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14
Sensor network localization by eigenvector synchronization over the Euclidean group
 In press
"... We present a new approach to localization of sensors from noisy measurements of a subset of their Euclidean distances. Our algorithm starts by finding, embedding and aligning uniquely realizable subsets of neighboring sensors called patches. In the noisefree case, each patch agrees with its global ..."
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Cited by 23 (14 self)
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We present a new approach to localization of sensors from noisy measurements of a subset of their Euclidean distances. Our algorithm starts by finding, embedding and aligning uniquely realizable subsets of neighboring sensors called patches. In the noisefree 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. 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. While our approach is applicable to higher dimensions, in the current paper we focus on the two dimensional case.
Localization from Incomplete Noisy Distance Measurements,”
 in IEEE ISIT
, 2011
"... AbstractWe consider the problem of positioning a cloud of points in the Euclidean space R d , 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 ..."
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Cited by 20 (0 self)
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AbstractWe consider the problem of positioning a cloud of points in the Euclidean space R d , 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 dimension d, and the average degree of the nodes in the graph.
Beyond convex relaxation: A polynomial–time non–convex optimization approach to network localization
, 2013
"... AbstractThe successful deployment and operation of locationaware networks, which have recently found many applications, depends crucially on the accurate localization of the nodes. Currently, a powerful approach to localization is that of convex relaxation. In a typical application of this approa ..."
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Cited by 9 (2 self)
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AbstractThe successful deployment and operation of locationaware networks, which have recently found many applications, depends crucially on the accurate localization of the nodes. Currently, a powerful approach to localization is that of convex relaxation. In a typical application of this approach, the localization problem is first formulated as a rankconstrained semidefinite program (SDP), where the rank corresponds to the target dimension in which the nodes should be localized. Then, the nonconvex rank constraint is either dropped or replaced by a convex surrogate, thus resulting in a convex optimization problem. In this paper, we explore the use of a nonconvex surrogate of the rank function, namely the socalled Schatten quasinorm, in network localization. Although the resulting optimization problem is nonconvex, we show, for the first time, that a firstorder critical point can be approximated to arbitrary accuracy in polynomial time by an interiorpoint algorithm. Moreover, we show that such a firstorder point is already sufficient for recovering the node locations in the target dimension if the input instance satisfies certain established uniqueness properties in the literature. Finally, our simulation results show that in many cases, the proposed algorithm can achieve more accurate localization results than standard SDP relaxations of the problem.
Fast and Near–Optimal Matrix Completion via Randomized Basis Pursuit
, 2009
"... Motivated by the philosophy and phenomenal success of compressed sensing, the problem of reconstructing a matrix from a sampling of its entries has attracted much attention recently. Such a problem can be viewed as an information–theoretic variant of the well–studied matrix completion problem, and t ..."
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Motivated by the philosophy and phenomenal success of compressed sensing, the problem of reconstructing a matrix from a sampling of its entries has attracted much attention recently. Such a problem can be viewed as an information–theoretic variant of the well–studied matrix completion problem, and the main objective is to design an efficient algorithm that can reconstruct a matrix by inspecting only a small number of its entries. Although this is an impossible task in general, Candès and co–authors have recently shown that under a so–called incoherence assumption, a rank
1Second Order Cone Programming for Sensor Network Localization with Anchor Position Uncertainty
"... Abstract—We consider the problem of node localization in sensor networks, and we focus on networks in which the ranging measurements are subject to errors and anchor positions are subject to uncertainty. We consider a statistical model for the uncertainty in the anchor positions and formulate the ro ..."
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Abstract—We consider the problem of node localization in sensor networks, and we focus on networks in which the ranging measurements are subject to errors and anchor positions are subject to uncertainty. We consider a statistical model for the uncertainty in the anchor positions and formulate the robust localization problem that finds a maximum likelihood estimation of the node positions. To overcome the nonconvexity of the resulting optimization problem, we obtain a convex relaxation that is based on the second order cone programming (SOCP). We also propose a possible distributed implementation using the SOCP convex relaxation. We present numerical studies that compare the presented approach to other existing convex relaxations for the robust localization problem in terms of positioning error and computational complexity. I.
