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Variational Flow Matching for Graph Generation
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Reformulates flow matching as variational inference (VFM) and derives CatFlow, a mean-field cross-entropy flow matching method for categorical graph generation that reports state-of-the-art results on abstract, QM9 and ZINC250k benchmarks.
Problem
Flow matching is usually defined on continuous vector fields; the paper asks how to build efficient continuous normalizing flows for categorical data such as graphs and molecules.
Approach
Parameterize the learned vector field as an expectation under a variational approximation of the posterior endpoint distribution, minimize KL to it (equivalent to cross-entropy). Under linear conditional fields a fully factorized (mean-field) approximation suffices, so for categorical data training reduces to per-component cross-entropy over endpoint classes, yielding CatFlow.
Key results
- On QM9, CatFlow reaches 99.81% validity, 99.95% uniqueness and 0.441 FCD, exceeding Dirichlet FM (99.10, 98.15, 0.888).
- On ZINC250k, CatFlow reaches 99.21% validity and FCD 13.211 versus 14.222 for Dirichlet FM.
- On Community-small, CatFlow achieves degree MMD 0.018 versus 0.045 for GDSS, with clustering and orbit matched to GDSS.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CatFlow instead of GDSS | Abstract graph generation · Ego-small | Degree MMD: 0.021 -> 0.013 | Helped |
| Use CatFlow instead of GDSS | Abstract graph generation · Community-small | Degree MMD: 0.045 -> 0.018 | Helped |
| Use CatFlow instead of Dirichlet FM | Molecular generation · QM9 | Valid: 99.10 -> 99.81 | Helped |
| Use CatFlow instead of Dirichlet FM | Molecular generation · QM9 | FCD: 0.888 -> 0.441 | Helped |
| Use CatFlow instead of Dirichlet FM | Molecular generation · ZINC250k | Valid: 97.52 -> 99.21 | Helped |
| Use CatFlow instead of Dirichlet FM | Molecular generation · ZINC250k | FCD: 14.222 -> 13.211 | Helped |
| Reduce model size (ablation) | CatFlow · Molecular generation | Valid-and-unique score | Helped |
| Reduce training data (ablation) | CatFlow · Molecular generation | Valid-and-unique score | Helped |
| Replace CatFlow with standard flow matching (ablation) | Molecular generation · Full-data, large-model setting | Valid-and-unique score | Hurt |
| Use CatFlow instead of standard flow matching (ablation) | Molecular generation · Training | Convergence speed | Helped |
Surprising findings
- CatFlow attains 0.441 FCD on QM9 and near-perfect validity, indicating saturation of the QM9 benchmark.
Limitations
- Reasoning about possible edges scales quadratically with number of nodes, so it does not scale well to large proteins.
- Only applied to categorical graph generation; other VFM use cases left to future work.
Experimental setup
Datasets: Ego-small (Citeseer) Community-small Enzymes Grid QM9 ZINC250k Stochastic Block Model Planar graphs
Benchmarks: Ego-small Community-small Enzymes Grid QM9 ZINC250k Stochastic Block Model Planar graphs
Metrics: MMD degree MMD clustering MMD orbit validity uniqueness FCD V.U.N.
Baselines: GraphVAE GNF EDP-GNN GDSS MoFlow GraphEBM DiGress Flow Matching Dirichlet FM SPECTRE ConGress
Builds on: flow matching DiGress Dirichlet flow matching score-based SDEs
Closest papers by meaning
- DeFoG: Discrete Flow Matching for Graph Generation (ICML 2025 · similarity 0.89)
- Categorical Flow Matching on Statistical Manifolds (NeurIPS 2024 · similarity 0.89)
- Controlled Generation with Equivariant Variational Flow Matching (ICML 2025 · similarity 0.89)
- Exponential Family Variational Flow Matching for Tabular Data Generation (ICML 2025 · similarity 0.88)
- Flow matching achieves almost minimax optimal convergence (ICLR 2025 · similarity 0.88)
- Flowette: Flow Matching with Graphette Priors for Graph Generation (NeurIPS 2026 · similarity 0.87)
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