AI 中文总结
该研究证明了Wasserstein距离与切片、最大切片Wasserstein距离的比较结果，确定了最大切片距离Hölder指数的最优性，给出了离散测度下的尖锐比较界并证明其线性依赖关系无法改进。
AI 中文摘要
我们证明了Wasserstein距离与其切片和最大切片对应形式之间的新比较结果。首先，我们证明Bobkov和Götze得到的单位球上最大切片1-Wasserstein距离的Hölder指数2/(d+2)对每个d≥2都是最优的，解决了他们工作中提出的一个问题。其次，我们证明在更强的结构假设下可以得到更尖锐的比较结果：若ν是离散测度，且μ与ν之间的最优耦合将每个点运输到ν的最近原子，则W_p(μ,ν)≤C√d K SW_{p,1}(μ,ν)，其中C为通用常数，复杂度参数K始终不超过原子数N，且可能小得多。这补充了Park和Slepčev得到的类似界。基于k维投影的切片Wasserstein距离也存在类似界。最后，利用Chen和Travaglini提出的几何偏差理论中的构造，我们证明该界中对K的线性依赖关系除多对数因子外无法改进。
英文摘要
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.
