Localization from Incomplete Noisy Distance Measurements
 Adel Javanmard,
 Andrea Montanari
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We consider the problem of positioning a cloud of points in the Euclidean space ℝ^{ d }, using noisy measurements of a subset of pairwise distances. This task has applications in various areas, such as sensor network localization and reconstruction of protein conformations from NMR measurements. It is also closely related to dimensionality reduction problems and manifold learning, where the goal is to learn the underlying global geometry of a data set using local (or partial) metric information. Here we propose a reconstruction algorithm based on semidefinite programming. 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 r _{0} 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 dimension d, and the average degree of the nodes in the graph.
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Within this Article
 Introduction
 Preliminaries
 Discussion
 Proof of Theorem 1.1 (Upper Bound)
 Proof of Lemma 4.1
 Proof of Lemma 4.2
 Proof of Theorem 1.1 (Lower Bound)
 Numerical Experiments
 References
 References
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 Title
 Localization from Incomplete Noisy Distance Measurements
 Journal

Foundations of Computational Mathematics
Volume 13, Issue 3 , pp 297345
 Cover Date
 20130601
 DOI
 10.1007/s1020801291295
 Print ISSN
 16153375
 Online ISSN
 16153383
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Rigidity theory
 Global rigidity
 Stress matrix
 Graph realization
 Network localization
 Manifold learning
 Semidefinite programming
 51K05
 52C25
 60D05
 68Q25
 58A99
 94A12
 Industry Sectors
 Authors

 Adel Javanmard ^{(1)}
 Andrea Montanari ^{(1)} ^{(2)}
 Author Affiliations

 1. Department of Electrical Engineering, Stanford University, Packard 239, Stanford, CA, 94305, USA
 2. Department of Statistics, Stanford University, Packard 272, Stanford, CA, 94305, USA