v2006.03.09 - Convex Optimization

292 CHAPTER 4. EUCLIDEAN DISTANCE MATRIXλ(−V T N OV N) j90080070060050040030020010001 2 3 4 5 6 7 8 9 10jFigure 71: Largest ten eigenvalues of −VN TOV N for map of USA, sorted bynonincreasing value. In the code (G.3.2), we normalize O by (N(N −1)/2) 2 .Any process of reconstruction that leaves comparative distanceinformation intact is called ordinal multidimensional scaling or isotonicreconstruction. Beyond rotation, reflection, and translation error, (4.5)list reconstruction by isotonic reconstruction is subject to error in absolutescale (dilation) and distance ratio. Yet Borg & Groenen argue: [34,2.2]reconstruction from complete comparative distance information for a largenumber of points is as highly constrained as reconstruction from an EDM;the larger the number, the better.4.13.2.1 Isotonic map of the USATo test Borg & Groenen’s conjecture, suppose we make a complete sort-indexmatrix O ∈ S N h ∩ R N×N+ for the map of the USA and then substitute O in placeof EDM D in the reconstruction process of4.12. Whereas EDM D returnedonly three significant eigenvalues (687), the sort-index matrix O is generallynot an EDM (certainly not an EDM with corresponding affine dimension 3)so returns many more. The eigenvalues, calculated with absolute numericalerror approximately 5E-7, are plotted in Figure 71:λ(−V T N OV N ) = [880.1 463.9 186.1 46.20 17.12 9.625 8.257 1.701 0.7128 0.6460 · · · ] T(692)