# Stability of Schr"odinger Potentials and Convergence of Sinkhorn's Algorithm

arXiv: https://arxiv.org/abs/2201.10059 · HTML: https://mlsanity.com/paper/2201.10059/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jan 2022.

## Built on by (most cited first, 13 of 13)

- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputation](https://mlsanity.com/paper/2305.07247/) (2023)
- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)
- [Convergence Rates for Regularized Optimal Transport via Quantization](https://mlsanity.com/paper/2208.14391/) (2022)
- [Reflected Schr"odinger Bridge for Constrained Generative Modeling](https://mlsanity.com/paper/2401.03228/) (2024)
- [Riemannian Diffusion Schr"odinger Bridge](https://mlsanity.com/paper/2207.03024/) (2022)
- [Wasserstein Mirror Gradient Flow as the limit of the Sinkhorn Algorithm](https://mlsanity.com/paper/2307.16421/) (2023)
- [Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2311.04041/) (2023)
- [Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approach](https://mlsanity.com/paper/2304.06549/) (2023)
- [Variational Online Mirror Descent for Robust Learning in Schr"odinger Bridge](https://mlsanity.com/paper/2504.02618/) (2025)
- [On the Expressive Power of Transformers for Contextual Relations](https://mlsanity.com/paper/2603.25860/) (2026)
- [Uniform Statistical Convergence of Empirical Sinkhorn Potentials with Exponential and Polynomial Dependence on the Regularization Parameter](https://mlsanity.com/paper/2608.29152/) (2026)
- [Low-Budget Active Learning through Entropic Optimal Transport](https://mlsanity.com/paper/2610.01199/) (2026)

## Abstract (arXiv)

We study the stability of entropically regularized optimal transport with respect to the marginals. Given marginals converging weakly, we establish a strong convergence for the Schr"odinger potentials describing the density of the optimal couplings. When the marginals converge in total variation, the optimal couplings also converge in total variation. This is applied to show that Sinkhorn's algorithm converges in total variation when costs are quadratic and marginals are subgaussian, or more generally for all continuous costs satisfying an integrability condition.