We consider structured approximation of measures in Wasserstein space $W_p(\mathbb{R}^d)$ for $p\in[1,\infty)$ by discrete and piecewise constant measures based on a scaled Voronoi partition of $\mathbb{R}^d$. We show that if a full rank lattice $\Lambda$ is scaled by a factor of $h\in(0,1]$, then approximation of a measure based on the Voronoi partition of $h\Lambda$ is $O(h)$ regardless of $d$ or $p$. We then use a covering argument to show that $N$-term approximations of compactly supported measures is $O(N^{-\frac1d})$ which matches known rates for optimal quantizers and empirical measure approximation in most instances. Finally, we extend these results to noncompactly supported measures with sufficient decay.

翻译：我们考虑在Wasserstein空间$W_p(\mathbb{R}^d)$（其中$p\in[1,\infty)$）中，通过基于$\mathbb{R}^d$的缩放Voronoi划分的离散和分段常数测度对测度进行结构化逼近。我们证明，若满秩格点$\Lambda$按因子$h\in(0,1]$缩放，则基于$h\Lambda$的Voronoi划分的测度逼近误差为$O(h)$，与维度$d$或$p$无关。进一步，我们利用覆盖论证表明：紧支撑测度的$N$项逼近误差为$O(N^{-\frac1d})$，在多数情形下与最优量化器和经验测度逼近的已知收敛速率一致。最后，我们将这些结果推广至具有足够衰减性的非紧支撑测度。