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Unitary Convolutions for Learning on Graphs and Groups
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Proposes unitary (and Lie-algebra orthogonal/unitary) graph and group convolutions that are provably isometric and avoid over-smoothing, evaluated on LRGB, heterophilous graphs, and a ring distance toy task.
Problem
Deep graph neural networks suffer from over-smoothing and struggle to learn long-range dependencies, and group-convolutional networks suffer from vanishing/exploding gradients with depth.
Approach
Replace the message-passing operator with a unitary matrix exp(i t A) (UniConv) or an exponential of a skew-Hermitian graph convolution (Lie UniConv), making the layer norm-preserving and invertible so that the Rayleigh quotient is invariant and representations do not converge; generalize to unitary group convolutions over finite groups.
Key results
- Lie UniGCN achieves 0.7173 test AP on PEPTIDES-FUNC, the best among compared models in the LRGB table.
- Unitary GCN reaches 87.21 AP on Roman-empire and 55.34 AP on Amazon-ratings, outperforming the listed MP and graph transformer baselines.
- On the ring graph-distance toy task with 100 nodes, only unitary GCNs learn the task while conventional message passing fails.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Lie UniGCN instead of DRew | LRGB graph learning · PEPTIDES-FUNC test | Test AP: 0.7150 ± 0.0044 -> 0.7173 ± 0.0061 | Helped |
| Use UniGCN instead of Graph ViT | LRGB graph learning · PEPTIDES-STRUCT test | Test MAE: 0.2449 ± 0.0016 -> 0.2425 ± 0.0009 | Helped |
| Use Lie UniGCN instead of GPS | LRGB graph learning · COCO test | Test F1: 0.3884 ± 0.0055 -> 0.3153 ± 0.0035 | Hurt |
| Use Lie UniGCN instead of CRAWL | LRGB graph learning · PASCAL-VOC test | Test F1: 0.4588 ± 0.0079 -> 0.4005 ± 0.0067 | Hurt |
| Use Unitary GCN instead of GT | Heterophilous graph node classification · Roman-empire test | Test AP: 86.51 ± 0.73 -> 87.21 ± 0.76 | Helped |
| Use Unitary GCN instead of SAGE | Heterophilous graph node classification · Amazon-ratings test | Test AP: 53.63 ± 0.39 -> 55.34 ± 0.74 | Helped |
| Use Lie Unitary GCN instead of SAGE | Heterophilous graph node classification · Minesweeper test | ROC AUC: 93.51 ± 0.57 -> 96.11 ± 0.10 | Helped |
| Use Lie Unitary GCN instead of GAT | Heterophilous graph node classification · Tolokers test | ROC AUC: 83.70 ± 0.47 -> 85.18 ± 0.43 | Helped |
| Use Lie Unitary GCN instead of GT | Heterophilous graph node classification · Questions test | ROC AUC: 77.95 ± 0.68 -> 80.01 ± 0.43 | Helped |
Surprising findings
- Enforcing unitarity in the feature transformation U is not necessary and often hurts performance; the authors leave it unconstrained in experiments.
Limitations
- Overhead of maintaining unitarity via matrix exponential approximations or Fourier diagonalization.
- Complex-valued weights require different initialization and activation functions.
- Unitary layers may be unsuitable for tasks whose target is not invertible or isometric.
Experimental setup
Datasets: PEPTIDES-FUNC PEPTIDES-STRUCT COCO PASCAL-VOC Roman-empire Amazon-ratings Minesweeper Tolokers Questions TUDataset
Benchmarks: LRGB Heterophilous Graph Dataset TUDataset
Metrics: test AP test MAE test F1 ROC AUC
Baselines: GCN GINE GatedGCN GPS GRIT Exphormer GUMP GT
Builds on: GCN unitary RNN expRNN GPS
Closest papers by meaning
- Unitary Convolutions for Message-passing and Positional Encodings on Directed Graphs (ICML 2026 · similarity 0.90)
- On the Stability of Graph Convolutional Neural Networks: A Probabilistic Perspective (NeurIPS 2025 · similarity 0.89)
- Approximately Equivariant Graph Networks (NeurIPS 2023 · similarity 0.88)
- Equivariant Machine Learning on Graphs with Nonlinear Spectral Filters (NeurIPS 2024 · similarity 0.87)
- Generalizing CNNs to graphs with learnable neighborhood quantization (NeurIPS 2024 · similarity 0.87)
- Variational Partial Group Convolutions for Input-Aware Partial Equivariance of Rotations and Color-Shifts (ICML 2024 · similarity 0.87)
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