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EuLearn: A 3D database for learning Euler characteristics

Rodrigo Fritz, Pablo Suárez-Serrato, Victor Mijangos, Anayanzi D. Martinez-Hernandez, Eduardo Ivan Velazquez Richards

TL;DR

EuLearn addresses a core gap in 3D learning by providing a dataset with uniformly distributed genus across $g\in\{0,1,...,10\}$, enabling robust learning of Euler characteristics. Surfaces are generated from thickened singular Fourier knots via Marching Cubes, yielding well-defined, orientable manifolds with topology tracked by $\chi(S)=V-E+F=2-2g$. Vanilla 3D architectures struggle at genus classification, prompting a non-Euclidean sampling strategy that preserves adjacency and graph-informed variants of PointNet and Transformer models, which achieve higher accuracy (around $0.81$) and better stability. The dataset (3,300 surfaces, 3 files per surface, about $15.2$GB) provides a benchmark and potential pretraining resource for topology-aware learning in 3D geometry tasks.

Abstract

We present EuLearn, the first surface datasets equitably representing a diversity of topological types. We designed our embedded surfaces of uniformly varying genera relying on random knots, thus allowing our surfaces to knot with themselves. EuLearn contributes new topological datasets of meshes, point clouds, and scalar fields in 3D. We aim to facilitate the training of machine learning systems that can discern topological features. We experimented with specific emblematic 3D neural network architectures, finding that their vanilla implementations perform poorly on genus classification. To enhance performance, we developed a novel, non-Euclidean, statistical sampling method adapted to graph and manifold data. We also introduce adjacency-informed adaptations of PointNet and Transformer architectures that rely on our non-Euclidean sampling strategy. Our results demonstrate that incorporating topological information into deep learning workflows significantly improves performance on these otherwise challenging EuLearn datasets.

EuLearn: A 3D database for learning Euler characteristics

TL;DR

EuLearn addresses a core gap in 3D learning by providing a dataset with uniformly distributed genus across , enabling robust learning of Euler characteristics. Surfaces are generated from thickened singular Fourier knots via Marching Cubes, yielding well-defined, orientable manifolds with topology tracked by . Vanilla 3D architectures struggle at genus classification, prompting a non-Euclidean sampling strategy that preserves adjacency and graph-informed variants of PointNet and Transformer models, which achieve higher accuracy (around ) and better stability. The dataset (3,300 surfaces, 3 files per surface, about GB) provides a benchmark and potential pretraining resource for topology-aware learning in 3D geometry tasks.

Abstract

We present EuLearn, the first surface datasets equitably representing a diversity of topological types. We designed our embedded surfaces of uniformly varying genera relying on random knots, thus allowing our surfaces to knot with themselves. EuLearn contributes new topological datasets of meshes, point clouds, and scalar fields in 3D. We aim to facilitate the training of machine learning systems that can discern topological features. We experimented with specific emblematic 3D neural network architectures, finding that their vanilla implementations perform poorly on genus classification. To enhance performance, we developed a novel, non-Euclidean, statistical sampling method adapted to graph and manifold data. We also introduce adjacency-informed adaptations of PointNet and Transformer architectures that rely on our non-Euclidean sampling strategy. Our results demonstrate that incorporating topological information into deep learning workflows significantly improves performance on these otherwise challenging EuLearn datasets.
Paper Structure (37 sections, 3 theorems, 12 equations, 19 figures, 7 tables, 2 algorithms)

This paper contains 37 sections, 3 theorems, 12 equations, 19 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Previous Dataset Work
  3. Construction of our EuLearn Dataset
  4. Selection of Curve Parameters
  5. Experiments
  6. Sampling
  7. Deep Learning Models
  8. Results
  9. Hardware Used
  10. Conclusions
  11. Acknowledgments
  12. Proofs
  13. Proof of \ref{['thm:FAUST-S2']}
  14. Proof of \ref{['thm:not-unif-dist-data']}
  15. Proof of \ref{['thm:EuLearn-unif-dist']}
  16. ...and 22 more sections

Key Result

Theorem 1

Every file in the FAUST dataset is homeomorphic to a $2$-dimensional sphere. Therefore, the FAUST dataset only has a single topological type.

Figures (19)

  • Figure 1: Examples of embedded, knotted, surfaces for each genus in our EuLearn database.
  • Figure 2: Examples of meshes generated from tubular neighborhoods around the following molecules: indole ($C_6H_4CCNH_3$), pyridine ($C_5H_5N$), 2-methyltetrahydrofuran ($C_5H_{10}O$), ethylene oxide ($C_2H_4O$), flavopereirin ($C_{17}H_{15}N_{2}$), respectively from left to right. QSAR molecular analysis Drugdesign models and predicts chemical properties using descriptors such as ring count (topologically related to genus), molecular shape, area, and interatomic distances toporingcountdrugdiscovery.
  • Figure 3: A sample of 4 examples for each genus from 0 to 5 from the EuLearn dataset.
  • Figure 4: Another sample of 4 examples for each genus from 6 to 10 from the EuLearn dataset.
  • Figure 5: Detail of the curve thickening process.
  • ...and 14 more figures

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • proof
  • proof
  • proof
  • Definition 3.4
  • ...and 3 more