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Computational and Applied Mathematics

, Volume 37, Issue 4, pp 4989–4999 | Cite as

A novel low-rank matrix completion approach to estimate missing entries in Euclidean distance matrix

  • Nilson J. M. Moreira
  • Leonardo T. Duarte
  • Carlile Lavor
  • Cristiano Torezzan
Article

Abstract

A Euclidean distance matrix (EDM) is a table of distance-square between points on a k-dimensional Euclidean space, with applications in many fields (e.g., engineering, geodesy, economics, genetics, biochemistry, and psychology). A problem that often arises is the absence (or uncertainty) of some EDM elements. In many situations, only a subset of all pairwise distances is available and it is desired to have some procedure to estimate the missing distances. In this paper, we address the problem of missing data in EDM through low-rank matrix completion techniques. We exploit the fact that the rank of a EDM is at most \(k+2\) and does not depend on the number of points, which is, in general, much bigger then k. We use a singular value decomposition approach that considers the rank of the matrix to be completed and computes, in each iteration, a parameter that controls the convergence of the method. After performing a number of computational experiments, we could observe that our proposal was able to recover, with high precision, random EDMs with more than 1000 points and up to 98% of missing data in few minutes. In addition, our method required a smaller number of iterations when compared to other competitive state-of-art technique.

Keywords

Euclidean distance matrix Low-rank Matrix completion 

Mathematics Subject Classification

15A83 Matrix completion problems 51Kxx Distance geometry 

Notes

Acknowledgements

The authors would like to thank the Brazilian research agencies CAPES, CNPq, and FAPESP for their financial support.

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Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics, IMECCState University of CampinasCampinasBrazil
  2. 2.School of Applied SciencesState University of CampinasLimeiraBrazil

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