Euclidean Distance Matrix Completion Research Articles

Efficient Rigid Body Localization Based on Euclidean Distance Matrix Completion for AGV Positioning Under Harsh Environment

In real-world applications for automatic guided vehicle (AGV) navigation, the positioning system based on the time-of-flight (TOF) measurements between anchors and tags is confronted with the problem of insufficient measurements caused by blockages to radio signals or lasers, etc. Mounting multiple tags at different positions of the AGV to collect more TOFs is a feasible solution to tackle this difficulty. Vehicle localization by exploiting the measurements between multiple tags and anchors is a rigid body localization (RBL) problem, which estimates both the position and attitude of the vehicle. However, the state-of-the-art solutions to the RBL problem do not deal with missing measurements, and thus will result in degraded localization availability and accuracy in harsh environments. In this paper, different from these existing solutions for RBL, we model this problem as a sensor network localization problem with missing TOFs. To solve this problem, we propose a new efficient RBL solution based on Euclidean distance matrix (EDM) completion, abbreviated as ERBL-EDMC. Firstly, we develop a method to determine the upper and lower bounds of the missing measurements to complete the EDM reliably, using the known relative positions between tags and the statistics of the TOF measurements. Then, based on the completed EDM, the global tag positions are obtained from a coarse estimation followed by a refinement step assisted with inter-tag distances. Finally, the optimal vehicle position and attitude are obtained iteratively based on the estimated tag positions from the previous step. Theoretical analysis and simulation results show that the proposed ERBL-EDMC method effectively solves the RBL problem with incomplete measurements. It obtains the optimal positioning results while maintaining low computational complexity compared with the existing RBL methods based on semi-definite relaxation.

A facial reduction approach for the single source localization problem

The single source localization problem (SSLP) appears in several fields such as signal processing and global positioning systems. The optimization problem of SSLP is nonconvex and difficult to find its globally optima solution. It can be reformulated as a rank constrained Euclidean distance matrix (EDM) completion problem with a number of equality constraints. In this paper, we propose a facial reduction approach to solve such an EDM completion problem. For the constraints of fixed distances between sensors, we reduce them to a face of the EDM cone and derive the closed formulation of the face. We prove constraint nondegeneracy for each feasible point of the resulting EDM optimization problem without a rank constraint, which guarantees the quadratic convergence of semismooth Newton's method. To tackle the nonconvex rank constraint, we apply the majorized penalty approach developed by Zhou et al. (IEEE Trans Signal Process 66(3):4331-4346, 2018). Numerical results verify the fast speed of the proposed approach while giving comparable quality of solutions as other methods.

Vision-Based Localization in Multi-Agent Networks With Communication Constraints

Among the diversified applications of Internet of Vehicles (IoV), vision-based localization has raised increasing concern for its extraordinary performance in terms of accuracy and flexibility. The ultra-reliability and low-latency requirements of IoV have posed urgent demand for the provision of precise relative network geometry and efficient data transmission. In this paper, we propose a vision-based relative localization scheme in the presence of communication constraints. First, the relative squared position error bound (SPEB) is derived via subspace projection of the Fisher information matrix. Next, we determine the local convexity of the relative SPEB with respect to bit allocation vectors and propose two bit allocation algorithms. Furthermore, we exploit the vision-based relative geometry and develop two localization algorithms based on Euclidean distance matrix completion. The simulation results validate that the proposed vision-based localization algorithms achieve near-optimal performance in terms of the relative SPEB under bandwidth constraints, as well as demonstrate the superiority of adopting cooperation among mobile agents.

Illinois-Type Methods for Noisy Euclidean Distance Realization

In this work, we introduce an iterative algorithm for the Euclidean distance matrix completion (EDMC) problem with noisy and incomplete distance measurements. The proposed method is based on semidefinite programming, utilizes a Pareto iterative approach, and performs a projection-free convex optimization over the spectrahedron to solve a level-set problem relevant to EDMC problems. The optimality trade-off between the trace of a positive semidefinite matrix and a loss function is pursued over Pareto optimal points with simple, derivative-free, costly efficient nonlinear equation root finding iterations called Illinois-type methods. We evaluate our approach numerically in a scenario where distance measurements are affected by multiplicative noise.

