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Computing the nearest Euclidean distance matrix with low embedding dimensions

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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.

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Notes

  1. Available from http://pdos.lcs.mit.edu/p2psim/Kingdata.

  2. Available from http://www.cs.cornell.edu/People/egs/meridian/data.php.

  3. Available from http://www.cs.rice.edu/~eugeneng/research/gnp.

  4. Data available from http://people.sc.fsu.edu/~jburkardt/datasets/cities/cities.html.

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Acknowledgments

The authors would like to thank Prof. Kim-Chuan Toh for his help on using the SDP solver as well as on the PDB problems. They also like to thank Bamdev Mishra for useful discussions on the manifold-based-optimization solver [40] (denoted by MBO solver in this paper). The authors also wish to thank the three referees as well as the AE for their valuable comments and constructive suggestions, which have greatly improved the quality and the presentation of the paper.

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Authors and Affiliations

  1. School of Mathematics, The University of Southampton, Highfield, Southampton, SO17 1BJ, UK

    Hou-Duo Qi

  2. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Xiaoming Yuan

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  1. Hou-Duo Qi
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  2. Xiaoming Yuan
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Correspondence to Hou-Duo Qi.

Additional information

This research was supported by the EPSRC grant EP/K007645/1.

This author was supported by the General Research Fund from Hong Kong Research Grants Council: 203712.

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Qi, HD., Yuan, X. Computing the nearest Euclidean distance matrix with low embedding dimensions. Math. Program. 147, 351–389 (2014). https://doi.org/10.1007/s10107-013-0726-0

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