# On the Contraction Coefficient of the Schr"odinger Bridge for Stochastic Linear Systems

arXiv: https://arxiv.org/abs/2309.06622 · HTML: https://mlsanity.com/paper/2309.06622/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 28 references that are in the map.
- First posted on arXiv 12 Sep 2023.

## Builds on (most cited first, 7 of 7)

- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [An Optimal Transport approach for the Schr"odinger bridge problem and convergence of Sinkhorn algorithm](https://mlsanity.com/paper/1911.06850/) (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)
- [Prediction and Optimal Feedback Steering of Probability Density Functions for Safe Automated Driving](https://mlsanity.com/paper/2009.09055/) (2020)

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

- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [Solution of the Probabilistic Lambert Problem: Connections with Optimal Mass Transport, Schr"odinger Bridge and Reaction-Diffusion PDEs](https://mlsanity.com/paper/2401.07961/) (2024)
- [Mean-Field Path-Integral Diffusion: From Samples to Interacting Agents](https://mlsanity.com/paper/2605.00007/) (2026)

## Abstract (arXiv)

Schr"odinger bridge is a stochastic optimal control problem to steer a given initial state density to another, subject to controlled diffusion and deadline constraints. A popular method to numerically solve the Schr"odinger bridge problems, in both classical and in the linear system settings, is via contractive fixed point recursions. These recursions can be seen as dynamic versions of the well-known Sinkhorn iterations, and under mild assumptions, they solve the so-called Schr"odinger systems with guaranteed linear convergence. In this work, we study a priori estimates for the contraction coefficients associated with the convergence of respective Schr"odinger systems. We provide new geometric and control-theoretic interpretations for the same. Building on these newfound interpretations, we point out the possibility of improved computation for the worst-case contraction coefficients of linear SBPs by preconditioning the endpoint support sets.