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Original Articles

Euclidean distance matrix completion problems

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Pages 695-717 | Received 15 Nov 2010, Accepted 17 Nov 2011, Published online: 22 Dec 2011

References

  • Alfakih , A. Y. 2003 . On the uniqueness of Euclidean distance matrix completions . Linear Algebra Appl. , 370 : 1 – 14 .
  • Alfakih , A. , Anjos , M. , Piccialli , V. and Wolkowicz , H. 2011 . Euclidean distance matrices, semidefinite programming, and sensor network localization . Port. Math. , 68 : 53 – 102 .
  • Alfakih , A. Y. , Khandani , A. and Wolkowicz , H. 1999 . Solving Euclidean distance matrix completion problems via semidefinite programming . Comput. Optim. Appl. , 12 : 13 – 30 . Available at citeseer.ist.psu.edu/alfakih97solving.html.
  • Alfakih , A. Y. and Wolkowicz , H. 1998 . “ On the Embeddability of Weighted Graphs in Euclidean Spaces ” . In Tech. Rep. CORR 98-12 , Ontario : Computer Science Department, University of Waterloo .
  • Bakonyi , M. and Johnson , C. R. 1995 . The Euclidean distance matrix completion problem . SIAM J. Matrix Anal. Appl. , 16 : 646 – 654 .
  • Berman , H. M. , Westbrook , J. , Feng , Z. , Gilliland , G. , Bhat , T. , Weissig , H. , Shindyalov , I. N. and Bourne , P. E. 2000 . The protein data bank . Nucleic Acids Res. , 28 : 235 – 242 .
  • Crippen , G. M. 1982 . Conformational analysis by energy embedding . J. Comput. Chem. , 3 : 471 – 476 .
  • Crippen , G. M. 1984 . Conformational analysis by scaled energy embedding . J. Comput. Chem. , 5 : 548 – 554 .
  • Fang , H.-r. 2006 . Matrix factorization, triadic matrices, and modified Cholesky factorizations for optimization , MD : University of Maryland . Ph.D. thesis
  • Fang , H.-r. and O'Leary , D. P. 2008 . Modified Cholesky factorizations: A catalog with new approaches . Math. Program. A , 115 : 319 – 349 .
  • Floyd , R. W. 1962 . Algorithm 97: Shortest path . Commun. ACM , 5 : 345
  • Glunt , W. , Hayden , T. L. and Raydan , M. 1993 . Molecular conformation from distance matrices . J. Comput. Chem. , 14 : 114 – 120 .
  • Gower , J. C. 1966 . Some distance properties of latent root and vector methods used in multivariate analysis . Biometrika , 53 : 325 – 338 .
  • Gower , J. C. 1985 . Properties of Euclidean and non-Euclidean distance matrices . Linear Algebra Appl. , 67 : 81 – 97 .
  • Grooms , I. G. , Lewis , R. M. and Trosset , M. W. 2009 . Molecular embedding via a second order dissimilarity parameterized approach . SIAM J. Sci. Comput. , 31 : 2733 – 2756 .
  • Havel , T. F. 1991 . An evaluation of computational strategies for use in the determination of protein structure from distance geometry constraints obtained by nuclear magnetic resonance . Prog. Biophys. Mol. Biol. , 56 : 43 – 78 .
  • Hendrickson , B. 1992 . Conditions for unique graph realizations . SIAM J. Sci. Comput. , 21 : 65 – 84 .
  • Hendrickson , B. 1995 . The molecule problem: exploiting structure in global optimization . SIAM J. Optim. , 5 : 835 – 857 .
  • Johnson , C. R. 1995 . Connections between the real positive semidefinite and distance matrix completion problems . Linear Algebra Appl. , 223–224 : 375 – 391 .
  • Kokiopoulou , E. and Saad , Y. 2005.113 . “ Orthogonal Neighborhood Preserving Projections ” . In Proceedings of the 5th IEEE International Conference on Data Mining Vol. 2005 , 234 – 241 . DOI:10.1109/ICDM.
  • Kokiopoulou , E. and Saad , Y. 2007 . Orthogonal neighborhood preserving projections: A projection-based dimensionality reduction technique . IEEE Trans. Pattern Anal. Mach. Intell. , 29 : 2143 – 2156 .
  • Kolda , T. G. , O'Leary , D. P. and Nazareth , L. 1998 . BFGS with update skipping and varying memory . SIAM J. Optim. , 8 : 1060 – 1083 .
  • Krislock , N. and Wolkowicz , H. 2010 . Explicit sensor network localization using semidefinite representations and facial reductions . SIAM J. Optim. , 20 : 2679 – 2708 .
  • Meza , J. C. , Oliva , R. A. , Hough , P. D. and Williams , P. J. 2007 . OPT++: An object-oriented toolkit for nonlinear optimization . ACM Trans. Math. Softw. , 33 : 12
  • Moré , J. J. and Wu , Z. 1995 . “ ε-optimal solutions to distance geometry problems via global continuation ” . Mathematics and Computer Science Division, Argonne National Laboratory . Technical Report MCS-P520-0595
  • Moré , J. J. and Wu , Z. 1997 . Global continuation for distance geometry problems . SIAM J. Optim. , 7 : 814 – 836 .
  • Moré , J. J. and Wu , Z. 1999 . Distance geometry optimization for protein structures . J. Global Optim. , 15 : 219 – 234 .
  • Purisima , E. O. and Scheraga , H. A. 1986 . An approach to the multiple-minima problems by relaxing dimensionality . Proc. Natl. Acad. Sci. USA , 83 : 2782 – 2786 .
  • Schnabel , R. B. and Eskow , E. 1990 . A new modified Cholesky factorization . SIAM J. Sci. Stat. Comput. , 11 : 1136 – 1158 .
  • Schnabel , R. B. and Eskow , E. 1999 . A revised modified Cholesky factorization algorithm . SIAM J. Optim. , 9 : 1135 – 1148 .
  • Schoenberg , I. J. 1935 . Remarks to Maurice Fréchet's article: ‘Sur la définition axiomatique d'une classe d'espace distanciés vectoriellement applicable sur l'espace de Hilbert’ . Ann. Math. , 36 : 724 – 732 .
  • Tenenbaum , J. B. , deSilva , V. and Langford , J. C. 2000 . A global geometric framework for nonlinear dimensionality reduction . Science , 290 : 2319 – 2323 .
  • Torgerson , W. S. 1952 . Multidimensional scaling: I. Theory and method . Psychometrika , 30 : 333 – 367 .
  • Trosset , M. W. 1998 . Applications of multidimensional scaling to molecular conformation . Comput. Sci. Statist. , 29 : 148 – 152 .
  • Trosset , M. W. 2000 . Distance matrix completion by numerical optimization . Comput. Optim. Appl. , 17 : 11 – 22 .
  • Warshall , S. 1962 . A theorem on boolean matrices . J. ACM , 9 : 11 – 12 .
  • Young , G. and Householder , A. S. 1938 . Discussion of a set of points in terms of their mutual distances . Psychometrika , 3 : 19 – 22 .
  • Zou , Z. , Byrd , R. H. and Schnabel , R. B. 1997 . A stochastic/perturbation global optimization algorithm for distance geometry problems . J. Global Optim. , 11 : 91 – 105 . Available at citeseer.ist.psu.edu/243345.html, mistakenly indexed by the Journal under ‘Bird’ instead of ‘Byrd’.

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