发表机构
Graduate School of Science, Tohoku University（东北大学理学研究科）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于固定有界坐标的可观测距离，使几何数据集类别可分离且测地线，并构造紧致屏幕与金字塔空间，实现原始类别的紧化，同时保证收敛检测与非扩张性质。
AI 中文摘要
我们引入一种可观测距离，通过固定的有界坐标来比较实值特征。该坐标在保留有限特征值之间区分度的同时，压缩了它们各自向无穷远的逃逸——这是经典可观测距离在所有几何数据集上不可分离性的根源。新距离使整个类别成为可分离且测地线的：每一对点都由一条常速路径连接，并且在度量测度空间上，诱导拓扑与集中拓扑一致。利用同一坐标，我们构造了紧致屏幕，其特征取值于一个固定的紧致区间。筛选后的类别在Box距离下是Polish且测地线的。按特征顺序组织其有限特征商空间，可得到一个紧致金字塔空间。由单个紧致屏幕生成的金字塔构成稠密子空间，因此该金字塔空间在过渡到紧致屏幕后紧化了原始类别。收敛性通过每个有限测量层的Box-Hausdorff行为来检测。最后，与一个公共度量测度因子取和度量乘积对新距离是非扩张的。更精确地说，由原始空间的指定耦合与公共因子的对角耦合所确定的比较得以保持，而对所有乘积耦合进行优化则得到非扩张不等式。
英文摘要
We introduce an observable distance that compares realvalued features through a fixed bounded coordinate. The coordinate retains the distinction between finite feature values while compressing their independent escape to infinity, the source of nonseparability for the classical observable distance on all geometric data sets. The new distance makes the full class separable and geodesic: every pair is joined by a constant-speed path, and on metric measure spaces the induced topology agrees with the concentration topology. From the same coordinate we construct compact screens, whose features take values in one fixed compact interval. The screened class is Polish and geodesic for the Box distance. Organizing its finite-feature quotients by the feature order yields a compact pyramid space. Pyramids generated by single compact screens form a dense subspace, so this pyramid space compactifies the original class after passage to compact screens. Convergence is detected through the Box-Hausdorff behavior of every finite measurement layer. Finally, taking sum-metric products with a common metric measure factor is nonexpansive for the new distance. More precisely, the comparison determined by a prescribed coupling of the original spaces and the diagonal coupling of the common factor is preserved, whereas optimization over all product couplings yields the nonexpansive inequality.
Comments39 pages
