# Schr"odinger bridge for generative AI: Soft-constrained formulation and convergence analysis

arXiv: https://arxiv.org/abs/2510.11829 · HTML: https://mlsanity.com/paper/2510.11829/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 69 references that are in the map.
- First posted on arXiv 13 Oct 2025.

## Builds on (most cited first, 30 of 40)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Fine-Tuning of Continuous-Time Diffusion Models as Entropy-Regularized Control](https://mlsanity.com/paper/2402.15194/) (2024)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [On Memorization in Diffusion Models](https://mlsanity.com/paper/2310.02664/) (2023)
- [Understanding and Improving Early Stopping for Learning with Noisy Labels](https://mlsanity.com/paper/2106.15853/) (2021)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [Improved sampling via learned diffusions](https://mlsanity.com/paper/2307.01198/) (2023)
- [Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modeling](https://mlsanity.com/paper/2304.00917/) (2023)
- [COT-GAN: Generating Sequential Data via Causal Optimal Transport](https://mlsanity.com/paper/2006.08571/) (2020)
- [On the Generalization Properties of Diffusion Models](https://mlsanity.com/paper/2311.01797/) (2023)
- [Recent Advances in Optimal Transport for Machine Learning](https://mlsanity.com/paper/2306.16156/) (2023)
- [Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory](https://mlsanity.com/paper/2403.11968/) (2024)
- [Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization](https://mlsanity.com/paper/2401.15604/) (2024)
- [Fine-tuning of diffusion models via stochastic control: entropy regularization and beyond](https://mlsanity.com/paper/2403.06279/) (2024)
- [Entropic Optimal Transport: Convergence of Potentials](https://mlsanity.com/paper/2104.11720/) (2021)
- [Stochastic Control for Fine-tuning Diffusion Models: Optimality, Regularity, and Convergence](https://mlsanity.com/paper/2412.18164/) (2024)
- [(De)-regularized Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/2409.14980/) (2024)
- [Displacement smoothness of entropic optimal transport](https://mlsanity.com/paper/2210.00225/) (2022)
- [Diffusion Factor Models: Generating High-Dimensional Returns with Factor Structure](https://mlsanity.com/paper/2504.06566/) (2025)
- [Transfer Learning for Diffusion Models](https://mlsanity.com/paper/2405.16876/) (2024)

## Built on by (most cited first, 3 of 3)

- [One-Step Generative Modeling via Wasserstein Gradient Flows](https://mlsanity.com/paper/2605.11755/) (2026)
- [Learning Generative Dynamics with Soft Law Constraints: A McKean-Vlasov FBSDE Approach](https://mlsanity.com/paper/2605.08928/) (2026)
- [Scalable Bi-causal Optimal Transport via KL Relaxation and Policy Gradients](https://mlsanity.com/paper/2605.17271/) (2026)

## Abstract (arXiv)

Generative AI can be framed as the problem of learning a model that maps simple reference measures into complex data distributions, and it has recently found a strong connection to the classical theory of the Schr"odinger bridge problems (SBPs) due partly to their common nature of interpolating between prescribed marginals via entropy-regularized stochastic dynamics. However, the classical SBP enforces hard terminal constraints, which often leads to instability in practical implementations, especially in high-dimensional or data-scarce regimes. To address this challenge, we follow the idea of the so-called soft-constrained Schr"odinger bridge problem (SCSBP), in which the terminal constraint is replaced by a general penalty function. This relaxation leads to a more flexible stochastic control formulation of McKean-Vlasov type. We establish the existence of optimal solutions for all penalty levels and prove that, as the penalty grows, both the controls and value functions converge to those of the classical SBP at a linear rate. Our analysis builds on Doob's h-transform representations, the stability results of Schr"odinger potentials, Gamma-convergence, and a novel fixed-point argument that couples an optimization problem over the space of measures with an auxiliary entropic optimal transport problem. These results not only provide the first quantitative convergence guarantees for soft-constrained bridges but also shed light on how penalty regularization enables robust generative modeling, fine-tuning, and transfer learning.