v2004.06.19 - 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 [84, §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) [85] askswhat facts can be deduced given only distance information. What can weknow about the underlying points that the distance information purports todescribe? We also ask what happens when the given distance informationis incomplete; or suppose the distance information is not reliable, available,or specified only by certain tolerances. These questions motivate a study ofinter-point distance, well represented in any spatial dimension by a simplematrix from linear algebra. 4.1 In what follows, we will answer some of thesequestions via Euclidean distance matrices.4.0 c○ 2001 Jon Dattorro, all rights reserved.4.1 e.g., √ D ∈ R N×N , a classical two-dimensional matrix representation of absolute interpointdistance because its entries (in ordered rows and columns) can be written neatly ona piece of paper. Matrix D will be reserved throughout to hold distance-square.129