Sequential lateration is a class of methods for multidimensional scaling where a suitable subset of nodes is first embedded by some method, e.g., a clique embedded by classical scaling, and then the remaining nodes are recursively embedded by lateration. A graph is a lateration graph when it can be embedded by such a procedure. We provide a stability result for a particular variant of sequential lateration. We do so in a setting where the dissimilarities represent noisy Euclidean distances between nodes in a geometric lateration graph. We then deduce, as a corollary, a perturbation bound for stress minimization. To argue that our setting applies broadly, we show that a (large) random geometric graph is a lateration graph with high probability under mild condition. This extends a previous result of Aspnes et al (2006).

翻译：序贯定位是一类多维缩放方法，该方法先通过某种方式嵌入合适的节点子集（例如，通过经典缩放嵌入一个团），再通过定位递归嵌入其余节点。若一个图能通过此类过程嵌入，则称为定位图。我们针对序贯定位的特定变体给出了一个稳定性结果。该结果设定中，不相似性表示几何定位图中节点间带有噪声的欧氏距离。随后，我们推导出应力最小化的一个扰动界作为推论。为论证该设定具有广泛适用性，我们证明：在温和条件下，一个（大型）随机几何图以高概率成为定位图。这扩展了Aspnes等人（2006）先前的结果。