Quartic First-Order Methods for Low-Rank Minimization

We study a general nonconvex formulation for low-rank minimization problems. We use recent results on non-Euclidean first-order methods to provide efficient and scalable algorithms. Our approach uses the geometry induced by the Bregman divergence of well-chosen kernel functions; for unconstrained problems, we introduce a novel family of Gram quartic kernels that improve numerical performance. Numerical experiments on Euclidean distance matrix completion and symmetric nonnegative matrix factorization show that our algorithms scale well and reach state-of-the-art performance when compared to specialized methods.

On the estimation of unknown distances for a class of Euclidean distance matrix completion problems with interval data

We consider some Euclidean distance matrix completion problems whose structure is inspired by molecular conformation problems. Some matrix distances are given precisely or in terms of intervals and other values are unknown. We present theoretical results that can be useful in methods to estimate the possible values of the unknown matrix distances.

Deep Learning Based Data Recovery for Localization

In this paper, we study the problem of euclidean distance matrix (EDM) recovery aiming to tackle the problem of received signal strength indicator sparsity and fluctuations in indoor environments for localization purposes. This problem is addressed under the constraints required by the internet of things communications ensuring low energy consumption and reduced online complexity compared to classical completion schemes. We propose EDM completion methods based on neural networks that allow an efficient distance recovery and denoising. A trilateration process is then applied to recovered distances to estimate the target’s position. The performance of different deep neural networks (DNN) and convolutional neural networks schemes proposed for matrix reconstruction are evaluated in a simulated indoor environment, using a realistic propagation model, and compared with traditional completion method based on the adaptative moment estimation algorithm. Obtained results show the superiority of the proposed DNN based completion systems in terms of localization mean error and online complexity compared to the classical completion.

Multi-dimensional scaling (MDS), Euclidean distance matrices (EDMs), matrix completion and outlier identification in array calibration and beamforming

The matrix of inter-element distances, a Euclidean Distance Matrix (EDM), is symmetric with zero diagonal. It thus has N(N-1)/2 distinct values, which we know are a function of 3N-6 xyz coordinates. Indeed, the EDM for the elements of a 3-D array has rank 3. As a result, it is possible to reconstruct missing elements of the EDM, using a process called matrix completion. Array calibration (aka array element localization or AEL) is determining the relative 3-D coordinates of all array elements, so that they can be incorporated into various beamforming processes. The eigenvectors of a complete and noise-free EDM immediately yield the array element xyz coordinates, using a method called Multi-Dimensional Scaling (MDS). If an EDM has missing measurements and/or is noisy, its matrix completion can recover a complete and more accurate EDM that then yields better relative xyz coordinates. Alternatively, testing a noisy EDM for its know properties can be used to identify outlier measurements (among the inter-element distances), before AEL is performed. We will present the mathematics underlying EDMs, MDS, and matrix completion, and demonstrate AEL results for incomplete and noisy EDMs, using these concepts.

Noisy Euclidean distance matrix completion with a single missing node

We present several solution techniques for the noisy single source localization problem, i.e. the Euclidean distance matrix completion problem with a single missing node to locate under noisy data. For the case that the sensor locations are fixed, we show that this problem is implicitly convex, and we provide a purification algorithm along with the SDP relaxation to solve it efficiently and accurately. For the case that the sensor locations are relaxed, we study a model based on facial reduction. We present several approaches to solve this problem efficiently, and we compare their performance with existing techniques in the literature. Our tools are semidefinite programming, Euclidean distance matrices, facial reduction, and the generalized trust region subproblem. We include extensive numerical tests.

Localization From Incomplete Euclidean Distance Matrix: Performance Analysis for the SVD–MDS Approach

Localizing a cloud of points from noisy measurements of a subset of pairwise distances has applications in various areas, such as sensor network localization and reconstruction of protein conformations from NMR measurements. In [1], Drineas et al. proposed a natural two-stage approach, named SVD-MDS, for this purpose. This approach consists of a low-rank matrix completion algorithm, named SVD-Reconstruct, to estimate random missing distances, and the classic multidimensional scaling (MDS) method to estimate the positions of nodes. In this paper, we present a detailed analysis for this method. More specifically, we first establish error bounds for Euclidean distance matrix (EDM) completion in both expectation and tail forms. Utilizing these results, we then derive the error bound for the recovered positions of nodes. In order to assess the performance of SVD-Reconstruct, we present the minimax lower bound of the zero-diagonal, symmetric, low-rank matrix completion problem by Fano's method. This result reveals that when the noise level is low, the SVD-Reconstruct approach for Euclidean distance matrix completion is suboptimal in the minimax sense; when the noise level is high, SVD-Reconstruct can achieve the optimal rate up to a constant factor.

