# Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm

arXiv: https://arxiv.org/abs/2110.06798 · HTML: https://mlsanity.com/paper/2110.06798/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Oct 2021.

## Built on by (most cited first, 22 of 22)

- [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)
- [Lower Complexity Adaptation for Empirical Entropic Optimal Transport](https://mlsanity.com/paper/2306.13580/) (2023)
- [Progressive Entropic Optimal Transport Solvers](https://mlsanity.com/paper/2406.05061/) (2024)
- [Distributional Reinforcement Learning with Regularized Wasserstein Loss](https://mlsanity.com/paper/2202.00769/) (2022)
- [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)
- [Efficient and Accurate Optimal Transport with Mirror Descent and Conjugate Gradients](https://mlsanity.com/paper/2307.08507/) (2023)
- [Optimal Transport on the Lie Group of Roto-translations](https://mlsanity.com/paper/2402.15322/) (2024)
- [A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2412.09235/) (2024)
- [On Rank Energy Statistics via Optimal Transport: Continuity, Convergence, and Change Point Detection](https://mlsanity.com/paper/2302.07964/) (2023)
- [Fast Tree-Field Integrators: From Low Displacement Rank to Topological Transformers](https://mlsanity.com/paper/2406.15881/) (2024)
- [Fitted value iteration methods for bicausal optimal transport](https://mlsanity.com/paper/2306.12658/) (2023)
- [Stability of Entropic Wasserstein Barycenters and application to random geometric graphs](https://mlsanity.com/paper/2210.10535/) (2022)
- [On the Expressive Power of Transformers for Contextual Relations](https://mlsanity.com/paper/2603.25860/) (2026)
- [Convergence rate of Tsallis entropic regularized optimal transport](https://mlsanity.com/paper/2304.06616/) (2023)
- [Hessian stability and convergence rates for entropic and Sinkhorn potentials via semiconcavity](https://mlsanity.com/paper/2504.11133/) (2025)
- [Robust Optimization in Causal Models and G-Causal Normalizing Flows](https://mlsanity.com/paper/2510.15458/) (2025)
- [QDSB: Quantized Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2605.11983/) (2026)
- [Uniform Statistical Convergence of Empirical Sinkhorn Potentials with Exponential and Polynomial Dependence on the Regularization Parameter](https://mlsanity.com/paper/2608.29152/) (2026)
- [General Quantification of Covariate and Concept Shifts](https://mlsanity.com/paper/2609.11918/) (2026)
- [Scaling Online Optimal Transport Minibatches for Flow Matching](https://mlsanity.com/paper/2610.18838/) (2026)

## Abstract (arXiv)

We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and H"older continuity of the optimal coupling in p-Wasserstein distance are obtained under general conditions including quadratic costs and unbounded marginals. The results for the value extend to regularization by an arbitrary divergence. As an application, we show convergence of Sinkhorn's algorithm in Wasserstein sense, including for quadratic cost. Two techniques are presented: The first compares an optimal coupling with its so-called shadow, a coupling induced on other marginals by an explicit construction. The second transforms one set of marginals by a change of coordinates and thus reduces the comparison of differing marginals to the comparison of differing cost functions under the same marginals.