention via Channel-wise Sample Permutationcited by 2
  8. 2025On the Existence of Universal Simulators of Attentioncited by 1
  9. 2026ExLipBaB: Exact Lipschitz Constant Computation for Piecewise Linear Neural Networkscited by 1
  10. 2021Approximating Probability Distributions by using Wasserstein Generative Adversarial Networkscited by 0
  11. 2024Active Learning for Regression based on Wasserstein distance and GroupSort Neural Networkscited by 0
  12. 2024Modeling Sampling Distributions of Test Statistics with Autogradcited by 0
  13. 2026Neural Networks With Dense Weights Are Not Universal Approximatorscited by 0
  14. 2026Self-Review Reinforcement Learning (SRRL) with Cross-Episode Memory and Policy Distillationcited by 0
  15. 2026Mutual Equilibrium: Multimodal Representation Learning through Reciprocal Feedbackcited by 0