v2007.09.13 - Convex Optimization

20 CHAPTER 1. OVERVIEWFigure 1: Orion nebula. (astrophotography by Massimo Robberto)Euclidean distance geometry is, fundamentally, a determination of pointconformation (configuration, relative position or location) by inference frominterpoint distance information. By inference we mean: e.g., given onlydistance information, determine whether there corresponds a realizableconformation of points; a list of points in some dimension that attains thegiven interpoint distances. Each point may represent simply location or,abstractly, any entity expressible as a vector in finite-dimensional Euclideanspace; e.g., distance geometry of music [72].It is a common misconception to presume that some desired pointconformation cannot be recovered in the absence of complete interpointdistance information. We might, for example, want to realize a constellationgiven only interstellar distance (or, equivalently, distance from Earth andrelative angular measurement; the Earth as vertex to two stars). At first itmay seem O(N 2 ) data is required, yet there are many circumstances wherethis can be reduced to O(N).If we agree that a set of points can have a shape (three points can forma triangle and its interior, for example, four points a tetrahedron), then