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Momentum Multi-Marginal Schrödinger Bridge Matching
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Proposes 3MSBM, a simulation-free momentum (phase-space) multi-marginal Schrödinger bridge matching method that uses closed-form multi-marginal Brownian bridges to learn smooth trajectories through multiple snapshots.
Problem
Inferring smooth, temporally coherent trajectories from sparse snapshot marginals, where pairwise bridge methods ignore long-range dependencies and global mmSB solvers scale poorly.
Approach
Lifts the stochastic dynamics to phase space (position and velocity) and derives closed-form multi-marginal momentum Brownian bridges conditioned on several pinned positions, then fits a parameterized acceleration by simulation-free bridge matching in an alternating scheme converging to the mmSB solution.
Key results
- On Gulf of Mexico, 3MSBM achieves the lowest rest-SWD (0.05) and lowest SWD at t1, t5, t7 among compared methods.
- On Embryoid Body, 3MSBM reduces MMD at t1 to 0.18 versus 0.37 for MMFM and 0.38 for DMSB.
- On Beijing air quality, 3MSBM attains rest SWD 21.25 versus 30.25 for MMFM and 33.42 for DMSB, and is ~2.5x faster to train than DMSB on EB.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use 3MSBM instead of the reported baselines | Multi-marginal trajectory inference · Lotka-Volterra | SWD: 0.16 -> 0.13; 0.11 -> 0.15 | Mixed |
| Use 3MSBM instead of the reported baselines | Multi-marginal trajectory inference · Gulf of Mexico | SWD: 0.15 -> 0.14; 0.11 -> 0.14 | Mixed |
| Use 3MSBM instead of MMFM and DMSB | Multi-marginal trajectory inference · Beijing air quality | SWD: 17.51 -> 17.70; 21.92 -> 9.78 | Mixed |
| Use 3MSBM instead of the reported baselines | Multi-marginal trajectory inference · Embryoid Body | MMD and SWD | Helped |
| Add multiple future pinned points rather than conditioning only on the next point (ablation) | 3MSBM · Gulf of Mexico | Mean W2 distance of left-out marginals | Helped |
| Increase the number of future pinned points beyond K=n+3 (ablation) | 3MSBM · Gulf of Mexico | Mean W2 distance of left-out marginals | No clear effect |
| Increase the total number of marginals while holding other hyperparameters fixed (ablation) | 3MSBM · Embryoid Body and Gulf of Mexico | Per-epoch training time | No clear effect |
Surprising findings
- On Lotka-Volterra, SBIRR achieves better imputation at some left-out points than 3MSBM, and MMFM is best at t2 on Beijing data.
Limitations
- Effectiveness in densely sampled high-dimensional image spaces (e.g., video interpolation) is unexplored.
- Pairwise and global baselines are compared mostly on low-dimensional or PCA-reduced data.
Experimental setup
Datasets: Lotka-Volterra synthetic Gulf of Mexico ocean currents Beijing multi-site air quality Embryoid Body single-cell (EB-100 PCA)
Benchmarks: Lotka-Volterra Gulf of Mexico Beijing air quality Embryoid Body
Metrics: Sliced Wasserstein Distance Maximum Mean Discrepancy wall-clock training time allocated memory
Baselines: DMSB SBIRR MMFM Smooth SB MIOFlow MMSFM
Builds on: Schrödinger Bridge Matching Deep Momentum Multi-Marginal Schrödinger Bridge Multi-Marginal Flow Matching Momentum Brownian Bridge
Closest papers by meaning
- Generalized Schrödinger Bridge Matching (ICLR 2024 · similarity 0.92)
- Feedback Schrödinger Bridge Matching (ICLR 2025 · similarity 0.91)
- Go With the Flow: Fast Diffusion for Gaussian Mixture Models (NeurIPS 2025 · similarity 0.91)
- Trajectory Inference with Smooth Schrödinger Bridges (ICML 2025 · similarity 0.90)
- Learning non-equilibrium diffusions with Schrödinger bridges: from exactly solvable to simulation-free (NeurIPS 2025 · similarity 0.90)
- Light and Optimal Schrödinger Bridge Matching (ICML 2024 · similarity 0.89)
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