We proposed a new criterion \textit{noise-stability}, which revised the classical rigidity theory, for evaluation of MDS algorithms which can truthfully represent the fidelity of global structure reconstruction; then we proved the noise-stability of the cMDS algorithm in generic conditions, which provides a rigorous theoretical guarantee for the precision and theoretical bounds for Euclidean embedding and its application in fields including wireless sensor network localization and satellite positioning. Furthermore, we looked into previous work about minimum-cost globally rigid spanning subgraph, and proposed an algorithm to construct a minimum-cost noise-stable spanning graph in the Euclidean space, which enabled reliable localization on sparse graphs of noisy distance constraints with linear numbers of edges and sublinear costs in total edge lengths. Additionally, this algorithm also suggests a scheme to reconstruct point clouds from pairwise distances at a minimum of $O(n)$ time complexity, down from $O(n^3)$ for cMDS.

翻译：我们提出了一个新的标准 \ textit{ noise-stable},用于评估MDS算法的经典僵硬理论,该算法可以真实地代表全球结构重建的忠诚性;然后,我们证明了在通用条件下的cMDS算法的噪声稳定性,它为欧几里德嵌入及其在包括无线传感器网络本地化和卫星定位在内的领域的应用提供了严格的理论保障。此外,我们研究了以前关于全球硬性透射子图的最低成本的工作,并提出了在欧几里德空间建立一个最低成本的噪音分布图的算法,该算法使得以边缘线数和总体边缘线性成本为线性距离限制的微小图块得以可靠地本地化。此外,这一算法还提出了一个计划,将点云从双向距离重建至少为$O(n)美元的时间复杂度,从用于光量度的美元(n)美元(n)美元)美元(n)美元(cMDS)。