CantorNet: A Sandbox for Testing Geometrical and Topological Complexity Measures

Michal Lewandowski, Hamid Eghbalzadeh, Bernhard A. Moser
Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations, PMLR 282:981-995, 2026.

Abstract

Many natural phenomena are characterized by self-similarity, for example the symmetry of human faces, or a repetitive motif of a song. Studying of such symmetries will allow us to gain deeper insights into the underlying mechanisms of complex systems. Recognizing the importance of understanding these patterns, we propose a geometrically inspired framework to study such phenomena in artificial neural networks. To this end, we introduce \emph{CantorNet}, inspired by the triadic construction of the Cantor set, which was introduced by Georg Cantor in the $19^\text{th}$ century. In mathematics, the Cantor set is a set of points lying on a single line that is self-similar and has a counter intuitive property of being an uncountably infinite null set. Similarly, we introduce CantorNet as a sandbox for studying self-similarity by means of novel topological and geometrical complexity measures. CantorNet constitutes a family of ReLU neural networks that spans the whole spectrum of possible Kolmogorov complexities, including the two opposite descriptions (linear and exponential as measured by the description length). CantorNet’s decision boundaries can be arbitrarily ragged, yet are analytically known. Besides serving as a testing ground for complexity measures, our work may serve to illustrate potential pitfalls in geometry-ignorant data augmentation techniques and adversarial attacks.

Cite this Paper

BibTeX
@InProceedings{pmlr-v282-lewandowski26a, title = {CantorNet: A Sandbox for Testing Geometrical and Topological Complexity Measures}, author = {Lewandowski, Michal and Eghbalzadeh, Hamid and Moser, Bernhard A.}, booktitle = {Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations}, pages = {981--995}, year = {2026}, editor = {Acosta, Francisco and Azeglio, Simone and Tolooshams, Bahareh and van de Geijn, Chase and Shewmake, Christian and Sanborn, Sophia and Miolane, Nina}, volume = {282}, series = {Proceedings of Machine Learning Research}, month = {14 Dec 2024--07 Dec 2025}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v282/main/assets/lewandowski26a/lewandowski26a.pdf}, url = {https://proceedings.mlr.press/v282/lewandowski26a.html}, abstract = {Many natural phenomena are characterized by self-similarity, for example the symmetry of human faces, or a repetitive motif of a song. Studying of such symmetries will allow us to gain deeper insights into the underlying mechanisms of complex systems. Recognizing the importance of understanding these patterns, we propose a geometrically inspired framework to study such phenomena in artificial neural networks. To this end, we introduce \emph{CantorNet}, inspired by the triadic construction of the Cantor set, which was introduced by Georg Cantor in the $19^\text{th}$ century. In mathematics, the Cantor set is a set of points lying on a single line that is self-similar and has a counter intuitive property of being an uncountably infinite null set. Similarly, we introduce CantorNet as a sandbox for studying self-similarity by means of novel topological and geometrical complexity measures. CantorNet constitutes a family of ReLU neural networks that spans the whole spectrum of possible Kolmogorov complexities, including the two opposite descriptions (linear and exponential as measured by the description length). CantorNet’s decision boundaries can be arbitrarily ragged, yet are analytically known. Besides serving as a testing ground for complexity measures, our work may serve to illustrate potential pitfalls in geometry-ignorant data augmentation techniques and adversarial attacks.} }
Endnote
%0 Conference Paper %T CantorNet: A Sandbox for Testing Geometrical and Topological Complexity Measures %A Michal Lewandowski %A Hamid Eghbalzadeh %A Bernhard A. Moser %B Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations %C Proceedings of Machine Learning Research %D 2026 %E Francisco Acosta %E Simone Azeglio %E Bahareh Tolooshams %E Chase van de Geijn %E Christian Shewmake %E Sophia Sanborn %E Nina Miolane %F pmlr-v282-lewandowski26a %I PMLR %P 981--995 %U https://proceedings.mlr.press/v282/lewandowski26a.html %V 282 %X Many natural phenomena are characterized by self-similarity, for example the symmetry of human faces, or a repetitive motif of a song. Studying of such symmetries will allow us to gain deeper insights into the underlying mechanisms of complex systems. Recognizing the importance of understanding these patterns, we propose a geometrically inspired framework to study such phenomena in artificial neural networks. To this end, we introduce \emph{CantorNet}, inspired by the triadic construction of the Cantor set, which was introduced by Georg Cantor in the $19^\text{th}$ century. In mathematics, the Cantor set is a set of points lying on a single line that is self-similar and has a counter intuitive property of being an uncountably infinite null set. Similarly, we introduce CantorNet as a sandbox for studying self-similarity by means of novel topological and geometrical complexity measures. CantorNet constitutes a family of ReLU neural networks that spans the whole spectrum of possible Kolmogorov complexities, including the two opposite descriptions (linear and exponential as measured by the description length). CantorNet’s decision boundaries can be arbitrarily ragged, yet are analytically known. Besides serving as a testing ground for complexity measures, our work may serve to illustrate potential pitfalls in geometry-ignorant data augmentation techniques and adversarial attacks.
APA
Lewandowski, M., Eghbalzadeh, H. & Moser, B.A.. (2026). CantorNet: A Sandbox for Testing Geometrical and Topological Complexity Measures. Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations, in Proceedings of Machine Learning Research 282:981-995 Available from https://proceedings.mlr.press/v282/lewandowski26a.html.

Related Material