mlsanityData to 7 Oct 2026

ReLU Neural Networks, Polyhedral Decompositions, and Persistent Homolog

arXiv 2306.17418 ↗First posted 30 Jun 2023math.AT cs.LG math.OCcited by 10

Not on the map. Why?

Abstract (arXiv)

A ReLU neural network leads to a finite polyhedral decomposition of input space and a corresponding finite dual graph. We show that while this dual graph is a coarse quantization of input space, it is sufficiently robust that it can be combined with persistent homology to detect homological signals of manifolds in the input space from samples. This property holds for a variety of networks trained for a wide range of purposes that have nothing to do with this topological application. We found this feature to be surprising and interesting; we hope it will also be useful.

Key facts. Cited by 10 papers in the map (data through 7 Oct 2026). Builds on 16 of its 25 references that are in the map. First posted on arXiv 30 Jun 2023.

Builds on

16 of 25 references.

References · 16

  1. 2014On the Number of Linear Regions of Deep Neural Networkscited by 378
  2. 2016On the Expressive Power of Deep Neural Networkscited by 302
  3. 2016Understanding Deep Neural Networks with Rectified Linear Unitscited by 253
  4. 2017Bounding and Counting Linear Regions of Deep Neural Networkscited by 161
  5. 2019Deep ReLU Networks Have Surprisingly Few Activation Patternscited by 160
  6. 2019PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signaturescited by 125
  7. 2018Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topologycited by 73
  8. 2021Intrinsic Dimension, Persistent Homology and Generalization in Neural Networkscited by 70
  9. 2020Reachable Polyhedral Marching (RPM): A Safety Verification Algorithm for Robotic Systems with Deep Neural Network Componentscited by 36
  10. 2022Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networkscited by 21
  11. 2018Reachable Set Estimation and Safety Verification for Piecewise Linear Systems with Neural Network Controllerscited by 15
  12. 2021Traversing the Local Polytopes of ReLU Neural Networks: A Unified Approach for Network Verificationcited by 13
4 more
  1. 2020Reachability Analysis for Feed-Forward Neural Networks using Face Latticescited by 11
  2. 2020Locally Linear Attributes of ReLU Neural Networkscited by 9
  3. 2022Local and global topological complexity measures OF ReLU neural network functionscited by 8
  4. 2023Hamming Similarity and Graph Laplacians for Class Partitioning and Adversarial Image Detectioncited by 4

Built on by

Cited by year (2026 to 7 Oct)

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

10 papers.

  1. 2024Defining Neural Network Architecture through Polytope Structures of Datasetcited by 2
  2. 2025Topological Signatures of ReLU Neural Network Activation Patternscited by 2
  3. 2023Piecewise Polynomial Regression of Tame Functions via Integer Programmingcited by 1
  4. 2024A rank decomposition for the topological classification of neural representationscited by 1
  5. 2024A Computational Tropical Geometry Framework for Neural Networkscited by 1
  6. 2026Characterizing the Discrete Geometry of ReLU Networkscited by 1
  7. 2025A Quotient Homology Theory of Representation in Neural Networkscited by 0
  8. 2025Persistent Topological Structures and Cohomological Flows as a Mathematical Framework for Brain-Inspired Representation Learningcited by 0
  9. 2026Causal Explanations from the Geometric Properties of ReLU Neural Networkscited by 0
  10. 2026Message Tuning Outshines Graph Prompt Tuning: A Prismatic Space Perspectivecited by 0