Localization from Incomplete Noisy Distance Measurements

—We consider the problem of positioning a cloud of points in the Euclidean space Rd , from noisy measurements of a subset of pairwise distances. This task has applications in various areas, such as sensor network localizations, NMR spectroscopy of proteins, and molecular conformation. Also, it is closely related to dimensionality reduction problems and manifold learning, where the goal is to learn the underlying global geometry of a data set using measured local (or partial) metric information. Here we propose a reconstruction algorithm based on a semidefinite programming approach. For a random geometric graph model and uniformly bounded noise, we provide a precise characterization of the algorithm’s performance: In the noiseless case, we find a radius r0 beyond which the algorithm reconstructs the exact positions (up to rigid transformations). In the presence of noise, we obtain upper and lower bounds on the reconstruction error that match up to a factor that depends only on the...

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Adel Javanmard, Andrea Montanari
• Keywords: CORR 2011, Dimensionality Reduction Problems, Education, Euclidean Space Rd, Sensor Network Localizations