# Momentum Multi-Marginal Schr"odinger Bridge Matching

arXiv: https://arxiv.org/abs/2506.10168 · HTML: https://mlsanity.com/paper/2506.10168/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 61 references that are in the map.
- First posted on arXiv 11 Jun 2025.

## Builds on (most cited first, 30 of 38)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [Denoising Diffusion Bridge Models](https://mlsanity.com/paper/2309.16948/) (2023)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Flow Matching in Latent Space](https://mlsanity.com/paper/2307.08698/) (2023)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Let us Build Bridges: Understanding and Extending Diffusion Generative Models](https://mlsanity.com/paper/2208.14699/) (2022)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modeling](https://mlsanity.com/paper/2304.00917/) (2023)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Manifold Interpolating Optimal-Transport Flows for Trajectory Inference](https://mlsanity.com/paper/2206.14928/) (2022)
- [Light and Optimal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2402.03207/) (2024)
- [Deep Momentum Multi-Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2303.01751/) (2023)
- [Generalized Schr"odinger Bridge Matching](https://mlsanity.com/paper/2310.02233/) (2023)
- [Aligned Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2302.11419/) (2023)
- [Learning stochastic dynamics from snapshots through regularized unbalanced optimal transport](https://mlsanity.com/paper/2410.00844/) (2024)
- [Multi-marginal Schr"odinger Bridges with Iterative Reference Refinement](https://mlsanity.com/paper/2408.06277/) (2024)
- [The data-driven Schroedinger bridge](https://mlsanity.com/paper/1806.01364/) (2018)
- [Stability of Entropic Optimal Transport and Schr"odinger Bridges](https://mlsanity.com/paper/2106.03670/) (2021)

## Built on by (most cited first, 11 of 11)

- [WFR-FM: Simulation-Free Dynamic Unbalanced Optimal Transport](https://mlsanity.com/paper/2601.06810/) (2026)
- [Is Flow Matching Just Trajectory Replay for Sequential Data?](https://mlsanity.com/paper/2602.08318/) (2026)
- [Generalized Schr"odinger Bridge on Graphs](https://mlsanity.com/paper/2602.04675/) (2026)
- [ABC: Any-Subset Autoregression via Non-Markovian Diffusion Bridges in Continuous Time and Space](https://mlsanity.com/paper/2604.27443/) (2026)
- [Free energy Estimation on Any State Space](https://mlsanity.com/paper/2605.31063/) (2026)
- [Trajectory inference via Acceleration Matching](https://mlsanity.com/paper/2608.03916/) (2026)
- [TracingFlow: A Simulation-Free Trajectory Inference Framework Based on Second-Order Dynamics](https://mlsanity.com/paper/2608.21070/) (2026)
- [Multi-marginal temporal Schr"odinger Bridge Matching from unpaired data](https://mlsanity.com/paper/2510.01894/) (2025)
- [Simulation-free Structure Learning for Stochastic Population Dynamics](https://mlsanity.com/paper/2510.16656/) (2025)
- [Contact Wasserstein Geodesics for Non-Conservative Schr"odinger Bridges](https://mlsanity.com/paper/2511.06856/) (2025)
- [Neural Wasserstein Splines: Representation-Invariant Second-Order Priors for Trajectory Inference](https://mlsanity.com/paper/2610.19878/) (2026)

## Abstract (arXiv)

Understanding complex systems by inferring trajectories from sparse sample snapshots is a fundamental challenge in a wide range of domains, e.g., single-cell biology, meteorology, and economics. Despite advancements in Bridge and Flow matching frameworks, current methodologies rely on pairwise interpolation between adjacent snapshots. This hinders their ability to capture long-range temporal dependencies and potentially affects the coherence of the inferred trajectories. To address these issues, we introduce Momentum Multi-Marginal Schr"odinger Bridge Matching (3MSBM), a novel matching framework that learns smooth measure-valued splines for stochastic systems that satisfy multiple positional constraints. This is achieved by lifting the dynamics to phase space and generalizing stochastic bridges to be conditioned on several points, forming a multi-marginal conditional stochastic optimal control problem. The underlying dynamics are then learned by minimizing a variational objective, having fixed the path induced by the multi-marginal conditional bridge. As a matching approach, 3MSBM learns transport maps that preserve intermediate marginals throughout training, significantly improving convergence and scalability. Extensive experimentation in a series of real-world applications validates the superior performance of 3MSBM compared to existing methods in capturing complex dynamics with temporal dependencies, opening new avenues for training matching frameworks in multi-marginal settings.