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Distance Matrix Completion by Numerical Optimization

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Abstract

Consider the problem of determining whether or not a partial dissimilarity matrix can be completed to a Euclidean distance matrix. The dimension of the distance matrix may be restricted and the known dissimilarities may be permitted to vary subject to bound constraints. This problem can be formulated as an optimization problem for which the global minimum is zero if and only if completion is possible. The optimization problem is derived in a very natural way from an embedding theorem in classical distance geometry and from the classical approach to multidimensional scaling. It belongs to a general family of problems studied by Trosset (Technical Report 97-3, Department of Computational & Applied Mathematics—MS 134, Rice University, Houston, TX 77005-1892, 1997) and can be formulated as a nonlinear programming problem with simple bound constraints. Thus, this approach provides a constructive technique for obtaining approximate solutions to a general class of distance matrix completion problems.

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References

  1. A. Alfakih, A. Khandani, and H. Wolkowicz, “An interior-point method for approximate distance matrix completions,” CORR, University of Waterloo, Technical Report 97-9, 1997.

  2. M. Bakonyi and C.R. Johnson, “The Euclidean distance matrix completion problem,” SIAM Journal on Matrix Analysis and Applications, vol. 16, pp. 646-654, 1995.

    Google Scholar 

  3. W. Braun, “Distance geometry and related methods for protein structure determination from NMR data,” Quarterly Reviews of Biophysics, vol. 19, pp. 115-157, 1987.

    Google Scholar 

  4. A.T. Brünger and M. Nilges, “Computational challenges for macromolecular structure determination by Xray crystallography and solution NMR spectroscopy,” Quarterly Reviews of Biophysics, vol. 26, pp. 49-125, 1993.

    Google Scholar 

  5. R.H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm for bound constrained optimization,” SIAM Journal on Scientific Computing, vol. 16, pp. 1190-1208, 1995.

    Google Scholar 

  6. G.M. Crippen and T.F. Havel, Distance Geometry and Molecular Conformation, John Wiley & Sons: New York, 1988.

    Google Scholar 

  7. D.M. Gay, “Algorithm 611. Subroutines for unconstrained minimization using a model/trust-region approach,” ACM Transactions on Mathematical Software, vol. 9, pp. 503-524, 1983.

    Google Scholar 

  8. D.M. Gay, “Atrust region approach to linearly constrained optimization,” in Numerical Analysis, F.A. Lootsma (Ed.), Springer: Berlin, 1984, pp. 171-189. Proceedings, Dundee, 1983.

    Google Scholar 

  9. W. Glunt, T.L. Hayden, and M. Raydan, “Molecular conformations from distance matrices,” Journal of Computational Chemistry, vol. 14, pp. 114-120, 1993.

    Google Scholar 

  10. R. Grone, C.R. Johnson, E.M. Sá, and H. Wolkowicz, “Positive definite completions of partial Hermitian matrices,” Linear Algebra and Its Applications, vol. 58, pp. 109-124, 1984.

    Google Scholar 

  11. T.F. Havel, “An evaluation of computational strategies for use in the determination of protein structure from distance constraints obtained by nuclear magnetic resonance,” Progress in Biophysics and Molecular Biology, vol. 56, pp. 43-78, 1991.

    Google Scholar 

  12. T.F. Havel and K. Wüthrich, “An evaluation of the combined use of nuclear magnetic resonance and distance geometry for the determination of protein conformations in solution,” Journal of Molecular Biology, vol. 182, pp. 281-294, 1985.

    Google Scholar 

  13. C.R. Johnson, “Matrix completion problems: A survey,” in Matrix Theory and Applications, C.R. Johnson (Ed.), American Mathematical Society: Providence, RI, 1990, pp. 171-198. Proceedings of Symposia in Applied Mathematics, Vol. 40.

    Google Scholar 

  14. C.R. Johnson, C. Jones, and B. Kroschel, “The distance matrix completion problem: Cycle completability,” Linear and Multilinear Algebra, vol. 39, pp. 195-207, 1995.

