mlsanityData to 14 Oct 2026

Local and global topological complexity measures OF ReLU neural network functions

arXiv 2204.06062 ↗First posted 12 Apr 2022math.AT cs.CG cs.LGcited by 8

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Abstract (arXiv)

We apply a generalized piecewise-linear (PL) version of Morse theory due to Grunert-Kuhnel-Rote to define and study new local and global notions of topological complexity for fully-connected feedforward ReLU neural network functions, F: R^n -> R. Along the way, we show how to construct, for each such F, a canonical polytopal complex K(F) and a deformation retract of the domain onto K(F), yielding a convenient compact model for performing calculations. We also give a construction showing that local complexity can be arbitrarily high.

Key facts. Cited by 8 papers in the map (data through 14 Oct 2026). Builds on 4 of its 19 references that are in the map. First posted on arXiv 12 Apr 2022.

Builds on

4 of 19 references.

References · 4

  1. 2016Understanding Deep Neural Networks with Rectified Linear Unitscited by 257
  2. 2019Deep ReLU Networks Have Surprisingly Few Activation Patternscited by 161
  3. 2018On Characterizing the Capacity of Neural Networks using Algebraic Topologycited by 38
  4. 2020On transversality of bent hyperplane arrangements and the topological expressiveness of ReLU neural networkscited by 35

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 1; 2023: 4; 2024: 2; 2025: 0; 2026: 1.

8 papers.

  1. 2023Hidden symmetries of ReLU networkscited by 39
  2. 2022Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networkscited by 22
  3. 2024Activation degree thresholds and expressiveness of polynomial neural networkscited by 14
  4. 2023ReLU Neural Networks, Polyhedral Decompositions, and Persistent Homologcited by 10
  5. 2023Topological Expressivity of ReLU Neural Networkscited by 10
  6. 2023Exploring the Complexity of Deep Neural Networks through Functional Equivalencecited by 4
  7. 2026Characterizing the Discrete Geometry of ReLU Networkscited by 1
  8. 2024Combinatorial Regularity for Relatively Perfect Discrete Morse Gradient Vector Fields of ReLU Neural Networkscited by 0