Graph realization and lowrank matrix completion
, 2012
"... This thesis consists of five chapters, and focuses on two main problems: the graph realization problem with its applications to localization of sensor network and structural biology, and the lowrank matrix completion problem. Chapter 1 is a brief introduction to rigidity theory and supplies the bac ..."
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This thesis consists of five chapters, and focuses on two main problems: the graph realization problem with its applications to localization of sensor network and structural biology, and the lowrank matrix completion problem. Chapter 1 is a brief introduction to rigidity theory and supplies the background needed for the subsequent chapters. Chapter 2 introduces the graph realization problem in dimension two, and its application to sensor network localization. Chapter 3 considers the three dimensional graph realization problem and its application to the molecule problem from structural biology. Chapter 4 focuses on the group synchronization problem, and provides a more indepth analysis of the synchronization methods used in our algorithms for the graph realization problem in R2 and R3. Finally, Chapter 5 investigates the problem of uniqueness of lowrank matrix completion, building on tools from rigidity theory. Rigidity theory tries to answer if a given partial set of distances dij = ‖pi−pj ‖ between n points in Rd uniquely determines the coordinates of the points p1,..., pn up to rigid transformations (translations, rotations, reflections). Chapter 1 is a self contained but extremely
Second Order Cone Programming for Sensor Network Localization with Anchor Position Uncertainty
"... Abstract Node localization is a difficult task in sensor networks in which the ranging measurements are subject to errors and anchor positions are subject to uncertainty. In this paper, the robust localization problem is formulated using the maximum likelihood criterion under an unbounded uncertain ..."
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Abstract Node localization is a difficult task in sensor networks in which the ranging measurements are subject to errors and anchor positions are subject to uncertainty. In this paper, the robust localization problem is formulated using the maximum likelihood criterion under an unbounded uncertainty model for the anchor positions. To overcome the nonconvexity of the resulting optimization problem, a convex relaxation leading to second order cone programming (SOCP) is devised. Furthermore, an analysis is performed in order to identify the set of nodes which are accurately positioned using robust SOCP, and to establish a relation between the solution of the proposed robust SOCP optimization and the existing robust optimization using semidefinite programming (SDP). Based on this analysis, a mixed robust SDPSOCP localization framework is proposed which benefits from the better accuracy of SDP and the lower complexity of SOCP. Since the centralized optimization involves a high computational complexity in large networks, we also derive the distributed implementation of the proposed robust SOCP convex relaxation. Finally, we propose an iterative optimization based on the expectation maximization algorithm for the cases where anchor uncertainty parameters are unavailable. Simulations confirm that the robust SOCP and mixed robust SDPSOCP provide tradeoffs between localization accuracy and computational complexity that render them attractive solutions, especially in networks with a large number of nodes.
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"... Over the few next decades, the personal vehicle may evolve into a device distinct from what exists today. This thesis considers energy and communication systems among vehicles that utilize two developing technologies: batterypowered drive trains and accurate sensors. We use linear optimization to c ..."
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Over the few next decades, the personal vehicle may evolve into a device distinct from what exists today. This thesis considers energy and communication systems among vehicles that utilize two developing technologies: batterypowered drive trains and accurate sensors. We use linear optimization to construct separate algorithms for networks of Plugin Electric Vehicles (PEVs) and networks of Sensorequipped Vehicles (SVs). Plugin electric vehicles will have flexible charging options, and may be capable of transmitting electricity back to the grid (i.e., discharging). We construct an automated mechanism for a fleet of PEVs that efficiently organizes distributed energy trading to benefit both the consumers and the electric utilities. A linear programming model of the fleet provides a composite valuation, which can be used in an online environment managed by a fleet aggregatorto allocate feasible energy exchange schedules that decrease the peak electricity demand and reduce the cost to consumers. The resulting charging and discharging schedules are assigned to tens of thousands of vehicles instantly as they plug into the grid and are robust to unexpected events in driving patterns. We give empirical results based on electricity and gasoline pricing, electricity demand, vehicle characteristics, and driving behaviors.
permission. SemiDefinite Programming Relaxation for NonLineofSight Localization
"... personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires pri ..."
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personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific