Orthogonal Graph Neural Networks
Abstract
Graph neural networks (GNNs) have received tremendous attention due to their superiority in learning node representations. These models rely on message passing and feature transformation functions to encode the structural and feature information from neighbors. However, stacking more convolutional layers significantly decreases the performance of GNNs. Most recent studies attribute this limitation to the over-smoothing issue, where node embeddings converge to indistinguishable vectors. Through a number of experimental observations, we argue that the main factor degrading the performance is the unstable forward normalization and backward gradient resulted from the improper design of the feature transformation, especially for shallow GNNs where the over-smoothing has not happened. Therefore, we propose a novel orthogonal feature transformation, named Ortho-GConv, which could generally augment the existing GNN backbones to stabilize the model training and improve the model's generalization performance. Specifically, we maintain the orthogonality of the feature transformation comprehensively from three perspectives, namely hybrid weight initialization, orthogonal transformation, and orthogonal regularization. By equipping the existing GNNs (e.g. GCN, JKNet, GCNII) with Ortho-GConv, we demonstrate the generality of the orthogonal feature transformation to enable stable training, and show its effectiveness for node and graph classification tasks.
Benchmark Results
Tested on 5 datasets 17 baseline values reported too low
Tables show presented results in global context (best 3 papers + GCN + MLP).
Classic
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | GraphSAGE+CNA | 0.9418
±0.0033 | |
| 2 | GCN+CNA | 0.9366
±0.0048 | |
| 3 | TransformerConv | 0.9350
±0.0058 | |
| 21 | GCN | 0.9004 best reported | best at pub: SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels (2017) best today: Understanding Heterophily for Graph Neural Networks (ICML 2024) |
| 39 | SAGE | 0.8911
±0.0012 | |
| 44 | GAT | 0.8897
±0.0014 best reported | best at pub: Learned Low Precision Graph Neural Networks (2020) |
| 45 | GCNII | 0.8849 | prev best: Graph Neural Networks Inspired by Classical Iterative Algorithms (ICML 2021) |
| 52 | Ortho-GCNII | 0.8881 | This paper |
| 55 | JKNet | 0.8870 best reported | best at pub: Learned Low Precision Graph Neural Networks (2020) |
| 70 | DAGNN | 0.8840 best reported | best at pub: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) |
| 108 | APPNP | 0.8787 | prev best: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) |
| 153 | Incep(DropEdge) | 0.8686 | This paper |
| 207 | Geom-GCN | 0.8540
±0.0040 best reported | best at pub: Non-Local Graph Neural Networks (2020) best today: The Expressive Power of Graph Neural Networks: A Survey (2023) |
| 223 | SGC | 0.8509
±0.0648 best reported | best at pub: Learning Conjoint Attentions for Graph Neural Nets (NeurIPS 2021) |
| 724 | MLP | 0.7768 |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | CNA | 0.9575
±0.0058 | |
| 2 | IS-GIB | 0.9390
±0.0187 | |
| 3 | EERM | 0.9112
±0.0145 | |
| 20 | APPNP | 0.8142
±0.0026 best reported | best at pub: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) best today: Feature Expansion for Graph Neural Networks (ICML 2023) |
| 35 | GAT | 0.8080
±0.0026 best reported | best at pub: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) best today: Feature Expansion for Graph Neural Networks (ICML 2023) |
| 36 | GCNII | 0.8076
±0.0030 best reported | best at pub: Extract the Knowledge of Graph Neural Networks and Go Beyond it: An Effective Knowledge Distillation Framework (WWW 2021) best today: Feature Expansion for Graph Neural Networks (ICML 2023) |
| 44 | GCN | 0.8033
±0.0088 best reported | best at pub: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) |
| 61 | GraphSAGE | 0.7923
±0.0053 | |
| 75 | DAGNN | 0.7860 best reported | best at pub: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) |
| 96 | SGC | 0.7765
±0.0012 best reported | best at pub: Deep Learning on Graphs: A Survey (2018) |
| 217 | Ortho-GCNII | 0.7420
±0.0040 | This paper |
| 350 | MLP | 0.7289
±0.0010 | |
| 372 | Incep(Drop) | 0.7270 | This paper |
| 380 | JKNet(Drop) | 0.7260 | This paper |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | LGTL | 0.9518
±0.0011 | |
| 2 | MixHop (LO) | 0.9513
±0.0038 | |
| 3 | GraphSAGE (LO) | 0.9511
±0.0036 | |
| 17 | APPNP | 0.9355
±0.0060 best reported | best at pub: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) |
| 18 | GCN | 0.9296
±0.0015 best reported | best at pub: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) |
| 20 | GAT | 0.9233
±0.0018 best reported | best at pub: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) |
| 58 | GCNII | 0.9030
±0.0043 best reported | best at pub: Graph Neural Networks Inspired by Classical Iterative Algorithms (ICML 2021) |
| 116 | SAGE | 0.8881
±0.0015 | |
| 141 | SGC | 0.8832
±0.0055 best reported | best at pub: Learning Conjoint Attentions for Graph Neural Nets (NeurIPS 2021) |
| 210 | MLP | 0.8690
±0.0074 | |
| 226 | DAGNN | 0.8640 best reported | best at pub: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) |
| 443 | Ortho-GCNII | 0.8010
±0.0030 | This paper |
| 515 | Incep(Drop) | 0.7950 | This paper |
| 566 | JKNet(Drop) | 0.7920 | This paper |
TU Dortmund
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | MSH-GNN | 0.9410
±0.0330 | |
| 2 | GRDL | 0.8260
±0.0120 | |
| 3 | GMN | 0.8225 | |
| 7 | DiffPool | 0.8161
±0.0232 best reported | best at pub: Memory-Based Graph Networks (ICLR 2020) |
| 32 | GIN | 0.7880
±0.0410 | |
| 34 | Ortho-g-U-Nets | 0.7878 | This paper |
| 59 | g-U-Nets | 0.7768 | This paper |
| 132 | DGCNN | 0.7626 | |
| 149 | MLP | 0.7600
±0.0320 | |
| 161 | PSCN | 0.7589
±0.0276 best reported | best today: Semi-Supervised Hierarchical Graph Classification (2022) Difference within reported uncertainty; not counted as underperformance. |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | GC-GNN (GCN) | 0.9210
±0.0810 | |
| 2 | SUGAR | 0.8403
±0.0133 | |
| 3 | Ortho-g-U-Nets | 0.8387 | This paper |
| 4 | g-U-Nets | 0.8300 | prev best: Graph U-Nets (2019) |
| 15 | SAGE | 0.8088
±0.0233 | |
| 18 | DiffPool | 0.8064 | This paper |
| 37 | DGCNN | 0.7937 | This paper |
| 101 | PSCN | 0.7712
±0.0241 best reported | best at pub: Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs (2017) best today: Semi-Supervised Hierarchical Graph Classification (2022) |
