Topological Signatures of ReLU Neural Network Activation Patterns

Vicente Bosca, Tatum Rask, Sunia Tanweer, Andrew R. Tawfeek, Branden Stone
Proceedings of the 1st Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2025), PMLR 321:287-301, 2026.

Abstract

This paper explores the topological signatures of ReLU neural network activation patterns. We consider feedforward neural networks with ReLU activation functions and analyze the polytope decomposition of the feature space induced by the network. Mainly, we investigate how the Fiedler partition of the dual graph and show that it appears to correlate with the decision boundary—in the case of binary classification. Additionally, we compute the homology of the cellular decomposition—in a regression task—to draw similar patterns in behavior between the training loss and polyhedral cell-count, as the model is trained.

Cite this Paper

BibTeX
@InProceedings{pmlr-v321-bosca26a, title = { Topological Signatures of ReLU Neural Network Activation Patterns}, author = {Bosca, Vicente and Rask, Tatum and Tanweer, Sunia and Tawfeek, Andrew R. and Stone, Branden}, booktitle = {Proceedings of the 1st Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2025)}, pages = {287--301}, year = {2026}, editor = {Bernardez Gil, Guillermo and Black, Mitchell and Cloninger, Alexander and Doster, Timothy and Emerson, Tegan and Garcı́a-Rodondo, Ińes and Holtz, Chester and Kotak, Mit and Kvinge, Henry and Mishne, Gal and Papillon, Mathilde and Pouplin, Alison and Rainey, Katie and Rieck, Bastian and Telyatnikov, Lev and Yeats, Eric and Wang, Qingsong and Wang, Yusu and Wayland, Jeremy}, volume = {321}, series = {Proceedings of Machine Learning Research}, month = {01--02 Dec}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v321/main/assets/bosca26a/bosca26a.pdf}, url = {https://proceedings.mlr.press/v321/bosca26a.html}, abstract = {This paper explores the topological signatures of ReLU neural network activation patterns. We consider feedforward neural networks with ReLU activation functions and analyze the polytope decomposition of the feature space induced by the network. Mainly, we investigate how the Fiedler partition of the dual graph and show that it appears to correlate with the decision boundary—in the case of binary classification. Additionally, we compute the homology of the cellular decomposition—in a regression task—to draw similar patterns in behavior between the training loss and polyhedral cell-count, as the model is trained.} }
Endnote
%0 Conference Paper %T Topological Signatures of ReLU Neural Network Activation Patterns %A Vicente Bosca %A Tatum Rask %A Sunia Tanweer %A Andrew R. Tawfeek %A Branden Stone %B Proceedings of the 1st Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2025) %C Proceedings of Machine Learning Research %D 2026 %E Guillermo Bernardez Gil %E Mitchell Black %E Alexander Cloninger %E Timothy Doster %E Tegan Emerson %E Ińes Garcı́a-Rodondo %E Chester Holtz %E Mit Kotak %E Henry Kvinge %E Gal Mishne %E Mathilde Papillon %E Alison Pouplin %E Katie Rainey %E Bastian Rieck %E Lev Telyatnikov %E Eric Yeats %E Qingsong Wang %E Yusu Wang %E Jeremy Wayland %F pmlr-v321-bosca26a %I PMLR %P 287--301 %U https://proceedings.mlr.press/v321/bosca26a.html %V 321 %X This paper explores the topological signatures of ReLU neural network activation patterns. We consider feedforward neural networks with ReLU activation functions and analyze the polytope decomposition of the feature space induced by the network. Mainly, we investigate how the Fiedler partition of the dual graph and show that it appears to correlate with the decision boundary—in the case of binary classification. Additionally, we compute the homology of the cellular decomposition—in a regression task—to draw similar patterns in behavior between the training loss and polyhedral cell-count, as the model is trained.
APA
Bosca, V., Rask, T., Tanweer, S., Tawfeek, A.R. & Stone, B.. (2026). Topological Signatures of ReLU Neural Network Activation Patterns. Proceedings of the 1st Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2025), in Proceedings of Machine Learning Research 321:287-301 Available from https://proceedings.mlr.press/v321/bosca26a.html.

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