# Foundations of Schr"odinger Bridges for Generative Modeling

arXiv: https://arxiv.org/abs/2603.18992 · HTML: https://mlsanity.com/paper/2603.18992/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- First posted on arXiv 19 Mar 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Built on by (most cited first, 1 of 1)

- [Bridge Graphical Models: Coupling, Projection, and Current-Preserving Dynamics for Generative Modeling](https://mlsanity.com/paper/2608.19144/) (2026)

## Abstract (arXiv)

At the core of modern generative modeling frameworks, including diffusion models, score-based models, and flow matching, is the task of transforming a simple prior distribution into a complex target distribution through stochastic paths in probability space. Schr"odinger bridges provide a unifying principle underlying these approaches, framing the problem as determining an optimal stochastic bridge between marginal distribution constraints with minimal-entropy deviations from a pre-defined reference process. This guide develops the mathematical foundations of the Schr"odinger bridge problem, drawing on optimal transport, stochastic control, and path-space optimization, and focuses on its dynamic formulation with direct connections to modern generative modeling. We build a comprehensive toolkit for constructing Schr"odinger bridges from first principles, and show how these constructions give rise to generalized and task-specific computational methods.