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. Here we are concerned with embedding dissimilarities by minimizing Kruskal's (1964) raw stress criterion. Standard results from the theory of point-to-set maps can be used to establish that, if $n$ is fixed and a sequence of dissimilarity matrices converges, then the limit of their embedded structures is the embedded structure of the limiting dissimilarity matrix. 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, {\em continuous MDS}. Within the continuous MDS framework, we derive two $L^p$ consistency results, one for embedding without constraints on the configuration, the other for embedding subject to {\em approximate Lipschitz constraints}\/ that encourage smoothness of the embedding function. The latter approach, {\em Approximate Lipschitz Embedding}\/ (ALE) is new. Finally, we demonstrate that embedded structures produced by ALE can be interpolated in a way that results in uniform convergence.

翻译：多维标度（MDS）是将一组 $n$ 个对象之间的邻近性信息嵌入到 $d$ 维欧几里得空间中的行为。正如心理测量学界最初构想的那样，MDS 关注的是嵌入与一组固定对象相关联的固定邻近性。而现代的关注点，例如在随机图统计推断的渐近理论发展中出现的，更典型地涉及研究与一个不断增长的对象集合相关联的邻近性序列的极限行为。本文关注通过最小化 Kruskal（1964）的原始应力准则来嵌入相异性。点集映射理论的标准结果可用于证明，如果 $n$ 固定且一系列相异性矩阵收敛，则其嵌入结构的极限就是极限相异性矩阵的嵌入结构。但如果 $n$ 增加呢？此时有必要重新表述 MDS，使得整个嵌入问题序列可以被视为一个固定空间中的优化问题序列。我们提出了这样一种重新表述，即{\em 连续多维标度}。在连续 MDS 框架内，我们推导了两个 $L^p$ 一致性结果：一个用于对配置不加约束的嵌入，另一个用于在鼓励嵌入函数平滑性的{\em 近似利普希茨约束}下的嵌入。后一种方法，即{\em 近似利普希茨嵌入}（ALE），是新颖的。最后，我们证明了由 ALE 产生的嵌入结构可以通过插值实现一致收敛。