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Toward a Unified Geometry Understanding : Riemannian Diffusion Framework for Graph Generation and Prediction
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Proposes GeoMancer, a Riemannian latent graph diffusion framework that uses a gyrokernel (generalized Fourier) mapping on product manifolds, task-specific manifold decoupling, and self-guided manifold-constrained (CFG++) diffusion to unify graph generation and prediction.
Problem
Latent graph diffusion models embed multi-level graph features (node, edge, graph) in a single Euclidean latent space, ignoring their heterogeneous curvature; exponential-map-based Riemannian embeddings are numerically unstable and diffusion sampling drifts off the data manifold, especially for unco
Approach
Replace exponential/logarithmic maps with an isometry-invariant generalized Fourier (gyrokernel) feature map on a learnable-curvature product manifold, decouple multi-level features onto task-specific component manifolds, and turn unconditional generation into conditional generation via k-means pseudo-labels from latent graph embeddings, with manifold-constrained classifier-free guidance.
Key results
- Unconditional QM9 generation: GeoMancer reaches 100.00% validity, FCD 0.09 and NSPDK 0.0002, best on these metrics among the listed baselines.
- Conditional QM9 generation: GeoMancer obtains the best MAE on mu (0.832), HOMO (304) and gap (581), with C_V tied at 1.002 for best.
- Node classification: GeoMancer reports 97.05% on Photo, 98.78% on Physics, 93.10% on PubMed and 83.50% on Cora, the best of the listed methods on those four datasets.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add self-guidance (ablation) | GeoMancer · QM9 unconditional | Novelty (%): 54.06 -> 90.43 | Helped |
| add self-guidance (ablation) | GeoMancer · QM9 unconditional | Validity (%): 98.99 -> 100.00 | Helped |
| add self-guidance (ablation) | GeoMancer · QM9 unconditional | Uniqueness (%): 97.61 -> 95.74 | Hurt |
| add self-guidance (ablation) | GeoMancer · QM9 unconditional | FCD: 0.12 -> 0.09 | Helped |
| add self-guidance (ablation) | GeoMancer · QM9 unconditional | NSPDK: 0.0003 -> 0.0002 | Helped |
| add CFG++ manifold-constrained guidance (ablation) | GeoMancer · QM9 unconditional | Uniqueness (%): 91.74 -> 95.74 | Helped |
| add CFG++ manifold-constrained guidance (ablation) | GeoMancer · QM9 unconditional | NSPDK: 0.0010 -> 0.0002 | Helped |
| add CFG++ manifold-constrained guidance (ablation) | GeoMancer · QM9 unconditional | Novelty (%): 76.43 -> 90.43 | Helped |
| add the Riemannian component (ablation) | GeoMancer · QM9 unconditional | FCD: 0.25 -> 0.09 | Helped |
| add the Riemannian component (ablation) | GeoMancer · QM9 unconditional | NSPDK: 0.0004 -> 0.0002 | Helped |
| use GeoMancer instead of LGD | QM9 conditional generation · QM9 μ | MAE: 0.879 -> 0.832 | Helped |
| use GeoMancer instead of LGD | Node classification · Photo | Accuracy: 96.94 ± 0.14 -> 97.05 ± 0.13 | Helped |
Surprising findings
- Removing the cfg++ component raises validity-related uniqueness drop: w/o cfg++ yields uniqueness 91.74% versus 95.74% for the full model, and novelty 76.43% versus 90.43%.
- Self-guidance is credited with the large validity gain, while the Riemannian component mainly improves FCD and NSPDK.
Limitations
- Model is not large enough to verify diffusion scaling laws on graph tasks, due to limited compute.
- Node classification gains over LGD are small, since node-level manifold modeling is simpler.
- Some reported numbers are not clearly described as means with standard deviations for generation tasks; error bars are reported only for node classification.
Experimental setup
Datasets: QM9 ZINC12K Cora PubMed Amazon Photo Physics Citeseer
Benchmarks: QM9 ZINC12K Cora PubMed Amazon Photo Physics Citeseer
Metrics: validity uniqueness novelty FCD NSPDK MAE accuracy
Baselines: MoFlow GraphAF GraphDF GDSS DiGress HGGT GruM LGD EDM GeoLDM GIN PNA
Builds on: LGD HyLa MotifRGC CFG++ GraphGPS
Closest papers by meaning
- Hyperbolic Geometric Latent Diffusion Model for Graph Generation (ICML 2024 · similarity 0.90)
- Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes (ICML 2024 · similarity 0.89)
- Efficient Diffusion Models for Symmetric Manifolds (ICML 2025 · similarity 0.89)
- Graph Generation with Diffusion Mixture (ICML 2024 · similarity 0.88)
- Manifold Diffusion Fields (ICLR 2024 · similarity 0.88)
- Motif-Aware Riemannian Graph Neural Network with Generative-Contrastive Learning (AAAI 2024 · similarity 0.88)
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