一种用于近似凸体中一点半空间深度的（希尔伯特）几何算法
A (Hilbert) geometric algorithm for approximating the halfspace depth of a point in a convex body
AI总结：
本文提出了首个用于近似开凸体中一点半空间深度的确定性算法，基于Hilbert度量设计数据结构回答深度修剪区域的近似成员查询，并给出了计算开平面凸多边形精确深度的算法。
摘要：
半空间（或 Tukey）深度是衡量多元数据集中数据点中心性的基本且稳健的度量。计算一个点相对于 $\mathbb{R}^d$ 中开凸体上均匀分布的深度是一个自然的算法问题。虽然测试凸体成员资格这一较粗略的任务已被广泛研究，但评估深度这一精细问题在文献中却相对较少受到关注。在这项工作中，我们提出了一种用于近似开凸体 $K \subset \mathbb{R}^d$ 中一点的半空间深度的算法。据我们所知，这是针对该问题的首个确定性算法。作为我们方法的一部分，我们设计了一种算法来回答 $K$ 的深度修剪区域（即深度函数的超水平集）的近似成员查询。我们的数据结构受 Abdelkader 和 Mount [SOSA 2024] 近期工作的启发，其中使用源自 $K$ 上 Hilbert 度量的几何结构来回答对 $K$ 的近似成员查询。我们数据结构的一个关键组成部分是深度修剪区域与 $K$ 的 Hilbert 度量球之间的一种新颖的定量比较。最后，为了突出该问题的计算开销，我们提出了一种算法，用于确定作为有限个半平面交集给出的开平面凸多边形中一个点的精确深度。
英文摘要：
Halfspace (or Tukey) depth is a fundamental and robust measure of centrality of data points in multivariate datasets. Computing the depth of a point with respect to the uniform distribution on an open convex body in $\mathbb{R}^d$ is a natural algorithmic problem. While the coarser task of testing membership in convex bodies has been extensively studied, the refined problem of evaluating depth has received comparatively little attention in the literature. In this work, we present an algorithm for approximating the halfspace depth of a point in an open convex body $K \subset \mathbb{R}^d.$ To the best of our knowledge, this is the first deterministic algorithm for this problem. As part of our approach, we design an algorithm for answering approximate membership queries for the depth-trimmed regions of $K$ (i.e., the superlevel sets of the depth function). Our data structure is inspired by recent work of Abdelkader and Mount [SOSA 2024], wherein approximate membership queries for $K$ are answered using geometric structures derived from the Hilbert metric on $K.$ A key component underlying our data structure is a novel quantitative comparison between the depth-trimmed regions and the Hilbert metric balls of $K$. Lastly, to highlight the computational expense of the problem, we present an algorithm for determining the exact depth of a point in an open planar convex polygon presented as the intersection of finitely many halfplanes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
