# Geometry-Aware Optimal Transport: Fast Intrinsic Dimension and Wasserstein Distance Estimation

arXiv: https://arxiv.org/abs/2602.04335 · HTML: https://mlsanity.com/paper/2602.04335/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 31 references that are in the map.
- First posted on arXiv 4 Feb 2026.

## Builds on (most cited first, 15 of 15)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm](https://mlsanity.com/paper/1802.04367/) (2018)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [Faster Wasserstein Distance Estimation with the Sinkhorn Divergence](https://mlsanity.com/paper/2006.08172/) (2020)
- [Scikit-dimension: a Python package for intrinsic dimension estimation](https://mlsanity.com/paper/2109.02596/) (2021)
- [On Efficient Optimal Transport: An Analysis of Greedy and Accelerated Mirror Descent Algorithms](https://mlsanity.com/paper/1901.06482/) (2019)
- [Statistical optimal transport](https://mlsanity.com/paper/2407.18163/) (2024)
- [Intrinsic Dimension Estimation Using Wasserstein Distances](https://mlsanity.com/paper/2106.04018/) (2021)
- [Minimax estimation of discontinuous optimal transport maps: The semi-discrete case](https://mlsanity.com/paper/2301.11302/) (2023)
- [A gradual, semi-discrete approach to generative network training via explicit Wasserstein minimization](https://mlsanity.com/paper/1906.03471/) (2019)
- [Lower Complexity Adaptation for Empirical Entropic Optimal Transport](https://mlsanity.com/paper/2306.13580/) (2023)
- [Towards Optimal Running Times for Optimal Transport](https://mlsanity.com/paper/1810.07717/) (2018)
- [Minimum intrinsic dimension scaling for entropic optimal transport](https://mlsanity.com/paper/2306.03398/) (2023)

## Abstract (arXiv)

Solving large scale Optimal Transport (OT) in machine learning typically relies on sampling measures to obtain a tractable discrete problem. While the discrete solver's accuracy is controllable, the rate of convergence of the discretization error is governed by the intrinsic dimension of our data. Therefore, the true bottleneck is the knowledge and control of the sampling error. In this work, we tackle this issue by introducing novel estimators for both sampling error and intrinsic dimension. The key finding is a simple, tuning-free estimator of OTc(ρ, hatρ) that utilizes the semi-dual OT functional and, remarkably, requires no OT solver. Furthermore, we derive a fast intrinsic dimension estimator from the multi-scale decay of our sampling error estimator. This framework unlocks significant computational and statistical advantages in practice, enabling us to (i) quantify the convergence rate of the discretization error, (ii) calibrate the entropic regularization of Sinkhorn divergences to the data's intrinsic geometry, and (iii) introduce a novel, intrinsic-dimension-based Richardson extrapolation estimator that strongly debiases Wasserstein distance estimation. Numerical experiments demonstrate that our geometry-aware pipeline effectively mitigates the discretization error bottleneck while maintaining computational efficiency.