Euclidean Distance Geometry Research Articles

A dual basis approach to multidimensional scaling

Classical multidimensional scaling (CMDS) is a technique that embeds a set of objects in a Euclidean space given their pairwise Euclidean distances. The main part of CMDS involves double centering a squared distance matrix and using a truncated eigendecomposition to recover the point coordinates. In this paper, motivated by a study in Euclidean distance geometry, we explore a dual basis approach to CMDS. We give an explicit formula for the dual basis vectors and fully characterize the spectrum of an essential matrix in the dual basis framework. We make connections to a related problem in metric nearness.

Identifying 3D Genome Organization in Diploid Organisms via Euclidean Distance Geometry

The spatial organization of the DNA in the cell nucleus plays an important role for gene regulation, DNA replication, and genomic integrity. Through the development of chromosome conformation capture experiments (such as 3C, 4C, Hi-C) it is now possible to obtain the contact frequencies of the DNA at the whole-genome level. In this paper, we study the problem of reconstructing the 3D organization of the genome from such whole-genome contact frequencies. A standard approach is to transform the contact frequencies into noisy distance measurements and then apply semidefinite programming (SDP) formulations to obtain the 3D configuration. However, neglected in such reconstructions is the fact that most eukaryotes including humans are diploid and therefore contain two copies of each genomic locus. We prove that the 3D organization of the DNA is not identifiable from distance measurements derived from contact frequencies in diploid organisms. In fact, there are infinitely many solutions even in the noise-free setting. We then discuss various additional biologically relevant and experimentally measurable constraints (including distances between neighboring genomic loci and higher-order interactions) and prove identifiability under these conditions. Furthermore, we provide SDP formulations for computing the 3D embedding of the DNA with these additional constraints and show that we can recover the true 3D embedding with high accuracy from both noiseless and noisy measurements. Finally, we apply our algorithm to real pairwise and higher-order contact frequency data and show that we can recover known genome organization patterns.

Latency and Reliability of mmWave Multi-Hop V2V Communications Under Relay Selections

Timely and reliable information sharing among autonomous vehicles (AVs) provides a promising approach for reducing traffic congestion and improving traffic efficiency in future intelligent transportation systems. In this paper, we consider millimeter-wave (mmWave) based multi-hop vehicle-to-vehicle (V2V) communications to facilitate ultra-reliable low-latency information sharing among AVs. We propose a novel framework for performance analysis and design of relay selection schemes in mmWave multi-hop V2V communications, while taking into account the mmWave signal propagation characteristics, road topology, and traffic conditions. In particular, considering the minimum tracking distance requirement of road traffic, the headway, i.e., the distance between adjacent AVs, is modeled as shifted-exponential distribution. Moreover, we model the communication path losses using the Manhattan distance metric in the taxicab geometry, which can more accurately capture the characteristics of mmWave signal propagation in urban grid roads than conventional Euclidean distance geometry. Based on the proposed model, we investigate the latency and reliability of mmWave multi-hop V2V communications for three widely adopted relay selection schemes, i.e., random with forward progress (RFP), most forward with fixed radius (MFR), and nearest with forward progress (NFP), respectively. Furthermore, we propose a novel relay selection scheme for joint optimization of the single-hop forward progress (FP) and single-hop latency according to the AVs' instantaneous locations and an estimate of the residual multi-hop latency. Simulation results show that, by balancing the current single-hop latency and the residual multi-hop latency for the multi-hop V2V network, the proposed relay selection scheme significantly outperforms the MFR, NFP and RFP in both multi-hop transmission latency and reliability of mmWave V2V communications.

Microphone array positioning technique with Euclidean distance geometry

Source localization and quantification by an acoustic array of microphones depend to a great extent on an accurate knowledge of the antenna position towards the radiating device. The present work details a methodology to determine the location of the microphones in relation to an object of study, starting from its geometric shape and that of the array, in order to reproduce an experimental configuration in any retro-propagating method. A set of reference sources are placed on several prominent locations of the device to estimate the times of flight (ToF) (and distances) between them and the microphones, connecting the array and the object together. The overall geometric configuration is thus defined by an Euclidean Distance Matrix (EDM), which is basically the matrix of squared distances between points. First, MultiDimensional Unfolding (MDU) technique is used to reconstruct the point set from distances. Second, this point set is then aligned with the device, using reference sources as anchor nodes. This orthogonal Procustes problem is solved by the Kabsch algorithm to obtain the optimal rotation and translation matrices between the coordinate system of the array and that of the object of study. The methodology is detailed, validated first by a numerical simulation of a typical experimental set-up. An experimental campaign is finally carried out to assess the robustness of the method in a typical test case.

Exact Reconstruction of Euclidean Distance Geometry Problem Using Low-Rank Matrix Completion

The Euclidean distance geometry problem arises in a wide variety of applications, from determining molecular conformations in computational chemistry to localization in sensor networks. When the distance information is incomplete, the problem can be formulated as a nuclear norm minimization problem. In this paper, this minimization program is recast as a matrix completion problem of a low-rank $r$ Gram matrix with respect to a suitable basis. The well-known restricted isometry property cannot be satisfied in this scenario. Instead, a dual basis approach is introduced to theoretically analyze the reconstruction problem. If the Gram matrix satisfies certain coherence conditions with parameter $\nu $ , the main result shows that the underlying configuration of $n$ points can be recovered with very high probability from $O(nr\nu \log ^{2}(n))$ uniformly random samples. Computationally, simple and fast algorithms are designed to solve the Euclidean distance geometry problem. Numerical tests on different 3-D data and protein molecules validate the effectiveness and efficiency of the proposed algorithms.

