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<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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> on a Euclidean space and its symmetrized empirical distribution in terms of the operator norm of the covariance matrix of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> and the diameter of the support of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>.</p>

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Sharp Bounds for Max-sliced Wasserstein Distances

  • 期刊论文

  • 发表日期：

    2025年1月22日

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  • March T. Boedihardjo

摘要

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 \(\mu \) μ on a Euclidean space and its symmetrized empirical distribution in terms of the operator norm of the covariance matrix of \(\mu \) μ and the diameter of the support of \(\mu \) μ .