ADMM-Based Sensor Network Localization Using Low-Rank Approximation

In this paper, we present an efficient localization algorithm for sensor networks using range information. Since each sensor can only communicate with its neighbors, the Euclidean distance matrix (EDM), composed by squared distances between each pair of sensors, is incomplete. The first step of the proposed algorithm is to fulfill the EDM completion and relative maps of sensor networks are then achieved by the multidimensional scaling technique. Besides the EDM, the centralized Gram matrix of sensors’ coordinates is also used to model the localization problem. Compared with other EDM-based localization algorithms, mathematical properties of both the EDM and the Gram matrix are appropriately exploited so as to improve the estimation accuracy. The resulting localization model is formulated as a semidefinite programming problem. An alternating direction method of multipliers is further developed to enhance the scalability of the proposed algorithm. The numerical experiments demonstrate that the proposed algorithm can effectively improve the estimation accuracy of both the EDM completion and the final localization.

Euclidean Distance Matrix Completion and Point Configurations from the Minimal Spanning Tree

The paper introduces a special case of the Euclidean distance matrix completion problem of interest in statistical data analysis where only the minimal spanning tree distances are given and the matrix completion must preserve the minimal spanning tree. Two solutions are proposed: one an adaptation of a more general method based on a dissimilarity parameterized formulation and the other an entirely novel method which constructs the point configuration directly through a guided random search. These methods as well as three standard edcmp methods are described and compared experimentally on real and synthetic data. It is found that the constructive method given by the guided random search algorithm clearly outperforms all others considered here. Notably, standard methods including the adaptation force peculiar and generally unwanted geometric structure on the point configurations their completions produce.

Noisy Euclidean Distance Realization: Robust Facial Reduction and the Pareto Frontier

We present two algorithms for large-scale noisy low-rank Euclidean distance matrix completion problems, based on semidefinite optimization. Our first method works by relating cliques in the graph of the known distances to faces of the positive semidefinite cone, yielding a combinatorial procedure that is provably robust and partly parallelizable. Our second algorithm is a first-order method for maximizing the trace---a popular low-rank inducing regularizer---in the formulation of the problem with a constrained misfit. Both of the methods output a point configuration that can serve as a high-quality initialization for local optimization techniques. Numerical experiments on large-scale sensor localization problems illustrate the two approaches.

Matrix Completion Optimization for Localization in Wireless Sensor Networks for Intelligent IoT.

Localization in wireless sensor networks (WSNs) is one of the primary functions of the intelligent Internet of Things (IoT) that offers automatically discoverable services, while the localization accuracy is a key issue to evaluate the quality of those services. In this paper, we develop a framework to solve the Euclidean distance matrix completion problem, which is an important technical problem for distance-based localization in WSNs. The sensor network localization problem is described as a low-rank dimensional Euclidean distance completion problem with known nodes. The task is to find the sensor locations through recovery of missing entries of a squared distance matrix when the dimension of the data is small compared to the number of data points. We solve a relaxation optimization problem using a modification of Newton’s method, where the cost function depends on the squared distance matrix. The solution obtained in our scheme achieves a lower complexity and can perform better if we use it as an initial guess for an interactive local search of other higher precision localization scheme. Simulation results show the effectiveness of our approach.

Decomposition Methods for Sparse Matrix Nearness Problems

We discuss three types of sparse matrix nearness problems: given a sparse symmetric matrix, find the matrix with the same sparsity pattern that is closest to it in Frobenius norm and (1) is positive semidefinite, (2) has a positive semidefinite completion, or (3) has a Euclidean distance matrix completion. Several proximal splitting and decomposition algorithms for these problems are presented and their performance is compared on a set of test problems. A key feature of the methods is that they involve a series of projections on small dense positive semidefinite or Euclidean distance matrix cones, corresponding to the cliques in a triangulation of the sparsity graph. The methods discussed include the dual block coordinate ascent algorithm (or Dykstra's method), the dual projected gradient and accelerated projected gradient algorithms, and a primal and a dual application of the Douglas--Rachford splitting algorithm.

