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, 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：多维尺度分析（MDS）是将一组涉及$n$个对象的邻近信息嵌入到$d$维欧氏空间中的方法。正如心理测量学界最初构想的那样，MDS关注的是将固定对象集的固定邻近信息进行嵌入。然而，在现代研究中（例如为随机图的统计推断发展渐近理论时），通常需要研究伴随对象集不断扩大的一系列邻近信息的极限行为。根据点对集映射理论的标准结果，若$n$固定，则嵌入结构的极限即为极限邻近信息的嵌入结构。但当$n$逐渐增大时又该如何处理？此时有必要对MDS进行重新表述，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。本文提出这种重新表述并推导出相关结论。