# Minimum intrinsic dimension scaling for entropic optimal transport

arXiv: https://arxiv.org/abs/2306.03398 · HTML: https://mlsanity.com/paper/2306.03398/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 58 references that are in the map.
- First posted on arXiv 6 Jun 2023.

## Builds on (most cited first, 19 of 19)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [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)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [An Optimal Transport approach for the Schr"odinger bridge problem and convergence of Sinkhorn algorithm](https://mlsanity.com/paper/1911.06850/) (2019)
- [Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm](https://mlsanity.com/paper/1905.13194/) (2019)
- [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)
- [A Study of Performance of Optimal Transport](https://mlsanity.com/paper/2005.01182/) (2020)
- [Multivariate rank via entropic optimal transport: sample efficiency and generative modeling](https://mlsanity.com/paper/2111.00043/) (2021)
- [Stability and Sample Complexity of Divergence Regularized Optimal Transport](https://mlsanity.com/paper/2212.00367/) (2022)
- [On Rank Energy Statistics via Optimal Transport: Continuity, Convergence, and Change Point Detection](https://mlsanity.com/paper/2302.07964/) (2023)

## Built on by (most cited first, 11 of 11)

- [Lower Complexity Adaptation for Empirical Entropic Optimal Transport](https://mlsanity.com/paper/2306.13580/) (2023)
- [Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps](https://mlsanity.com/paper/2411.07154/) (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)
- [Multivariate Stochastic Dominance via Optimal Transport and Applications to Models Benchmarking](https://mlsanity.com/paper/2406.06425/) (2024)
- [Training generative models from privatized data](https://mlsanity.com/paper/2306.09547/) (2023)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Linear Bandits beyond Inner Product Spaces, the case of Bandit Optimal Transport](https://mlsanity.com/paper/2502.07397/) (2025)
- [Statistical Analysis of the Sinkhorn Iterations for Two-Sample Schr"odinger Bridge Estimation](https://mlsanity.com/paper/2510.22560/) (2025)
- [Geometry-Aware Optimal Transport: Fast Intrinsic Dimension and Wasserstein Distance Estimation](https://mlsanity.com/paper/2602.04335/) (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)

Motivated by the manifold hypothesis, which states that data with a high extrinsic dimension may yet have a low intrinsic dimension, we develop refined statistical bounds for entropic optimal transport that are sensitive to the intrinsic dimension of the data. Our bounds involve a robust notion of intrinsic dimension, measured at only a single distance scale depending on the regularization parameter, and show that it is only the minimum of these single-scale intrinsic dimensions which governs the rate of convergence. We call this the Minimum Intrinsic Dimension scaling (MID scaling) phenomenon, and establish MID scaling with no assumptions on the data distributions so long as the cost is bounded and Lipschitz, and for various entropic optimal transport quantities beyond just values, with stronger analogs when one distribution is supported on a manifold. Our results significantly advance the theoretical state of the art by showing that MID scaling is a generic phenomenon, and provide the first rigorous interpretation of the statistical effect of entropic regularization as a distance scale.