Convex Euclidean distance embedding for collaborative position localization with NLOS mitigation

One of the challenging problems in collaborative position localization arises when the distance measurements contain non-line-of-sight (NLOS) biases. Convex optimization has played a major role in modelling such problems and numerical algorithm developments. One of the successful examples is the semi-definite programming (SDP), which translates Euclidean distances into the constraints of positive semidefinite matrices, leading to a large number of constraints in the case of NLOS biases. In this paper, we propose a new convex optimization model that is built upon the concept of Euclidean distance matrix (EDM). The resulting EDM optimization has an advantage that its Lagrangian dual problem is well structured and hence is conducive to algorithm developments. We apply a recently proposed 3-block alternating direction method of multipliers to the dual problem and tested the algorithm on some real as well as simulated data of large scale. In particular, the EDM model significantly outperforms the existing SDP model and several others.

[1]  Zhaosong Lu,et al.  Penalty Decomposition Methods for $L0$-Norm Minimization , 2010, ArXiv.

[2]  Paul Tseng,et al.  Second-Order Cone Programming Relaxation of Sensor Network Localization , 2007, SIAM J. Optim..

[3]  R.L. Moses,et al.  Locating the nodes: cooperative localization in wireless sensor networks , 2005, IEEE Signal Processing Magazine.

[4]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[5]  Luca Geretti,et al.  A Mixed Convex/Nonconvex Distributed Localization Approach for the Deployment of Indoor Positioning Services , 2008, IEEE Transactions on Mobile Computing.

[6]  Stephen P. Boyd,et al.  Further Relaxations of the Semidefinite Programming Approach to Sensor Network Localization , 2008, SIAM J. Optim..

[7]  R. Michael Buehrer,et al.  NLOS Mitigation Using Linear Programming in Ultrawideband Location-Aware Networks , 2007, IEEE Transactions on Vehicular Technology.

[8]  Peter Stoica,et al.  Source localization from range-difference measurements , 2006 .

[9]  Ting Kei Pong.  Edge-based semidefinite programming relaxation of sensor network localization with lower bound constraints , 2012, Comput. Optim. Appl..

[10]  Patrick J. F. Groenen,et al.  Modern Multidimensional Scaling: Theory and Applications , 2003 .

[11]  R. Mathar,et al.  A cyclic projection algorithm via duality , 1989 .

[12]  R. M. Buehrer,et al.  Non-line-of-sight identification in ultra-wideband systems based on received signal statistics , 2007 .

[13]  Alfred O. Hero,et al.  Relative location estimation in wireless sensor networks , 2003, IEEE Trans. Signal Process..

[14]  Ting Kei Pong,et al.  Comparing SOS and SDP relaxations of sensor network localization , 2012, Comput. Optim. Appl..

[15]  S. Kay.  Fundamentals of statistical signal processing: estimation theory , 1993 .

[16]  B. Ripley,et al.  Robust Statistics , 2018, Encyclopedia of Mathematical Geosciences.

[17]  Ismail Güvenç,et al.  A Survey on TOA Based Wireless Localization and NLOS Mitigation Techniques , 2009, IEEE Communications Surveys & Tutorials.

[18]  Michael C. Hout,et al.  Multidimensional Scaling , 2003, Encyclopedic Dictionary of Archaeology.

[19]  R. Michael Buehrer,et al.  Collaborative position location with NLOS mitigation , 2010, 2010 IEEE 21st International Symposium on Personal, Indoor and Mobile Radio Communications Workshops.

[20]  Houduo Qi,et al.  A Semismooth Newton Method for the Nearest Euclidean Distance Matrix Problem , 2013, SIAM J. Matrix Anal. Appl..

[21]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .

[22]  R. Rockafellar.  Convex Analysis: (pms-28) , 1970 .

[23]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[24]  H. Vincent Poor,et al.  Non-Line-of-Sight Node Localization Based on Semi-Definite Programming in Wireless Sensor Networks , 2009, IEEE Transactions on Wireless Communications.

[25]  Jian Li,et al.  Exact and Approximate Solutions of Source Localization Problems , 2008, IEEE Transactions on Signal Processing.

[26]  Anthony Man-Cho So,et al.  Robust Convex Approximation Methods for TDOA-Based Localization Under NLOS Conditions , 2016, IEEE Transactions on Signal Processing.

