ASAP: An Eigenvector Synchronization Algorithm for the Graph Realization Problem

We review a recent algorithm for localization of points in Euclidean space from a sparse and noisy subset of their pairwise distances. Our approach starts by extracting and embedding uniquely realizable subsets of neighboring sensors called patches. In the noise-free case, each patch agrees with its global positioning up to an unknown rigid motion of translation, rotation, and possibly reflection. The reflections and rotations are estimated using the recently developed eigenvector synchronization algorithm, while the translations are estimated by solving an overdetermined linear system. In other words, to every patch, there corresponds an element of the Euclidean group Euc(3) of rigid transformations in $${\mathbb{R}}^{3}$$ , and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. The algorithm is scalable as the number of nodes increases, and can be implemented in a distributed fashion. Extensive numerical experiments show that it compares favorably to other existing algorithms in terms of robustness to noise, sparse connectivity and running time.