We consider structured approximation of measures in Wasserstein space $\mathrm{W}_p(\mathbb{R}^d)$ for $p\in[1,\infty)$ using general measure approximants compactly supported on Voronoi regions derived from 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. Additionally, we generalize our construction to nonuniform Voronoi partitions, highlighting the flexibility and robustness of our approach for various measure approximation scenarios. Finally, we extend these results to noncompactly supported measures with sufficient decay. Our findings are pertinent to applications in computer vision and machine learning where measures are used to represent structured data such as images.

翻译：本文研究Wasserstein空间$\mathrm{W}_p(\mathbb{R}^d)$（其中$p\in[1,\infty)$）中测度的结构化逼近问题，采用基于$\mathbb{R}^d$缩放Voronoi划分所导出Voronoi区域的紧支集一般测度逼近子。我们证明：若将满秩格点$\Lambda$缩放$h\in(0,1]$倍，则基于$h\Lambda$的Voronoi划分的测度逼近误差为$O(h)$，该结果与维度$d$及参数$p$无关。进一步通过覆盖论证得到紧支集测度的$N$项逼近误差为$O(N^{-\frac1d})$，该结果在多数情况下与最优量化器及经验测度逼近的已知速率一致。此外，我们将构造推广至非均匀Voronoi划分，凸显了该方法在不同测度逼近场景下的灵活性与鲁棒性。最后，我们将结果拓展至具有充分衰减性的非紧支集测度。本研究结论对计算机视觉与机器学习领域具有重要意义，其中测度常被用于表示图像等结构化数据。