[27]  Yinyu Ye,et al.  Semidefinite programming based algorithms for sensor network localization , 2006, TOSN.

[28]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[29]  W. Glunt,et al.  An alternating projection algorithm for computing the nearest euclidean distance matrix , 1990 .

[30]  Kim-Chuan Toh,et al.  A Convergent 3-Block SemiProximal Alternating Direction Method of Multipliers for Conic Programming with 4-Type Constraints , 2014, SIAM J. Optim..

[31]  Kim-Chuan Toh,et al.  An inexact primal–dual path following algorithm for convex quadratic SDP , 2007, Math. Program..

[32]  Marcos Raydan,et al.  Molecular conformations from distance matrices , 1993, J. Comput. Chem..

[33]  Nathan Krislock,et al.  Explicit Sensor Network Localization using Semidefinite Representations and Facial Reductions , 2010, SIAM J. Optim..

[34]  Yinyu Ye,et al.  Semidefinite programming for ad hoc wireless sensor network localization , 2004, Third International Symposium on Information Processing in Sensor Networks, 2004. IPSN 2004.

[35]  Georgios B. Giannakis,et al.  Sparsity-Exploiting Robust Multidimensional Scaling , 2012, IEEE Transactions on Signal Processing.

[36]  I. J. Schoenberg.  Remarks to Maurice Frechet's Article ``Sur La Definition Axiomatique D'Une Classe D'Espace Distances Vectoriellement Applicable Sur L'Espace De Hilbert , 1935 .

[37]  Peter J. Rousseeuw,et al.  Robust Regression and Outlier Detection , 2005, Wiley Series in Probability and Statistics.

[38]  Jon C. Dattorro,et al.  Convex Optimization & Euclidean Distance Geometry , 2004 .

[39]  R. Michael Buehrer,et al.  NLOS mitigation in TOA-based localization using semidefinite programming , 2013, 2013 10th Workshop on Positioning, Navigation and Communication (WPNC).

[40]  Thomas L. Hayden,et al.  Approximation by matrices positive semidefinite on a subspace , 1988 .

[41]  Andreu Urruela,et al.  A non-line-of-sight mitigation technique based on ML-detection , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[42]  Kim-Chuan Toh,et al.  A Convergent Proximal Alternating Direction Method of Multipliers for Conic Programming with 4-Block Constraints , 2014 .

[43]  Chia-Chin Chong,et al.  NLOS Identification and Weighted Least-Squares Localization for UWB Systems Using Multipath Channel Statistics , 2008, EURASIP J. Adv. Signal Process..

[44]  Jiawang Nie,et al.  Sum of squares method for sensor network localization , 2006, Comput. Optim. Appl..

[45]  Xiaoming Yuan,et al.  A Lagrangian Dual Approach to the Single-Source Localization Problem , 2013, IEEE Transactions on Signal Processing.

[46]  Henry Wolkowicz,et al.  Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming , 1999, Comput. Optim. Appl..

[47]  M. Fréchet.  Sur La Definition Axiomatique D'Une Classe D'Espaces Vectoriels Distancies Applicables Vectoriellement Sur L'Espace de Hilbert , 1935 .

[48]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[49]  Frederick R. Forst,et al.  On robust estimation of the location parameter , 1980 .

[50]  Jian Li,et al.  Lecture Notes - Source Localization from Range-Difference Measurements , 2006, IEEE Signal Processing Magazine.

[51]  A. Householder,et al.  Discussion of a set of points in terms of their mutual distances , 1938 .

[52]  Xiaoming Yuan,et al.  Computing the nearest Euclidean distance matrix with low embedding dimensions , 2014, Math. Program..

[53]  R. Michael Buehrer,et al.  Cooperative sensor localization with NLOS mitigation using semidefinite programming , 2012, 2012 9th Workshop on Positioning, Navigation and Communication.

[54]  Peter J. Rousseeuw,et al.  Robust regression and outlier detection , 1987 .

[55]  Benoît Champagne,et al.  Distributed cooperative localization in wireless sensor networks without NLOS identification , 2014, 2014 11th Workshop on Positioning, Navigation and Communication (WPNC).