Protein structure prediction from inaccurate and sparse NMR data using an enhanced genetic algorithm

Nuclear Magnetic Resonance Spectroscopy (most commonly known as NMR Spectroscopy) is used to generate approximate and partial distances between pairs of atoms of the native structure of a protein. To predict protein structure from these partial distances by solving the Euclidean distance geometry problem from the partial distances obtained from NMR Spectroscopy, we can predict three-dimensional (3D) structure of a protein. In this paper, a new genetic algorithm is proposed to efficiently address the Euclidean distance geometry problem towards building 3D structure of a given protein applying NMR's sparse data. Our genetic algorithm uses (i) a greedy mutation and crossover operator to intensify the search; (ii) a twin removal technique for diversification in the population; (iii) a random restart method to recover from stagnation; and (iv) a compaction factor to reduce the search space. Reducing the search space drastically, our approach improves the quality of the search. We tested our algorithms on a set of standard benchmarks. Experimentally, we show that our enhanced genetic algorithms significantly outperforms the traditional genetic algorithms and a previously proposed state-of-the-art method. Our method is capable of producing structures that are very close to the native structures and hence, the experimental biologists could adopt it to determine more accurate protein structures from NMR data.

Distance geometry for kissing spheres

A kissing sphere is a sphere that is tangent to a fixed reference ball. We develop in this paper a distance geometry for kissing spheres, which turns out to be a generalization of the classical Euclidean distance geometry.

Optimal Location Determination of Some Public Facilities within Minna Metropolis: A Geospatial Technique Approach

Theoretical frameworks used to explain the location of services include central place theory, aspects of industrial location theory and spatial diffusion theory. They are all described as normative theories which optimize with respect to defined criteria operating in prescribed environmental conditions (Rushton, 1979). However, recent advancement in geospatial technologies has led to several applications in geographically orientated challenges, hence, the adoption of an effective decision tool like Geographic Information System (GIS), high resolution products of satellite remote sensing as well as the Global Positioning System (GPS) in solving the rather challenging task of optimal location for facilities with respect to necessary criteria. Minna metropolis is being affected adversely by the problem of appropriate location of public facilities. These facilities are either too far from their market zone or they are too congested in a particular location and in some cases, political consideration to the siting of these facilities dominate without given considerations to the necessary criteria for demands and public interest. The study is an investigation into the “optimal determination of the locations of some public facilities in Minna metropolis of Niger State, using geospatial techniques”. The fusion of remote sensing, geographic information system (GIS) and GPS techniques was explored. Geodata base of existing facilities was created and Euclidean-distance geometry used to spatially analyze the appropriate locations with regards to the set of standard criteria. The results showed the haphazard and uncontrolled pattern of development of schools location. However, petrol stations met the evaluated criteria and optimal location indexes.

Euclidean Distance Geometry and Applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set 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 the most important applications: molecular conformation, localization of sensor networks and statics.

How to project onto an isotone projection cone

The solution of the complementarity problem defined by a mapping f : R n → R n and a cone K ⊂ R n consists of finding the fixed points of the operator P K ∘ ( I - f ) , where P K is the projection onto the cone K and I stands for the identity mapping. For the class of isotone projection cones (cones admitting projections isotone with respect to the order relation they generate) and f satisfying certain monotonicity properties, the solution can be obtained by iterative processes (see G. Isac, A.B. Németh, Projection methods, isotone projection cones, and the complementarity problem, J. Math. Anal. Appl. 153(1) (1990) 258–275 and S.Z. Németh, Iterative methods for nonlinear complementarity problems on isotone projection cones, J. Math. Anal. Appl. 350(1) (2009) 340–347). These algorithms require computing at each step the projection onto the cone K . In general, computing the projection mapping onto a cone K is a difficult and computationally expensive problem. In this note it is shown that the projection of an arbitrary point onto an isotone projection cone in R n can be obtained by projecting recursively at most n - 1 times into subspaces of decreasing dimension. This emphasizes the efficiency of the algorithms mentioned above and furnishes a handy tool for some problems involving special isotone projection cones, as for example the non-negative monotone cones occurring in reconstruction problems (see e.g. Section 5.13 in J. Dattorro, Convex Optimization and Euclidean Distance Geometry, Meboo, 2005, v2009.04.11).

Theory of semidefinite programming for Sensor Network Localization

We analyze the semidefinite programming (SDP) based model and method for the position estimation problem in sensor network localization and other Euclidean distance geometry applications. We use SDP duality and interior-point algorithm theories to prove that the SDP localizes any network or graph that has unique sensor positions to fit given distance measures. Therefore, we show, for the first time, that these networks can be localized in polynomial time. We also give a simple and efficient criterion for checking whether a given instance of the localization problem has a unique realization in $$\mathcal{R}^2$$ using graph rigidity theory. Finally, we introduce a notion called strong localizability and show that the SDP model will identify all strongly localizable sub-networks in the input network.