# Max-sliced Wasserstein concentration and uniform ratio bounds of empirical measures on RKHS

arXiv: https://arxiv.org/abs/2405.13153 · HTML: https://mlsanity.com/paper/2405.13153/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 67 references that are in the map.
- First posted on arXiv 21 May 2024.

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

- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [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)
- [Faster Wasserstein Distance Estimation with the Sinkhorn Divergence](https://mlsanity.com/paper/2006.08172/) (2020)
- [Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces](https://mlsanity.com/paper/1804.10556/) (2018)
- [Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances](https://mlsanity.com/paper/2210.09160/) (2022)
- [On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification](https://mlsanity.com/paper/2006.12301/) (2020)
- [The Radon cumulative distribution transform and its application to image classification](https://mlsanity.com/paper/1511.03206/) (2015)
- [Fast Optimal Transport Averaging of Neuroimaging Data](https://mlsanity.com/paper/1503.08596/) (2015)
- [Minimax Confidence Intervals for the Sliced Wasserstein Distance](https://mlsanity.com/paper/1909.07862/) (2019)
- [Sharp bounds for max-sliced Wasserstein distances](https://mlsanity.com/paper/2403.00666/) (2024)
- [Statistical Optimal Transport posed as Learning Kernel Embedding](https://mlsanity.com/paper/2002.03179/) (2020)
- [Sliced Wasserstein distance between probability measures on Hilbert spaces](https://mlsanity.com/paper/2307.05802/) (2023)

## Built on by (most cited first, 1 of 1)

- [Contrast-Free ICA and Causal Inference via Wasserstein Distances to the Gaussian](https://mlsanity.com/paper/2607.12832/) (2026)

## Abstract (arXiv)

Optimal transport and the Wasserstein distance Wp have recently seen a number of applications in the fields of statistics, machine learning, data science, and the physical sciences. These applications are however severely restricted by the curse of dimensionality, meaning that the number of data points needed to estimate these problems accurately increases exponentially in the dimension. To alleviate this problem, a number of variants of Wp have been introduced. We focus here on one of these variants, namely the max-sliced Wasserstein metric overlineWp. This metric reduces the high-dimensional minimization problem given by Wp to a maximum of one-dimensional measurements in an effort to overcome the curse of dimensionality. In this note we derive concentration results and upper bounds on the expectation of overlineWp between the true and empirical measure on unbounded reproducing kernel Hilbert spaces. We show that, under quite generic assumptions, probability measures concentrate uniformly fast in one-dimensional subspaces, at (nearly) parametric rates. Our results rely on an improvement of currently known bounds for overlineWp in the finite-dimensional case.