Localization from Incomplete Noisy Distance Measurements
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
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.
- A.Y. Alfakih, A. Khandani, H. Wolkowicz, Solving Euclidean distance matrix completion problems via semidefinite programming, Comput. Optim. Appl. 12, 13–30 (1999). CrossRef
- L. Asimow, B. Roth, The rigidity of graphs, Trans. Am. Math. Soc. 245, 279–289 (1978). CrossRef
- J. Aspnes, T. Eren, D.K. Goldenberg, A.S. Morse, W. Whiteley, Y.R. Yang, B.D.O. Anderson, P.N. Belhumeur, A theory of network localization, IEEE Trans. Mob. Comput. 5(12), 1663–1678 (2006). CrossRef
- M. Belkin, P. Niyogi, Laplacian eigenmaps for dimensionality reduction and data representation, Neural Comput. 15, 1373–1396 (2002). CrossRef
- M. Bernstein, V. de Silva, J. Langford, J. Tenenbaum, Graph approximations to geodesics on embedded manifolds. Technical Report, Stanford University, Stanford, CA, 2000.
- P. Biswas, Y. Ye, Semidefinite programming for ad hoc wireless sensor network localization, in Proceedings of the 3rd International Symposium on Information Processing in Sensor Networks, IPSN ’04 (ACM, New York, 2004), pp. 46–54. CrossRef
- S.P. Boyd, A. Ghosh, B. Prabhakar, D. Shah, Mixing times for random walks on geometric random graphs, in Proceedings of the 7th Workshop on Algorithm Engineering and Experiments and the 2nd Workshop on Analytic Algorithmics and Combinatorics, ALENEX/ANALCO 2005, Vancouver, BC, Canada, 22 January 2005 (SIAM, Philadelphia, 2005), pp. 240–249.
- S. Butler, Eigenvalues and structures of graphs. Ph.D. thesis, University of California, San Diego, CA, 2008.
- R. Connelly, Generic global rigidity, Discrete Comput. Geom. 33, 549–563 (2005). CrossRef
- T. Cox, M. Cox, Multidimensional Scaling, Monographs on Statistics and Applied Probability, vol. 88 (Chapman & Hall, London, 2001).
- P. Diaconis, L. Saloff-Coste, Comparison theorems for reversible Markov chains, Ann. Appl. Probab. 3(3), 696–730 (1993). CrossRef
- D.L. Donoho, C. Grimes, Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data, Proc. Natl. Acad. Sci. USA 100(10), 5591–5596 (2003). CrossRef
- S.J. Gortler, A.D. Healy, D.P. Thurston, Characterizing generic global rigidity, Am. J. Math. 132, 897–939 (2010). CrossRef
- F. Lu, S.J. Wright, G. Wahba, Framework for kernel regularization with application to protein clustering, Proc. Natl. Acad. Sci. USA 102(35), 12332–12337 (2005). CrossRef
- G. Mao, B. Fidan, B.D.O. Anderson, Wireless sensor network localization techniques, Comput. Netw. ISDN Syst. 51, 2529–2553 (2007).
- S. Oh, A. Karbasi, A. Montanari, Sensor network localization from local connectivity: performance analysis for the MDS-MAP algorithm, in IEEE Information Theory Workshop 2010 (ITW 2010) (2010).
- N. Patwari, J.N. Ash, S. Kyperountas, R. Moses, N. Correal, Locating the nodes: cooperative localization in wireless sensor networks, IEEE Signal Process. Mag. 22, 54–69 (2005). CrossRef
- M. Penrose, Random Geometric Graphs (Oxford University Press, Oxford, 2003). CrossRef
- L.K. Saul, S.T. Roweis, Y. Singer, Think globally, fit locally: unsupervised learning of low dimensional manifolds, J. Mach. Learn. Res. 4, 119–155 (2003).
- D. Shamsi, Y. Ye, N. Taheri, On sensor network localization using SDP relaxation. arXiv:1010.2262 (2010).
- A. Singer, A remark on global positioning from local distances, Proc. Natl. Acad. Sci. USA 105(28), 9507–9511 (2008). CrossRef
- A.M.-C. So, Y. Ye, Theory of semidefinite programming for sensor network localization, in Symposium on Discrete Algorithms (2005), pp. 405–414.
- J.B. Tenenbaum, V. Silva, J.C. Langford, A global geometric framework for nonlinear dimensionality reduction, Science 290(5500), 2319–2323 (2000). CrossRef
- K.Q. Weinberger, L.K. Saul, An introduction to nonlinear dimensionality reduction by maximum variance unfolding, in Proceedings of the 21st National Conference on Artificial Intelligence, 2 (AAAI Press, Menlo Park, 2006), pp. 1683–1686.
- Z. Zhu, A.M.-C. So, Y. Ye, Universal rigidity: towards accurate and efficient localization of wireless networks, in IEEE International Conference on Computer Communications (2010), pp. 2312–2320.
- Localization from Incomplete Noisy Distance Measurements
Foundations of Computational Mathematics
Volume 13, Issue 3 , pp 297-345
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Rigidity theory
- Global rigidity
- Stress matrix
- Graph realization
- Network localization
- Manifold learning
- Semidefinite programming