Multidimensional scaling (MDS) is the act of embedding proximity information about a set of $n$ objects in $d$-dimensional Euclidean space. As originally conceived by the psychometric community, MDS was concerned with embedding a fixed set of proximities associated with a fixed set of objects. Modern concerns, e.g., that arise in developing asymptotic theories for statistical inference on random graphs, more typically involve studying the limiting behavior of a sequence of proximities associated with an increasing set of objects. Standard results from the theory of point-to-set maps imply that, if $n$ is fixed and a sequence of proximities converges, then the limit of the embedded structures is the embedded structure of the limiting proximities. But what if $n$ increases? It then becomes necessary to reformulate MDS so that the entire sequence of embedding problems can be viewed as a sequence of optimization problems in a fixed space. We present such a reformulation and derive some consequences.

翻译：多维缩放（MDS）是将关于$n$个对象的邻近信息嵌入至$d$维欧几里得空间的过程。该概念最初由心理测量学界提出，专注于嵌入与固定对象集合相关联的固定邻近信息。现代研究——例如发展随机图统计推断的渐近理论时遇到的问题——更典型地涉及研究随对象集合增大而变化的邻近序列的极限行为。点-集映射理论的标准结果表明：若$n$固定且邻近序列收敛，则嵌入结构的极限即为极限邻近信息的嵌入结构。但当$n$增大时如何？此时必须重构MDS，使得整个嵌入问题序列可被视为固定空间中的优化问题序列。本文提出此类重构并推导其若干推论。