# Convergence Rates for Regularized Optimal Transport via Quantization

arXiv: https://arxiv.org/abs/2208.14391 · HTML: https://mlsanity.com/paper/2208.14391/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 63 references that are in the map.
- First posted on arXiv 30 Aug 2022.

## Builds on (most cited first, 23 of 23)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [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 estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)
- [On Efficient Optimal Transport: An Analysis of Greedy and Accelerated Mirror Descent Algorithms](https://mlsanity.com/paper/1901.06482/) (2019)
- [Multi-marginal optimal transport: theory and applications](https://mlsanity.com/paper/1406.0026/) (2014)
- [Entropic Optimal Transport: Convergence of Potentials](https://mlsanity.com/paper/2104.11720/) (2021)
- [Stability of Entropic Optimal Transport and Schr"odinger Bridges](https://mlsanity.com/paper/2106.03670/) (2021)
- [An explicit analysis of the entropic penalty in linear programming](https://mlsanity.com/paper/1806.01879/) (2018)
- [Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm](https://mlsanity.com/paper/2110.06798/) (2021)
- [Entropic Optimal Transport: Geometry and Large Deviations](https://mlsanity.com/paper/2102.04397/) (2021)
- [Quadratically-Regularized Optimal Transport on Graphs](https://mlsanity.com/paper/1704.08200/) (2017)
- [Quadratically regularized optimal transport](https://mlsanity.com/paper/1903.01112/) (2019)
- [Polynomial-time algorithms for Multimarginal Optimal Transport problems with structure](https://mlsanity.com/paper/2008.03006/) (2020)
- [Asymptotics for semi-discrete entropic optimal transport](https://mlsanity.com/paper/2106.11862/) (2021)
- [Stability of Schr"odinger Potentials and Convergence of Sinkhorn's Algorithm](https://mlsanity.com/paper/2201.10059/) (2022)
- [Computational methods for adapted optimal transport](https://mlsanity.com/paper/2203.05005/) (2022)
- [Towards Optimal Running Times for Optimal Transport](https://mlsanity.com/paper/1810.07717/) (2018)
- [Optimal Transport losses and Sinkhorn algorithm with general convex regularization](https://mlsanity.com/paper/2007.00976/) (2020)
- [Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm](https://mlsanity.com/paper/1710.08234/) (2017)
- [Convergence rate of general entropic optimal transport costs](https://mlsanity.com/paper/2206.03347/) (2022)
- [Optimal transport with f-divergence regularization and generalized Sinkhorn algorithm](https://mlsanity.com/paper/2105.14337/) (2021)

## Built on by (most cited first, 12 of 12)

- [Lower Complexity Adaptation for Empirical Entropic Optimal Transport](https://mlsanity.com/paper/2306.13580/) (2023)
- [Reflected Schr"odinger Bridge for Constrained Generative Modeling](https://mlsanity.com/paper/2401.03228/) (2024)
- [Minimum intrinsic dimension scaling for entropic optimal transport](https://mlsanity.com/paper/2306.03398/) (2023)
- [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)
- [Estimating the Rate-Distortion Function by Wasserstein Gradient Descent](https://mlsanity.com/paper/2310.18908/) (2023)
- [Fitted value iteration methods for bicausal optimal transport](https://mlsanity.com/paper/2306.12658/) (2023)
- [Multivariate Stochastic Dominance via Optimal Transport and Applications to Models Benchmarking](https://mlsanity.com/paper/2406.06425/) (2024)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Convergence rate of Tsallis entropic regularized optimal transport](https://mlsanity.com/paper/2304.06616/) (2023)
- [Quadratically Regularized Optimal Transport: Localization Bounds and Affine Case Analysis](https://mlsanity.com/paper/2605.24644/) (2026)
- [Randomized Optimal Switching Problem and Related Mirror Descent Flow](https://mlsanity.com/paper/2606.12875/) (2026)

## Abstract (arXiv)

We study the convergence of divergence-regularized optimal transport as the regularization parameter vanishes. Sharp rates for general divergences including relative entropy or Lp regularization, general transport costs and multi-marginal problems are obtained. A novel methodology using quantization and martingale couplings is suitable for non-compact marginals and achieves, in particular, the sharp leading-order term of entropically regularized 2-Wasserstein distance for all marginals with finite (2+δ)-moment.