mlsanityData to 14 Oct 2026

Gamma-convergence of a nonlocal perimeter arising in adversarial machine learning

arXiv 2211.15223 ↗First posted 28 Nov 2022math.AP cs.LG math.OCcited by 10

Not on the map. Why?

Abstract (arXiv)

In this paper we prove Gamma-convergence of a nonlocal perimeter of Minkowski type to a local anisotropic perimeter. The nonlocal model describes the regularizing effect of adversarial training in binary classifications. The energy essentially depends on the interaction between two distributions modelling likelihoods for the associated classes. We overcome typical strict regularity assumptions for the distributions by only assuming that they have bounded BV densities. In the natural topology coming from compactness, we prove Gamma-convergence to a weighted perimeter with weight determined by an anisotropic function of the two densities. Despite being local, this sharp interface limit reflects classification stability with respect to adversarial perturbations. We further apply our results to deduce Gamma-convergence of the associated total variations, to study the asymptotics of adversarial training, and to prove Gamma-convergence of graph discretizations for the nonlocal perimeter.

Key facts. Cited by 10 papers in the map (data through 14 Oct 2026). Builds on 20 of its 53 references that are in the map. First posted on arXiv 28 Nov 2022.

Builds on

20 of 53 references.

References · 20

  1. 2017Towards Deep Learning Models Resistant to Adversarial Attackscited by 6,415
  2. 2013Intriguing properties of neural networkscited by 5,155
  3. 2021Recent Advances in Adversarial Training for Adversarial Robustnesscited by 279
  4. 2019Lower Bounds on Adversarial Robustness from Optimal Transportcited by 53
  5. 2019Scaleable input gradient regularization for adversarial robustnesscited by 47
  6. 2021Calibration and Consistency of Adversarial Surrogate Lossescited by 47
  7. 2019Adversarial Risk via Optimal Transport and Optimal Couplingscited by 36
  8. 2022The Many Faces of Adversarial Riskcited by 28
  9. 2021On the Existence of the Adversarial Bayes Classifier (Extended Version)cited by 25
  10. 2014Continuum limit of total variation on point cloudscited by 24
  11. 2022The Multimarginal Optimal Transport Formulation of Adversarial Multiclass Classificationcited by 21
  12. 2021The Geometry of Adversarial Training in Binary Classificationcited by 20
8 more
  1. 2020Adversarial Classification: Necessary conditions and geometric flowscited by 19
  2. 2020Rates of Convergence for Laplacian Semi-Supervised Learning with Low Labeling Ratescited by 18
  3. 2021On the regularized risk of distributionally robust learning over deep neural networkscited by 8
  4. 2014Consistency of Cheeger and Ratio Graph Cutscited by 6
  5. 2020From graph cuts to isoperimetric inequalities: Convergence rates of Cheeger cuts on data cloudscited by 6
  6. 2020Continuum Limit of Lipschitz Learning on Graphscited by 6
  7. 2022The Consistency of Adversarial Training for Binary Classificationcited by 3
  8. 2021Variational Regularization in Inverse Problems and Machine Learningcited by 2

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 4; 2024: 3; 2025: 2; 2026: 1.

10 papers.

  1. 2023On the existence of solutions to adversarial training in multiclass classificationcited by 9
  2. 2024An Optimal Transport Approach for Computing Adversarial Training Lower Bounds in Multiclass Classificationcited by 5
  3. 2023On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce itcited by 4
  4. 2024A mean curvature flow arising in adversarial trainingcited by 4
  5. 2023It begins with a boundary: A geometric view on probabilistically robust learningcited by 3
  6. 2025Consensus-based optimization for closed-box adversarial attacks and a connection to evolution strategiescited by 2
  7. 2024Uniform Convergence of Adversarially Robust Classifierscited by 1
  8. 2025Lower Bounds on Adversarial Robustness for Multiclass Classification with General Loss Functionscited by 1
  9. 2023Can We Rely on AI?cited by 0
  10. 2026Duality for the Adversarial Total Variationcited by 0