# Distributionally Robust Schr"odinger Bridge

arXiv: https://arxiv.org/abs/2610.02043 · HTML: https://mlsanity.com/paper/2610.02043/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 38 references that are in the map.
- First posted on arXiv 1 Oct 2026.

## Builds on (most cited first, 22 of 22)

- [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)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Adversarial Latent Autoencoders](https://mlsanity.com/paper/2004.04467/) (2020)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Unbalanced minibatch Optimal Transport; applications to Domain Adaptation](https://mlsanity.com/paper/2103.03606/) (2021)
- [Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation](https://mlsanity.com/paper/2010.05862/) (2020)
- [Deep Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2209.09893/) (2022)
- [Outlier-Robust Optimal Transport](https://mlsanity.com/paper/2012.07363/) (2020)
- [Light and Optimal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2402.03207/) (2024)
- [Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport](https://mlsanity.com/paper/2305.14777/) (2023)
- [Sinkhorn Distributionally Robust Optimization](https://mlsanity.com/paper/2109.11926/) (2021)
- [Adjoint Schr"odinger Bridge Sampler](https://mlsanity.com/paper/2506.22565/) (2025)
- [A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/2205.00362/) (2022)
- [Soft-constrained Schrodinger Bridge: a Stochastic Control Approach](https://mlsanity.com/paper/2403.01717/) (2024)
- [Feedback Schr"odinger Bridge Matching](https://mlsanity.com/paper/2410.14055/) (2024)

## Abstract (arXiv)

Schr"odinger bridge (SB) learns stochastic transport between prescribed initial and target distributions. When the initial distribution shifts at test time, the learned dynamics can fail to recover the target distribution. We introduce the Distributionally Robust Schr"odinger Bridge (DRSB), which learns a single controller that accounts for uncertainty in the initial distribution. The DRSB objective consists of control energy and a KL penalty between the resulting terminal distribution and the target distribution. DRSB seeks a single controller that minimizes the worst-case value of this objective as the initial distribution varies within an ambiguity set around the nominal distribution. We derive an exact variational formulation of this objective and connect its fixed-terminal-cost subproblem to stochastic optimal control and distributionally robust optimization. This formulation motivates an alternating algorithm that updates the adversarial initial distribution, estimates the terminal log-density ratio, and trains the controller. We develop Wasserstein and Sinkhorn variants using stochastic control optimality conditions to approximate the gradients required for adversarial updates. Experiments on two-dimensional transport tasks and image-to-image translation show improved robustness to input perturbations relative to standard SB, with a tradeoff in nominal performance. On Gaussian mixture transport, Sinkhorn DRSB also achieves lower mean sliced Wasserstein distance than fixed-level noise augmentation at both tested unseen noise levels.