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Unsupervised Parameter-free Simplicial Representation Learning with Scattering Transforms
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Proposes simplicial scattering networks (SSN), a parameter-free, label-free scattering-transform model built on random-walk matrices over the four simplicial adjacencies, evaluated on node, graph, closure, trajectory, and simplex tasks.
Problem
Simplicial neural networks have high training complexity and need task-specific labels, so there is a need for task-agnostic, unsupervised representations of higher-order simplicial complex data.
Approach
Extends geometric scattering transforms to simplicial complexes by replacing non-square powers of random walk matrices with a sequential message-passing filter bank over boundary, coboundary, upper and lower adjacencies, taking differences across scales to retain high-frequency information; no learnable parameters.
Key results
- SSN reaches 99.39% on high-school node classification and 91.43% on primary-school, beating the best supervised graph baseline by about 5% on average.
- On trajectory prediction, SSN reaches 49% on Ocean and 61% on Synthetic, outperforming SCoNe (30.0 and 55.4).
- SSN is reported to be up to hundreds of times faster than SNNs in runtime, but it trails MPSN on graph classification (e.g., REDDIT-B 88.4 vs 92.2).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| include boundary and coboundary adjacency information rather than only Hodge Laplacians (ablation) | SSN · primary-school | accuracy: 81.63 ± 3.29 -> 91.43 ± 2.20 | Helped |
| include boundary and coboundary adjacency information rather than only Hodge Laplacians (ablation) | SSN · senate-bills | accuracy: 80.85 ± 1.86 -> 83.22 ± 3.9 | Helped |
| include boundary and coboundary adjacency information rather than only Hodge Laplacians (ablation) | SSN · email-Enron | AUC-PR: 17.20 ± 0.97 -> 17.26 ± 0.87 | No clear effect |
| use scale-wise diffusion features without consecutive-scale differences (ablation) | SSN · primary-school | accuracy: 91.43 ± 2.20 vs 85 ± 3.13 | Hurt |
| use scale-wise diffusion features without consecutive-scale differences (ablation) | SSN · madison-reviews | accuracy: 96.3 ± 1 vs 91.7 ± 2.5 | Hurt |
| use higher-order simplices for feature aggregation (ablation) | SSN · simplicial representations under structural perturbations | stability | Helped |
| increase maximum scale J (ablation) | SSN · downstream tasks | performance | Helped |
Surprising findings
- A parameter-free model outperforms several supervised simplicial neural networks on node and simplex classification.
- On graph classification, SSN underperforms supervised MPSN on most datasets, with a 1.5 percentage point drop on NCI1 vs MPSN and no clique structure in NCI1.
Limitations
- Performance lags supervised MPSN on graph classification and on the Mesh trajectory task.
- Gains depend on the presence of higher-order structure (cliques); on NCI1 SSN performs on par with GIN.
- Hyperparameter J must be chosen, and the paper uses a ridge or MLP classifier trained on top of the features.
Experimental setup
Datasets: primary-school high-school senate-bills PROTEINS NCI1 REDDIT-B REDDIT-M IMDB-B email-Enron Ocean Synthetic Mesh
Benchmarks: senate-bills primary-school high-school PROTEINS NCI1 REDDIT-B REDDIT-M IMDB-B email-Enron Ocean Synthetic Mesh
Metrics: accuracy AUC-PR
Baselines: GCN GAT GIN GraphSage GSN SCNN MPSN ScGCN GSAN SCoNe DGCNN Log. Reg.
Builds on: geometric scattering diffusion wavelets MPSN SCNN
Closest papers by meaning
- Exploitation of a Latent Mechanism in Graph Contrastive Learning: Representation Scattering (NeurIPS 2024 · similarity 0.88)
- Learning From Simplicial Data Based on Random Walks and 1D Convolutions (ICLR 2024 · similarity 0.87)
- Towards Scalable and Deep Graph Neural Networks via Noise Masking (AAAI 2025 · similarity 0.85)
- Higher-Order Graph Convolutional Network with Flower-Petals Laplacians on Simplicial Complexes (AAAI 2024 · similarity 0.85)
- Simplicial Representation Learning with Neural $k$-Forms (ICLR 2024 · similarity 0.85)
- Graph and Simplicial Complex Prediction Gaussian Process via Hodgelet Representations (NeurIPS 2026 · similarity 0.84)
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