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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 25 (15 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
"... Abstract—We 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 21 (0 self)
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Abstract—We 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. I.
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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Cited by 3 (0 self)
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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.
FOR THE DEGREE OF
"... 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.
POINT LOCALIZATION AND DENSITY ESTIMATION FROM ORDINAL KNN GRAPHS USING SYNCHRONIZATION
"... We consider the problem of embedding unweighted, directed knearest neighbor graphs in lowdimensional Euclidean space. The knearest neighbors of each vertex provide ordinal information on the distances between points, but not the distances themselves. Relying only on such ordinal information, al ..."
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We consider the problem of embedding unweighted, directed knearest neighbor graphs in lowdimensional Euclidean space. The knearest neighbors of each vertex provide ordinal information on the distances between points, but not the distances themselves. Relying only on such ordinal information, along with the lowdimensionality, we recover the coordinates of the points up to arbitrary similarity transformations (rigid transformations and scaling). Furthermore, we also illustrate the possibility of robustly recovering the underlying density via the Total Variation Maximum Penalized Likelihood Estimation (TVMPLE) method. We make existing approaches scalable by using an instance of a localtoglobal algorithm based on group synchronization, recently proposed in the literature in the context of sensor network localization, and structural biology, which we augment with a scale synchronization step. We show our approach compares favorably to the recently proposed Local Ordinal Embedding (LOE) algorithm even in the case of smaller sized problems, and also demonstrate its scalability on large graphs. The above divideandconquer paradigm can be of independent interest to the machine learning community when tackling geometric embeddings problems. Index Terms — knearestneighbor graphs, ordinal constraints, graph embeddings, eigenvector synchronization 1.
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
Graph realization and lowrank . . .
, 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