mlsanityData to 7 Oct 2026

On the Geometry of Adversarial Examples

arXiv 1811.00525 ↗First posted 1 Nov 2018cs.LG stat.MLcited by 43

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Abstract (arXiv)

Adversarial examples are a pervasive phenomenon of machine learning models where seemingly imperceptible perturbations to the input lead to misclassifications for otherwise statistically accurate models. We propose a geometric framework, drawing on tools from the manifold reconstruction literature, to analyze the high-dimensional geometry of adversarial examples. In particular, we highlight the importance of codimension: for low-dimensional data manifolds embedded in high-dimensional space there are many directions off the manifold in which to construct adversarial examples. Adversarial examples are a natural consequence of learning a decision boundary that classifies the low-dimensional data manifold well, but classifies points near the manifold incorrectly. Using our geometric framework we prove (1) a tradeoff between robustness under different norms, (2) that adversarial training in balls around the data is sample inefficient, and (3) sufficient sampling conditions under which nearest neighbor classifiers and ball-based adversarial training are robust.

Key facts. Cited by 43 papers in the map (data through 7 Oct 2026). First posted on arXiv 1 Nov 2018.

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Citing papers per year: 2021: 9; 2022: 12; 2023: 13; 2024: 3; 2025: 2; 2026: 4.

43 papers.

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  1. 2021Adversarial Examples Make Strong Poisonscited by 115
  2. 2021Relating Adversarially Robust Generalization to Flat Minimacited by 57
  3. 2021The Dimpled Manifold Model of Adversarial Examples in Machine Learningcited by 41
  4. 2022Probabilistically Robust Learning: Balancing Average- and Worst-case Performancecited by 36
  5. 2022DISCO: Adversarial Defense with Local Implicit Functionscited by 33
  6. 2023Why Does Little Robustness Help? A Further Step Towards Understanding Adversarial Transferabilitycited by 32
  7. 2022Gradient Methods Provably Converge to Non-Robust Networkscited by 27
  8. 2021Towards Robust Neural Networks via Close-loop Controlcited by 23
  9. 2023The Double-Edged Sword of Implicit Bias: Generalization vs. Robustness in ReLU Networkscited by 23
  10. 2022Adversarial Rademacher Complexity of Deep Neural Networkscited by 18
  11. 2021Shift Invariance Can Reduce Adversarial Robustnesscited by 16
  12. 2022On Robust Prefix-Tuning for Text Classificationcited by 15
31 more
  1. 2022Evolution of Neural Tangent Kernels under Benign and Adversarial Trainingcited by 14
  2. 2025Adversarial Examples Are Not Bugs, They Are Superpositioncited by 13
  3. 2022On the Robustness of Bayesian Neural Networks to Adversarial Attackscited by 12
  4. 2022Understanding Adversarial Robustness Against On-manifold Adversarial Examplescited by 12
  5. 2023Adversarial Examples Might be Avoidable: The Role of Data Concentration in Adversarial Robustnesscited by 11
  6. 2022How many perturbations break this model? Evaluating robustness beyond adversarial accuracycited by 9
  7. 2022A law of adversarial risk, interpolation, and label noisecited by 8
  8. 2023Adversarial Examples Exist in Two-Layer ReLU Networks for Low Dimensional Linear Subspacescited by 8
  9. 2023Deep Reinforcement Learning for Autonomous Cyber Defence: A Surveycited by 6
  10. 2022Towards Out-of-Distribution Adversarial Robustnesscited by 5
  11. 2023How many dimensions are required to find an adversarial example?cited by 5
  12. 2022Learning Globally Smooth Functions on Manifoldscited by 4
  13. 2023How adversarial attacks can disrupt seemingly stable accurate classifierscited by 4
  14. 2026Mitigating Manifold Departure: Uncertainty-Aware Subspace Rectification for Trustworthy MLLM Decodingcited by 4
  15. 2024Model X-ray:Detecting Backdoored Models via Decision Boundarycited by 3
  16. 2024Persistent Classification: A New Approach to Stability of Data and Adversarial Examplescited by 3
  17. 2021Benford's law: what does it say on adversarial images?cited by 1
  18. 2021Random Projections for Improved Adversarial Robustnesscited by 1
  19. 2021Understanding Robustness in Teacher-Student Setting: A New Perspectivecited by 1
  20. 2021Localized Uncertainty Attackscited by 1
  21. 2023Boosting Adversarial Attacks by Leveraging Decision Boundary Informationcited by 1
  22. 2023Adversarial Sample Detection Through Neural Network Transport Dynamicscited by 1
  23. 2024MALT Powers Up Adversarial Attackscited by 1
  24. 2026Solving adversarial examples requires solving exponential misalignmentcited by 1
  25. 2023Kernels, Data & Physicscited by 0
  26. 2023F2AT: Feature-Focusing Adversarial Training via Disentanglement of Natural and Perturbed Patternscited by 0
  27. 2023On The Relationship Between Universal Adversarial Attacks And Sparse Representationscited by 0
  28. 2023On robust overfitting: adversarial training induced distribution matterscited by 0
  29. 2025Ensemble Distribution Distillation for Self-Supervised Human Activity Recognitioncited by 0
  30. 2026Uncovering and Understanding FPR Manipulation Attack in Industrial IoT Networkscited by 0
  31. 2026On Adversarial Vulnerability of Vision-Language Models through the Lens of Intermediate Spectral Subspacescited by 0