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MANTRA: The Manifold Triangulations Assemblage
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Introduces MANTRA, a large purely combinatorial dataset of ~43k surface and ~250k 3-manifold triangulations, and benchmarks graph- and simplicial-complex-based models on Betti number, homeomorphism type, and orientability prediction, including barycentric subdivision invariance tests.
Problem
Topological deep learning lacks large, intrinsically higher-order datasets for benchmarking models; existing higher-order datasets are annotated or derived from graphs/time series, and the only purely higher-order dataset (Torus) is tiny and low-diversity.
Approach
A combinatorial dataset of closed surface and 3-manifold triangulations (from Lutz's collection) labeled with topological invariants, with no vertex/edge features, used to test whether models learn topology and remain invariant under barycentric subdivision.
Key results
- Simplicial complex-based models outperform graph-based models on beta_1, beta_2, orientability and homeomorphism type (e.g., best T-model homeomorphism AUROC ~0.91 for CT on full surfaces).
- Orientability is the hardest task; models perform near random (AUROC ~0.56 for simplicial models on full surfaces) and worse than random for 3-manifolds.
- Barycentric subdivision of test sets degrades performance of all models that beat random, showing lack of topological invariance.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use simplicial complex-based models instead of graph-based models | MANTRA surface prediction · 2-M surfaces | AUROC (β1): 0.5 -> 0.68 | Helped |
| Use simplicial complex-based models instead of graph-based models | MANTRA surface prediction · 2-M surfaces | AUROC (homeomorphism type): 0.47 -> 0.69 | Helped |
| Use graph-based models instead of simplicial complex-based models | MANTRA β0 prediction · 2-M surfaces | accuracy (β0): 0.73 -> 1.0 | Helped |
| Use simplicial complex-based models instead of graph-based models | MANTRA surface prediction with known homeomorphism type · 2-M surfaces, known homeomorphism type | AUROC (homeomorphism type): 0.49 -> 0.66 | Helped |
| Use simplicial complex-based models instead of graph-based models | MANTRA 3-manifold prediction · 3-manifolds | AUROC (β3): 0.14 -> 0.16 | Helped |
| Apply barycentric subdivision to test triangulations | Models trained on known-type surfaces · 2-M surfaces, known homeomorphism type | AUROC (homeomorphism type): 0.49 -> 0.49; 0.66 -> 0.6 | Hurt |
| Apply barycentric subdivision to test triangulations | Models trained on known-type surfaces · 2-M surfaces, known homeomorphism type | AUROC (β2): 0.5 -> 0.5; 0.52 -> 0.51 | Hurt |
| Apply barycentric subdivision to test triangulations | Models trained on known-type surfaces · 2-M surfaces, known homeomorphism type | accuracy (β0): 1.0 -> 0.47; 0.57 -> 0.21 | Hurt |
| Weight cross-entropy loss to penalize errors in under-represented classes | Homeomorphism type and orientability classifiers · MANTRA classification tasks | AUROC | No clear effect |
| Use simplicial complex-based models instead of graph-based models | MANTRA surface prediction · 2-M surfaces | AUROC (orientability): 0.5 -> 0.56 | No clear effect |
Surprising findings
- Higher-order models that predict better than random degrade under one barycentric subdivision, indicating they are not invariant to topology-preserving refinements.
- Graph-based models can predict beta_0 perfectly while simplicial models often fail, and orientability is near-random or below random for most models.
Limitations
- Triangulations limited to at most 10 vertices, restricting transfer to high-resolution meshes.
- Models trained only 6 epochs with non-optimized hyperparameters; only one barycentric subdivision tested.
- Mostly message-passing models evaluated; many other topological architectures omitted.
Experimental setup
Datasets: MANTRA (surfaces and 3-manifold triangulations from F. H. Lutz's collection)
Benchmarks: MANTRA
Metrics: AUROC accuracy
Baselines: MLP GCN GAT Graph Transformer TopologyAdaptive GCN SAN SCCN SCCNN SCN CT DECT
Builds on: TopoBenchmarkX SAN SCCN SCCNN SCN
Closest papers by meaning
- Topological Blindspots: Understanding and Extending Topological Deep Learning Through the Lens of Expressivity (ICLR 2025 · similarity 0.88)
- Higher-Order Graph Convolutional Network with Flower-Petals Laplacians on Simplicial Complexes (AAAI 2024 · similarity 0.85)
- Cover learning for large-scale topology representation (ICML 2025 · similarity 0.85)
- The Geometric Wall: Manifold Structure Predicts Layerwise Sparse Autoencoder Scaling Laws (NeurIPS 2026 · similarity 0.85)
- Point-Level Topological Representation Learning on Point Clouds (ICML 2025 · similarity 0.85)
- TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks (ICML 2025 · similarity 0.85)
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