mlsanityData to 14 Oct 2026

Topology and geometry of data manifold in deep learning

arXiv 2204.08624 ↗First posted 19 Apr 2022cs.LG cs.CV math.ATcited by 12

Not on the map. Why?

Abstract (arXiv)

Despite significant advances in the field of deep learning in applications to various fields, explaining the inner processes of deep learning models remains an important and open question. The purpose of this article is to describe and substantiate the geometric and topological view of the learning process of neural networks. Our attention is focused on the internal representation of neural networks and on the dynamics of changes in the topology and geometry of the data manifold on different layers. We also propose a method for assessing the generalizing ability of neural networks based on topological descriptors. In this paper, we use the concepts of topological data analysis and intrinsic dimension, and we present a wide range of experiments on different datasets and different configurations of convolutional neural network architectures. In addition, we consider the issue of the geometry of adversarial attacks in the classification task and spoofing attacks on face recognition systems. Our work is a contribution to the development of an important area of explainable and interpretable AI through the example of computer vision.

Key facts. Cited by 12 papers in the map (data through 14 Oct 2026). Builds on 40 of its 82 references that are in the map. First posted on arXiv 19 Apr 2022.

Builds on

40 of 82 references.

References · 40

  1. 2015Deep Residual Learning for Image Recognitioncited by 52,096
  2. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  3. 2014Very Deep Convolutional Networks for Large-Scale Image Recognitioncited by 15,364
  4. 2014Explaining and Harnessing Adversarial Examplescited by 7,364
  5. 2018MobileNetV2: Inverted Residuals and Linear Bottleneckscited by 4,795
  6. 2017Squeeze-and-Excitation Networkscited by 4,789
  7. 2018Large Scale GAN Training for High Fidelity Natural Image Synthesiscited by 2,626
  8. 2019Fantastic Generalization Measures and Where to Find Themcited by 513
  9. 2021The Intrinsic Dimension of Images and Its Impact on Learningcited by 315
  10. 2018Characterizing Adversarial Subspaces Using Local Intrinsic Dimensionalitycited by 282
  11. 2019Intrinsic dimension of data representations in deep neural networkscited by 282
  12. 2018Estimating the intrinsic dimension of datasets by a minimal neighborhood informationcited by 250
28 more
  1. 2017Searching for Activation Functionscited by 169
  2. 2016Technical Report on the CleverHans v2.1.0 Adversarial Examples Librarycited by 124
  3. 2019Ripser: efficient computation of Vietoris-Rips persistence barcodescited by 117
  4. 2018Predicting the Generalization Gap in Deep Networks with Margin Distributionscited by 113
  5. 2020Topology of deep neural networkscited by 112
  6. 2020CelebA-Spoof: Large-Scale Face Anti-Spoofing Dataset with Rich Annotationscited by 109
  7. 2018Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topologycited by 73
  8. 2021On the geometry of generalization and memorization in deep neural networkscited by 73
  9. 2021Intrinsic Dimension, Persistent Homology and Generalization in Neural Networkscited by 70
  10. 2021Scikit-dimension: a Python package for intrinsic dimension estimationcited by 68
  11. 2021Deep Learning for Face Anti-Spoofing: A Surveycited by 65
  12. 2020Hausdorff Dimension, Heavy Tails, and Generalization in Neural Networkscited by 62
  13. 2018On the Intrinsic Dimensionality of Image Representationscited by 57
  14. 2020Computing the Testing Error without a Testing Setcited by 54
  15. 2018Generalization Error in Deep Learningcited by 47
  16. 2021The Dimpled Manifold Model of Adversarial Examples in Machine Learningcited by 41
  17. 2018On Characterizing the Capacity of Neural Networks using Algebraic Topologycited by 38
  18. 2018Attacks on State-of-the-Art Face Recognition using Attentional Adversarial Attack Generative Networkcited by 38
  19. 2020Empirical Studies on the Properties of Linear Regions in Deep Neural Networkscited by 36
  20. 2020Deep Networks and the Multiple Manifold Problemcited by 35
  21. 2020Hierarchical nucleation in deep neural networkscited by 31
  22. 2018Topological Approaches to Deep Learningcited by 29
  23. 2018Topological Data Analysis of Decision Boundaries with Application to Model Selectioncited by 19
  24. 2021Activation Landscapes as a Topological Summary of Neural Network Performancecited by 18
  25. 2016Efficient Representation of Low-Dimensional Manifolds using Deep Networkscited by 13
  26. 2017Adversary Detection in Neural Networks via Persistent Homologycited by 12
  27. 2018Curvature-based Comparison of Two Neural Networkscited by 6
  28. 2019Abstraction Mechanisms Predict Generalization in Deep Neural Networkscited by 5

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 4; 2024: 5; 2025: 1; 2026: 2.

12 papers.

  1. 2023Intrinsic Dimension Estimation for Robust Detection of AI-Generated Textscited by 112
  2. 2024A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Modelscited by 42
  3. 2023Metric Space Magnitude and Generalisation in Neural Networkscited by 15
  4. 2024Persistent Topological Features in Large Language Modelscited by 12
  5. 2024Deep Learning as Ricci Flowcited by 4
  6. 2023Singular value decomposition based matrix surgerycited by 3
  7. 2024Defining Neural Network Architecture through Polytope Structures of Datasetcited by 2
  8. 2023Data Topology-Dependent Upper Bounds of Neural Network Widthscited by 1
  9. 2025Neural Feature Geometry Evolves as Discrete Ricci Flowcited by 1
  10. 2024Quantifying Manifolds: Do the manifolds learned by Generative Adversarial Networks converge to the real data manifoldcited by 0
  11. 2026Self-Directed Task Identificationcited by 0
  12. 2026DCRL: Decoupling and Coupling Reinforcement Learning via Policy-Reward Manifold Alignmentcited by 0