Nearly sharp comparison results for sliced and max-sliced Wasserstein distances

Abstract

We prove new comparison results between the Wasserstein distance and its sliced and max-sliced counterparts. First, we show that the Hölder exponent~$\frac{2}{d+2}$ obtained by Bobkov and Götze for the max-sliced 1-Wasserstein distance on the unit ball is optimal for every $d \geq 2$, settling a question raised in their work. Second, we show that sharper comparisons are possible under stronger structural assumptions: if $ν$ is a discrete measure and the optimal coupling between $μ$ and $ν$ transports each point to a nearest atom of $ν$, then $W_p(μ, ν) \leq C \sqrt{d}\, K \, \mathrm{SW}_{p,1}(μ, ν)$ for a universal constant $C$, where the complexity parameter $K$ is always at most the number of atoms $N$ and can be substantially smaller. This complements a similar bound due to Park and Slepčev. An analogous bound holds for the sliced Wasserstein distance based on $k$-dimensional projections. Finally, using a construction from geometric discrepancy theory due to Chen and Travaglini, we prove that the linear dependence on $K$ in this bound cannot be improved, up to polylogarithmic factors.

Disclosure

“3) SW1,1 (µ, ν) Cd (log N )d /N 8Cd (log N )d (log(K + 1))d which is the claimed bound. □ AI Disclosure Claude Fable 5 and ChatGPT 5.6 were used in editing the manuscript, checking for errors, and filling in routine technical details; in particular, these models proved useful in correcting an error in a previous version of Theorem 1.3. The authors developed the argume”

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2026-08-23T05:13:59.956078Z

Structural counts