<i>Ab-initio</i>reconstruction of complex Euclidean networks in two dimensions

Gujarathi, S. R.; Farrow, Christopher L.; Glosser, Connor; Granlund, L.; Duxbury, Phillip M.

doi:10.1103/physreve.89.053311

Cited by 10 publications

(21 citation statements)

References 32 publications

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“…Another example of unlabeled distances is in sparse phase retrieval, where the distances between the unknown non-zero lags in a signal are revealed in its autocorrelation function [7]. Recently, motivated by problems in crystallog-raphy, Gujarahati and co-authors published an algorithm for reconstruction of Euclidean networks from unlabeled distance data [31].…”

Section: A Prior Artmentioning

confidence: 99%

See 1 more Smart Citation

Euclidean Distance Matrices: Essential theory, algorithms, and applications

Dokmanić

Parhizkar²,

Ranieri

et al. 2015

IEEE Signal Process. Mag.

468

276

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Euclidean distance matrices (EDMs) are matrices of the squared distances between points. The definition is deceivingly simple; thanks to their many useful properties, they have found applications in psychometrics, crystallography, machine learning, wireless sensor networks, acoustics, and more. Despite the usefulness of EDMs, they seem to be insufficiently known in the signal processing community. Our goal is to rectify this mishap in a concise tutorial. We review the fundamental properties of EDMs, such as rank or (non)definiteness, and show how the various EDM properties can be used to design algorithms for completing and denoising distance data. Along the way, we demonstrate applications to microphone position calibration, ultrasound tomography, room reconstruction from echoes, and phase retrieval. By spelling out the essential algorithms, we hope to fast-track the readers in applying EDMs to their own problems. The code for all of the described algorithms and to generate the figures in the article is available online at http://lcav.epfl.ch/ivan.dokmanic. Finally, we suggest directions for further research. IntroductIonImagine that you land at Geneva International Airport with the Swiss train schedule but no map. Perhaps surprisingly, this may be sufficient to reconstruct a rough (or not so rough) map of the Alpine country, even if the train times poorly translate to distances or if some of the times are unknown. The way to do it is by using EDMs; for an example, see "Swiss Trains (Swiss Map Reconstruction)."We often work with distances because they are convenient to measure or estimate. In wireless sensor networks, for example, the sensor nodes measure the received signal strengths of the packets sent by other nodes or the time of arrival (TOA) of pulses emitted by their neighbors [1]. Both of these proxies allow for distance estimation between pairs of nodes; thus, we can attempt to reconstruct the network topology. This is often termed self-localization [2]- [4]. The molecular conformation problem is another instance of a distance problem [5], and so is reconstructing a room's geometry from echoes [6]. Less obviously, sparse phase retrieval [7] can be converted to a distance problem and addressed using EDMs.Sometimes the data are not metric, but we seek a metric representation, as it happens commonly in psychometrics [8]. As a matter of fact, the psychometrics community is at the root of the development of a number of tools related to EDMs, including multidimensional scaling (MDS)-the problem of finding the best point set representation of a given set of distances. More abstractly, we can study EDMs for objects such as images, which live in highdimensional vector spaces [9].EDMs are a useful description of the point sets and a starting point for algorithm design. A typical task is to retrieve the original point configuration: it may initially come as a surprise that this requires no more than an eigenvalue decomposition (EVD) of a symmetric matrix. In fact, the majority of Euclidean distance problems requi...

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Section: A Prior Artmentioning

confidence: 99%

“…What can we say about uniqueness of incomplete unlabeled distance sets? Some of the questions have been answered by Gujarathi [31], but many remain. The quest is on for faster algorithms, as well as algorithms that can handle noisy distances.…”

Section: Ideas For Future Researchmentioning

confidence: 99%

Euclidean Distance Matrices: Essential theory, algorithms, and applications

Dokmanić

Parhizkar²,

Ranieri

et al. 2015

IEEE Signal Process. Mag.

468

276

View full text Add to dashboard Cite

show abstract

“…Usually, the phenomena (e.g., phase information) of coupled agents occurred in a network are observed, but the topology of the network is unknown. Estimating the topology of the network from those phenomena becomes a key in a multidisciplinary research field [1][2][3]. Network reconstruction is a reverse engineering problem and a number of methods have been proposed to address this problem [4][5][6][7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of networks from one-step data by matching positions

Dang

Jiao

2018

Physica A: Statistical Mechanics and its Applications

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h i g h l i g h t s• A method for reconstructing the topology of networks with only one step data. • This algorithm utilizes the natural sparsity and degree distribution of real world networks.• The problem of matching position is defined and a simple algorithm is developed.• The matching position can be used in network reconstruction and other problems.• The proposed method is verified in networks with ten thousands of vertices. a b s t r a c tIt is a challenge in estimating the topology of a network from short time series data. In this paper, matching positions is developed to reconstruct the topology of a network from only one-step data. We consider a general network model of coupled agents, in which the phase transformation of each node is determined by its neighbors. From the phase transformation information from one step to the next, the connections of the tail vertices are reconstructed firstly by the matching positions. Removing the already reconstructed vertices, and repeatedly reconstructing the connections of tail vertices, the topology of the entire network is reconstructed. For sparse scale-free networks with more than ten thousands nodes, we almost obtain the actual topology using only the one-step data in simulations.

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“…The major bottleneck in PDF analysis is the extraction of nanostructures from the data, as standard techniques are based on either refinement from a good initial guess [3]; or they are based on simulated annealing [4][5][6]. A more systematic approach to finding good starting structures is a high priority in the field and provides the motivation for developing ab-initio distance geometry approaches [7,8]. Alternative experimental approaches such as high resolution transmission electron microscopy (see Fig.…”

Section: Introductionmentioning

confidence: 99%

“…For this reason we call this the unassigned distance geometry (UDG) problem [7][8][9] and it is clearly harder than the distance geometry (DG) problem that arises in protein structure determination from NMR data [12][13][14][15][16] where the graph structure is given. Here we refer to this problem as the assigned distance geometry (ADG) problem to distinguish it from the UDG problem.…”

Section: Introductionmentioning

confidence: 99%

Assigned and unassigned distance geometry: applications to biological molecules and nanostructures

et al. 2016

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Distance geometry approaches to finding the structure of proteins from nuclear magnetic resonance (NMR) data are standard, however it is only recently that distance geometry methods have been applied to ab-initio structure determination from the pair distribution function (PDF). The interatomic distances extracted from the PDF data are unassigned, whereas NMR experiments are designed to enable peak assignment leading to the assigned distance geometry problem. The difference between assigned and unassigned distance geometry problems are described first. We then survey recent work on the unassigned distance geometry problem and its application to the nanostructure problem, and place this work in the context of algorithms for nanostructure determination from PDF data. Necessary theory including elements of graph rigidity and homometric sets are discussed. Algorithms for generic and non-generic cases are outlined and computational results are presented.

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Ab-initioreconstruction of complex Euclidean networks in two dimensions

Cited by 10 publications

References 32 publications

Euclidean Distance Matrices: Essential theory, algorithms, and applications

Euclidean Distance Matrices: Essential theory, algorithms, and applications

Reconstruction of networks from one-step data by matching positions

Assigned and unassigned distance geometry: applications to biological molecules and nanostructures

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