Computing the nearest Euclidean distance matrix with low embedding dimensions

Hou Duo Qi*, Xiaoming YUAN

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

23 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)351-389
Number of pages39
JournalMathematical Programming
Volume147
Issue number1-2
DOIs
Publication statusPublished - Oct 2013

Scopus Subject Areas

  • Software
  • Mathematics(all)

User-Defined Keywords

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

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