# Langevin Diffusion Approximation to Same Marginal Schr"odinger Bridge

arXiv: https://arxiv.org/abs/2505.07647 · HTML: https://mlsanity.com/paper/2505.07647/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 38 references that are in the map.
- First posted on arXiv 12 May 2025.

## Builds on (most cited first, 9 of 9)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Entropic Optimal Transport: Convergence of Potentials](https://mlsanity.com/paper/2104.11720/) (2021)
- [Manifold learning with bi-stochastic kernels](https://mlsanity.com/paper/1711.06711/) (2017)
- [Entropic Optimal Transport: Geometry and Large Deviations](https://mlsanity.com/paper/2102.04397/) (2021)
- [Iterated Schr"odinger bridge approximation to Wasserstein Gradient Flows](https://mlsanity.com/paper/2406.10823/) (2024)
- [The Riemannian geometry of Sinkhorn divergences](https://mlsanity.com/paper/2405.04987/) (2024)

## Abstract (arXiv)

We introduce a novel approximation to the same marginal Schr"odinger bridge using the Langevin diffusion. As varepsilon downarrow 0, it is known that the barycentric projection (also known as the entropic Brenier map) of the Schr"odinger bridge converges to the Brenier map, which is the identity. Our diffusion approximation is leveraged to show that, under suitable assumptions, the difference between the two is varepsilon times the gradient of the marginal log density (i.e., the score function), in L2. More generally, we show that the family of Markov operators, indexed by varepsilon > 0, derived from integrating test functions against the conditional density of the static Schr"odinger bridge at temperature varepsilon, admits a derivative at varepsilon=0 given by the generator of the Langevin semigroup. Hence, these operators satisfy an approximate semigroup property at low temperatures.