Euclidean Distance Matrix Completion Problem Research Articles

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.

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.

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.

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.

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.

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.

Approximate and exact completion problems for Euclidean distance matrices using semidefinite programming

A partial pre-distance matrix A is a matrix with zero diagonal and with certain elements fixed to given nonnegative values; the other elements are considered free. The Euclidean distance matrix completion problem chooses nonnegative values for the free elements in order to obtain a Euclidean distance matrix, EDM. The nearest (or approximate) Euclidean distance matrix problem is to find a Euclidean distance matrix, EDM, that is nearest in the Frobenius norm to the matrix A, when the free variables are discounted. In this paper we introduce two algorithms: one for the exact completion problem and one for the approximate completion problem. Both use a reformulation of EDM into a semidefinite programming problem, SDP. The first algorithm is based on an implicit equation for the completion that for many instances provides an explicit solution. The other algorithm is based on primal–dual interior-point methods that exploit the structure and sparsity. Included are results on maps that arise that keep the EDM and SDP cones invariant. We briefly discuss numerical tests.

On the uniqueness of Euclidean distance matrix completions: the case of points in general position

The Euclidean distance matrix completion problem (EDMCP) is the problem of determining whether or not a given partial matrix can be completed into a Euclidean distance matrix (EDM). In this paper, we investigate the necessary and sufficient conditions for the uniqueness of a given EDM completion in the case where this EDM completion is generated by points in general position. We also show that the problem of checking the validity of these conditions can be formulated as a semidefinite programming problem.

On the uniqueness of Euclidean distance matrix completions

The Euclidean distance matrix completion problem (EDMCP) is the problem of determining whether or not a given partial matrix can be completed into a Euclidean distance matrix. In this paper, we present necessary and sufficient conditions for a given solution of the EDMCP to be unique.

Some Properties for the Euclidean Distance Matrix and Positive Semidefinite Matrix Completion Problems

The Euclidean distance matrix (EDM) completion problem and the positive semidefinite (PSD) matrix completion problem are considered in this paper. Approaches to determine the location of a point in a linear manifold are studied, which are based on a referential coordinate set and a distance vector whose components indicate the distances from the point to other points in the set. For a given referential coordinate set and a corresponding distance vector, sufficient and necessary conditions are presented for the existence of such a point that the distance vector can be realized. The location of the point (if it exists) given by the approaches in a linear manifold is independent of the coordinate system, and is only related to the referential coordinate set and the corresponding distance vector. An interesting phenomenon about the complexity of the EDM completion problem is described. Some properties about the uniqueness and the rigidity of the conformation for solutions to the EDM and PSD completion problems are presented.

Semidefinite programming for discrete optimization and matrix completion problems

Semidefinite programming (SDP) is currently one of the most active areas of research in optimization. SDP has attracted researchers from a wide variety of areas because of its theoretical and numerical elegance as well as its wide applicability. In this paper we present a survey of two major areas of application for SDP, namely discrete optimization and matrix completion problems. In the first part of this paper we present a recipe for finding SDP relaxations based on adding redundant constraints and using Lagrangian relaxation. We illustrate this with several examples. We first show that many relaxations for the max-cut problem (MC) are equivalent to both the Lagrangian and the well-known SDP relaxation. We then apply the recipe to obtain new strengthened SDP relaxations for MC as well as known SDP relaxations for several other hard discrete optimization problems. In the second part of this paper we discuss two completion problems, the positive semidefinite matrix completion problem and the Euclidean distance matrix completion problem. We present some theoretical results on the existence of such completions and then proceed to the application of SDP to find approximate completions. We conclude this paper with a new application of SDP to find approximate matrix completions for large and sparse instances of Euclidean distance matrices.

Polynomial Instances of the Positive Semidefinite and Euclidean Distance Matrix Completion Problems

Given an undirected graph G=(V,E) with node set V=[1,n], a subset $S\subseteq V$, and a rational vector $a\in {\rm\bf Q}^{S\cup E}$, the positive semidefinite matrix completion problem consists of determining whether there exists a real symmetric n × n positive semidefinite matrix X=(xij) satisfying xii=ai ($i\in S$) and xij=aij ($ij\in E$). Similarly, the Euclidean distance matrix completion problem asks for the existence of a Euclidean distance matrix completing a partially defined given matrix. It is not known whether these problems belong to NP. We show here that they can be solved in polynomial time when restricted to the graphs having a fixed minimum fill-in, the minimum fill-in of graph G being the minimum number of edges needed to be added to G in order to obtain a chordal graph. A simple combinatorial algorithm permits us to construct a completion in polynomial time in the chordal case. We also show that the completion problem is polynomially solvable for a class of graphs including wheels of fixed length (assuming all diagonal entries are specified). The running time of our algorithms is polynomially bounded in terms of n and the bitlength of the input a. We also observe that the matrix completion problem can be solved in polynomial time in the real number model for the class of graphs containing no homeomorph K4.

Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming

Given a partial symmetric matrix A with only certain elements specified, the Euclidean distance matrix completion problem (EDMCP) is to find the unspecified elements of A that make A a Euclidean distance matrix (EDM). In this paper, we follow the successful approach in [20] and solve the EDMCP by generalizing the completion problem to allow for approximate completions. In particular, we introduce a primal-dual interior-point algorithm that solves an equivalent (quadratic objective function) semidefinite programming problem (SDP). Numerical results are included which illustrate the efficiency and robustness of our approach. Our randomly generated problems consistently resulted in low dimensional solutions when no completion existed.

A connection between positive semidefinite and euclidean distance matrix completion problems

The positive semidefinite and Euclidean distance matrix completion problems have received a lot of attention in the literature. Results have been obtained for these two problems that are very similar in their formulations. Although there is a strong relationship between positive semidefinite matrices and Euclidean distance matrices, it was not clear (as noted by Johnson et al.) how to link the two completion problems. The purpose of this note is twofold. First, we show how the results for the Euclidean distance matrix completion problem can be derived from the corresponding results for the positive semidefinite completion problem, using a functional transform introduced by Schoenberg. Second, we introduce a new set of necessary conditions that are stronger than some previously known ones and we identify the graphs for which these conditions suffice for ensuring completability.

A connection between positive semidefinite and Euclidean distance matrix completion problems

The positive semidefinite and Euclidean distance matrix completion problems have received a lot of attention in the literature. Results have been obtained for these two problems that are very similar in their formulations. Although there is a strong relationship between positive semidefinite matrices and Euclidean distance matrices, it was not clear (as noted by Johnson et al.) how to link the two completion problems. The purpose of this note is twofold. First, we show how the results for the Euclidean distance matrix completion problem can be derived from the corresponding results for the positive semidefinite completion problem, using a functional transform introduced by Schoenberg. Second, we introduce a new set of necessary conditions that are stronger than some previously known ones and we identify the graphs for which these conditions suffice for ensuring completability.