# Lower Complexity Adaptation for Empirical Entropic Optimal Transport arXiv: https://arxiv.org/abs/2306.13580 · HTML: https://mlsanity.com/paper/2306.13580/ ## Key facts - Cited by 12 papers in the map (data through 14 Oct 2026). - Builds on 32 of its 103 references that are in the map. - First posted on arXiv 23 Jun 2023. ## Builds on (most cited first, 30 of 32) - [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013) - [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017) - [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018) - [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015) - [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017) - [Image reconstruction by domain transform manifold learning](https://mlsanity.com/paper/1704.08841/) (2017) - [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017) - [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018) - [Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm](https://mlsanity.com/paper/1802.04367/) (2018) - [Unsupervised Alignment of Embeddings with Wasserstein Procrustes](https://mlsanity.com/paper/1805.11222/) (2018) - [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021) - [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019) - [Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems](https://mlsanity.com/paper/1610.06519/) (2016) - [Faster Wasserstein Distance Estimation with the Sinkhorn Divergence](https://mlsanity.com/paper/2006.08172/) (2020) - [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021) - [Multilevel Clustering via Wasserstein Means](https://mlsanity.com/paper/1706.03883/) (2017) - [An Optimal Transport approach for the Schr"odinger bridge problem and convergence of Sinkhorn algorithm](https://mlsanity.com/paper/1911.06850/) (2019) - [Entropy-Regularized 2-Wasserstein Distance between Gaussian Measures](https://mlsanity.com/paper/2006.03416/) (2020) - [Entropic Optimal Transport: Convergence of Potentials](https://mlsanity.com/paper/2104.11720/) (2021) - [Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm](https://mlsanity.com/paper/2110.06798/) (2021) - [Inference for Empirical Wasserstein Distances on Finite Spaces](https://mlsanity.com/paper/1610.03287/) (2016) - [Entropic Optimal Transport: Geometry and Large Deviations](https://mlsanity.com/paper/2102.04397/) (2021) - [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022) - [The statistical effect of entropic regularization in optimal transportation](https://mlsanity.com/paper/2006.05199/) (2020) - [Asymptotics for semi-discrete entropic optimal transport](https://mlsanity.com/paper/2106.11862/) (2021) - [Convergence Rates for Regularized Optimal Transport via Quantization](https://mlsanity.com/paper/2208.14391/) (2022) - [Minimum intrinsic dimension scaling for entropic optimal transport](https://mlsanity.com/paper/2306.03398/) (2023) - [Improved Complexity Analysis of the Sinkhorn and Greenkhorn Algorithms for Optimal Transport](https://mlsanity.com/paper/2305.14939/) (2023) - [Nearly Tight Convergence Bounds for Semi-discrete Entropic Optimal Transport](https://mlsanity.com/paper/2110.12678/) (2021) - [Entropic Gromov-Wasserstein Distances: Stability and Algorithms](https://mlsanity.com/paper/2306.00182/) (2023) ## Built on by (most cited first, 12 of 12) - [Distributional Preference Alignment of LLMs via Optimal Transport](https://mlsanity.com/paper/2406.05882/) (2024) - [Plug-in estimation of Schr"odinger bridges](https://mlsanity.com/paper/2408.11686/) (2024) - [Estimation of entropy-regularized optimal transport maps between non-compactly supported measures](https://mlsanity.com/paper/2311.11934/) (2023) - [Trade-off Between Dependence and Complexity for Nonparametric Learning -- an Empirical Process Approach](https://mlsanity.com/paper/2401.08978/) (2024) - [Multivariate Stochastic Dominance via Optimal Transport and Applications to Models Benchmarking](https://mlsanity.com/paper/2406.06425/) (2024) - [Neural Estimation for Scaling Entropic Multimarginal Optimal Transport](https://mlsanity.com/paper/2506.00573/) (2025) - [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025) - [Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximization](https://mlsanity.com/paper/2410.02628/) (2024) - [On Robust Cross Domain Alignment](https://mlsanity.com/paper/2412.15861/) (2024) - [Geometry-Aware Optimal Transport: Fast Intrinsic Dimension and Wasserstein Distance Estimation](https://mlsanity.com/paper/2602.04335/) (2026) - [Multimodal Alignment Through Joint Kernel Entropic Gromov--Wasserstein Optimal Transport](https://mlsanity.com/paper/2608.04234/) (2026) - [Statistical Inference for Adversarial Training: Central Limit Theorems via Optimal Transport](https://mlsanity.com/paper/2609.22240/) (2026) ## Abstract (arXiv) Entropic optimal transport (EOT) presents an effective and computationally viable alternative to unregularized optimal transport (OT), offering diverse applications for large-scale data analysis. In this work, we derive novel statistical bounds for empirical plug-in estimators of the EOT cost and show that their statistical performance in the entropy regularization parameter ε and the sample size n only depends on the simpler of the two probability measures. For instance, under sufficiently smooth costs this yields the parametric rate n-1/2 with factor ε-d/2, where d is the minimum dimension of the two population measures. This confirms that empirical EOT also adheres to the lower complexity adaptation principle, a hallmark feature only recently identified for unregularized OT. As a consequence of our theory, we show that the empirical entropic Gromov-Wasserstein distance and its unregularized version for measures on Euclidean spaces also obey this principle. Additionally, we comment on computational aspects and complement our findings with Monte Carlo simulations. Our techniques employ empirical process theory and rely on a dual formulation of EOT over a single function class. Crucial to our analysis is the observation that the entropic cost-transformation of a function class does not increase its uniform metric entropy by much.