Multidimensional scaling localization with anchors

Multidimensional Scaling (MDS) is a widely used technique for visualizing a set of objects in an n-dimensional space. It has been extensively applied in wireless sensor networks for deriving the coordinates of a set of nodes in distance-based Localization. Many variants of MDS have been proposed to overcome issues such as partial connectivity and different types of noise in the measurements. In particular, some works adapted and modified the MDS technique to include the notion of anchors. However, in order to maintain the original formulation of MDS, the algorithm was twisted by adding constraints to the minimization function or adapting the final result through roto-translations. Unfortunately, however, these adaptions do not fully solve the problem, because they try to align the relative positions of the nodes to the global reference system provided by the anchors only after the MDS algorithm. This paper provides a theoretical generalization of the classical MDS algorithm when some of the coordinates of some elements (e.g., anchors in the case of localization) are known. The proposed generalization can be applied to any of the many MDS variants (e.g., classical MDS, ordinal MDS, MDS-MAP, GM-MDS) that minimize the stress function with the SMACOF technique. The formulation is proved to be correct and does not add any constraints to MDS.