# Entropic Optimal Transport: Convergence of Potentials

arXiv: https://arxiv.org/abs/2104.11720 · HTML: https://mlsanity.com/paper/2104.11720/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Apr 2021.

## Built on by (most cited first, 30 of 31)

- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [LAVA: Data Valuation without Pre-Specified Learning Algorithms](https://mlsanity.com/paper/2305.00054/) (2023)
- [One-Step Generative Modeling via Wasserstein Gradient Flows](https://mlsanity.com/paper/2605.11755/) (2026)
- [Minimax estimation of discontinuous optimal transport maps: The semi-discrete case](https://mlsanity.com/paper/2301.11302/) (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)
- [Lower Complexity Adaptation for Empirical Entropic Optimal Transport](https://mlsanity.com/paper/2306.13580/) (2023)
- [Wasserstein Mirror Gradient Flow as the limit of the Sinkhorn Algorithm](https://mlsanity.com/paper/2307.16421/) (2023)
- [Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps](https://mlsanity.com/paper/2411.07154/) (2024)
- [Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2311.04041/) (2023)
- [Plug-in estimation of Schr"odinger bridges](https://mlsanity.com/paper/2408.11686/) (2024)
- [Nearly Tight Convergence Bounds for Semi-discrete Entropic Optimal Transport](https://mlsanity.com/paper/2110.12678/) (2021)
- [Matching for causal effects via multimarginal unbalanced optimal transport](https://mlsanity.com/paper/2112.04398/) (2021)
- [A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2412.09235/) (2024)
- [Wasserstein Distributionally Robust Optimization with Wasserstein Barycenters](https://mlsanity.com/paper/2203.12136/) (2022)
- [Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasets](https://mlsanity.com/paper/2407.01718/) (2024)
- [Schr"odinger bridge for generative AI: Soft-constrained formulation and convergence analysis](https://mlsanity.com/paper/2510.11829/) (2025)
- [Sinkhorn Flow: A Continuous-Time Framework for Understanding and Generalizing the Sinkhorn Algorithm](https://mlsanity.com/paper/2311.16706/) (2023)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Hierarchical Maximum Entropy via the Renormalization Group](https://mlsanity.com/paper/2509.01424/) (2025)
- [Geometric Regularization for Long-Tailed Semi-Supervised Learning via Gaussian Feature Bridges](https://mlsanity.com/paper/2608.20710/) (2026)
- [Estimation of Stationary Optimal Transport Plans](https://mlsanity.com/paper/2107.11858/) (2021)
- [Semi-Discrete Optimal Transport: Nearly Minimax Estimation With Stochastic Gradient Descent and Adaptive Entropic Regularization](https://mlsanity.com/paper/2405.14459/) (2024)
- [Hessian stability and convergence rates for entropic and Sinkhorn potentials via semiconcavity](https://mlsanity.com/paper/2504.11133/) (2025)
- [Langevin Diffusion Approximation to Same Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2505.07647/) (2025)
- [Designing Algorithms for Entropic Optimal Transport from an Optimisation Perspective](https://mlsanity.com/paper/2507.12246/) (2025)
- [E-ROBOT: a dimension-free method for robust statistics and machine learning via Schr"odinger bridge](https://mlsanity.com/paper/2509.11532/) (2025)
- [Decreasing Entropic Regularization Averaged Gradient for Semi-Discrete Optimal Transport](https://mlsanity.com/paper/2510.27340/) (2025)
- [Multimodal Alignment Through Joint Kernel Entropic Gromov--Wasserstein Optimal Transport](https://mlsanity.com/paper/2608.04234/) (2026)
- [Uniform Statistical Convergence of Empirical Sinkhorn Potentials with Exponential and Polynomial Dependence on the Regularization Parameter](https://mlsanity.com/paper/2608.29152/) (2026)

## Abstract (arXiv)

We study the potential functions that determine the optimal density for varepsilon-entropically regularized optimal transport, the so-called Schr"odinger potentials, and their convergence to the counterparts in classical optimal transport, the Kantorovich potentials. In the limit varepsilon→0 of vanishing regularization, strong compactness holds in L1 and cluster points are Kantorovich potentials. In particular, the Schr"odinger potentials converge in L1 to the Kantorovich potentials as soon as the latter are unique. These results are proved for all continuous, integrable cost functions on Polish spaces. In the language of Schr"odinger bridges, the limit corresponds to the small-noise regime.