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27
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.
On a discretizable . . . molecular distance geometry problem
, 2009
"... The molecular distance geometry problem can be formulated as the problem of finding an immersion in R 3 of a given undirected, nonnegatively weighted graph G. In this paper, we discuss a set of graphs G for which the problem may also be formulated as a combinatorial search in discrete space. This is ..."
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Cited by 37 (29 self)
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The molecular distance geometry problem can be formulated as the problem of finding an immersion in R 3 of a given undirected, nonnegatively weighted graph G. In this paper, we discuss a set of graphs G for which the problem may also be formulated as a combinatorial search in discrete space. This is theoretically interesting as an example of “combinatorialization” of a continuous nonlinear problem. It is also algorithmically interesting because the natural combinatorial solution algorithm performs much better than a global optimization approach on the continuous formulation. We present a Branch and Prune algorithm which can be used for obtaining a set of positions of the atoms of protein conformations when only some of the distances between the atoms are known.
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,
Molecular distance geometry methods: From continuous to discrete
, 2009
"... Distance geometry problems arise from the need to position entities in the Euclidean Kspace given some of their respective distances. Entities may be atoms (molecular distance geometry), wireless sensors (sensor network localization), or abstract vertices of a graph (graph drawing). In the context ..."
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Cited by 21 (20 self)
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Distance geometry problems arise from the need to position entities in the Euclidean Kspace given some of their respective distances. Entities may be atoms (molecular distance geometry), wireless sensors (sensor network localization), or abstract vertices of a graph (graph drawing). In the context of molecular distance geometry, the distances are usually known because of chemical properties and Nuclear Magnetic Resonance experiments; sensor networks can estimate their relative distance by recording the power loss during a twoway exchange; finally, when drawing graphs in 2D or 3D, the graph to be drawn is given, and therefore distances between vertices can be computed. Distance geometry problems involve a search in a continuous Euclidean space, but sometimes the problem structure helps reduce the search to a discrete set of points. In this paper we survey some continuous and discrete methods for solving some problems of molecular distance geometry.
(Robust) EdgeBased Semidefinite Programming Relaxation of Sensor Network Localization
 MATH PROGRAM
"... Recently Wang, Zheng, Boyd, and Ye (SIAM J Optim 19:655–673, 2008) proposed a further relaxation of the semidefinite programming (SDP) relaxation of the sensor network localization problem, named edgebased SDP (ESDP). In simulation, the ESDP is solved much faster by interiorpoint method than SDP r ..."
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Cited by 19 (2 self)
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Recently Wang, Zheng, Boyd, and Ye (SIAM J Optim 19:655–673, 2008) proposed a further relaxation of the semidefinite programming (SDP) relaxation of the sensor network localization problem, named edgebased SDP (ESDP). In simulation, the ESDP is solved much faster by interiorpoint method than SDP relaxation, and the solutions found are comparable or better in approximation accuracy. We study some key properties of the ESDP relaxation, showing that, when distances are exact, zero individual trace is not only sufficient, but also necessary for a sensor to be correctly positioned by an interior solution. We also show via an example that, when distances are inexact, zero individual trace is insufficient for a sensor to be accurately positioned by an interior solution. We then propose a noiseaware robust version of ESDP relaxation for which small individual trace is necessary and sufficient for a sensor to be accurately positioned by a certain analytic center solution, assuming the noise level is sufficiently small. For this analytic center solution, the position error for each sensor is shown to be in the order of the square root of its trace. Lastly, we propose a logbarrier penalty coordinate gradient descent method to find such an analytic center solution. In simulation, this method is much faster than interiorpoint method for solving ESDP, and the solutions found are comparable in approximation accuracy. Moreover, the method can distribute its computation over the sensors via local communication, making it practical for positioning and tracking in real time.
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
On the Computation of Protein Backbones by using Artificial Backbones of Hydrogens
"... NMR experiments provide information from which some of the distances between pairs of hydrogen atoms of a protein molecule can be estimated. Such distances can be exploited in order to identify the threedimensional conformation of the molecule: this problem is known in the literature as the Molecu ..."
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Cited by 13 (12 self)
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NMR experiments provide information from which some of the distances between pairs of hydrogen atoms of a protein molecule can be estimated. Such distances can be exploited in order to identify the threedimensional conformation of the molecule: this problem is known in the literature as the Molecular Distance Geometry Problem (MDGP). In this paper, we show how an artificial backbone of hydrogens can be defined which allows the reformulation of the MDGP as a combinatorial problem. This is done with the aim of solving the problem by the Branch and Prune (BP) algorithm, which is able to solve it efficiently. Moreover, we show how the real backbone of a protein conformation can be computed by using the coordinates of the hydrogens found by the BP algorithm. Formal proofs of the presented results are provided, as well as computational experiences on a set of instances whose size ranges from 60 to 6000 atoms.
Computing artificial backbones of hydrogen atoms in order to discover protein backbones
 In Proceedings of the International Multiconference on Computer Science and Information Technology
, 2009
"... Abstract—NMR experiments are able to provide some of the distances between pairs of hydrogen atoms in molecular conformations. The problem of finding the coordinates of such atoms is known as the molecular distance geometry problem. This problem can be reformulated as a combinatorial optimization pr ..."
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Cited by 11 (11 self)
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Abstract—NMR experiments are able to provide some of the distances between pairs of hydrogen atoms in molecular conformations. The problem of finding the coordinates of such atoms is known as the molecular distance geometry problem. This problem can be reformulated as a combinatorial optimization problem and efficiently solved by an exact algorithm. To this purpose, we show how an artificial backbone of hydrogens can be generated that satisfies some assumptions needed for having the combinatorial reformulation. Computational experiments show that the combinatorial approach to this problem is very promising. I.
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.
Global registration of multiple point clouds using semidefinite programming. arXiv:1306.5226 [cs.CV
, 2013
"... ABSTRACT. Consider N points in R d and M local coordinate systems that are related through unknown rigid transforms. For each point we are given (possibly noisy) measurements of its local coordinates in some of the coordinate systems. Alternatively, for each coordinate system, we observe the coordin ..."
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Cited by 8 (4 self)
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ABSTRACT. Consider N points in R d and M local coordinate systems that are related through unknown rigid transforms. For each point we are given (possibly noisy) measurements of its local coordinates in some of the coordinate systems. Alternatively, for each coordinate system, we observe the coordinates of a subset of the points. The problem of estimating the global coordinates of the N points (up to a rigid transform) from such measurements comes up in distributed approaches to molecular conformation and sensor network localization, and also in computer vision and graphics. The leastsquares formulation, though nonconvex, has a well known closedform solution for the case M = 2 (based on the singular value decomposition). However, no closed form solution is known for M ≥ 3. In this paper, we propose a semidefinite relaxation of the leastsquares formulation, and prove conditions for exact and stable recovery for both this relaxation and for a previously proposed spectral relaxation. In particular, using results from rigidity theory and the theory of semidefinite programming, we prove that the semidefinite relaxation can guarantee recovery under more adversarial measurements compared to the spectral counterpart. We perform numerical experiments on simulated data to confirm the theoretical findings. We empirically demonstrate that (a) unlike the spectral relaxation, the relaxation gap is mostly zero for the semidefinite program (i.e., we are able to solve the original nonconvex problem) up to a certain noise threshold, and (b) the semidefinite program performs significantly better than spectral and manifoldoptimization methods, particularly at large noise levels.