# Multi-Marginal Schr"odinger Bridge Matching

arXiv: https://arxiv.org/abs/2510.16587 · HTML: https://mlsanity.com/paper/2510.16587/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 62 references that are in the map.
- First posted on arXiv 18 Oct 2025.

## Builds on (most cited first, 30 of 31)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [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)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [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)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modeling](https://mlsanity.com/paper/2304.00917/) (2023)
- [Non-Denoising Forward-Time Diffusions](https://mlsanity.com/paper/2312.14589/) (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)
- [Deep Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2209.09893/) (2022)
- [Schr"odinger Bridge Flow for Unpaired Data Translation](https://mlsanity.com/paper/2409.09347/) (2024)
- [A Computational Framework for Solving Wasserstein Lagrangian Flows](https://mlsanity.com/paper/2310.10649/) (2023)
- [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)
- [Alternating the Population and Control Neural Networks to Solve High-Dimensional Stochastic Mean-Field Games](https://mlsanity.com/paper/2002.10113/) (2020)
- [Multi-marginal Schr"odinger Bridges with Iterative Reference Refinement](https://mlsanity.com/paper/2408.06277/) (2024)
- [The most likely evolution of diffusing and vanishing particles: Schrodinger Bridges with unbalanced marginals](https://mlsanity.com/paper/2108.02879/) (2021)
- [Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series](https://mlsanity.com/paper/2410.05602/) (2024)
- [Variational Schr"odinger Diffusion Models](https://mlsanity.com/paper/2405.04795/) (2024)
- [Multi-marginal Schrodinger bridges](https://mlsanity.com/paper/1902.08319/) (2019)
- [Minimizing relative entropy of path measures under marginal constraints](https://mlsanity.com/paper/2001.10920/) (2020)

## Abstract (arXiv)

Understanding the continuous evolution of populations from discrete temporal snapshots is a critical research challenge, particularly in fields like developmental biology and systems medicine where longitudinal tracking of individual entities is often impossible. Such trajectory inference is vital for unraveling the mechanisms of dynamic processes. While Schr"odinger Bridge (SB) offer a potent framework, their traditional application to pairwise time points can be insufficient for systems defined by multiple intermediate snapshots. This paper introduces Multi-Marginal Schr"odinger Bridge Matching (MSBM), a novel algorithm specifically designed for the multi-marginal SB problem. MSBM extends iterative Markovian fitting (IMF) to effectively handle multiple marginal constraints. This technique ensures robust enforcement of all intermediate marginals while preserving the continuity of the learned global dynamics across the entire trajectory. Empirical validations on synthetic data and real-world single-cell RNA sequencing datasets demonstrate the competitive or superior performance of MSBM in capturing complex trajectories and respecting intermediate distributions, all with notable computational efficiency.