# Forward Reverse Kernel Regression for the Schr"odinger bridge problem

arXiv: https://arxiv.org/abs/2507.00640 · HTML: https://mlsanity.com/paper/2507.00640/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 26 references that are in the map.
- First posted on arXiv 1 Jul 2025.

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

- [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)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Entropic and displacement interpolation: a computational approach using the Hilbert metric](https://mlsanity.com/paper/1506.04255/) (2015)
- [The data-driven Schroedinger bridge](https://mlsanity.com/paper/1806.01364/) (2018)
- [On the Convergence Rate of Sinkhorn's Algorithm](https://mlsanity.com/paper/2212.06000/) (2022)
- [Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps](https://mlsanity.com/paper/2411.07154/) (2024)
- [Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2311.04041/) (2023)
- [Plug-in estimation of Schr"odinger bridges](https://mlsanity.com/paper/2408.11686/) (2024)

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

- [Direct Estimation of Schr"odinger Bridge Time-Series Drifts: Finite-Sample, Asymptotic, and Adaptive Guarantees](https://mlsanity.com/paper/2605.05432/) (2026)

## Abstract (arXiv)

In this paper, we study the Schr"odinger Bridge Problem (SBP), which is central to entropic optimal transport. For general reference processes and begin--endpoint distributions, we propose a forward-reverse iterative Monte Carlo procedure to approximate the Schr"odinger potentials in a nonparametric way. In particular, we use kernel based Monte Carlo regression in the context of Picard iteration of a corresponding fixed point problem. By preserving in the iteration positivity and contractivity in a Hilbert metric sense, we develop a provably convergent algorithm. Furthermore, we provide convergence rates for the potential estimates and prove their optimality. Finally, as an application, we propose a non-nested Monte Carlo procedure for the final dimensional distributions of the Schr"odinger Bridge process, based on the constructed potentials and the forward-reverse simulation method for conditional diffusions.