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Generalized Schrödinger Bridge Matching
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Proposes Generalized Schrödinger Bridge Matching (GSBM), a simulation-free matching algorithm that solves generalized Schrödinger bridge problems with task-specific state costs via conditional stochastic optimal control and Gaussian path approximation with optional path-integral resampling.
Problem
Learning diffusion/flow models that transport between two boundary distributions while optimizing a task-specific state cost (Generalized Schrödinger Bridge), where existing methods either relax feasibility or lack stable convergence in high dimensions.
Approach
Alternate between a Stage 1 drift matching step (implicit or explicit matching loss) and a Stage 2 conditional stochastic optimal control problem solved per endpoint pair with a Gaussian path approximation parameterized by splines, so the learned marginals stay near the boundary distributions throughout training.
Key results
- On mean-field crowd navigation tasks, GSBM maintains feasibility throughout training while DeepGSB shows instability and occasional divergence.
- On high-dimensional (1000-d) opinion depolarization, GSBM achieves close to half the GSB objective value of DeepGSB.
- On AFHQ dog-to-cat translation, GSBM achieves lower FID than DSBM while adding only about 0.5% wallclock overhead.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add the task-specific state cost and solve CondSOC | DSBM on AFHQ dog-to-cat · AFHQ dog-to-cat | FID | Helped |
| Add the task-specific state cost and solve CondSOC | DSBM on AFHQ dog-to-cat · AFHQ dog-to-cat | wallclock time: 0.5% | Hurt |
| Enable path integral resampling (ablation) | GSBM on Stunnel · Stunnel | objective value | Helped |
| Enable path integral resampling (ablation) | GSBM on Stunnel · Stunnel | runtime: 8% | Hurt |
| Use implicit instead of explicit matching | GSBM on Stunnel · Stunnel | runtime: ×2.7 | Hurt |
| Add the task-specific state cost and solve CondSOC | DSBM on opinion depolarization · Opinion depolarization | objective value | Helped |
| Maintain feasibility throughout training with GSBM rather than using DeepGSB | DeepGSB · Stunnel, Vneck, and GMM | feasibility | Helped |
| Use GSBM rather than DeepGSB | DeepGSB on LiDAR navigation · LiDAR manifold navigation | objective value | Helped |
| Use the task-specific latent-space state cost rather than the degenerate cost | DSBM · AFHQ dog-to-cat | coupling quality | Helped |
Surprising findings
- GSBM remains functional with sigma=0, reducing to action matching and flow matching with a generalized geodesic path.
Limitations
- Requires differentiability of the state cost V_t.
- Convergence analysis relies on quadratic control cost, leaving necessary conditions.
- Feasibility is approximate for the explicit matching loss, which does not share the same minimizer as the implicit loss in general.
Experimental setup
Datasets: AFHQ (dog-cat) Mt. Rainier LiDAR scans (34,183 points) synthetic 2D mean-field crowd navigation tasks
Benchmarks: Stunnel Vneck GMM AFHQ dog-to-cat translation
Metrics: W2 distance to target GSB objective value FID directional similarity
Baselines: DeepGSB DSBM NLSB
Builds on: Schrödinger bridge matching DSBM DeepGSB entropic action matching flow matching
Closest papers by meaning
- Feedback Schrödinger Bridge Matching (ICLR 2025 · similarity 0.92)
- Momentum Multi-Marginal Schrödinger Bridge Matching (NeurIPS 2025 · similarity 0.92)
- Go With the Flow: Fast Diffusion for Gaussian Mixture Models (NeurIPS 2025 · similarity 0.92)
- Light and Optimal Schrödinger Bridge Matching (ICML 2024 · similarity 0.92)
- Variational Schrödinger Diffusion Models (ICML 2024 · similarity 0.90)
- Exponential Convergence Guarantees for Iterative Markovian Fitting (NeurIPS 2025 · similarity 0.90)
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