Document Type

Journal Article

Department/Unit

Department of Mathematics

Title

Computing the nearest Euclidean distance matrix with low embedding dimensions

Language

English

Abstract

© 2013, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society. Euclidean distance embedding appears in many high-profile applications including wireless sensor network localization, where not all pairwise distances among sensors are known or accurate. The classical Multi-Dimensional Scaling (cMDS) generally works well when the partial or contaminated Euclidean Distance Matrix (EDM) is close to the true EDM, but otherwise performs poorly. A natural step preceding cMDS would be to calculate the nearest EDM to the known matrix. A crucial condition on the desired nearest EDM is for it to have a low embedding dimension and this makes the problem nonconvex. There exists a large body of publications that deal with this problem. Some try to solve the problem directly and some are the type of convex relaxations of it. In this paper, we propose a numerical method that aims to solve this problem directly. Our method is strongly motivated by the majorized penalty method of Gao and Sun for low-rank positive semi-definite matrix optimization problems. The basic geometric object in our study is the set of EDMs having a low embedding dimension. We establish a zero duality gap result between the problem and its Lagrangian dual problem, which also motivates the majorization approach adopted. Numerical results show that the method works well for the Euclidean embedding of Network coordinate systems and for a class of problems in large scale sensor network localization and molecular conformation.

Keywords

Euclidean distance matrix, Lagrangian duality, Low-rank approximation, Majorization method, Semismooth Newton-CG method

Publication Date

2014

Source Publication Title

Mathematical Programming

Volume

147

Issue

1

Start Page

351

End Page

389

Publisher

Springer Verlag

DOI

10.1007/s10107-013-0726-0

Link to Publisher's Edition

http://dx.doi.org/10.1007/s10107-013-0726-0

ISSN (print)

00255610

ISSN (electronic)

14364646

This document is currently not available here.

Share

COinS