Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming

Alfakih, Abdo Y.; Khandani, Amir K.; Wolkowicz, Henry

doi:10.1007/978-1-4615-5197-3_2

Cited by 122 publications

(213 citation statements)

References 24 publications

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“…If lim n→∞ nr 2 log n > 8, with high probability, G n (r) is a trilateration graph. Proof: Partition [0, 1] 2 into cn log n squares of equal size where c < 1.…”

Section: Theorem 12mentioning

confidence: 99%

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A Theory of Network Localization

Aspnes

Eren

Goldenberg

et al. 2006

IEEE Trans. on Mobile Comput.

563

396

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In this paper we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigidity theory to test the conditions for unique localizability and to construct uniquely localizable networks.We further study the computational complexity of network localization and investigate a subclass of grounded graphs where localization can be computed efficiently. We conclude with a discussion of localization in sensor networks where the sensors are placed randomly.

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“…If lim n→∞ nr 2 log n > 8, with high probability, G n (r) is a trilateration graph. Proof: Partition [0, 1] 2 into cn log n squares of equal size where c < 1.…”

Section: Theorem 12mentioning

confidence: 99%

“…A related problem called molecular conformation has been studied in the chemistry community, e.g., [2], [25], [38]. However, the focus of these studies is on 3D.…”

Section: Related Workmentioning

confidence: 99%

A Theory of Network Localization

Aspnes

Eren

Goldenberg

et al. 2006

IEEE Trans. on Mobile Comput.

563

396

View full text Add to dashboard Cite

show abstract

“…Since K 1 = + , the case of linear inequality constraints of the form A(X) ≥ b fits the analysis in our framework by considering Q = + × · · · × + . The NNM problem (2) often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science, see, e.g., [1,3,17,18,21,27]. The rank minimization problem refers to finding a matrix X ∈ n 1 ×n 2 to minimize rank(X) subject to linear constraints, i.e., min rank(X) : A(X) = b, X ∈ n 1 ×n 2 .…”

Section: Introductionmentioning

confidence: 99%

An implementable proximal point algorithmic framework for nuclear norm minimization

2011

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The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In this paper, we study inexact proximal point algorithms in the primal, dual and primal-dual forms for solving the nuclear norm minimization with linear equality and second order cone constraints. We design efficient implementations of these algorithms and present comprehensive convergence results. In particular, we investigate the performance of our proposed algorithms in which the inner sub-problems are approximately solved by the gradient projection method or the accelerated proximal gradient method. Our numerical results for solving randomly generated matrix completion problems and real matrix completion problems show that our algorithms perform favorably in comparison to several recently proposed state-of-the-art algorithms. Interestingly, our proposed algorithms are connected with other algorithms that have been studied in the literature.

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“…Also, we are able to derive a variance-based active learning method for efficiently selecting which distances to calculate. This arises as a natural product of the probabilistic, Bayesian framework, and is in contrast with the abilities of other proposed models [1] [14].…”

Section: Introductionmentioning

confidence: 99%

A nonparametric Bayesian model for kernel matrix completion

Paisley

Carin

2010

2010 IEEE International Conference on Acoustics, Speech and Signal Processing

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We present a nonparametric Bayesian model for completing low-rank, positive semidefinite matrices. Given an N × N matrix with underlying rank r, and noisy measured values and missing values with a symmetric pattern, the proposed Bayesian hierarchical model nonparametrically uncovers the underlying rank from all positive semidefinite matrices, and completes the matrix by approximating the missing values. We analytically derive all posterior distributions for the fully conjugate model hierarchy and discuss variational Bayes and MCMC Gibbs sampling for inference, as well as an efficient measurement selection procedure. We present results on a toy problem, and a music recommendation problem, where we complete the kernel matrix of 2,250 pieces of music.

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Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming

Cited by 122 publications

References 24 publications

A Theory of Network Localization

A Theory of Network Localization

An implementable proximal point algorithmic framework for nuclear norm minimization

A nonparametric Bayesian model for kernel matrix completion

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