# A Generalized Sinkhorn Algorithm for Mean-Field Schr"odinger Bridge

arXiv: https://arxiv.org/abs/2604.06531 · HTML: https://mlsanity.com/paper/2604.06531/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 34 references that are in the map.
- First posted on arXiv 8 Apr 2026.

## Builds on (most cited first, 14 of 14)

- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [Entropic and displacement interpolation: a computational approach using the Hilbert metric](https://mlsanity.com/paper/1506.04255/) (2015)
- [Reflected Schr"odinger Bridge: Density Control with Path Constraints](https://mlsanity.com/paper/2003.13895/) (2020)
- [Optimal steering of a linear stochastic system to a final probability distribution, part II](https://mlsanity.com/paper/1410.3447/) (2014)
- [Density control of interacting agent systems](https://mlsanity.com/paper/2108.07342/) (2021)
- [Optimal steering of a linear stochastic system to a final probability distribution, Part III](https://mlsanity.com/paper/1608.03622/) (2016)
- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [Finite Horizon Density Steering for Multi-input State Feedback Linearizable Systems](https://mlsanity.com/paper/1909.12511/) (2019)
- [Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly](https://mlsanity.com/paper/2307.14442/) (2023)
- [Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2311.04041/) (2023)
- [On the Hopf-Cole Transform for Control-affine Schr"odinger Bridge](https://mlsanity.com/paper/2503.17640/) (2025)
- [Schr"odinger's control and estimation paradigm with spatio-temporal distributions on graphs](https://mlsanity.com/paper/2312.05679/) (2023)
- [An excursion onto Schr"odinger's bridges: Stochastic flows with spatio-temporal marginals](https://mlsanity.com/paper/2404.07402/) (2024)
- [Markov Kernels, Distances and Optimal Control: A Parable of Linear Quadratic Non-Gaussian Distribution Steering](https://mlsanity.com/paper/2504.15753/) (2025)

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

- [Nonlocal Mean Field Schr"odinger Bridge with Learned Interactions](https://mlsanity.com/paper/2606.04265/) (2026)
- [Schr"odinger Bridges over Kinetic Swarming Models](https://mlsanity.com/paper/2608.25281/) (2026)
- [Cost-augmented Schr"odinger bridges on graphs are exactly solvable: a Feynman-Kac tilt replaces learned control](https://mlsanity.com/paper/2610.02195/) (2026)

## Abstract (arXiv)

The mean-field Schr"odinger bridge (MFSB) problem concerns designing a minimum-effort controller that guides a diffusion process with nonlocal interaction to reach a given distribution from another by a fixed deadline. Unlike the standard Schr"odinger bridge, the dynamical constraint for MFSB is the mean-field limit of a population of interacting agents with controls. It serves as a natural model for large-scale multi-agent systems. The MFSB is computationally challenging because the nonlocal interaction makes the problem nonconvex. We propose a generalization of the Hopf-Cole transform for MFSB and, building on it, design a Sinkhorn-type recursive algorithm to solve the associated system of integro-PDEs. Under mild assumptions on the interaction potential, we discuss convergence guarantees for the proposed algorithm. We present numerical examples with repulsive and attractive interactions to illustrate the theoretical contributions.