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On Characterizing the Capacity of Neural Networks using Algebraic Topology

arXiv 1802.04443 ↗First posted 13 Feb 2018cs.LG cs.CG cs.NEcited by 37

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Abstract (arXiv)

The learnability of different neural architectures can be characterized directly by computable measures of data complexity. In this paper, we reframe the problem of architecture selection as understanding how data determines the most expressive and generalizable architectures suited to that data, beyond inductive bias. After suggesting algebraic topology as a measure for data complexity, we show that the power of a network to express the topological complexity of a dataset in its decision region is a strictly limiting factor in its ability to generalize. We then provide the first empirical characterization of the topological capacity of neural networks. Our empirical analysis shows that at every level of dataset complexity, neural networks exhibit topological phase transitions. This observation allowed us to connect existing theory to empirically driven conjectures on the choice of architectures for fully-connected neural networks.

Key facts. Cited by 37 papers in the map (data through 7 Oct 2026). First posted on arXiv 13 Feb 2018.

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Citing papers per year: 2021: 10; 2022: 10; 2023: 6; 2024: 3; 2025: 4; 2026: 4.

37 papers.

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  1. 2024Position: Categorical Deep Learning is an Algebraic Theory of All Architecturescited by 32
  2. 2022Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networkscited by 21
  3. 2021Topological Uncertainty: Monitoring trained neural networks through persistence of activation graphscited by 20
  4. 2022RipsNet: a general architecture for fast and robust estimation of the persistent homology of point cloudscited by 20
  5. 2021Activation Landscapes as a Topological Summary of Neural Network Performancecited by 17
  6. 2022Topology and geometry of data manifold in deep learningcited by 12
  7. 2022Experimental Observations of the Topology of Convolutional Neural Network Activationscited by 10
  8. 2021Topological Measurement of Deep Neural Networks Using Persistent Homologycited by 9
  9. 2022Predicting the generalization gap in neural networks using topological data analysiscited by 9
  10. 2022Lower and Upper Bounds for Numbers of Linear Regions of Graph Convolutional Networkscited by 9
  11. 2023Topological Expressivity of ReLU Neural Networkscited by 9
  12. 2021Characterizing and Measuring the Similarity of Neural Networks with Persistent Homologycited by 8
25 more
  1. 2022Local and global topological complexity measures OF ReLU neural network functionscited by 8
  2. 2021Quantitative Performance Assessment of CNN Units via Topological Entropy Calculationcited by 6
  3. 2023Deep neural networks architectures from the perspective of manifold learningcited by 5
  4. 2023On Characterizing the Evolution of Embedding Space of Neural Networks using Algebraic Topologycited by 5
  5. 2021Persistent Homology Captures the Generalization of Neural Networks Without A Validation Setcited by 4
  6. 2022Topological data analysis on noisy quantum computerscited by 4
  7. 2025The Geometry of ReLU Networks through the ReLU Transition Graphcited by 4
  8. 2026Representation Geometry as a Diagnostic for Out-of-Distribution Robustnesscited by 3
  9. 2021Dive into Layers: Neural Network Capacity Bounding using Algebraic Geometrycited by 2
  10. 2021Abstraction, Reasoning and Deep Learning: A Study of the "Look and Say" Sequencecited by 2
  11. 2022Zeroth-Order Topological Insights into Iterative Magnitude Pruningcited by 2
  12. 2024A Theoretical Study of Neural Network Expressive Power via Manifold Topologycited by 2
  13. 2022Isometric Representations in Neural Networks Improve Robustnesscited by 1
  14. 2023Data Topology-Dependent Upper Bounds of Neural Network Widthscited by 1
  15. 2023Topological Parallax: A Geometric Specification for Deep Perception Modelscited by 1
  16. 2026TopoGeoScore: A Self-Supervised Source-Only Geometric Framework for OOD Checkpoint Selectioncited by 1
  17. 2021Leveraging The Topological Consistencies of Learning in Deep Neural Networkscited by 0
  18. 2021Improving the Behaviour of Vision Transformers with Token-consistent Stochastic Layerscited by 0
  19. 2023A Structural Approach to the Design of Domain Specific Neural Network Architecturescited by 0
  20. 2024Combinatorial Regularity for Relatively Perfect Discrete Morse Gradient Vector Fields of ReLU Neural Networkscited by 0
  21. 2025A Quotient Homology Theory of Representation in Neural Networkscited by 0
  22. 2025History Matching under Uncertainty of Geological Scenarios with Implicit Geological Realism Control with Generative Deep Learning and Graph Convolutionscited by 0
  23. 2025nDNA -- the Semantic Helix of Artificial Cognitioncited by 0
  24. 2026On the Relationship Between Representation Geometry and Generalization in Deep Neural Networkscited by 0
  25. 2026On the Topology of Neural Network Superlevel Setscited by 0