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18
Approximation Accuracy, Gradient Methods, and Error Bound for Structured Convex Optimization
, 2009
"... Convex optimization problems arising in applications, possibly as approximations of intractable problems, are often structured and large scale. When the data are noisy, it is of interest to bound the solution error relative to the (unknown) solution of the original noiseless problem. Related to this ..."
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Cited by 38 (1 self)
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Convex optimization problems arising in applications, possibly as approximations of intractable problems, are often structured and large scale. When the data are noisy, it is of interest to bound the solution error relative to the (unknown) solution of the original noiseless problem. Related to this is an error bound for the linear convergence analysis of firstorder gradient methods for solving these problems. Example applications include compressed sensing, variable selection in regression, TVregularized image denoising, and sensor network localization.
Explicit Sensor Network Localization Using Semidefinite Representations and Clique Reductions
 Department of Combinatorics and Optimization, University of Waterloo
, 2009
"... AMS Subject Classification: The sensor network localization, SNL, problem in embedding dimension r, consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solu ..."
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Cited by 28 (10 self)
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AMS Subject Classification: The sensor network localization, SNL, problem in embedding dimension r, consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solution techniques relax this problem to a weighted, nearest, (positive) semidefinite programming, SDP,completion problem, by using the linear mapping between Euclidean distance matrices, EDM, and semidefinite matrices. The resulting SDP is solved using primaldual interior point solvers, yielding an expensive and inexact solution. This relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the SDP cone, implying that the Slater constraint qualification fails. Cliques in the graph of the SNL problem give rise to this degeneracy in the SDP relaxation. In this paper, we take advantage of the absence of the Slater constraint qualification and derive a technique for the SNL problem, with exact data, that explicitly solves the corresponding rank restricted SDP problem. No SDP solvers are used. For randomly generated instances,
Euclidean Distance Matrices and Applications
"... Over the past decade, Euclidean distance matrices, or EDMs, have been receiving increased attention for two main reasons. The first reason is that the many applications of EDMs, such as molecular conformation in bioinformatics, dimensionality reduction in machine learning and statistics, and especia ..."
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Cited by 14 (0 self)
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Over the past decade, Euclidean distance matrices, or EDMs, have been receiving increased attention for two main reasons. The first reason is that the many applications of EDMs, such as molecular conformation in bioinformatics, dimensionality reduction in machine learning and statistics, and especially
Distributed Maximum Likelihood Sensor Network Localization
 IEEE Transactions on Signal Processing
, 2014
"... Abstract—We propose a class of convex relaxations to solve the sensor network localization problem, based on a maximum likelihood (ML) formulation. This class, as well as the tightness of the relaxations, depends on the noise probability density function (PDF) of the collected measurements.We deri ..."
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Cited by 7 (3 self)
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Abstract—We propose a class of convex relaxations to solve the sensor network localization problem, based on a maximum likelihood (ML) formulation. This class, as well as the tightness of the relaxations, depends on the noise probability density function (PDF) of the collected measurements.We derive a computational efficient edgebased version of this ML convex relaxation class and we design a distributed algorithm that enables the sensor nodes to solve these edgebased convex programs locally by communicating only with their close neighbors. This algorithm relies on the alternating direction method of multipliers (ADMM), it converges to the centralized solution, it can run asynchronously, and it is computation errorresilient. Finally, we compare our proposed distributed scheme with other available methods, both analytically and numerically, and we argue the added value of ADMM, especially for largescale networks. Index Terms—Distributed optimization, convex relaxations, sensor network localization, distributed algorithms, ADMM, distributed localization, sensor networks, maximum likelihood. I.
EUCLIDEAN DISTANCE GEOMETRY AND APPLICATIONS
"... Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We surv ..."
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Cited by 6 (1 self)
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Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of its most important applications, including molecular conformation, localization of sensor networks and statics. Key words. Matrix completion, barandjoint framework, graph rigidity, inverse problem, protein conformation, sensor network.
On bar frameworks, stress matrices and semidefinite programming
 Math. Program. Ser. B
"... A bar framework G(p) in rdimensional Euclidean space is a graph G = (V,E) on the vertices 1, 2,..., n, where each vertex i is located at point pi in Rr. Given a framework G(p) in Rr, a problem of great interest is that of determining whether or not there exists another framework G(q), not obtained ..."
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Cited by 5 (2 self)
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A bar framework G(p) in rdimensional Euclidean space is a graph G = (V,E) on the vertices 1, 2,..., n, where each vertex i is located at point pi in Rr. Given a framework G(p) in Rr, a problem of great interest is that of determining whether or not there exists another framework G(q), not obtained from G(p) by a rigid motion, such that qi−qj 2 = pi−pj 2 for all (i, j) ∈ E. This problem is known as either the global rigidity problem or the universal rigidity problem depending on whether such a framework G(q) is restricted to be in the same rdimensional space or not. The stress matrix S of a bar framework G(p) plays a key role in these and other related problems. In this paper, we show that semidefinite programming (SDP) can be effectively used to address the universal rigidity problem. In particular, we use the notion of nondegeneracy of SDP to obtain a sufficient condition for universal rigidity, and to rederive the known sufficient condition for generic universal rigidity. We present new results concerning positive semidefinite stress matrices and we use a semidefinite version of Farkas lemma to characterize bar frameworks that admit a nonzero positive semidefinite stress matrix S.
Global optimal solutions to a general sensor network localization problem
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Noisy Sensor Network Localization using Semidefinite Representations and Facial Reduction
, 2010
"... In this paper we extend a recent algorithm for solving the sensor network localization problem (SNL) to include instances with noisy data. In particular, we continue to exploit the implicit degeneracy in the semidefinite programming (SDP) relaxation of SNL. An essential step involves finding good in ..."
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Cited by 3 (0 self)
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In this paper we extend a recent algorithm for solving the sensor network localization problem (SNL) to include instances with noisy data. In particular, we continue to exploit the implicit degeneracy in the semidefinite programming (SDP) relaxation of SNL. An essential step involves finding good initial estimates for a noisy Euclidean distance matrix, EDM, completion problem. After finding the EDM completion from the noisy data, we rotate the problem using the original positions of the anchors. This is a preliminary working paper, and is a work in progress. Tests are currently ongoing.
Canonical primaldual method for solving nonconvex minimization problems
, 2014
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Comparing SOS and SDP relaxations of sensor network localization
, 2010
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