# Sharp bounds for max-sliced Wasserstein distances

arXiv: https://arxiv.org/abs/2403.00666 · HTML: https://mlsanity.com/paper/2403.00666/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 51 references that are in the map.
- First posted on arXiv 1 Mar 2024.

## Builds on (most cited first, 17 of 17)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019)
- [Generative Modeling using the Sliced Wasserstein Distance](https://mlsanity.com/paper/1803.11188/) (2018)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [Sliced Wasserstein Generative Models](https://mlsanity.com/paper/1904.05408/) (2019)
- [Projection Robust Wasserstein Distance and Riemannian Optimization](https://mlsanity.com/paper/2006.07458/) (2020)
- [Asymptotic Guarantees for Learning Generative Models with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1906.04516/) (2019)
- [Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances](https://mlsanity.com/paper/2210.09160/) (2022)
- [Energy-Based Sliced Wasserstein Distance](https://mlsanity.com/paper/2304.13586/) (2023)
- [On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification](https://mlsanity.com/paper/2006.12301/) (2020)
- [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022)
- [Two-sample Test using Projected Wasserstein Distance](https://mlsanity.com/paper/2010.11970/) (2020)
- [Amortized Projection Optimization for Sliced Wasserstein Generative Models](https://mlsanity.com/paper/2203.13417/) (2022)
- [Covariance Estimation: Optimal Dimension-free Guarantees for Adversarial Corruption and Heavy Tails](https://mlsanity.com/paper/2205.08494/) (2022)
- [The out-of-sample prediction error of the square-root-LASSO and related estimators](https://mlsanity.com/paper/2211.07608/) (2022)

## Built on by (most cited first, 10 of 10)

- [Lightspeed Geometric Dataset Distance via Sliced Optimal Transport](https://mlsanity.com/paper/2501.18901/) (2025)
- [Statistical and Computational Guarantees of Kernel Max-Sliced Wasserstein Distances](https://mlsanity.com/paper/2405.15441/) (2024)
- [Summarizing Bayesian Nonparametric Mixture Posterior -- Sliced Optimal Transport Metrics for Gaussian Mixtures](https://mlsanity.com/paper/2411.14674/) (2024)
- [Max-sliced Wasserstein concentration and uniform ratio bounds of empirical measures on RKHS](https://mlsanity.com/paper/2405.13153/) (2024)
- [Minimax-Optimal Two-Sample Test with Sliced Wasserstein](https://mlsanity.com/paper/2510.27498/) (2025)
- [Bayesian Multiple Multivariate Density-Density Regression](https://mlsanity.com/paper/2601.02640/) (2026)
- [Sliced Inner Product Gromov-Wasserstein Distances](https://mlsanity.com/paper/2605.08546/) (2026)
- [Max-sliced 2-Wasserstein distance](https://mlsanity.com/paper/2403.02142/) (2024)
- [Streaming Sliced Optimal Transport](https://mlsanity.com/paper/2505.06835/) (2025)
- [Revisiting the Sliced Wasserstein Kernel for persistence diagrams: a Figalli-Gigli approach](https://mlsanity.com/paper/2602.06539/) (2026)

## Abstract (arXiv)

We obtain essentially matching upper and lower bounds for the expected max-sliced 1-Wasserstein distance between a probability measure on a separable Hilbert space and its empirical distribution from n samples. By proving a Banach space version of this result, we also obtain an upper bound, that is sharp up to a log factor, for the expected max-sliced 2-Wasserstein distance between a symmetric probability measure μ on a Euclidean space and its symmetrized empirical distribution in terms of the operator norm of the covariance matrix of μ and the diameter of the support of μ.