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Topological Schrödinger Bridge Matching
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Proposes the Topological Schrödinger Bridge Problem (TSBP) with topology-aware linear SDE references, derives a closed-form Gaussian topological SB, and builds TSB-based generative/matching models with topological neural policies trained by likelihood.
Problem
Schrödinger bridge matching of signal distributions on topological domains (graphs, simplicial complexes), where classical Euclidean SB methods ignore the underlying topology.
Approach
Set the SB reference process to a linear topology-aware SDE (topological convolution drift, e.g., topological heat diffusion), derive closed-form Gaussian topological SB marginals and SDE, show forward-backward topological SDE optimality for general cases, and parameterize the policies with graph/simplicial neural networks trained by likelihood.
Key results
- TSB-VE reduces 1-Wasserstein distance on seismic magnitudes to 7.69 versus 11.49 for SB-VE and on traffic flows to 10.51 versus 19.04 for SB-VE.
- TSB-BM achieves 7.51 on brain signals versus 12.08 for SB-BM, and 0.14 versus 0.33 on single-cell data.
- Across all five datasets, the best TSB variant beats the best SB variant in the reported 1-Wasserstein distances over five runs.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use topology-aware TSB references instead of Euclidean SB references | Topological signal generation and matching · Seismic magnitudes | 1-Wasserstein distance: 11.49 -> 7.69 | Helped |
| Use topology-aware TSB references instead of Euclidean SB references | Topological signal generation and matching · Traffic flows | 1-Wasserstein distance: 18.22 -> 9.92 | Helped |
| Use topology-aware TSB references instead of Euclidean SB references | Topological signal generation and matching · Brain signals | 1-Wasserstein distance: 12.08 -> 7.51 | Helped |
| Use topology-aware TSB references instead of Euclidean SB references | Topological signal generation and matching · Single-cell data | 1-Wasserstein distance: 0.33 -> 0.14 | Helped |
| Use topology-aware TSB-VP instead of Euclidean SB-VP | Ocean current matching · Ocean currents | 1-Wasserstein distance: 0.83 -> 0.53 | Helped |
| Use GCN rather than ResBlock as the policy model | TSB-BM/VE · Training curves on seismic and traffic datasets | training convergence | Helped |
| Use the closed-form Gaussian TSB prior rather than TSB-BM reference dynamics | Ocean current matching · Ocean currents | final-state fidelity | Helped |
Limitations
- Closed-form Gaussian solution only for Gaussian boundary distributions; general case relies on unknown policies approximated by networks.
- Reference dynamics restricted to linear drift up to order one in the main experiments.
- Baselines limited to Euclidean SB models without topology awareness.
Experimental setup
Datasets: Human Connectome Project fMRI brain signals single-cell embryoid body data ocean currents drifter data IRIS seismic events PeMSD4 traffic flows
Benchmarks: Human Connectome Project brain signals single-cell embryoid body ocean currents IRIS seismic PeMSD4 traffic
Metrics: 1-Wasserstein distance Bures-Wasserstein distance
Baselines: SB-BM SB-VE SB-VP
Builds on: Schrödinger bridge score-based generative models stochastic interpolants likelihood training of SB models
Closest papers by meaning
- Generalized Schrödinger Bridge Matching (ICLR 2024 · similarity 0.89)
- Momentum Multi-Marginal Schrödinger Bridge Matching (NeurIPS 2025 · similarity 0.89)
- Discrete Diffusion Schrödinger Bridge Matching for Graph Transformation (ICLR 2025 · similarity 0.88)
- Light and Optimal Schrödinger Bridge Matching (ICML 2024 · similarity 0.88)
- Feedback Schrödinger Bridge Matching (ICLR 2025 · similarity 0.88)
- Categorical Schrödinger Bridge Matching (ICML 2025 · similarity 0.87)
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