{"title":"Sharp Bounds for Max-sliced Wasserstein Distances","authors":"March T. Boedihardjo","doi":"10.1007/s10208-025-09690-1","DOIUrl":null,"url":null,"abstract":"<p>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 <i>n</i> 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 <span>\\(\\mu \\)</span> on a Euclidean space and its symmetrized empirical distribution in terms of the operator norm of the covariance matrix of <span>\\(\\mu \\)</span> and the diameter of the support of <span>\\(\\mu \\)</span>.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"22 1","pages":""},"PeriodicalIF":2.5000,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Foundations of Computational Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10208-025-09690-1","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
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Abstract
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 \(\mu \) on a Euclidean space and its symmetrized empirical distribution in terms of the operator norm of the covariance matrix of \(\mu \) and the diameter of the support of \(\mu \).
期刊介绍：
Foundations of Computational Mathematics (FoCM) will publish research and survey papers of the highest quality which further the understanding of the connections between mathematics and computation. The journal aims to promote the exploration of all fundamental issues underlying the creative tension among mathematics, computer science and application areas unencumbered by any external criteria such as the pressure for applications. The journal will thus serve an increasingly important and applicable area of mathematics. The journal hopes to further the understanding of the deep relationships between mathematical theory: analysis, topology, geometry and algebra, and the computational processes as they are evolving in tandem with the modern computer.
With its distinguished editorial board selecting papers of the highest quality and interest from the international community, FoCM hopes to influence both mathematics and computation. Relevance to applications will not constitute a requirement for the publication of articles.
The journal does not accept code for review however authors who have code/data related to the submission should include a weblink to the repository where the data/code is stored.
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