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Robust Angular Synchronization via Directed Graph Neural Networks
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Proposes GNNSync, an end-to-end trainable directed-GNN (DIMPA) framework with projected gradient refinement and unsupervised upset/cycle-consistency losses for angular synchronization and k-synchronization, evaluated on synthetic graphs and sensor network localization.
Problem
Angular (SO(2)) synchronization and its heterogeneous k-synchronization extension perform poorly under high measurement noise and outliers, and standard NN architectures do not handle directed, group-structured measurement graphs.
Approach
Combine a directed GNN (DIMPA) producing initial angles with several projected gradient steps (inspired by the Generalized Power Method) inside an end-to-end pipeline, trained without ground-truth angles using an upset loss and a reweighted cycle-consistency loss that downweights noisy edges.
Key results
- GNNSync attains generally the best MSE against baselines for angular synchronization (k=1) across noise levels up to 0.9.
- For k>1, GNNSync-cycle attains leading performance, with the margin growing as noise increases, density decreases, or k increases.
- Results are averaged over 10 runs with error bars on synthetic data sets.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use GNNSync for angular synchronization | Angular synchronization (k=1) · Synthetic angular synchronization datasets | MSE | Helped |
| Use GNNSync-cycle with cycle inconsistency loss | k-synchronization · Synthetic k-synchronization datasets (k=2, 3, 4) | MSE | Helped |
| Use cycle consistency loss rather than upset loss | GNNSync for k-synchronization · Harder k-synchronization problems | MSE | Helped |
| Apply projected gradient steps to the initial angle estimates (ablation) | GNNSync · Synthetic synchronization datasets | MSE | Helped |
| Train end-to-end with projected gradient steps instead of applying them only as post-processing (ablation) | GNNSync · Synthetic synchronization datasets | MSE | Helped |
| Make the projected gradient step sizes trainable instead of fixed (ablation) | GNNSync · Synthetic synchronization datasets | MSE | No clear effect |
| Use separate H matrices for estimated edge groups instead of one shared H (ablation) | GNNSync for k-synchronization · k-synchronization datasets (k>1) | MSE | Hurt |
| Use baseline-predicted angles as input features to GNNSync (ablation) | GNNSync · Synthetic synchronization datasets | MSE | Helped |
Surprising findings
- Competitive baselines are outperformed by trivial all-equal-angle guesses at high noise levels, underscoring difficulty of the regime.
Limitations
- Framework currently limited to SO(2); extension to Euc(2) or SO(3) left open.
- Loss is non-differentiable at some points; theoretical guarantees only cover upset loss, not the general-k cycle loss.
- Training relies on noise/outlier structure of synthetic models; real-world evaluation limited to SNL.
Experimental setup
Datasets: synthetic angular synchronization graphs (ERO, BAO, RGGO outlier models) U.S. cities map (n=1097) PACM point cloud data set
Benchmarks: synthetic ERO/BAO/RGGO angular synchronization k-synchronization for k in {2,3,4} U.S. cities sensor network localization PACM point cloud localization
Metrics: MSE with angle correction Average Normalized Error
Baselines: Spectral Spectral_RN GPM TranSync CEMP_GCW CEMP_MST TAS
Builds on: DIMPA GNNRank Generalized Power Method Spectral_RN
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