Ad hoc microphone array calibration: Euclidean distance matrix completion algorithm and theoretical guarantees

This paper addresses the problem of ad hoc microphone array calibration where only partial information about the distances between microphones is available. We construct a matrix consisting of the pairwise distances and propose to estimate the missing entries based on a novel Euclidean distance matrix completion algorithm by alternative low-rank matrix completion and projection onto the Euclidean distance space. This approach confines the recovered matrix to the EDM cone at each iteration of the matrix completion algorithm. The theoretical guarantees of the calibration performance are obtained considering the random and locally structured missing entries as well as the measurement noise on the known distances. This study elucidates the links between the calibration error and the number of microphones along with the noise level and the ratio of missing distances. Thorough experiments on real data recordings and simulated setups are conducted to demonstrate these theoretical insights. A significant improvement is achieved by the proposed Euclidean distance matrix completion algorithm over the state-of-the-art techniques for ad hoc microphone array calibration.

Euclidean distance matrix completion problems

A Euclidean distance matrix (EDM) is one in which the (i, j) entry specifies the squared distance between particle i and particle j. Given a partially specified symmetric matrix A with zero diagonal, the Euclidean distance matrix completion problem (EDMCP) is to determine the unspecified entries to make A an EDM. We survey three different approaches to solving the EDMCP. We advocate expressing the EDMCP as a non-convex optimization problem using the particle positions as variables and solving using a modified Newton or quasi-Newton method. To avoid local minima, we develop a randomized initialization technique that involves a nonlinear version of the classical multidimensional scaling, and a dimensionality relaxation scheme with optional weighting. Our experiments show that the method easily solves the artificial problems introduced by Moré and Wu. It also solves the 12 much more difficult protein fragment problems introduced by Hendrickson, and the six larger protein problems introduced by Grooms, Lewis and Trosset.

Robust Localization of Nodes and Time-Recursive Tracking in Sensor Networks Using Noisy Range Measurements

Simultaneous localization and tracking (SLAT) in sensor networks aims to determine the positions of sensor nodes and a moving target in a network, given incomplete and inaccurate range measurements between the target and each of the sensors. One of the established methods for achieving this is to iteratively maximize a likelihood function (ML) of positions given the observed ranges, which requires initialization with an approximate solution to avoid convergence towards local extrema. This paper develops methods for handling both Gaussian and Laplacian noise, the latter modeling the presence of outliers in some practical ranging systems that adversely affect the performance of localization algorithms designed for Gaussian noise. A modified Euclidean distance matrix (EDM) completion problem is solved for a block of target range measurements to approximately set up initial sensor/target positions, and the likelihood function is then iteratively refined through majorization-minimization (MM). To avoid the computational burden of repeatedly solving increasingly large EDM problems in time-recursive operation, an incremental scheme is exploited whereby a new target/node position is estimated from previously available node/target locations to set up the iterative ML initial point for the full spatial configuration. The above methods are first derived under Gaussian noise assumptions, and modifications for Laplacian noise are then considered. Analytically, the main challenges to overcome in the Laplacian case stem from the non-differentiability of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\ell_1$</tex> </formula> norms that arise in the various cost functions. Simulation results show that the proposed algorithms significantly outperform existing localization methods in the presence of outliers, while providing comparable performance for Gaussian noise.

Deriving Finite Sphere Packings

Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for deriving all packings of n spheres in R^3 satisfying minimal rigidity constraints (>= 3 contacts per sphere and >= 3n-6 total contacts). We derive such packings for n <= 10, and provide a preliminary set of maximal contact packings for 10 < n <= 20. The resultant set of packings has some striking features, among them: (i) all minimally rigid packings for n <= 9 have 3n-6 contacts, (ii) non-rigid packings satisfying minimal rigidity constraints arise for n >= 9, (iii) the number of ground states (i.e. packings with the maximal number of contacts) oscillates with respect to n, (iv) for 10 <= n <= 20 there are only a small number of packings with the maximal number of contacts, and for 10 <= n < 13 these are all commensurate with the HCP lattice. The general method presented here may have applications to other related problems in mathematics, such as the Erdos repeated distance problem and Euclidean distance matrix completion problems.

One-step organic phosphating

We consider some Euclidean distance matrix completion problems whose structure is inspired by molecular conformation problems. Some matrix distances are given precisely or in terms of intervals and other values are unknown. We present theoretical results that can be useful in methods to estimate the possible values of the unknown matrix distances.