v2006.03.09 - Convex Optimization

Chapter 4Euclidean Distance MatrixThese results were obtained by Schoenberg (1935), a surprisinglylate date for such a fundamental property of Euclidean geometry.−John Clifford Gower [90,3]By itself, distance information between many points in Euclidean space islacking. We might want to know more; such as, relative or absolute positionor dimension of some hull. A question naturally arising in some fields(e.g., geodesy, economics, genetics, psychology, biochemistry, engineering)[57] asks what facts can be deduced given only distance information. Whatcan we know about the underlying points that the distance informationpurports to describe? We also ask what it means when given distanceinformation is incomplete; or suppose the distance information is not reliable,available, or specified only by certain tolerances (affine inequalities). Thesequestions motivate a study of interpoint distance, well represented in anyspatial dimension by a simple matrix from linear algebra. 4.1 In what follows,we will answer some of these questions via Euclidean distance matrices.4.1 e.g.,◦√D ∈ R N×N , a classical two-dimensional matrix representation of absoluteinterpoint distance because its entries (in ordered rows and columns) can be written neatlyon a piece of paper. Matrix D will be reserved throughout to hold distance-square.2001 Jon Dattorro. CO&EDG version 03.09.2006. All rights reserved.Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry,Meboo Publishing USA, 2005.215