Realizing Euclidean distance matrices by sphere intersection

Abstract This paper presents the theoretical properties of an algorithm to find a realization of a (full) n × n Euclidean distance matrix in the smallest possible embedding dimension. Our algorithm performs linearly in n , and quadratically in the minimum embedding dimension, which is an improvement w.r.t. other algorithms.

[1]  Leo Liberti,et al.  On the number of realizations of certain Henneberg graphs arising in protein conformation , 2014, Discret. Appl. Math..

[2]  Carlile Lavor,et al.  A note on computing the intersection of spheres in \(\mathbb{R}^{n}\) , 2018 .

[3]  Michael C. Hout,et al.  Multidimensional Scaling , 2003, Encyclopedic Dictionary of Archaeology.

[4]  Martin Vetterli,et al.  Euclidean Distance Matrices: Essential theory, algorithms, and applications , 2015, IEEE Signal Processing Magazine.

[5]  Jon C. Dattorro,et al.  Convex Optimization & Euclidean Distance Geometry , 2004 .

[6]  Leo Liberti,et al.  Distance Geometry: Theory, Methods, and Applications , 2013, Distance Geometry.

[7]  Leo Liberti,et al.  Euclidean Distance Geometry and Applications , 2012, SIAM Rev..

[8]  H. Scheraga,et al.  Solution of the embedding problem and decomposition of symmetric matrices. , 1985, Proceedings of the National Academy of Sciences of the United States of America.

[9]  Tibérius O. Bonates,et al.  An algorithm for realizing Euclidean distance matrices , 2015, Electron. Notes Discret. Math..

[10]  Leo Liberti,et al.  Open research areas in distance geometry , 2016, ArXiv.

[11]  I. J. Schoenberg.  Remarks to Maurice Frechet's Article ``Sur La Definition Axiomatique D'Une Classe D'Espace Distances Vectoriellement Applicable Sur L'Espace De Hilbert , 1935 .

[12]  Leo Liberti,et al.  The discretizable molecular distance geometry problem , 2006, Computational Optimization and Applications.

[13]  James Demmel,et al.  Fast linear algebra is stable , 2006, Numerische Mathematik.

[14]  Antonio Mucherino,et al.  Assigned and unassigned distance geometry: applications to biological molecules and nanostructures , 2016, 4OR.

[15]  I. D. Coope,et al.  Reliable computation of the points of intersection of $n$ spheres in $R^n$ , 2000 .

[16]  Carlile Lavor,et al.  A NOTE ON COMPUTING THE INTERSECTION OF SPHERES IN , 2017 .