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Mixup Barcodes: Quantifying Geometric-Topological Interactions Between Point Clouds

Authors Hubert Wagner , Nickolas Arustamyan , Matthew Wheeler , Peter Bubenik

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LIPIcs.SoCG.2026.94.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • mixup barcode
  • persistent homology
  • persistence barcode
  • persistence diagram
  • image persistent homology
  • image persistence
  • deep learning
  • multilayer perceptron
  • topology of neural network embeddings
  • disentanglement

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Abstract

We propose a novel geometric-topological descriptor called a mixup barcode. Intuitively, it characterizes the shape of a point cloud as well as its spatial relationship with another point cloud embedded in the same ambient space. More technically, it enriches a standard persistence barcode with information on the image persistent homology. In three dimensions it captures natural spatial relationships like overlap and surrounding; in higher dimensions more intricate spatial relationships are captured. We provide a theoretical setup and a simple algorithm for mixup barcodes.
As a proof of concept, we explore data arising in a geometric-topological problem from machine learning. Specifically, we take first steps towards verifying a hypothesis stating that geometric-topological relationships within intermediate point cloud representations in an artificial neural network can hinder its training. More broadly, our experiments suggest that mixup barcodes are useful for characterizing spatial relationships and spatial interactions (i.e. the evolution of spatial relationships) that are hard to directly visualize or capture using standard methods.

Cite As Get BibTex

Hubert Wagner, Nickolas Arustamyan, Matthew Wheeler, and Peter Bubenik. Mixup Barcodes: Quantifying Geometric-Topological Interactions Between Point Clouds. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 94:1-94:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.94

Author Details

Hubert Wagner
  • University of Florida, Gainesville, FL, USA
Nickolas Arustamyan
  • University of Central Florida, Orlando, FL, USA
Matthew Wheeler
  • University of Florida, Gainesville, FL, USA
Peter Bubenik
  • University of Florida, Gainesville, FL, USA

Funding

  • Wagner, Hubert: Partially supported by the 2022 Google Research Scholar Award in Algorithms and Optimization.
  • Wheeler, Matthew: Supported by The American Association of Immunologists through an Intersect Fellowship for Computational Scientists and Immunologists.

Acknowledgements

The authors thank Ondřej Draganov, Teresa Heiss, Herbert Edelsbrunner and Yasu Hiraoka for helpful discussions.

Supplementary Materials

References

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