# Localized Schr"odinger Bridge Sampler

arXiv: https://arxiv.org/abs/2409.07968 · HTML: https://mlsanity.com/paper/2409.07968/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 49 references that are in the map.
- First posted on arXiv 12 Sep 2024.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation](https://mlsanity.com/paper/2006.00702/) (2020)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [Message Passing Stein Variational Gradient Descent](https://mlsanity.com/paper/1711.04425/) (2017)
- [A Good Score Does not Lead to A Good Generative Model](https://mlsanity.com/paper/2401.04856/) (2024)
- [Denoising: A Powerful Building-Block for Imaging, Inverse Problems, and Machine Learning](https://mlsanity.com/paper/2409.06219/) (2024)
- [Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation](https://mlsanity.com/paper/2007.07383/) (2020)
- [Spectral convergence of diffusion maps: improved error bounds and an alternative normalisation](https://mlsanity.com/paper/2006.02037/) (2020)
- [MALA-within-Gibbs samplers for high-dimensional distributions with sparse conditional structure](https://mlsanity.com/paper/1908.09429/) (2019)
- [Localization for MCMC: sampling high-dimensional posterior distributions with local structure](https://mlsanity.com/paper/1710.07747/) (2017)
- [Stable generative modeling using Schr"odinger bridges](https://mlsanity.com/paper/2401.04372/) (2024)

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

- [Learning dynamical systems with hit-and-run random feature maps](https://mlsanity.com/paper/2501.06661/) (2025)
- [A friendly introduction to triangular transport](https://mlsanity.com/paper/2503.21673/) (2025)
- [Digital Twins: McKean-Pontryagin Control for Partially Observed Physical Twins](https://mlsanity.com/paper/2510.00937/) (2025)

## Abstract (arXiv)

We consider the problem of sampling from an unknown distribution for which only a sufficiently large number of training samples are available. In this paper, we build on previous work combining Schr"odinger bridges and plug & play Langevin samplers. A key bottleneck of these approaches is the exponential dependence of the required training samples on the dimension, d, of the ambient state space. We propose a localization strategy which exploits conditional independence of conditional expectation values. Localization thus replaces a single high-dimensional Schr"odinger bridge problem by d low-dimensional Schr"odinger bridge problems over the available training samples. In this context, a connection to multi-head self attention transformer architectures is established. As for the original Schr"odinger bridge sampling approach, the localized sampler is stable and geometric ergodic. The sampler also naturally extends to conditional sampling and to Bayesian inference. We demonstrate the performance of our proposed scheme through experiments on a high-dimensional Gaussian problem, on a temporal stochastic process, and on a stochastic subgrid-scale parametrization conditional sampling problem. We also extend the idea of localization to plug & play Langevin samplers using kernel-based denoising in combination with Tweedie's formula.