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Equivariant Machine Learning on Graphs with Nonlinear Spectral Filters
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Proposes nonlinear spectral filters (NLSFs) that are exactly equivariant to graph functional shifts and transferable across graphs, via analysis/synthesis in a signal-dependent spectral domain, and reports gains over spectral GNNs on node and graph classification.
Problem
Standard spectral GNNs break equivariance to graph functional shifts (unitary operators commuting with the GSO) because pointwise nonlinearities act in the node domain, and standard graph Fourier representations depend on arbitrary eigenvector choices, so they are not transferable across graphs.
Approach
Define analysis and synthesis transforms that project signals onto eigenspaces (index-based) or spectral bands (value-based) and synthesize using the input signal itself as the basis, so the nonlinear frequency response acts on eigenspace-norm coefficients; this yields exact equivariance to functional shifts, eigenbasis independence, and transferability. A Laplacian attention mixes index-based (L)
Key results
- Att-Node-level NLSFs reach 85.37% on Cora and 75.41% on Citeseer in semi-supervised node classification, best among compared spectral GNNs.
- Att-Pooling-NLSFs achieve best graph classification accuracy on MUTAG (86.89), PTC (71.02), ENZYMES (69.94), PROTEINS (84.89), IMDB-M (55.28) and COLLAB (82.19).
- On filtered dense-split Chameleon and Squirrel, Att-Node-level NLSFs reach 42.53 and 42.66, above the listed baselines.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use att-Node-level NLSFs instead of the strongest listed baseline | Node classification · Cora | accuracy: 84.12 -> 85.37 | Helped |
| Use att-Node-level NLSFs instead of the strongest listed baseline | Node classification · Citeseer | accuracy: 73.45 -> 75.41 | Helped |
| Use att-Node-level NLSFs instead of the strongest listed baseline | Node classification · Pubmed | accuracy: 82.69 -> 82.22 | Hurt |
| Use att-Node-level NLSFs instead of the strongest listed baseline | Node classification · Chameleon | accuracy: 49.79 -> 50.58 | Helped |
| Use att-Node-level NLSFs instead of the strongest listed baseline | Node classification · Squirrel | accuracy: 38.24 -> 38.39 | Helped |
| Use att-Node-level NLSFs instead of the strongest listed baseline | Node classification · Actor | accuracy: 35.98 -> 35.13 | Hurt |
| Use att-Node-level NLSFs instead of the strongest listed baseline | Dense-split node classification · Chameleon and Squirrel, original and filtered | accuracy | Mixed |
| Use att-Pooling-NLSFs instead of the strongest listed baseline | Graph classification · MUTAG, PTC, ENZYMES, PROTEINS, IMDB-M, COLLAB | accuracy | Helped |
| Use att-Pooling-NLSFs instead of the strongest listed baseline | Graph classification · NCI1 and IMDB-B | accuracy | Hurt |
| Add the node-level NLSF representation and pooling/readout to graph-level NLSFs | att-Graph-level NLSFs · MUTAG, PTC, ENZYMES, PROTEINS, NCI1, IMDB-B, IMDB-M, COLLAB | accuracy | Helped |
Surprising findings
- Pooling-NLSFs consistently outperform graph-level NLSFs, which the authors attribute to higher expressivity from the pooling readout.
Limitations
- On large graphs, NLSFs depend only on leading eigenvectors and their complement, so information in intermediate spectral bands can be missed.
- Full eigendecomposition is costly for dense or large graphs.
Experimental setup
Datasets: Cora Citeseer Pubmed Chameleon Squirrel Actor MUTAG PTC NCI1 ENZYMES PROTEINS IMDB-B
Benchmarks: Cora Citeseer Pubmed Chameleon Squirrel Actor MUTAG PTC ENZYMES PROTEINS NCI1 IMDB-B
Metrics: accuracy
Baselines: GCN GAT SAGE ChebNet ChebNetII CayleyNet APPNP GPRGNN ARMA JacobiConv BernNet Specformer
Builds on: ChebNetII GPRGNN BernNet Specformer functional maps
Closest papers by meaning
- A Characterization Theorem for Equivariant Networks with Point-wise Activations (ICLR 2024 · similarity 0.89)
- HoloNets: Spectral Convolutions do extend to Directed Graphs (ICLR 2024 · similarity 0.89)
- Spectral Graph Neural Networks are Incomplete on Graphs with a Simple Spectrum (NeurIPS 2025 · similarity 0.89)
- On the Generalization of Equivariant Graph Neural Networks (ICML 2024 · similarity 0.89)
- On the hardness of learning under symmetries (ICLR 2024 · similarity 0.88)
- Equivariance Everywhere All At Once: A Recipe for Graph Foundation Models (NeurIPS 2025 · similarity 0.88)
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