# The data-driven Schroedinger bridge

arXiv: https://arxiv.org/abs/1806.01364 · HTML: https://mlsanity.com/paper/1806.01364/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2018.

## Built on by (most cited first, 26 of 26)

- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [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)
- [Manifold Interpolating Optimal-Transport Flows for Trajectory Inference](https://mlsanity.com/paper/2206.14928/) (2022)
- [Bayesian Learning via Neural Schr"odinger-F"ollmer Flows](https://mlsanity.com/paper/2111.10510/) (2021)
- [A mean-field games laboratory for generative modeling](https://mlsanity.com/paper/2304.13534/) (2023)
- [Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputation](https://mlsanity.com/paper/2305.07247/) (2023)
- [First Hitting Diffusion Models for Generating Manifold, Graph and Categorical Data](https://mlsanity.com/paper/2209.01170/) (2022)
- [Momentum Multi-Marginal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.10168/) (2025)
- [Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly](https://mlsanity.com/paper/2307.14442/) (2023)
- [DPBridge: Latent Diffusion Bridge for Dense Prediction](https://mlsanity.com/paper/2412.20506/) (2024)
- [FLAC: Maximum Entropy RL via Kinetic Energy Regularized Bridge Matching](https://mlsanity.com/paper/2602.12829/) (2026)
- [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)
- [One-Line-of-Code Data Mollification Improves Optimization of Likelihood-based Generative Models](https://mlsanity.com/paper/2305.18900/) (2023)
- [Simplified Diffusion Schr"odinger Bridge](https://mlsanity.com/paper/2403.14623/) (2024)
- [Feedback Schr"odinger Bridge Matching](https://mlsanity.com/paper/2410.14055/) (2024)
- [Path Structured Multimarginal Schr"odinger Bridge for Probabilistic Learning of Hardware Resource Usage by Control Software](https://mlsanity.com/paper/2310.00604/) (2023)
- [Solution of the Probabilistic Lambert Problem: Connections with Optimal Mass Transport, Schr"odinger Bridge and Reaction-Diffusion PDEs](https://mlsanity.com/paper/2401.07961/) (2024)
- [Forward Reverse Kernel Regression for the Schr"odinger bridge problem](https://mlsanity.com/paper/2507.00640/) (2025)
- [Incorporating Pre-trained Diffusion Models in Solving the Schr"odinger Bridge Problem](https://mlsanity.com/paper/2508.18095/) (2025)
- [Generalized Schr"odinger Bridge on Graphs](https://mlsanity.com/paper/2602.04675/) (2026)
- [MIOFlow 2.0: A unified framework for inferring cellular stochastic dynamics from single cell and spatial transcriptomics data](https://mlsanity.com/paper/2603.22564/) (2026)
- [Control, Transport and Sampling: Towards Better Loss Design](https://mlsanity.com/paper/2405.13731/) (2024)
- [Statistical Analysis of the Sinkhorn Iterations for Two-Sample Schr"odinger Bridge Estimation](https://mlsanity.com/paper/2510.22560/) (2025)
- [Path-Space Mirror Descent for On-Policy Reinforcement Learning under the Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2603.21621/) (2026)

## Abstract (arXiv)

Erwin Schroedinger posed, and to a large extent solved in 1931/32 the problem of finding the most likely random evolution between two continuous probability distributions. This article considers this problem in the case when only samples of the two distributions are available. A novel iterative procedure is proposed, inspired by Fortet-Sinkhorn type algorithms. Since only samples of the marginals are available, the new approach features constrained maximum likelihood estimation in place of the nonlinear boundary couplings, and importance sampling to propagate the functions varphi and hatvarphi solving the Schroedinger system. This method is well-suited to high-dimensional settings, where introducing grids leads to numerically unfeasible or unreliable methods. The methodology is illustrated in two applications: entropic interpolation of two-dimensional Gaussian mixtures, and the estimation of integrals through a variation of importance sampling.