Euclidean distance matrices: special subsets, systems of coordinates and multibalanced matrices

Tarazaga, Pablo; Sterba-Boatwright, Blair; Wijewardena, Kithsiri

doi:10.1590/s0101-82052007000300006

Cited by 4 publications

(2 citation statements)

References 8 publications

(14 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In turn, it is easy to see that S c ∩ S n + is a face of S n + isomorphic to S n−1 + ; see the discussion after Lemma 4.11 for more details. These and other related results have appeared in a number of publications; see for example [1,13,14,19,20,[35][36][37][38].…”

Section: 2mentioning

confidence: 56%

Coordinate Shadows of Semidefinite and Euclidean Distance Matrices

Pataki¹,

Wolkowicz

2015

SIAM J. Optim.

View full text Add to dashboard Cite

We consider the projected semidefinite and Euclidean distance cones onto a subset of the matrix entries. These two sets are precisely the input data defining feasible semidefinite and Euclidean distance completion problems. We classify when these sets are closed and use the boundary structure of these two sets to elucidate the Krislock-Wolkowicz facial reduction algorithm. In particular, we show that under a chordality assumption, the "minimal cones" of these problems admit combinatorial characterizations. As a by-product, we record a striking relationship between the complexity of the general facial reduction algorithm (singularity degree) and facial exposedness of conic images under a linear mapping. Introduction.To motivate the discussion, consider an undirected graph G = (V, E) with vertex set V = {1, . . . , n} and edge set E ⊂ V × V possibly containing self-loops. The classical positive semidefinite (PSD) completion problem asks whether given a data vector a indexed by E, there exists an n × n positive semidefinite matrix X completing a, meaning X ij = a ij for all ij ∈ E. Similarly, the Euclidean distance matrix (EDM) completion problem asks whether given such a data vector, there exists a Euclidean distance matrix completing it. For a survey of these two problems, see for example [22,2,24,25]. The semidefinite and Euclidean distance completion problems are often mentioned in the same light due to a number of parallel results; see, e.g., [21]. Here, we consider a related construction: projections of the PSD cone S n + and the EDM cone E n onto matrix entries indexed by E. These "coordinate shadows," denoted by P(S n + ) and P(E n ), respectively, appear naturally: they are precisely the sets of data vectors that render the corresponding completion problems feasible. We note that these sets are interesting types of "spectrahedral shadows"-an area of intensive research in recent years. For a representative sample of recent papers on spectrahedral shadows, we refer to [25,11,15,16,3] and references therein.In this paper, our goal is twofold: we will (1) highlight the geometry of the two sets P(S n + ) and P(E n ) and (2) illustrate how such geometric considerations yield a much simplified and transparent analysis of an EDM completion algorithm proposed in [18]. To this end, we begin by asking a basic question: Under what conditions are the coordinate shadows P(S n + ) and P(E n ) closed?

show abstract

Section: 2mentioning

confidence: 56%

Coordinate Shadows of Semidefinite and Euclidean Distance Matrices

Pataki¹,

Wolkowicz

2015

SIAM J. Optim.

View full text Add to dashboard Cite

show abstract

“…where J := I − 1 n ee T is the projection onto e ⊥ . These and other related constructions have appeared in a number of publications; see for example [1,19,20,24,25,[33][34][35][36].…”

mentioning

confidence: 99%

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

Krislock¹,

Voronin²,

Wolkowicz³

2017

SIAM J. Optim.

View full text Add to dashboard Cite

We present two algorithms for large-scale 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 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. AMS subject classifications. 90C22, 90C25 , 52A991 Introduction. A pervasive task in distance geometry is the inverse problem: given only local pairwise Euclidean distances among a set of points, recover their locations in space. More precisely, given a weighted undirected graph G = (V, E, d) on a vertex set {1, . . . , n} and an integer r, find (if possible) a set of points x 1 , . . . , x n in R r satisfyingwhere · denotes the usual Euclidean norm on R r . In most applications, the given squared distances d ij are inexact, and one then seeks points x 1 , . . . , x n satisfying the distance constraints only approximately. This problem appears under numerous names in the literature, such as Euclidean Distance Matrix (EDM) completion and graph realization [2,12,26], and is broadly applicable for example in wireless networks, statistics, robotics, protein reconstruction, and dimensionality reduction in data analysis; the recent survey [27] has an extensive list of relevant references. Fixing notation, we will refer to this problem as EDM completion, throughout. The EDM completion problem can be modeled as the nonconvex feasibility problem: find a symmetric n × n matrix X satisfying (1.1)for all ij ∈ E, Xe = 0, rank X ≤ r, X 0,where e stands for the vector of all ones. Indeed, if X = P P T is a maximal rank factorization of such a matrix X, then the rows of P yield a solution to the EDM

show abstract

Classes of EDMs

Alfakih

2018

Euclidean Distance Matrices and Their Applications in Rigidity Theory

View full text Add to dashboard Cite

Euclidean distance matrices: special subsets, systems of coordinates and multibalanced matrices

Cited by 4 publications

References 8 publications

Coordinate Shadows of Semidefinite and Euclidean Distance Matrices

Coordinate Shadows of Semidefinite and Euclidean Distance Matrices

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

Classes of EDMs

Contact Info

Product

Resources

About