发表机构
University of California, San Diego; Halıcıoğlu Data Science Institute, University of California, San Diego（加州大学圣地亚哥分校; 加州大学圣地亚哥分校哈奇奥卢数据科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究序数球面多维标度的唯一性，证明在大样本极限下，仅凭序数比较可唯一恢复单位球面上的对象（相差正交变换），并扩展至球面外部与内部展开的序数变体。
AI 中文摘要
球面约束嵌入引起了广泛关注，因为在许多应用中会出现具有固有圆形或球形结构的数据。虽然在度量方面已经提出了许多方法，但在序数方面，无论是方法论还是理论上，相关工作都很少。在此，我们聚焦于序数球面多维标度（MDS）的唯一性这一基本问题：给定一个可实现场景，其中底层对象位于单位球面上，且已知信息仅为形如“对象 $i$ 比对象 $k$ 更相似于对象 $j$”的序数比较，在大样本极限下，是否可能唯一恢复原始对象（相差一个正交变换）？我们对此给出肯定回答，不仅适用于上述场景，也适用于球面外部展开（又称后方交会）和球面内部展开（又称偏好映射）的序数变体。
英文摘要
There has been general interest in spherically constrained embeddings as data with an inherently circular or spherical structure arise in a number of applications. While many methods have been proposed on the metric side, little work has been done on the ordinal side in terms of methodology or theory. Here, we focus on the fundamental question of uniqueness of ordinal spherical MDS: Given a realizable setting in which underlying objects lie on the unit sphere and all that is known are their ordinal comparisons of the form `object $i$ is more similar to object $j$ than object $k$', is it possible to uniquely recover the original objects, up to an orthogonal transformation, in the large-sample limit? We answer affirmatively, both in the setting just described, as well as in the settings of spherical external unfolding (aka lateration) and spherical internal unfolding (aka preference mapping) in their ordinal variants.