    Google Scholar 

  15. C. Johnson, B. Kroschel, and H. Wolkowicz, “An interior-point method for approximate positive semidefinite completions,” Computational Optimization and Applications, vol. 9, pp. 175-190, 1998.

    Google Scholar 

  16. C.R. Johnson and P. Tarazaga, “Connections between the real positive semidefinite and distance matrix completion problems,” Linear Algebra and Its Applications, vol. 223/224, pp. 375-391, 1995.

    Google Scholar 

  17. I.D. Kuntz, J.F. Thomason, and C.M. Oshiro, “Distance geometry,” in Nuclear Magnetic Resonance, Part B: Structure and Mechanism, N.J. Oppenheimer and T.L. James (Eds.), Academic Press: New York, 1989, pp. 159-204. Methods in Enzymology, Vol. 177.

    Google Scholar 

  18. K.V. Mardia, J.T. Kent, and J.M. Bibby, Multivariate Analysis, Academic Press: Orlando, 1979.

    Google Scholar 

  19. J.J. Moré and Z. Wu, “∈-Optimal solutions to distance geometry problems via global continuation,” Mathematics & Computer Science Division, Argonne National Laboratory, Argonne, IL 60439-4844, Preprint MCSP520-0595, May 1995.

    Google Scholar 

  20. J.J. Moré and Z. Wu, “Distance geometry optimization for protein structures,” Mathematics & Computer Science Division, Argonne National Laboratory, Argonne, IL 60439-4844, Preprint ANL/MCS-P628-1296, Dec. 1996.

    Google Scholar 

  21. T.A. Parks, “Reducible nonlinear programming problems,” Technical Report 85-8, Department of Mathematical Sciences, Rice University, Houston, TX, 1985. Author's Ph.D. Thesis.

    Google Scholar 

  22. Statistical Sciences, S-PLUS for Windows Reference Manual, Version 3.1, Vol. 2, Statistical Sciences: Seattle, 1993.

  23. M.W. Trosset, “Numerical algorithms for multidimensional scaling,” in Classification and Knowledge Organization, R. Klar and P. Opitz (Eds.), Springer: Berlin, 1997, pp. 80-92. Proceedings of the 20th annual conference of the Gesellschaft für Klassifikation e.V., held March 6-8, 1996, in Freiburg, Germany.

    Google Scholar 

  24. M.W. Trosset, “Computing distances between convex sets and subsets of the positive semidefinite matrices,” Technical Report 97-3, Department of Computational & Applied Mathematics-MS 134, Rice University, Houston, TX 77005-1892, 1997.

    Google Scholar 

  25. M.W. Trosset, “Applications of multidimensional scaling to molecular conformation,” Computing Science and Statistics, vol. 29, pp. 148-152, 1998.

    Google Scholar 

  26. M.W. Trosset, “A new formulation of the nonmetric STRAIN problem in multidimensional scaling,” Journal of Classification, vol. 15, pp. 15-35, 1998.

    Google Scholar 

  27. M.W. Trosset and G.N. Phillips, “Deriving interatomic distance bounds from chemical structure,” in Statistics in Molecular Biology and Genetics, F. Seillier-Moiseiwitsch (Ed.), IMS Lecture Notes-Monograph Series, vol. 33, pp. 276-287, 1999.

  28. C. Zhu, R.H. Byrd, P. Lu, and J. Nocedal, “L-BFGS-B: Fortran subroutines for large-scale bound constrained optimization,” Technical Report NAM-11 (Revised), Department of Electrical Engineering & Computer Science, Northwestern University, Evanston, IL 60208, Dec. 1994.

    Google Scholar 

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  1. Department of Mathematics, College of William & Mary, P.O. Box 8795, Williamsburg, VA, 23187-8795, USA

    Michael W. Trosset

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  1. Michael W. Trosset
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Trosset, M.W. Distance Matrix Completion by Numerical Optimization. Computational Optimization and Applications 17, 11–22 (2000). https://doi.org/10.1023/A:1008722907820

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