发表机构
DGIST, Republic of Korea（大邱庆北科学技术院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明双曲空间最远点查询最优核集大小为Θ(ε^{-(D-1)/2})，通过理想边界包络实现精确归约，并给出上下界匹配的构造。
AI 中文摘要
我们研究双曲空间中用于最远点查询的核集。给定非空有限点集 $P \subset \mathbb{H}^D$ 和 $0<\varepsilon \le 1$，我们寻求一个核集 $P_{\varepsilon} \subseteq P$，使得对于每个查询点，其最远距离相对于 $P$ 的最远距离低估至多一个加性 $\varepsilon$，并且至少保留其 $1-\varepsilon$ 的比例。对于每个固定的 $D \ge 2$，我们证明最优最坏情况核集大小为 $\Theta\bigl(\varepsilon^{-(D-1)/2}\bigr)$。我们的主要几何工具是从双曲查询到理想边界上包络的精确归约。在双曲模型（hyperboloid model）中，每个输入点诱导一个正边界分数函数，其对数给出沿测地射线（geodesic rays）的渐近距离偏移。我们将\emph{理想边界包络}定义为这些函数的逐点最大值，并证明所有查询上的加性损失上确界等于输入核集与核集包络之间的最大对数差距。对于上界，我们将最小包围球（minimum-enclosing-ball）中心移至原点并归一化空间坐标，得到一个有界欧几里得点集，其边界包络远离零。标准欧几里得核（kernel）随后同时逼近所有方向分数最大值，结构定理给出两个保证。对于下界，固定半径双曲球面上的球面填充（spherical packing），结合对跖查询（antipodal queries）和双曲余弦定律，使每个输入点不可或缺，即使对两个保证分别而言也匹配上界。
英文摘要
We study coresets for farthest-point queries in hyperbolic space. Given a nonempty finite set $P \subset \mathbb{H}^D$ and $0<\varepsilon \le 1$, we seek a coreset $P_{\varepsilon} \subseteq P$ whose farthest distance from every query point underestimates that of $P$ by at most an additive $\varepsilon$ and retains at least a $1-\varepsilon$ fraction of it. For every fixed $D \ge 2$, we prove that the optimal worst-case coreset size is $Θ\bigl(\varepsilon^{-(D-1)/2}\bigr)$. Our main geometric ingredient is an exact reduction from hyperbolic queries to an upper envelope on the ideal boundary. In the hyperboloid model, each input point induces a positive boundary-score function whose logarithm gives its asymptotic distance offset along geodesic rays. We define the \emph{ideal-boundary envelope} as the pointwise maximum of these functions and prove that the supremum additive loss over all queries equals the maximum logarithmic gap between the input and coreset envelopes. For the upper bound, we move the minimum-enclosing-ball center to the origin and normalize the spatial coordinates, obtaining a bounded Euclidean point set whose boundary envelope is bounded away from zero. A standard Euclidean kernel then approximates all directional score maxima simultaneously, and the structure theorem yields both guarantees. For the lower bound, a spherical packing on a fixed-radius hyperbolic sphere, together with antipodal queries and the hyperbolic cosine law, makes every input point indispensable, matching the upper bound even for either guarantee separately.
