泊松超平面镶嵌零胞在方向扰动下的Wasserstein稳定性
Wasserstein stability of the zero cell of a Poisson hyperplane tessellation under directional perturbations
浏览论文内容
中文总结 AI 辅助
该研究分析泊松超平面镶嵌零胞在方向分布扰动下的Wasserstein稳定性，给出其局部Hölder连续性的定量界，并将结果应用于界定零胞密度估计器的期望总变差距离。
中文摘要 AI 辅助
d维欧氏空间中的平稳泊松超平面过程由强度参数与单位球面上的偶概率测度（称为方向分布）刻画。本研究探讨诱导超平面镶嵌中包含原点的随机凸多面体——零胞在方向分布扰动下的稳定性。研究结果给出定量界，确立了零胞分布关于随机凸体空间上的Wasserstein度量与单位球面上概率分布的局部Hölder连续性。作为应用，本研究建立了由泊松超平面镶嵌构造的密度估计器的稳定性界，具体而言，通过零胞估计得到的概率测度间的期望总变差距离，可由其方向分布间的Wasserstein距离界定。
英文摘要
A stationary Poisson hyperplane process in $\mathbb{R}^d$ is characterized by an intensity parameter and an even probability measure on the unit sphere called the directional distribution. In this work, we investigate the stability of the zero cell, i.e., the random convex polytope of the induced hyperplane tessellation containing the origin, under perturbations of the directional distribution. Our results provide quantitative bounds establishing local Hölder continuity of the distribution of the zero cell with respect to Wasserstein metrics on the space of random convex bodies and probability distributions on the unit sphere. As an application, we establish stability bounds for density estimators constructed from Poisson hyperplane tessellations. In particular, we bound the expected total variation distance between the zero-cell-based estimated probability measures in terms of the Wasserstein distance between their directional distributions.
