# Multi-marginal Schr"odinger Bridges with Iterative Reference Refinement

arXiv: https://arxiv.org/abs/2408.06277 · HTML: https://mlsanity.com/paper/2408.06277/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 82 references that are in the map.
- First posted on arXiv 12 Aug 2024.

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

- [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)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [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)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modeling](https://mlsanity.com/paper/2304.00917/) (2023)
- [Manifold Interpolating Optimal-Transport Flows for Trajectory Inference](https://mlsanity.com/paper/2206.14928/) (2022)
- [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022)
- [Score-based Generative Neural Networks for Large-Scale Optimal Transport](https://mlsanity.com/paper/2110.03237/) (2021)
- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [Light and Optimal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2402.03207/) (2024)
- [Scalable Unbalanced Optimal Transport using Generative Adversarial Networks](https://mlsanity.com/paper/1810.11447/) (2018)
- [Deep Momentum Multi-Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2303.01751/) (2023)
- [Towards a mathematical theory of trajectory inference](https://mlsanity.com/paper/2102.09204/) (2021)
- [Energy-guided Entropic Neural Optimal Transport](https://mlsanity.com/paper/2304.06094/) (2023)
- [Neural Structure Learning with Stochastic Differential Equations](https://mlsanity.com/paper/2311.03309/) (2023)
- [Partially Observed Trajectory Inference using Optimal Transport and a Dynamics Prior](https://mlsanity.com/paper/2406.07475/) (2024)

## Built on by (most cited first, 27 of 27)

- [Curly Flow Matching for Learning Non-gradient Field Dynamics](https://mlsanity.com/paper/2510.26645/) (2025)
- [Trajectory Inference with Smooth Schr"odinger Bridges](https://mlsanity.com/paper/2503.00530/) (2025)
- [Momentum Multi-Marginal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.10168/) (2025)
- [Diffusion & Adversarial Schr"odinger Bridges via Iterative Proportional Markovian Fitting](https://mlsanity.com/paper/2410.02601/) (2024)
- [Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme](https://mlsanity.com/paper/2506.01502/) (2025)
- [Joint trajectory and network inference via reference fitting](https://mlsanity.com/paper/2409.06879/) (2024)
- [Oh SnapMMD! Forecasting Stochastic Dynamics Beyond the Schr"odinger Bridge's End](https://mlsanity.com/paper/2505.16082/) (2025)
- [Branched Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.09007/) (2025)
- [Gradient-flow SDEs have unique transient population dynamics](https://mlsanity.com/paper/2505.21770/) (2025)
- [Learning non-equilibrium diffusions with Schr"odinger bridges: from exactly solvable to simulation-free](https://mlsanity.com/paper/2505.16644/) (2025)
- [Schr"odinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentres](https://mlsanity.com/paper/2506.17197/) (2025)
- [Beyond Continuity: Simulation-free Reconstruction of Discrete Branching Dynamics from Single-cell Snapshots](https://mlsanity.com/paper/2605.00545/) (2026)
- [Flows and Diffusions on the Neural Manifold](https://mlsanity.com/paper/2507.10623/) (2025)
- [Optimal Multimarginal Schr"odinger Bridge: Minimum Spanning Tree over Measure-valued Vertices](https://mlsanity.com/paper/2509.10626/) (2025)
- [Relative Entropy Estimation in Function Space: Theory and Applications to Trajectory Inference](https://mlsanity.com/paper/2604.20775/) (2026)
- [PACE: Geometry-Aware Bridge Transport for Single-Cell Trajectory Inference](https://mlsanity.com/paper/2605.18587/) (2026)
- [Leveraging Gauge Freedom for Learning Non-Gradient Population Dynamics of Stochastic Systems](https://mlsanity.com/paper/2605.25107/) (2026)
- [Two-Parameter Flows for Learning Population Dynamics of Physical Systems](https://mlsanity.com/paper/2605.26285/) (2026)
- [Wasserstein Residuals: Learning Gradient Flows from Population Dynamics](https://mlsanity.com/paper/2607.04738/) (2026)
- [Trajectory inference via Acceleration Matching](https://mlsanity.com/paper/2608.03916/) (2026)
- [Private Continuous-Time Synthetic Trajectory Generation via Mean-Field Langevin Dynamics](https://mlsanity.com/paper/2506.12203/) (2025)
- [Multi-marginal temporal Schr"odinger Bridge Matching from unpaired data](https://mlsanity.com/paper/2510.01894/) (2025)
- [Multi-Marginal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2510.16587/) (2025)
- [Contact Wasserstein Geodesics for Non-Conservative Schr"odinger Bridges](https://mlsanity.com/paper/2511.06856/) (2025)
- [Mutual Information Optimal Density Control of Linear Systems and Generalized Schr"odinger Bridges with Reference Refinement](https://mlsanity.com/paper/2605.09349/) (2026)
- [Multimarginal flow matching with optimal transport potentials](https://mlsanity.com/paper/2606.05327/) (2026)
- [Simulation-Free Learning of Population Dynamics with Wasserstein Lagrangian Residuals](https://mlsanity.com/paper/2610.03679/) (2026)

## Abstract (arXiv)

Practitioners often aim to infer an unobserved population trajectory using sample snapshots at multiple time points. E.g., given single-cell sequencing data, scientists would like to learn how gene expression changes over a cell's life cycle. But sequencing any cell destroys that cell. So we can access data for any particular cell only at a single time point, but we have data across many cells. The deep learning community has recently explored using Schr"odinger bridges (SBs) and their extensions in similar settings. However, existing methods either (1) interpolate between just two time points or (2) require a single fixed reference dynamic (often set to Brownian motion within SBs). But learning piecewise from adjacent time points can fail to capture long-term dependencies. And practitioners are typically able to specify a model family for the reference dynamic but not the exact values of the parameters within it. So we propose a new method that (1) learns the unobserved trajectories from sample snapshots across multiple time points and (2) requires specification only of a family of reference dynamics, not a single fixed one. We demonstrate the advantages of our method on simulated and real data.