Multi-Marginal Schrödinger Bridge Matching

2025-10-18

AI Summary

In developmental biology and systems medicine, continuous individual trajectories are experimentally inaccessible; only discrete-time population snapshots are obtainable. Conventional Schrödinger bridge methods support only two-time-point modeling and thus cannot infer continuous dynamics under constraints imposed by multiple intermediate-time marginal distributions. Method: We propose the multi-marginal Schrödinger bridge matching algorithm, extending the iterative Markovian fitting framework by integrating optimal transport with entropy regularization. This enables joint enforcement of distributional constraints at multiple intermediate time points while preserving global dynamical continuity. Contribution: Our method is the first to achieve exact marginal matching across multiple time points and consistent continuous-trajectory inference. It significantly outperforms state-of-the-art approaches on both synthetic benchmarks and single-cell RNA-seq data, achieving superior accuracy—faithfully recovering marginal distributions at all time points—while maintaining computational efficiency.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

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 Schrödinger 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 Schrödinger 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.

Problem

• Modeling continuous population evolution from discrete temporal snapshots
• Extending Schrödinger Bridge framework to handle multiple intermediate distributions
• Inferring trajectories when longitudinal individual tracking is impossible

Innovation

• Multi-Marginal Schrödinger Bridge Matching algorithm for multi-marginal constraints
• Extends iterative Markovian fitting to handle multiple intermediate snapshots
• Preserves trajectory continuity while enforcing all marginal distributions