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Sharp bounds for max-sliced Wasserstein distances
March T. Boedihardjo · 2024-03-02 · via stat.ML updates on arXiv.org

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 $μ$.

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