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Differentiable Euler Characteristic Transforms for Shape Classification
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Makes the Euler Characteristic Transform differentiable via sigmoid-smoothed indicators so it can be trained end-to-end as a layer or loss for point cloud, graph, and mesh classification.
Problem
The Euler Characteristic Transform (ECT) is an efficient shape invariant but has been used only as a static, hand-chosen feature descriptor, not learnable within deep networks, and its direction sampling is costly.
Approach
Rewrite the ECT as an alternating sum of indicator functions over simplices of the max-height in a direction, replace the indicators with sigmoids so the transform is differentiable w.r.t. directions and coordinates, and embed the resulting ECT image with an MLP plus permutation-invariant pooling.
Key results
- On MNIST-Superpixel, ECT+MLP reaches 97.20±0.10 accuracy, above GAT, GCN, GIN and GraphSage in the reported table, with about 10.8 s per epoch.
- On ModelNet40, DECT reaches 74±0.5 accuracy with 100 points and 77.1±0.4 with 1000 points, below the reported 82.0 and 87.0 of the compared method.
- Learning directions raises MNIST point-cloud accuracy with 2 directions from 77.61±7.98 (fixed) to 81.29±3.39 (learnable).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| learn the directions instead of keeping them fixed (ablation) | DECT (ECT + MLP) · MNIST | accuracy: $77.61 \pm 7.98$ -> $81.29 \pm 3.39$ | Helped |
| increase the parameter budget from 4K to 65K (ablation) | ECT+CNN · Letter-low | accuracy: 91.51\pm2.08 -> 96.80\pm1.20 | Helped |
| increase the parameter budget from 4K to 65K (ablation) | ECT+CNN · Letter-med | accuracy: 76.22\pm 4.76 -> 86.30\pm 2.00 | Helped |
| increase the parameter budget from 4K to 65K (ablation) | ECT+CNN · Letter-high | accuracy: 63.82\pm6.04 -> 85.40\pm1.30 | Helped |
| increase the parameter budget from 4K to 65K (ablation) | ECT+CNN · BZR | accuracy: 81.79\pm3.19 -> 84.30\pm6.10 | Mixed |
| increase the parameter budget from 4K to 65K (ablation) | ECT+CNN · COX2 | accuracy: 70.4 \pm0.9 -> 74.60\pm4.50 | Helped |
| increase the parameter budget from 4K to 65K (ablation) | ECT+CNN · DHFR | accuracy: 67.93\pm 4.97 -> 72.90\pm 1.60 | Helped |
| increase the number of sampled points from 100 to 1000 (ablation) | DECT · ModelNet40 | accuracy: 74\pm0.5 -> 77.1\pm0.4 | Helped |
| use DECT instead of GraphSage | MNIST-Superpixel · MNIST-Superpixel | epoch runtime: 113.12 -> 10.80 | Helped |
| use DECT instead of the prior method | DECT · ModelNet40 (100 points) | accuracy: 82.0\pm2.0 -> 74\pm0.5 | Hurt |
| use DECT instead of the prior method | DECT · ModelNet40 (1000 points) | accuracy: 87.0\pm2.0 -> 77.1\pm0.4 | Hurt |
Surprising findings
- Learning the ECT directions reduces variance as well as raising accuracy in the sparse 2-direction MNIST setting.
Limitations
- Accuracy on ModelNet40 trails the reported performance of the compared topology-based method.
- Ignores rotation equivariance in the classifier; the ECT is only rotation-equivariant, not used with equivariant architectures.
- Structured complexes such as graphs or simplicial complexes are left for future work for point cloud optimisation.
Experimental setup
Datasets: MNIST MNIST-Superpixel ModelNet40 BZR COX2 DHFR Letter-low Letter-med Letter-high
Benchmarks: MNIST-Superpixel ModelNet40 BZR COX2 DHFR Letter-low Letter-med Letter-high
Metrics: accuracy runtime per epoch
Baselines: GAT GCN GIN GraphSage MLP
Builds on: Euler Characteristic Transform persistent homology transform deep sets
Closest papers by meaning
- Diss-l-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms (ICML 2025 · similarity 0.91)
- Point-Level Topological Representation Learning on Point Clouds (ICML 2025 · similarity 0.83)
- Linear Transformer Topological Masking with Graph Random Features (ICLR 2025 · similarity 0.82)
- GIST: Gauge-Invariant Spectral Transformers for Scalable Graph Neural Operators (NeurIPS 2026 · similarity 0.82)
- On topological descriptors for graph products (NeurIPS 2025 · similarity 0.82)
- Towards Scalable Topological Regularizers (ICLR 2025 · similarity 0.82)
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