# Transport with Support: Data-Conditional Diffusion Bridges

arXiv: https://arxiv.org/abs/2301.13636 · HTML: https://mlsanity.com/paper/2301.13636/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 69 references that are in the map.
- First posted on arXiv 31 Jan 2023.

## Builds on (most cited first, 29 of 29)

- [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)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [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)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Multivariate Probabilistic Time Series Forecasting via Conditioned Normalizing Flows](https://mlsanity.com/paper/2002.06103/) (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)
- [Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation](https://mlsanity.com/paper/2006.00702/) (2020)
- [Conditional Simulation Using Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2202.13460/) (2022)
- [Differentiable Particle Filtering via Entropy-Regularized Optimal Transport](https://mlsanity.com/paper/2102.07850/) (2021)
- [Score-Based Diffusion meets Annealed Importance Sampling](https://mlsanity.com/paper/2208.07698/) (2022)
- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [Deep Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2209.09893/) (2022)
- [Reflected Schr"odinger Bridge: Density Control with Path Constraints](https://mlsanity.com/paper/2003.13895/) (2020)
- [A flexible state space model for learning nonlinear dynamical systems](https://mlsanity.com/paper/1603.05486/) (2016)
- [Deterministic particle flows for constraining stochastic nonlinear systems](https://mlsanity.com/paper/2112.05735/) (2021)
- [Neural Stochastic Control](https://mlsanity.com/paper/2209.07240/) (2022)

## Built on by (most cited first, 9 of 9)

- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Unpaired Image-to-Image Translation via Neural Schr"odinger Bridge](https://mlsanity.com/paper/2305.15086/) (2023)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Augmented Bridge Matching](https://mlsanity.com/paper/2311.06978/) (2023)
- [Departures: Distributional Transport for Single-Cell Perturbation Prediction with Neural Schr"odinger Bridges](https://mlsanity.com/paper/2511.13124/) (2025)
- [Geometric constraints improve inference of sparsely observed stochastic dynamics](https://mlsanity.com/paper/2304.00423/) (2023)
- [Uncertainty Quantification in Machine Learning Based Segmentation: A Post-Hoc Approach for Left Ventricle Volume Estimation in MRI](https://mlsanity.com/paper/2312.02167/) (2023)
- [Twisted Schr"odinger Bridge Matching](https://mlsanity.com/paper/2607.16987/) (2026)
- [Learning to Approximate Particle Smoothing Trajectories via Diffusion Generative Models](https://mlsanity.com/paper/2406.00561/) (2024)

## Abstract (arXiv)

The dynamic Schr"odinger bridge problem provides an appealing setting for solving constrained time-series data generation tasks posed as optimal transport problems. It consists of learning non-linear diffusion processes using efficient iterative solvers. Recent works have demonstrated state-of-the-art results (eg. in modelling single-cell embryo RNA sequences or sampling from complex posteriors) but are limited to learning bridges with only initial and terminal constraints. Our work extends this paradigm by proposing the Iterative Smoothing Bridge (ISB). We integrate Bayesian filtering and optimal control into learning the diffusion process, enabling the generation of constrained stochastic processes governed by sparse observations at intermediate stages and terminal constraints. We assess the effectiveness of our method on synthetic and real-world data generation tasks and we show that the ISB generalises well to high-dimensional data, is computationally efficient, and provides accurate estimates of the marginals at intermediate and terminal times.